Games, Scales and Suslin Cardinals: The Cabal Seminar Volume I (Lecture Notes in Logic, Volume 31)

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Games, Scales, and Suslin Cardinals: The Cabal Seminar, Volume I The proceedings of the Los Angeles Caltech–UCLA “Cabal Seminar” were originally published in the 1970s and 1980s. Games, Scales, and Suslin Cardinals is the first of a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics, and discussion of research developments since the publication of the original volumes. Focusing on the subjects of “Games and Scales” (Part I) and “Suslin Cardinals, Partition Properties, Homogeneity” (Part II), each of the two sections is preceded by an introductory survey putting the papers into present context. This volume will be an invaluable reference for anyone interested in higher set theory.

Alexander S. Kechris is Professor of Mathematics at the California Institute of Technology. He is the recipient of numerous honors, including the J. S. Guggenheim Memorial Foundation Fellowship and the Carol Karp Prize of the Association for Symbolic Logic, and is a member of the Scientific Research Board of the American Institute of Mathematics. Benedikt L¨owe is Universitair Docent in Logic and Scientific Director of the Graduate Programme in Logic at the Institute for Logic, Language and Computation of the Universiteit van Amsterdam. He is an editor of the Journal of Logic, Language and Information and managing editor of Tbilisi Mathematical Journal. He is a board member of the DVMLG and the EACSL. John R. Steel is Professor of Mathematics at the University of California, Berkeley. Prior to that, he was a professor in the mathematics department at UCLA. He is a recipient of the Carol Karp Prize of the Association for Symbolic Logic and of a Humboldt Prize. Steel is a former Fellow at the Wissenschaftskolleg zu Berlin and the Sloan Foundation.

LECTURE NOTES IN LOGIC

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LECTURE NOTES IN LOGIC 31

Games, Scales, and Suslin Cardinals: The Cabal Seminar, Volume I Edited by

ALEXANDER S. KECHRIS California Institute of Technology

¨ BENEDIKT L OWE Universiteit van Amsterdam

JOHN R. STEEL University of California, Berkeley

association for symbolic logic

v

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521899512 © The Association for Symbolic Logic 2008 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2008

ISBN-13

978-0-511-45552-0

eBook (EBL)

ISBN-13

978-0-521-89951-2

hardback

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

CONTENTS Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

PART I: GAMES AND S C ALES John R. Steel Games and scales. Introduction to Part I. . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Alexander S. Kechris and Yiannis N. Moschovakis Notes on the theory of scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

Itay Neeman Propagation of the scale property using games. . . . . . . . . . . . . . . . . . . . . .

75

John R. Steel Scales on Σ11 -sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

Yiannis N. Moschovakis Inductive scales on inductive sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

Yiannis N. Moschovakis Scales on coinductive sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Donald A. Martin and John R. Steel The extent of scales in L(R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Donald A. Martin The largest countable this, that, and the other . . . . . . . . . . . . . . . . . . . . . . 121 John R. Steel Scales in L(R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 John R. Steel Scales in K(R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Donald A. Martin The real game quantifier propagates scales . . . . . . . . . . . . . . . . . . . . . . . . . 209 John R. Steel Long games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 vii

viii

contents

John R. Steel The length-ù1 open game quantifier propagates scales . . . . . . . . . . . . . . 260 PART II: S US LIN C ARDINALS, PARTITION PROPERTIES, HOMOGENEITY Steve Jackson Suslin cardinals, partition properties, homogeneity. Introduction to Part II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Alexander S. Kechris Suslin cardinals, κ-Suslin sets, and the scale property in the hyperprojective hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 Alexander S. Kechris, Eugene M. Kleinberg, Yiannis N. Moschovakis, and W. Hugh Woodin The axiom of determinacy, strong partition properties, and nonsingular measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Alexander S. Kechris and W. Hugh Woodin The equivalence of partition properties and determinacy . . . . . . . . . . . . 355 Alexander S. Kechris and W. Hugh Woodin Generic codes for uncountable ordinals, partition properties, and elementary embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Alexander S. Kechris A coding theorem for measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 Donald A. Martin and John R. Steel The tree of a Moschovakis scale is homogeneous . . . . . . . . . . . . . . . . . . . 404 Donald A. Martin and W. Hugh Woodin Weakly homogeneous trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

PREFACE

This is the first of four volumes containing reprints of the papers in the original Cabal Seminar volumes of the Springer Lecture Notes in Mathematics series [Cabal i, Cabal ii, Cabal iii, Cabal iv], unpublished material, and new papers. We have grouped the papers of the original Cabal Seminar volumes according to their topics. This volume contains the papers on “Games and Scales” (Part I) and “Suslin Cardinals, Partition Properties, Homogeneity” (Part II). Each of the parts contains an introductory survey (written by John Steel and Steve Jackson, respectively) putting the papers into a present-day context. Table 1 gives an overview of the papers in this volume with their original references. This volume must not be understood as a historical edition of old papers. In the 1980s, there were a number of results obtained by the researchers associated with the Cabal Seminar, some of which were intended for a fifth Cabal volume that was never published. We include some of these papers in this volume, together with papers reporting on new developments related to the research of the Cabal Seminar. These papers are Steel’s “Scales in K(R)” and “The Length-ù1 Open Game Quantifier Propagates Scales” in Part Iand “The Equivalence of Partition Properties and Determinacy” and “Generic Codes for Uncountable Ordinals, Partition Properties, and Elementary Embeddings” by Kechris and Woodin, “The Tree of a Moschovakis Scale is Homogeneous” by Martin and Steel, and “Weakly Homogeneous Trees” by Martin and Woodin in Part II. We also added a new expository paper, “Propagation of the Scale Property Using Games” by Neeman, and modernized and made uniform the notation and layout, and we have given the authors the opportunity to make corrections to their original papers. The new LATEX layout has resulted in changes in the numbering of sections and theorems. The typing and design were partially funded by NSF Grant DMS-0100745; Johan van Benthem’s Spinoza project Logic in Action; the Institute for Logic, Language and Computation; the Association for Symbolic Logic; and the DFG-NWO collaborative project “Determinacy and Combinatorics” (DFG: KO 1353/3-1; NWO: DN 61-532). A lot of people were involved in typing, ix

x

PREFACE

Steel

Part I Games and scales Introduction to Part I

new

Kechris, Moschovakis

Notes on the theory of scales

[Cabal i, pp. 1–53]

Neeman

Propagation of the scale property using games

new

Steel

Scales on Σ11 -sets

[Cabal iii, pp. 72–76]

Moschovakis

Inductive scales on inductive sets

[Cabal i, pp. 185–192]

Moschovakis

Scales on coinductive sets

[Cabal iii, pp. 77–85]

Martin, Steel

The extent of scales in L(R)

[Cabal iii, pp. 86–96]

Martin

The largest countable this, that, and the other

[Cabal iii, pp. 97–106]

Steel

Scales in L(R)

[Cabal iii, pp. 107–156]

Steel

Scales in K(R)

new

Martin

The real game quantifier propagates scales

[Cabal iii, pp. 157–171]

Steel

Long games

[Cabal iv, pp. 56–97]

Steel

The length-ù1 open game quantifier propagates scales

new

Jackson

Part II Suslin cardinals, partition properties, homogeneity Introduction to Part II

new

Kechris

Suslin cardinals, κ-Suslin sets, and the scale property in the hyperprojective hierarchy

[Cabal ii, pp. 127–146]

Kechris, Kleinberg, Moschovakis, Woodin

The axiom of determinacy, strong partition properties, and nonsingular measures

[Cabal ii, pp. 75–100]

Kechris, Woodin

The equivalence of partition properties and determinacy

new

Kechris, Woodin

Generic codes for uncountable ordinals, partition properties, and elementary embeddings

new

Kechris

A coding theorem for measures

[Cabal iv, pp. 103–109]

Martin, Steel

The tree of a Moschovakis scale is homogeneous

new

Martin, Woodin

Weakly homogeneous trees

new

Table 1.

PREFACE

xi

laying out, and proofreading the papers. We should like to thank (in alphabetic order) Edgar Andrade, Stefan Bold, Samson de Jager, Leona Kershaw, Tomasz Polacik, Doroth´ee Reuther, and Philipp Rohde for their important contribution. Very special thanks are due to Samson de Jager, who coordinated the typesetting effort in the final two years of the project. REFERENCES

Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer, 1981. [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Alexander S. Kechris, Donald A. Martin, and John R. Steel [Cabal iv] Cabal seminar 81–85, Lecture Notes in Mathematics, no. 1333, Berlin, Springer, 1988. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978.

The Editors Alexander S. Kechris, Pasadena, CA ¨ Benedikt Lowe, Amsterdam John R. Steel, Berkeley, CA

PART I: GAMES AND SCALES

GAMES AND SCALES INTRODUCTION TO PART I

JOHN R. STEEL

The construction and use of Suslin representations for sets of reals lies at the heart of descriptive set theory. Indeed, virtually every paper in descriptive set theory in the Cabal Seminar volumes deals with such representations in one way or another. Most of the papers in the section to follow focus on the construction of optimally definable Suslin representations via gametheoretic methods. In this introduction, we shall attempt to put those papers in a broader historical and mathematical context. We shall also give a short synopsis of the papers themselves, and describe some of the work done later to which they are related. §1. Some definitions and history. A tree on a set X is a subset of X <ù closed under initial segments. If T is a tree on X × Y , then we regard T as a set of pairs (s, t) of sequences with dom(s) = dom(t). If T is a tree, we use [T ] for the set of infinite branches of T , and if T is on X × Y , we write p[T ] = {x ∈

ù

X : ∃y ∈

ù

Y ∀n < ù((x ↾ n, y ↾ n) ∈ T )}.

We call p[T ] the projection of T , and say that T is a Suslin representation of <ù p[T ], or that p[T S ] is Y -Suslin via T . For s ∈ X , let Ts = {u : (s, u) ∈ T }, and put Tx = n Tx↾n . Then x ∈ p[T ] iff [Tx ] 6= ∅ iff Tx is illfounded. Any set A ⊆ ù X is trivially A-Suslin. For the most part, useful Suslin representations come from trees on some X × Y such that Y is wellordered. Assuming (as we do) the Axiom of Choice (AC), this is no restriction on Y , but we can parlay it into an important and useful restriction by requiring in addition that T be definable in some way or other. A variant of this approach is to require that T belong to a model of AD. If T is definable, and X and Y are definably wellordered, and p[T ] is nonempty, then the leftmost branch (x, f) of T gives us a definable element x of p[T ]. (Here “leftmost” can be determined by the lexicographic order on X × Y .) The simplest nontrivial X to consider are the countable ones. This is by far the most well-studied case in the Cabal volumes. In this case, one may regard p[T ] as a subset of the Baire space ù ù, that is, as a set of “logician’s reals”. The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

3

4

JOHN R. STEEL

Thus if A is a nonempty set of reals, κ is an ordinal, and A is κ-Suslin via a definable tree, then A has a definable element. Suslin representations were first discovered in 1917 by Suslin [Sus17], who isolated the class of ù-Suslin sets of reals, showed that it properly includes the Borel sets, and showed that sets in this class have various regularity properties. (For example, they are all Baire and Lebesgue measurable, and have the perfect set property.) Suslin also found a beautiful characterization of the Borel sets of reals as those which are both ù-Suslin and have ù-Suslin complements. (The ù-Suslin sets of reals are precisely the Σ11 sets of reals, almost by definition.) e definable elements, and in the “boldDefinable Suslin representations yield face” setting of classical descriptive set theory, this comes out as a uniformization result. Here we say that a function f uniformizes a relation R iff dom(f) = {x : ∃yR(x, y)}, and ∀x ∈ dom(f)R(x, f(x)). If R is a Σ11 relation, say e R = p[T ] where T is a tree on (ù × ù) × ù, then we can use leftmost branches to uniformize R: let f(x) = y, where (y, h) is the leftmost branch of Tx . One can calculate that for any open set U , f −1 (U ) is in the ó-algebra generated by the Σ11 sets, and is therefore Lebesgue and Baire measurable. This classical e uniformization result was proved by Jankov and von Neumann around 1940 [vN49]. The “lightface”, effective refinement of a uniformization theorem is a basis theorem, where we say a pointclass Λ is a basis for a pointclass Γ just in case every nonempty set of reals in Γ has a member which is in Λ. Kleene [Kle55] proved the lightface version of the Jankov-von Neumann result. He observed that if A ⊆ ù ù is lightface Σ11 , then A = p[T ] for some recursive tree ¨ T , and that the leftmost branch of T is recursive in the set W of all Godel numbers of wellfounded trees on ù. Thus {x : x ≤T W } is a basis for Σ11 . In 1935–38, toward the end of the classical period, Novikoff and Kondoˆ constructed definable, ù1 -Suslin representations for arbitrary Σ12 sets, and used them to show every Σ12 relation has a Σ12 uniformization. e(See [LN35, e e Kon38].) The effective refinement of this landmark theorem is due to Addison, who showed that the ù1 -Suslin reprentations of nonempty lightface Σ12 sets constructed by Novikoff and Kondoˆ yield, via leftmost branches, lightface ∆12 elements for such sets. Logicians often meet Suslin representations through the Shoenfield Absoluteness theorem. Shoenfield [Sch61] showed that a certain tree T on ù × ù1 which comes from the Novikoff-Kondoˆ construction is in L. Because wellfoundedness is absolute to transitive models of ZF, he was able to conclude that the leftmost branch of T is in L, and thus, that every nonempty Σ12 set of reals has an element in L. From this it follows easily that L is Σ12 correct. This method of using definable Suslin representations to obtain correctness and absoluteness results for models of set theory is very important. In addition to definability, there is a second very useful property a Suslin representation might have. We call a tree T on X × Y homogeneous just in case there is a family hìs : s ∈ X <ù i such that

INTRODUCTION TO PART I

5

(1) for all s, ìs is a countably complete 2-valued measure (i.e. ultrafilter) on {u : (s, u) ∈ T }, (2) if s ⊆ t, and ìs (A) = 1, then ìt ({u : u ↾ dom(s) ∈ A}) = 1, and (3) for any x ∈ p[T ] and any hAi : i < ùi such that ìx↾i (Ai ) = 1 for all i, there is a f ∈ Y ù such that f ↾ i ∈ Ai for all i. We say T is κ-homogeneous if the measures ìs can be taken to be κ-additive. If T is κ-homogeneous, then we also call p[T ] a κ-homogeneously Suslin set. We write Hom T κ for the pointclass of κ-homogeneous sets, and Hom∞ for the pointclass κ Homκ . The concept of homogeneity is implicit in Martin’s 1968 proof [Mar70A] of Π11 determinacy, and was first explicitly isolated by Martin and Kechris. Martin showed that if κ is a measurable cardinal, then every Π11 set of reals is κ homogeneous, via a Shoenfield tree on ù × κ. He also showed that every homogeneously Suslin set of reals is determined. Martin’s proof became the template for all later proofs of definable determinacy from large cardinal hypotheses. Indeed, the standard characterization of descriptive set theory, as the study of the good behavior of definable sets of reals, would perhaps be more accurate if one replaced “definable” by “∞-homogeneously Suslin”. There are two natural weakenings of homogeneity. First, a tree T on X × (ù × Y ) is κ-weakly homogeneous just in case it is κ- homogeneous when viewed as a tree on (X × ù) × Y . Thus the weakly homogeneous subsets of ù X are just the existential real quantifications of a homogeneous subsets of ù X × ù ù, and Martin’s [Mar70A] shows in effect that whenever κ is measurable, all Σ12 sets of reals are κ-weakly homogeneous. Second, a pair of e X × Y and X × Z respectively, are κ-absolute complements trees S and T , on iff V[G] |= p[S] = ù X \ p[T ] whenever G is V-generic for a poset of cardinality < κ. The fundamental Martin-Solovay construction, also from 1968 (see [MS69]), shows that every κ-weakly homogeneous tree has a κ-absolute complement. The projection of a κ-absolutely complemented tree is said to be κ-universally Baire. This concept was first explicitly isolated and studied by Feng, Magidor, and Woodin in [FMW92]. Any universally Baire set has the Baire property and is Lebesgue measurable, but one cannot show in ZFC alone that such sets must be determined. (See [FMW92].) On the other hand, if there are arbitrarily large Woodin cardinals, then for any set of reals A, A is κ-homogeneous for all κ iff A is κ-weakly homogeneous for all κ iff A is κ-universally Baire for all κ. (This is work of Martin, Solovay, Steel, and Woodin; see [Lar04, Theorem 3.3.13] for one exposition, and [SteA] for another.) Although our discussion of homogeneity has focussed on its use in situations where the Axiom of Choice and the existence of large cardinals is assumed, the concept is also quite important in contexts in which full AD is assumed. AD

6

JOHN R. STEEL

gives us not just measures, but homogeneity measures; indeed, assuming AD, a set of reals is homogeneously Suslin iff both it and its complement are Θ-Suslin. (This result of Martin from the 80’s can be found in [MS89].) The analysis of homogeneity measures is a central theme in the work of Kunen, Martin, and Jackson [Sol78A, Jac88, Jac99] which located the projective ordinals among the alephs. The reader should see Jackson’s surveys [Jac07A] and [Jac07B] for more on homogeneity and the projective ordinals in the AD context. §2. Construction methods. One could group the methods for producing useful Suslin representations as follows: (1) the Martin-Solovay construction, (2) trees to produce an elementary submodel, and (3) scale constructions using comparison games. We discuss these methods briefly: 2.1. The Martin-Solovay construction. The Martin-Solovay construction makes use of homogeneity. If T on X × Y is is κ- weakly homogeneous via the system of measures ì ~ , and |X | < κ, then the construction produces a tree ms(T, ì ~ ) which is a κ-absolute complement for T . The construction of ms(T, ì ~ ) is effective, and its basic properties can be proved to hold in ZF + DC. Martin and Solovay [MS69] applied it with T the Shoenfield tree for Σ12 and ì ~ its weak homogeneity measures implicit in Martin’s [Mar70A]. They showed thereby that if κ is measurable, then for any Σ13 formula ϕ, there is a tree U such that p[U ] = {x ∈ ù ù : ϕ(x)} is true in every generic extension of V by a poset of size < κ. The Martin-Solovay tree ms(T, ì ~ ) is definable from T and ì ~ . Now suppose T be on ù × Y . There is a simple variant of ms(T, ì ~ ) which is definable from S T and the restrictions of the measures in ì ~ to {L[T, x] : x ∈ ù ù}. Let us call this variant ms∗ (T, ì ~ ). If T is the Shoenfield tree, so that T ∈ L, then one can define these restricted weak homogeneity measures, and hence ms∗ (T, ì ~) itself, from the sharp function on the reals. Martin and Solovay showed this way that ∆14 is a basis for Π12 , and Mansfield later improved their result by showing the class of Π13 singletons is a basis for Π12 . (See [Man71].) These results are not optimal, however. We do not know whether one can get the optimal basis and uniformization results in the projective hierarchy using the Martin-Solovay construction. Under appropriate large cardinal hypotheses, the Martin-Solovay tree is itself homogeneous. (See [MS07] for a precise statement.) Thus under the appropriate large cardinal hypotheses, one can show via the Martin-Solovay construction that the pointclass Hom∞ is closed under complements and real quantification. 2.2. The tree to produce an elementary submodel. If a set A of reals admits a definition with certain condensation and generic absoluteness properties,

INTRODUCTION TO PART I

7

then A is universally Baire. More precisely, let κ be a cardinal, and ϕ(v0 , v1 ) a formula in the language of set theory, and t any set. Let ô > κ, X ≺ Vô be countable, and let M be the transitive collapse of X , with κ¯ and t¯ the images of κ and t under collapse. We say X is generically hϕ, Ai-correct iff whenever g is M -generic for a poset of size < κ¯ in M , then for all reals y ∈ M [g], ¯ y ∈ A ⇔ M [g] |= ϕ[y, t]. If the set of generically hϕ, Ai correct X is club in ℘ù1 (Vô ), then A admits a κ-absolutely complemented Suslin representation T . The construction of T is relatively straightforward: if (y, f) ∈ [T ], then f will have built an X in our ¯ club of generically correct hulls, together with a proof that M [g] |= ϕ[y, t], for some g generic over the collapse M of X . (We are not certain as to the origin of this construction. Woodin made early use of it. See [FMW92] or [SteA].) One can use either stationary tower forcing (cf. the Tree Production Lemma, [Lar04] or [SteA]) or iterations to make reals generic [Ste07B, § 7] to obtain, for various interesting hϕ, Ai, a club of generically hϕ, Ai-correct X . If one replaces Vô by an appropriate direct limit of mice, then the tree to produce an elementary submodel becomes definable, at a level corresponding to the definability of the iteration strategies for the mice in question. See the concluding paragraphs of [Ste95A], and [Ste07B, § 8]. One can use this to get optimal basis and uniformization results for various pointclasses, for example (Σ21 )L(R) . It is difficult to obtain the optimal basis and uniformization results for Π13 by these methods, but, building on work of Hugh Woodin, Itay Neeman has succeeded in doing so. (This work is unpublished.) 2.3. Propagation of scales using comparison games. The simplest method for obtaining optimally definable Suslin representations makes direct use of the determinacy of certain infinite games. It was discovered in 1971 by Mosˆ chovakis, who used it to extend the Novikoff-Kondo-Addison theorems to the higher levels of the projective hierarchy. (The original paper is [Mos71A]; see also [KM78B] and [Mos80, Chapter 6].) As part of this work, Moschovakis introduced the basic notion of a scale, which we now describe. Let T be a tree on ù × ë, and A = p[T ]. One can get a “small” subtree of T which still projects to A by considering only ordinals < ë which appear in some leftmost branch. The scale of T does this, then records the resulting subtree as a sequence of norms, i.e. ordinal-valued functions, on A. More precisely, for x ∈ A and n < ù, put ϕn (x) = |hlx (0), ..., lx (n)i|lex , where for u ∈ ë ën . Then

n+1

, |u|lex is the ordinal rank of u in the lexicographic order on

ϕ ~ = hϕn : n < ùi is the scale of T . It has the properties:

8

JOHN R. STEEL

(a) Suppose that xi ∈ A for all i < ù, and xi → x as i → ∞, and for all n, ϕn (xi ) is eventually constant as n → ∞, then (i) (limit property) x ∈ A, and (ii) (lower semi-continuity) for all n, ϕn (x) ≤ the eventual value of ϕn (xi ) as i → ∞. (b) (refinement property) if x, y ∈ A and ϕn (x) < ϕn (y), then ϕm (x) < ϕm (y) for all m > n. A sequence of norms on A with property (a) is called a scale on A. Any scale on A can be easily transformed into a scale on A with the refinement property. If ϕ ~ is a scale on A, then we define the tree of ϕ ~ to be Tϕ~ = {(hx(0), ..., x(n − 1)i, hϕ0 (x), ..., ϕn−1 (x)i) : n < ù and x ∈ A}. It is not hard to see that p[Tϕ~ ] = A. If ϕ ~ has the refinement property, and ø ~ is the scale of Tϕ~ , then ø ~ is equivalent to ϕ, ~ in the sense that for all n, x and y, øn (x) ≤ øn (y) iff ϕn (x) ≤ ϕn (y). The reader should see [KM78B, 6B] and [Jac07B, § 2] for more on the relationship between scales and Suslin representations. There are least two benefits to considering the scale of a tree: first, it becomes easier to state and prove optimal definability results, and second, the construction of Suslin representations using comparison games becomes clearer. Concerning definability, we have Definition 2.1. Let Γ be a pointclass, and ϕ ~ a scale on A, where A ∈ Γ; then we call ϕ ~ a Γ-scale on A just in case the relations R(n, x, y) ⇔ x ∈ A ∧ (y 6∈ A ∨ ϕn (x) ≤ ϕn (y)), and S(n, x, y) ⇔ x ∈ A ∧ (y 6∈ A ∨ ϕn (x) < ϕ(y)) are each in Γ. We say Γ has the scale property just in case every set in Γ admits a Γ-scale, and write Scale(Γ) in this case. Moschovakis showed that if Γ is a pointclass which is closed under universal real quantification, has other mild closure properties, and has the scale property, then every Γ relation has a Γ uniformization, and the Γ singletons are a basis for Γ. [KM78B, 3A-1]. He also showed that assuming ∆12n determinacy, both Π12n+1 and Σ12n+2 have the scale property [KM78B, 3B,e3C]. From this, ˆ one gets the natural generalization of Novikoff-Kondo-Addison to the higher levels of the projective hierachy. Moschovakis’ construction of scales goes by propagating them from a set A to a set B obtained from A via certain logical operations. One starts with the fact that Σ01 has the scale property, and uses these propagation theorems to obtain definable scales on more complicated sets. The propagation works at the level of the individual norms in the scales.

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For example, if ϕ is a norm of A, where A ⊆ X × ù Y , and B(y) ⇔ ∃xA(x, y), then we obtain the “inf ” norm on B by setting ø(y) = inf{ϕ(x, y) : A(x, y)}. If either X is an ordinal, or X = ù ù, then inf norms can be used to transform a scale on A into a scale on B. (See [KM78B, 3B-2].) This transformation has a simple meaning in terms of the tree of the scale; if X = ù ù, it corresponds to regarding a tree on (Y × ù) × κ as a tree on Y × (ù × κ). Definable scales do not propagate under negation or universal quantification over ordinals. (Otherwise, it would be possible to assign to each countable ordinal α a scale on the set of wellorders of ù of order type α, in a definable way. This would then yield a definable function picking a codes for the countable ordinals.) Moschovakis’ main advance in [Mos71A] was to show that universal quantification over the reals propagates definable scales. Here it is definitely important to work with scales, rather than their associated trees. As before, the propagation takes place at the level of individual norms. Let ϕ be a norm on A, where A ⊆ R × Y , and let B(y) ⇔ ∀xA(x, y). To each y ∈ B, we associate fy : R → OR, where fy (x) = ϕ(x, y). Our norm on B records an ordinal measure of the growth rate of fy . Namely, given f, g : R → OR, we let G(f, g) be the game on ù: I plays out x0 , II plays out x1 , the players alternating moves as usual. Player II wins iff f(x0 ) ≤ g(x1 ). (Thus a winning strategy for II witnesses that g grows at least as fast as f, in an effective way.) Now put f ≤∗ g ⇔ II has a winning strategy in G(f, g). Granted full AD, one can show ≤∗ is a prewellorder of all the ordinal-valued functions on R, and granted only determinacy for sets simply definable from ϕ, one can show that ≤∗ prewellorders the fy for y ∈ B. Our norm on B is then given by ø(y) = ordinal rank of fy in ≤∗ ↾ {fz : B(z)}. (See [KM78B, 2C-1].) The norm ø is generally called the “fake sup” norm obtained from ϕ; the ordinal ø(y) measures how difficult it is to verify A(x, y) at arbitrary x. The fake-sup construction was first used in [AM68], to propagate the prewellordering property, which involves only one norm. Granted enough determinacy, the construction can be used to transform a scale on A into a scale on B, where B(y) ⇔ ∀xA(x, y). The key additional idea is to record, for each

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basic neighborhood Ns , the ordinal rank of fy ↾ Ns in ≤∗ ↾ {fz ↾ Ns : B(z)}. See [KM78B, 3C-1]. Using more sophisticated comparison games, one can combine the techniques for propagating scales under universal and existential real quantification, as well as existential ordinal quantification. This leads to the propagation of scales under various game quantifiers. We shall discuss these results in more detail in the next section. Although the fake-sup method of propagating scales was invented in order to obtain optimally definable scales, one can show that under AD, the tree of the scale it produces is very often homogeneous. (The tree of any scale is the surjective image of R, so it is too small to be homogeneous in V.) See [MS07], where it is also shown that the tree very often has the “generic codes” property of [KW07]. §3. Individual papers. We pass to an extended table of contents for the papers in the block to follow, together with pointers to some related results and literature. We also include a number of proof sketches. Some of these sketches will only make sense to readers with significant background knowledge. We have included references to fuller explanations in the literature when possible. Notes on the theory of scales [KM78B]. This is a survey paper, written in 1971. It is still an excellent starting point for anyone seeking basic information regarding the construction and use of scales under determinacy hypotheses. It is truly remarkable how much of the descriptive set theory that is founded on large cardinals and determinacy emerged in the early years of the subject. The paper begins in §2 – §4 with the inf and fake-sup constructions, and their corollaries regarding the scale property and uniformization in the projective hierarchy. Theorem 3.1 (Moschovakis 1970). Assume all ∆12n games are determined; e then (1) Π12n+1 and Σ12n+2 have the scale property, and hence (2) every Π12n+1 (respectively Σ12n+2 ) relation on R can be uniformized by a Π12n+1 (respectively Σ12n+2 ) function. In §6, the projective ordinals ä 1n := sup{α : α is the order type of a ∆1n prewellorder of R} e e are introduced. One can show that, assuming PD, any Π12n+1 -norm on a complete Π12n+1 set has length ä 12n+1 ; see [Mos80, 4C.14]. From the scale e all Π1 sets are ä 1 -Suslin, and thence property for Π12n+1 one then gets that 2n+1 2n+1 that all Σ12n+2 sets are ä 12n+1 -Suslin. (For n = 0, thisereduces to the classical e

INTRODUCTION TO PART I

11

Novikoff-Kondoˆ result that all Σ12 sets are ù1 -Suslin.) The size of the projective ordinals, both in inner models of AD, and in the full universe V, is therefore a very important topic. It is a classical result that ä 11 = ù1 , while the size of e much later work, some the larger projective ordinals has been the subject of of which will be collected in a block of papers in a later volume in this series. §7 proves the Kunen-Martin theorem: Theorem 3.2 (Kunen, Martin). Every κ-Suslin wellfounded relation on R has rank < κ + . This basic result has important corollaries concerning the sizes of the projective ordinals. For example, because all Σ12n+2 sets are ä 12n+1 -Suslin, we have e that ä 12n+2 ≤ (ä 12n+1 )+ , and in particular, ä 12 ≤ ù2 . e e e §8 investigates the way in which Suslin representations yield ∞-Borel representations. It is shown that κ-Suslin sets are κ ++ -Borel (i.e. can be built up from open sets using complementation and wellordered unions of length < κ ++ ). Of course, if CH holds, then every set of reals is a union of ù1 singletons; the true content of the result of §8 lies in the fact that the κ ++ -Borel representation is definable from the κ-Suslin representation. §8 also shows that, assuming PD, every ∆12n+1 set is ä 12n+1 -Borel. If n = 0, this is just Suslin’s original theorem. In ordereto obtain aeconverse when n > 0, we must impose a definability restriction on our ä 12n+1 -Borel representation, since again, it could be that every set of reals is ùe1 + 1-Borel. One way to do that is to assume full AD, and Martin showed that indeed, assuming AD, every ∆12n+1 set is e ä 12n+1 -Borel. So we have e Theorem 3.3 (Martin, Moschovakis). Assume AD; then the ∆12n+1 sets of e reals are precisely the ä 12n+1 -Borel sets. e See [Mos80, 7D.9]. This fully generalizes Suslin’s 1917 theorem to the higher levels of the projective hierarchy. §5 and §9 introduce inner models, obtained from Suslin representations, which have certain degrees of correctness. In §5, it is shown that for n ≥ 2, there is a unique, minimal Σ1n -correct inner model Mn∗ containing all the ordinals; the model is obtained by closing under constructibility and an optimally definable Skolem function for Σ1n . (Kechris and Moschovakis call this model Mn —not to be confused with Mn ; see below.) §9 considers the model L[T ], where T is the tree of a Π12n+1 scale on a complete Π12n+1 set. These models have proved more important in later work than the Mn∗ . It is shown that if n = 0, then L[T ] = L; in particular, L[T ] is independent of the Π12n+1 scale and complete set chosen. Moschovakis conjectured that L[T ] is independent of these choices if n > 0 as well, and more vaguely, that it is a “correct higher level analog of L”. Becker’s paper [Bec78] contains an excellent summary of what was known in 1977 about the models of §5 and §9. The independence conjecture, which

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inspired a great deal of work, became the third Victoria Delfino problem. Harrington and Kechris [HK81] made a significant advance by showing that the reals of L[T ], where T is the tree of any Π12n+1 scale on a complete Π12n+1 set, are the largest countable Σ12n+2 set of reals, and hence independent of the choice of T . Building on this work, Moschovakis made a step forward in the late 70’s with the introduction of the model HΓ, for Γ a pointclass which resembles Π11 in a certain technical sense, and has the scale property. (See [Mos80, 8G.17 ff.].) Assuming ∆12n -determinacy, the pointclass Π12n+1 is an example of such a Γ, e many more examples. The model H is of the form L[U ], where but there are Γ U is a universal ∃R Γ (in the codes) subset of the prewellordering ordinal of Γ, and one can think of it as a fragment of HOD corresponding to Γdefinability. Using the Harrington-Kechris work, Moschovakis showed that HΓ is independent of the universal set and Γ-norm used to define U , that it includes L[T ], for the tree T of a Γ scale on a complete Γ set, and that R ∩ HΓ is the largest countable ∃R Γ set of reals. (See [Mos80, 8G.17 ff.].) Moschovakis’ results require a bit more than Γ-determinacy. The independence of L[T ] was finally provedeby Becker and Kechris [BK84], who showed Theorem 3.4 (Becker, Kechris 1984). Let Γ be a pointclass which resembles Π11 and has the scale property, and suppose AD holds in L(Γ, R). Let T be the tree of any Γ-scale on a complete Γ set; then L[T ] = HΓ . e The Becker-Kechris proof makes heavy use of a class of games introduced by Martin in order to obtain an approximation to Theorem 3.4. Not long after the last of the Cabal Seminar volumes appeared, our understanding of the large cardinal side of the “equivalence” between large cardinals and determinacy caught up with our understanding of the determinacy side. This equivalence is mediated by the canonical inner models for large cardinal hypotheses, which are sometimes called extender models. We can now identify each of the models of §5 and §9 as an extender model, and thereby understand it much more deeply than we could using only pure descriptive set theory. For example, most nontrivial facts in the first order theory of L[T ] (e.g., that the GCH, and Jensen’s diamond and square principles, hold in L[T ]) seem to require its identification as an extender model for proof. The identifications are as follows: Here and in the rest of the paper, for 0 ≤ n ≤ ù, we let Mn be the minimal iterable proper class extender model with n Woodin cardinals. If L[M |ã] n ≥ 2 is even, then Mn∗ is L[Mn−2 |ã], where ã is least such that ã = ù1 n−2 M and L[Mn−2 |ã] is Σ1n -correct. (For n > 2, we have that ã < ù1 n−2 .) If n is ∗ odd, then Mn is the minimal proper class extender model Q such that if S is an initial segment of Q projecting to ù, then Mn−2 (S)# is an initial segment of Q. These identifications are implicit in [Ste95B]. Finally, if n ≥ 3 is odd, and T is the tree of a Π1n scale on a complete Π1n set, then there is an iterate Q

INTRODUCTION TO PART I

13

of Mn−1 such that L[T ] = L[Q|ä 1n ]. This identification is implicit in [Ste95A], e where the parallel fact with the pointclass Πn replaced by ΣL(R) , and Mn−1 1 replaced by Mù , is proved. So we have Theorem 3.5 (Steel 1994). Assume there are ù Woodin cardinals with a measurable above them all, and let Γ = Π12n+1 or Γ = ΣL(R) ; then HΓ is an 1 iterable extender model. In a similar vein, the prewellordering and scale theorems of §2 - §4 can now be proved using extender models. In the prewellordering case, the proof is due to Woodin, and in the scale case, to Neeman; in neither case is the proof published, but see [Ste95B]. These proofs require significantly more theory than the comparsion game approach, but in some ways they give deeper insight into the meaning of the norms being constructed. Finally, Suslin and ∞-Borel representations are related to Lebesgue measurability, the Baire property, and the perfect set property in §10 and §11. Solovay’s breakthrough results from 1966 on the regularity of Σ12 sets under large cardinal hypotheses [Sol66] are thereby extended to other pointclasses. A basic result on the existence of largest countable sets is proved (in effect): Theorem 3.6 (Kechris, Moschovakis). Suppose Γ is adequate, ù-parametrized, has the scale property, and is closed under ∃R , and suppose all Γ games e are determined; then there is a largest countable Γ set of reals. When it exists, the largest countable Γ set is called CΓ . The theorem is implicit in the proof of Theorem 11B-2, which proves the existence of CΓ for Γ = Σ12n . Kechris’ paper [Kec75] contains further basic information in this area. The sets CΓ are quite important, partly because many of them show up naturally as the set of reals in some canoncal inner model. For example, Solovay showed that CΣ12 = R ∩ L [KM78B, 11B-1], and we now know that for any n, CΣ12n+2 = R ∩ M2n . (See [Ste95B]. Note that M0 = L.) In general, under the hypotheses of Theorem 3.4, we have C∃R Γ = R ∩ HΓ . (See [Mos80, 8G.29].) Kechris [Kec75] shows that assuming Π12n+1 - determinacy, there is a largest e countable Π12n+1 set of reals CΠ12n+1 . This result is due to Guaspari, Kechris, and Sacks for n = 0, in which case CΠ12n+1 has an inner-model-theoretic meaning as the set of reals ∆12n+1 -equivalent to the first order theory of some level of M2n projecting to ù. It is open whether this characterization of CΠ12n+1 holds also for n > 0. Propagation of the scale property using games [Nee07]. Scales on Σ11 sets [Ste83B]. e Moschovakis unified his results on scale propagation under the real quantifiers into a single theorem on the propagation of scales under the game

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quantifier on ù. Letting A ⊆ R × R, we put ayA(x, y) ⇔ ∃n0 ∀n1 ∃n2 ∀n3 ...A(x, hni : i < ùi), where we interpret the right hand side as meaning its quantifier string has a Skolem function, that is, that player I wins the game on ù with payoff Ax = {y : A(x, y)}. We write aA for {x : ayA(x, y)}, and if Γ is a pointclass, we set aΓ = {aA : A ∈ Γ}. The following is often called the third periodicity theorem. It dates from approximately 1973; see [Mos73] or [Mos80, 6E]. Theorem 3.7 (Moschovakis). Let Γ be an adequate, ù-parameterized pointclass closed under quantification over ù, and suppose Γ(x)-determinacy holds for all reals x. Suppose Γ has the scale property; then (a) aΓ has the scale property, and (b) if G is a game on ù with payoff set in Γ, and the player whose payoff is Γ has a winning strategy in G, then that player has a aΓ winning strategy. The proof involves a more sophisticated comparison game: given a norm ϕ on A, one gets a norm on aA using comparison games in which the two players play out the games with payoff Ax1 and Ax2 simultaneously, in different roles on the two boards, each trying to win in his role as player I with lower ϕ-norm than the other. The first paper in the present pair gives a thorough exposition of the proof of this theorem. (See also [Mos80, 6E].) It is easy to see that aΠ1n = Σ1n+1 , and assuming Σ1n -determinacy, that e Thus Theorem 3.7 aΣ1n = Π1n+1 . Setting Σ10 = Σ01 , this is true for n = 0 as well. subsumes Theorem 3.1. Part (b) of Theorem 3.7, on the existence of canonical winning strategies, is very useful. In the special case of projective sets, we get Corollary 3.8 (Moschovakis). Assume ∆12n -determinacy, and let G be a e with Σ1 payoff has a winning game with Σ12n payoff, and suppose the player 2n strategy; then he has a ∆12n+1 winning strategy. Moschovakis’ proof used Σ12n -determinacy, but Martin later showed this e so we have stated the theorem in its sharper follows from ∆12n -determinacy, e form. Of course, we also get ∆12n+2 strategies for games won by a player with Π12n+1 payoff from Corollary 3.8, but this already follows easily from the basis theorem for Π12n+1 . It is easy to see that these definability bounds on winning strategies are optimal. It is natural to ask what are the optimally definable scales and winning strategies for the projective pointclasses which zig when they should have zagged, that is, for Σ12n+1 and Π12n+2 . The second paper in this pair gives part of the answer. Let α-Π1 be the α th level of the difference hierarchy over Π1 1

1

INTRODUCTION TO PART I

(see [Ste83B]). and let Λ0 =

[

15

ùk-Π11 .

k<ù

Steel gives a simple proof in [Ste83B] that every Σ11 set admits a very good scale whose associated prewellorders are each in Λ0 , and in fact, each set in Λ0 admits a very good scale whose associated prewellorders are all in Λ0 . (We are not demanding that the sequence of prewellorders be in Λ0 .) Now let a(n) = a....a be the n-fold composition of the game quantifier on ù; then the proof of third periodicity theorem easily gives Theorem 3.9 (Steel 1980). Let n ≥ 1, and suppose all a(n−1) Λ0 (x) games are determined, for all reals x; then (a) every a(n) Λ0 set admits a very good scale, all of whose norms are a(n) Λ0 , and (b) if G is a game with payoff in a(n−1) Λ0 , then there is a winning strategy ó for G such that for any k, ó ↾ {p : lh(p) ≤ k} is in a(n) Λ0 . It is easy to see that for n ≥ 1, (Σ1n ∪ Π1n ) ⊆ a(n−1)

[

ùk-Π11 ⊆ ∆1n+1 .

k<ù

The best bounds on the definability of very good scales and winning strategies for Σ1n sets (n odd) and Π1n sets (n even) are just those given by Theorem 3.9 and this inclusion. That the bounds cannot be improved follows from Martin [Mar83A]; see below. (We should note here that Busch [Bus76] showed that every Σ11 set admits a scale all of whose prewellorders are (ù + 3)-Π11 . However, the Busch scale is not very good, and transforming it to a very good scale involves taking intersections, which drives us up to Λ0 . The third periodicity propagation technique requires, in effect, that the input scale be very good.) The progress of inner model theory has shed some light on these results. Neeman [Nee95] gives an inner-model-theoretic proof that every Σ12n game won by the player with Σ12n payoff has as ∆12n+1 winning strategy, as a byproduct of his proof of Σ12n determinacy from the existence and iterability of M2n−1 . Neeman’s work also gives an insight into the pointclasses a(n) Λ0 . For n ≥ 0, let Tkn be the theory in Mn of its first k indiscernibles. Thus the reals in Mn are just those reals which are recursive in some Tkn . One can show that every an+1 ùk-Π11 real is recursive in Tkn , and that Tkn itself is a(n+1) ù(k + 1)-Π11 . The proof is an induction on n, with the base case n = 0 being due to Martin, as part of his proof of ùk-Π11 -determinacy from the existence of the sharp of M0 = L. ( Here is a proof sketch of the n > 0 case for experts: To reduce a an+1 ùk-Π11 real to Tkn , we ask questions about what is forced in collapse of the bottom Woodin of Mn about its first k indiscernibles. The answer we get

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will reflect an+1 ùk-Π11 truth because every real, and in particular a winning strategy witnessing or refuting the outer a quantifier, is generic over an iterate of Mn for this collapse. To show that Tkn itself is a(n+1) ù(k + 1)-Π11 , we use a game in which the players play a putative M#n ’s, say P and Q respectively, and then the two are coiterated inside Mn−1 (hP, Qi).) It follows that M#n is Turing equivalent to the set of true a(n+1) Λ0 sentences. From this we see that any game with a(n) Λ0 payoff has a winning strategy which is recursive in M#n . (By Theorem 3.14(b) below, no better definability bound is possible.) In particular, every nonempty Σ12n+1 set has a member recursive in M#2n , using the trivial game in which I must play a member of the set and a witness to the Π12n matrix, and II does nothing. This gives us an inner-model-theoretic proof of Martin and Solovay’s generalization of the Kleene Basis Theorem for Σ11 [KMS83, 5.6]. Inductive scales on inductive sets [Mos78]. Scales on coinductive sets [Mos83]. The extent of scales in L(R) [MS83]. It is natural to try to extend the civilizing influence of definable scales to more complicated sets. The remaining papers in this block use the comparison game construction to do that, while showing that, most of the time, the scales produced are definable in the simplest possible logical form. The papers in this group, which represent work done in late 1979, exploit the uniformities in the comparison game method of propagating scales. Let us use ∃Ord , ∃R , and ∀R to stand for existential quantification over the ordinals, over the reals, and universal quantification over the reals, respectively. Because the propagation of scales under these operations is uniform in the scales, one gets inductive scales on inductive sets; this is done in [Mos78]. (A set is Jκ (R) inductive iff it is Σ1 R , where κR is least κ such that Jκ (R) |= KP.) Since AD implies that the pointclass of inductive sets is closed under real quantification and wellordered unions, it seemed at first that one needed a radically new idea to go further. (One cannot hope to show that the class of scaled sets is closed under complement!) The existence of definable scales for coinductive sets became the second Victoria Delfino problem. However, it turned out that what was missing was more in the nature of a subtle observation: the comparison game propagation of scales under ∃or , ∃R , and ∀R acts at the level of individual norms–it corresponds, in each case, to a continuous operation on the input scale. Moschovakis realized this, and realized that it could be used to define scales on any set A definable in the form A(x) ⇔ ∃x0 ∃α0 ∀x1 ∃x2 ∃α1 ∀x3 ...∀nR(hx0 |n, ..., xn |ni, hα0 , ..., αn i), where R is definable, the α’s are ordinals, and the x’s are reals. (This is done by simultaneously defining scales on each of the ù-many sets defined by the

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formula on the right with some initial segment of its quantifiers removed. The scale on any such set is obtained from the scale on the set corresponding to removing one more quantifier by the continuous operation corresponding to that quantifier.) The expression displayed on the right hand side above gives what is called a closed game representation of A: it asserts that player I wins the infinite game in which he plays the even x’s and the α’s, while II plays the odd x’s, and the payoff indicated by the matrix is closed in the product of the Baire topology on R and the discrete topology on the ordinals. What [Mos83] shows, in effect, is that granted sufficient determinacy, any set with a closed game representation admits a definable scale. (The converse is trivial.) In the special case that the game involves no ordinal moves, one gets Theorem 3.10 (Moschovakis 1979). Suppose all games in JκR +1 (R) are determined, and let A be coinductive; then A admits a scale whose associated prewellorders are each in JκR +1 (R). Martin and Steel showed in [MS83] that in fact, every ΣL(R) set admits a 1 closed game representation in L(R), which together with Moschovakis’ work and some simple definability calculations implies Theorem 3.11 (Martin, Steel 1979). Assume ADL(R) ; then the pointclass has the scale property.

ΣL(R) 1

Kechris and Solovay had observed earlier that, assuming AD, the relation “x, y ∈ R and y is not ordinal definable from x” is ordinal definable, but admits no uniformization, and hence no scale, which is ordinal definable from a real. (Let f be a uniformizing function, and suppose f is ordinal definable from x; then f(x) is ordinal definable from x, a contradiction.) If V=L(R), then this relation is Π1 , while every set whatsoever is ordinal definable from a real, so we have a Π1 set which admits no scale at all. A simple Wadge argument then shows that assuming ADL(R) , the sets admitting scales in L(R) are precisely the ΣL(R) sets. 1 Under suitable large cardinal assumptions, one can construct natural models of AD properly larger than L(R). These models, and L(R) itself, all satisfy a certain strengthening of AD called AD+ . The theory AD+ was isolated by Woodin, and part of his work is the following far-reaching generalization of Theorem 3.11: Theorem 3.12 (Woodin, mid 90’s). Assume AD+ ; then the pointclass Σ21 has the scale property. Note here that ΣL(R) = (Σ21 )L(R) , so that Woodin’s theorem reduces to that 1 of Martin and Steel if our model of AD+ is L(R). No proof of Theorem 3.12 has been published as yet, but the theory AD+ is described in [Woo99, Section

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9.1], where Theorem 3.12 and related results are stated as Theorem 9.7. A proof that Σ21 has the scale property in those models of AD+ obtained from models with large cardinals via the standard means, i.e. the derived model construction, is exposited in [SteA, §7]. The largest countable this, that, and the other [Mar83A]. Moschovakis [Mos83] shows that the norms of the scale on a coinductive set it constructs are each first order definable over JκR (R). It is natural to ask whether one can do better: does every coinductive set admit a scale such that Jκ (R) for some fixed n < ù, all the norms are Σn R ? In [Mar83A], Martin proves e an important reflection result which implies that the answer is “no”. Let us α write y ∈ OD (x) to mean that y is ordinal definable from x over Jα (R). Theorem 3.13 (Martin 1980). Assume inductive determinacy, and suppose x, y ∈ R and y ∈ ODκR (x); then y ∈ ODα (x) for some α < κR . (Though not literally stated in [Mar83A], this is a fairly direct consequence of [Mar83A, Lemma 4.1].) Now the relation “x, y ∈ R and ∀α < κR (y 6∈ ODα (x))” is coinductive, and by the Kechris-Solovay argument, it cannot be uniformized by a function in JκR +1 (R), and hence it admits no scale whose sequence of associated prewellorders is in JκR (R). Thus one cannot improve Moschovakis’ definability bound. Martin’s reflection result is part of a characterization of CΓ , of the largest Jκ countable Γ set of reals, for various pointclasses Γ. Letting IND = Σ1 R (R) be the pointclass of (lightface) inductive sets, it is easy to see that assuming inductive determinacy, CIND = {y : y ∈ ODκR (∅)}. (See [Ste83A, 2.11].) This characterizes the members of CIND in terms of definability from ordinals, in a way which parallels Kechris’ characterization of CΣ12n as {y : y is ∆12n in a countable ordinal}. Martin found characterizations of CIND and CΣ12n in terms of definability wihout parameters: Theorem 3.14 (Martin 1980). Assume all games in JκR +1 (R) are determined; then for any real y (a) y ∈ CIND iff y is definable over JκR (R) from no parameters, and S (b) y ∈ CΣ12n iff y is a(2n−1) k<ù ùk-Π11 . The left-to-right directions make use of the existence of scales on coinductive sets in case (a), and on Π12n sets in case (b), which are definable in the appropriate way: each norm being definable over JκR (R) in case (a), each S norm being a(2n−1) k<ù ùk-Π11 in case (b). The right-to-left directions are reflection arguments, and they show that the definability bounds for scales used in the other direction are best possible. Soon after Martin proved Theorem 3.14, Kechris and Woodin used his technique, among other ideas, to prove a vaguely similar reflection result: if

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there is a non-determined game in JκR +1 (R), then there is a non-determined game in JκR (R). It follows that the hypothesis of Moschovakis’ scale existence result Theorem 3.10 can be reduced to JκR (R)-determinacy. See [KW83], which is reprinted in Part II of this volume. Somewhat more general results along the same lines were a key ingredient in the proof of Theorem 3.15 (Kechris, Woodin 1980). If there is a non-determined game in L(R), then there is a non-determined game whose payoff is Suslin in L(R). The structure of this proof has played an important role in later proofs of ADL(R) under various hypotheses. See below. Kechris-Woodin [KW83] also proves something along the lines of Theorem S 3.14(b): ∆12n -determinacy implies a2n−1 k<ù ùk-Π11 -determinacy. (Martin e ∆1 -determinacy implies Σ1 -determinacy earlier, by a different had proved 2n 2n e method. Seee[KS85].) Once again, the progress of inner model theory has given us deeper insight into these results of Martin and Kechris-Woodin. Itay Neeman found an inner-model-theoretic proof of Theorem 3.14(b). Let Mn be the minimal iterable proper class extender model with n Woodin cardinals, and Tkn be the theory in Mn of its first k indiscernibles. By [Ste95B], the reals in CΣ12n are precisely the reals in M2n , and hence are just those reals which are recursive in some Tk2n . But by [Nee95], every a2n−1 ùk-Π11 real is recursive in Tk , and that Tk itself is a(2n−1) ù(k + 1)-Π11 . (We sketched this proof above.) Theorem 3.14(b) now follows easily. Mitch Rudominer found an inner-model-theoretic proof of Theorem 3.14(a); in this case, the role of M2n is played by the minimal iterable extender model P having ù Woodin cardinals, which by minimality are cofinal in the ordinals of P, and the role of Tk2n is played by the Σ0 theory in P of its first k Woodin cardinals. See [Rud99]. It is worth noting that the set of reals in M2n+1 is also well known from descriptive set theory; it is the set Q2n+1 . See [KMS83] for the many characterizations of this set. In general, the reals of Mn , for any n, can be characterized in terms of ordinal definability as those reals which are ∆1n+2 in a countable ordinal. Neeman and Woodin have proved the Kechris-Woodin theorem within the projective hierarchy by the methods of inner model theory, and at the same time generalized it to the odd levels as well. Woodin (unpublished) showed that for any n ≥ 1, Π1n -determinacy implies that for all reals x, Mn (x)# exe ists, and S Neeman1 [Nee95] showed that the existence of these mice implies n−1 a k<ù ùk-Π1 -determinacy. For n = 1, these are results of Harrington [Har78] Martin (unpublished) respectively. It is known from work of Kechris and Solovay [KS85] that these “transfer results” for determinacy in the projective hierarchy cannot be improved.

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No one has as yet proved that JκR (R)-determinacy implies JκR +1 (R)-determinacy by purely inner-model-theoretic methods. Scales in L(R) [Ste83A]. Scales in K(R) [Ste07C]. Given that a set of reals admits a scale in L(R), it is natural to ask what is the least level of the Levy hierarchy for L(R) at which such a scale appears. The papers in the last group answered this question in some important cases. The paper [Ste83A], work of Steel from 1980, knits the arguments of those papers together into a complete answer. It turns out that the (lexicographically) least hã, ni such that A admits a J (R) J (R) Jã (R) scale is the lexicographically least hã, ni such that A ∈ Σnã and Σnã Σn e e e J (R) has the scale property. [Ste83A] characterizes those pointclasses Σnã which e have the scale property in terms of reflection properties. The key concept is that of a Σ1 -gap. Let us say M ≺R 1 N iff M is an elementary submodel of N with respect to Σ1 formulae about parameters from R ∪ {R}. Definition 3.16. The interval [α, â] is a Σ1 -gap iff 1. Jα (R) ≺R 1 Jâ (R), 2. ∀ã < α(Jã (R) 6≺R 1 Jα (R)), and 3. Jâ (R) 6≺R J (R). 1 â+1 With the convention that [(ä 21 )L(R) , ∞] is also a Σ1 gap, we have that Σ1 gaps e J (R) partition OR. In order to determine whether Σnã has the scale property, we e consider the unique Σ1 gap [α, â] to which ã belongs. [Ste83A] shows that, assuming enough determinacy, J (R)

does not have the scale property (Kechris, (1) if α < ã < â then Σnã e Solovay), J (R) (2) if ã = α and n = 1, then Σnã has the scale property, (3) if ã = α, n > 1, and α iseadmissible, or if ã = â and [α, â] is a strong J (R) gap, then Σnã does not have the scale property (Martin), e (4) if ã = α, n > 1, and α is inadmissible, or if ã = â and [α, â] is a J (R) weak gap, and ñn (Jâ (R)) = R, then Σnã or its dual class has the scale e property, according to the “zig-zag” pattern. The weak/strong distinction for gaps is motivated by Martin’s proof of Theorem 3.14; strong gaps have a reflection property which is used in this argument. The most important applications of this analysis occur in inductive proofs of ADL(R) . The first of these, which set the pattern, is the Kechris-Woodin proof of Theorem 3.15, that Suslin determinacy implies determinacy in L(R). Their argument goes roughly as follows: let â be least such that there is a non-determined game in Jâ+1 (R). The failure of determinacy is a Σ1 fact,

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so â ends a gap of the form [α, â]. Setting ã = â, and letting n be least J (R) such that there is a non-determined Σnâ game, we have enough determinacy e to prove (1)-(4) above. (This comes from inspecting the proof of [Ste83A], and using an observation of Kechris.) The Kechris-Woodin result that JκR (R)determinacy implies JκR +1 (R)-determinacy generalizes routinely so as to show that we cannot be in case (3). In all other cases, we have enough determinacy J (R) Jâ (R) to show that either Σnâ or Σn+1 has the scale property. Thus the payoff of e e our non-determined game is Suslin in L(R), a contradiction. Kechris and Woodin used Theorem 3.15 to show that if V=L(R) and there are arbitrarily large strong partition cardinals below Θ, then AD holds. With the proofs of determinacy from Woodin cardinals in the mid-80’s [MS89], and the advances in techniques for constructing correct inner models with Woodin cardinals in the late 80’s and 1990 [MS94, MS94, Ste96], it became possible to use this pattern of argument to prove ADL(R) under many different hypotheses. Woodin pioneered this core model induction method, in his 1990 proof that the Proper Forcing Axiom together with the existence of an inaccessible cardinal implies ADL(R) . In such an argument, one proves not just determinacy, but the existence of correct mice “certifying” the determinacy in question, by an induction. The scale analysis is used to get a definable scale on a set coding truth at the next level of correctness, and then core model theory is used to construct mice correct at that level. The method has been used a number of times since 1990, by Woodin and others. (See [Ket00, Zob00].) Indeed, if the large cardinal strength of a theory T is not close to the surface, it is highly likely that any proof that T implies ADL(R) will use the core model induction method. Only very recently has a paper describing and using the method been published; see [Ste05]. [Ste07C] extends the scale analysis of [Ste83A] to K(R). This is useful in constructing mice with more than infinitely many Woodin cardinals by the core model induction method. The real game quantifier propagates scales [Mar83B]. Long games [Ste88]. The length-ù1 open game quantifier propagates scales [Ste07A]. These papers push the comparison game construction of scales as far as it has been pushed to date. It is natural to ask whether there is a propagation theorem behind Moschovakis’ Theorem 3.10. Martin obtains a positive answer in [Mar83B], showing that the game quantifier corresponding to games of length ù on the reals propagates definable scales. In fact, he shows that the game quantifier corresponding to games of any fixed countable length propagates definable scales. (Note here that a game of length α on R can be simulated by a game of

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length ùα on ù.) He also proves the other half of third periodicity, that there are definable winning strategies for these games. Both theorems require the determinacy of the games in question as a hypothesis. Steel extends these results to the game quantifiers corresponding to clopen games of length ù1 in [Ste88], and then finally to the game quantifier corresponding to open games of length ù1 in [Ste07A]. These latter results represent the limit of what has been done in this direction, so let us state them more precisely. Let A ⊆ R×ù <ù1 . For x ∈ R, consider the game of length ù1 in which I and II play natural numbers, alternating as usual, with I going first at limit ordinals. We fix some natural way of representing runs of such games by p ∈ ù1 ù, and then declare that p is a winning run for I in GAx iff ∃α < ù1 A(x, p ↾ α). Thus GAx is open for I in the topology on ù1 ù whose basis consists of sets of the form {p ∈ ù1 ù : q ⊂ p}, where q ∈ ù <ù1 . We then put x ∈ aù1 A ⇔ I has a winning strategy in GAx . In order to calculate definability, we code countable sequences q of natural numbers by reals q ∗ in some simple way. Putting A∗ (x, y) ⇔ ∃q(y = q ∗ ∧ A(x, q)), we then define, for any pointclass Γ, aù1 open-Γ = {aù1 A : A∗ ∈ Γ}. The main result of [Ste07A] is Theorem 3.17 (Steel). Let Γ be an adequate pointclass with the scale property, and suppose that all (boldface) length ù1 open-Γ games are determined; e then (a) aù1 open-Γ has the scale property, and (b) if the player with open payoff has a winning strategy in a length ù1 open-Γ game, then he has a aù1 open Γ winning strategy. Thus the determinacy of length ù1 open-∆12 games implies that aù1 open − ∆12 has the scale property. It is not known ehow to construct definable scales on the complements of aù1 open-∆12 sets, or show that if the closed player wins such a game, he has a definable winning strategy, under any definable determinacy or large cardinal assumption. We should note here that assuming the determinacy of the long games involved, the complements of aù1 open-∆12 sets are just the aù1 closed-∆12 sets. in the natural meaning of this term. All aù1 closed-∆12 sets are Σ21 , and assuming CH, all Σ21 sets are aù1 closed-∆12 . Woodin’s Σ21 absoluteness theorem [Lar04, Theorem 3.2.1] can be formulated without referring to CH as follows: if there are arbitrarily large measurable Woodin cardinals, then aù1 closed-∆12 sentences are absolute between setgeneric extensions of V. [Ste88, §3] uses a weaker form of this theorem to show that the determinacy of games ending at the first Σ2 admissible relative to the play, with

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∆12 -in-the-codes payoff, implies that there is an inner model of ADR containing all reals and ordinals. This was the first proof that some form of definable determinacy implies that there is an inner model of ADR . In the early 90’s, Woodin showed how to obtain such a model directly, under the optimal large cardinal hypothesis. This work is unpublished, but is exposited in [SteA]. [Ste88, §4] also proves the determinacy of games ending at the first Σ1 admissible relative to the play in the theory ZF + AD + DC+ “every set of reals admits a scale” + “ù1 is ℘(R)-supercompact”. The latter had been proved to have a model containing all reals and ordinals assuming ZFC+ “there is a supercompact cardinal” by Woodin (unpublished, but see [SteA]), so [Ste88, §4] completed the proof of a small fragment of clopen determinacy beyond fixed countable length from large cardinals. Neeman [Nee06] has since proved this fragment of clopen determinacy directly, from essentially the optimal large cardinal hypotheses. Although it is not known how to construct definable scales on aù1 closed-∆12 sets under any definable determinacy or large cardinal assumption, one can do so under the assumption that there are sufficiently strong countable iterable structures. What one needs is essentially a countable, ù1 + 1-iterable model ZF− + “there is a measurable Woodin cardnal”. (A slightly weaker theory suffices.) Woodin, and later independently Steel, showed that aù1 closed-∆12 truth can be reduced to truth in any such model, by asking what is forced in the extender algebra at its measurable Woodin cardinal. This yields a recursive function t such that if ϕ is a aù1 closed-∆12 formula, and x is a real, then ϕ(x) ⇔ M |= t(ϕ)[x], where M is any iterable model as above such that x ∈ M . (Sadly, this work is still unpublished. The arguments of [Ste07B, §7] are a prototype lower down.) We can now use the tree-to-produce-an-elementary-submodel method to get a definable scale. For let M0 be the minimal extender model satisfying ZF− + “there is a measurable Woodin cardinal”. M0 exists and is iterable by our assumption. Because it is minimal, M0 is pointwise definable, and hence by [Ste07B, 4.10] it has a unique ù1 + 1 iteration strategy with the Dodd-Jensen property. We can then define the direct limit M∞ of all countable iterates of M0 under the iteration maps given by this strategy, as in [Ste95A]. Our Suslin representation T of the universal aù1 closed-∆12 set is definable from M∞ , and hence definable. Roughly speaking, T builds along each branch a putative aù1 closed-∆12 truth ϕ(x), a model N and an embedding of N into M∞ , and a proof that x is N -generic for the extender algebra at the bottom Woodin cardinal of N , and that N [x] |= t(ϕ)[x]. It is much harder to construct definable winning strategies for aù1 closed-∆12 games granted the existence of an iterable model of ZF− + “there is a measurable Woodin cardnal”. One must show that there are any strategies at

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all, in the first place! Itay Neeman [Nee04, Nee] has made great progress on the questions of the existence and definability of winning strategies in length ù1 open-Hom∞ games. He shows that these strategies can be obtained by a logically simple transformation from iteration strategies for appropriate mice. In the case of length ù1 closed-∆12 , the appropriate mouse is essentially the minimal iterable mouse with a measurable Woodin cardinal. Letting M0 be as above, Neeman’s work shows that any length ù1 closed-∆12 game has a winning strategy which is easily definable from the unique iteration strategy for M0 , which by its uniqueness is itself definable. The problem of constructing iteration strategies for mice at the level of M0 and beyond has been one of the fundamental open problems in pure set theory since the mid 1980’s. Carthago delenda est! REFERENCES

John W. Addison and Yiannis N. Moschovakis [AM68] Some consequences of the axiom of definable determinateness, Proceedings of the National Academy of Sciences of the United States of America, no. 59, 1968, pp. 708–712. James Baumgartner, Donald A. Martin, and Saharon Shelah [BMS84] Axiomatic set theory. Proceedings of the AMS-IMS-SIAM joint summer research conference held in Boulder, Colo., June 19–25, 1983, Contemporary Mathematics, vol. 31, Amer. Math. Soc., Providence, RI, 1984. Howard S. Becker [Bec78] Partially playful universes, In Kechris and Moschovakis [Cabal i], pp. 55–90. Howard S. Becker and Alexander S. Kechris [BK84] Sets of ordinals constructible from trees and the third Victoria Delfino problem, In Baumgartner et al. [BMS84], pp. 13–29. Douglas R. Busch [Bus76] ë-Scales, κ-Souslin sets and a new definition of analytic sets, The Journal of Symbolic Logic, vol. 41 (1976), p. 373. Qi Feng, Menachem Magidor, and W. Hugh Woodin [FMW92] Universally Baire sets of reals, In Judah et al. [JJW92], pp. 203–242. Leo A. Harrington [Har78] Analytic determinacy and 0# , The Journal of Symbolic Logic, vol. 43 (1978), pp. 685–693. Leo A. Harrington and Alexander S. Kechris [HK81] On the determinacy of games on ordinals, Annals of Mathematical Logic, vol. 20 (1981), pp. 109–154. Stephen Jackson [Jac88] AD and the projective ordinals, In Kechris et al. [Cabal iv], pp. 117–220. [Jac99] A computation of ä 15 , vol. 140, Memoirs of the AMS, no. 670, American Mathematical e Society, July 1999. [Jac07A] Structural consequences of AD, In Kanamori and Foreman [KF07]. [Jac07B] Suslin cardinals, partition properties, homogeneity. Introduction to Part II, this volume, 2007.

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H. Judah, W. Just, and W. Hugh Woodin [JJW92] Set theory of the continuum, MSRI publications, vol. 26, Springer-Verlag, 1992. Akihiro Kanamori and Matthew Foreman [KF07] Handbook of set theory into the 21st century, Springer, 2007. Alexander S. Kechris [Kec75] The theory of countable analytical sets, Transactions of the American Mathematical Society, vol. 202 (1975), pp. 259–297. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Alexander S. Kechris, Donald A. Martin, and Robert M. Solovay [KMS83] Introduction to Q-theory, In Kechris et al. [Cabal iii], pp. 199–282. Alexander S. Kechris, Donald A. Martin, and John R. Steel [Cabal iv] Cabal seminar 81–85, Lecture Notes in Mathematics, no. 1333, Berlin, Springer, 1988. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. [KM78B] Notes on the theory of scales, this volume, originally published in Cabal Seminar 76–77 [Cabal i], pp. 1–53. Alexander S. Kechris and Robert M. Solovay [KS85] On the relative consistency strength of determinacy hypotheses, Transactions of the American Mathematical Society, vol. 290 (1985), no. 1, pp. 179–211. Alexander S. Kechris and W. Hugh Woodin [KW83] Equivalence of determinacy and partition properties, Proceedings of the National Academy of Sciences of the United States of America, vol. 80 (1983), no. 6 i., pp. 1783–1786. [KW07] Generic codes for uncountable ordinals, this volume, originally circulated manuscript, 2007. Richard O. Ketchersid [Ket00] Toward ADR from the continuum hypothesis and an ù1 -dense ideal, Ph.D. thesis, Berkeley, 2000. Stephen C. Kleene [Kle55] Arithmetical predicates and function quantifiers, Transactions of the American Mathematical Society, vol. 79 (1955), pp. 312–340. Motokiti Kondoˆ [Kon38] Sur l’uniformization des complementaires analytiques et les ensembles projectifs de la seconde classe, Japanese Journal of Mathematics, vol. 15 (1938), pp. 197–230. Paul B. Larson [Lar04] The stationary tower: Notes on a course by W. Hugh Woodin, University Lecture Series (AMS), vol. 32, Providence, RI, 2004. N. N. Luzin and P. S. Novikov [LN35] Choix effectif d’un point dans un complemetaire analytique arbitraire, donne par un crible, Fundamenta Mathematicae, vol. 25 (1935), pp. 559–560. Richard Mansfield [Man71] A Souslin operation on Π12 , Israel Journal of Mathematics, vol. 9 (1971), no. 3, pp. 367– 379.

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Donald A. Martin [Mar70A] Measurable cardinals and analytic games, Fundamenta Mathematicae, (1970), no. LXVI, pp. 287–291. [Mar83A] The largest countable this, that, and the other, this volume, originally published in Kechris et al. [Cabal iii], pp. 97–106. [Mar83B] The real game quantifier propagates scales, this volume, originally published in Kechris et al. [Cabal iii], pp. 157–171. Donald A. Martin and Robert M. Solovay [MS69] A basis theorem for Σ13 sets of reals, Annals of Mathematics, vol. 89 (1969), pp. 138–160. Donald A. Martin and John R. Steel [MS83] The extent of scales in L(R), this volume, originally published in Kechris et al. [Cabal iii], pp. 86–96. [MS89] A proof of projective determinacy, Journal of the American Mathematical Society, vol. 2 (1989), pp. 71–125. [MS94] Iteration trees, Journal of the American Mathematical Society, vol. 7 (1994), pp. 1–73. [MS07] The tree of a Moschovakis scale is homogeneous, this volume, 2007. William J. Mitchell and John R. Steel [MS94] Fine structure and iteration trees, Lecture Notes in Logic, vol. 3, Springer-Verlag, Berlin, 1994. Yiannis N. Moschovakis [Mos71A] Uniformization in a playful universe, Bulletin of the American Mathematical Society, vol. 77 (1971), pp. 731–736. [Mos73] Analytical definability in a playful universe, Logic, methodology, and philosophy of science IV (Patrick Suppes, Leon Henkin, Athanase Joja, and Gr. C. Moisil, editors), North-Holland, 1973, pp. 77–83. [Mos78] Inductive scales on inductive sets, this volume, originally published in Kechris and Moschovakis [Cabal i], pp. 185–192. [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. [Mos83] Scales on coinductive sets, this volume, originally published in Kechris et al. [Cabal iii], pp. 77–85. Itay Neeman [Nee] Games of length ù1 , Journal of Mathematical Logic, to appear. [Nee95] Optimal proofs of determinacy, The Bulletin of Symbolic Logic, vol. 1 (1995), pp. 327–339. [Nee04] The determinacy of long games, de Gruyter Series in Logic and its Applications, vol. 7, Walter de Gruyter, Berlin, 2004. [Nee06] Determinacy for games ending at the first admissible relative to the play, The Journal of Symbolic Logic, vol. 71 (2006), no. 2, pp. 425– 459. [Nee07] Propagation of the scale property using games, this volume, 2007. Mitchell Rudominer [Rud99] The largest countable inductive set is a mouse set, The Journal of Symbolic Logic, vol. 64 (1999), pp. 443– 459. Joseph R. Schoenfield [Sch61] The problem of predicativity, Essays on the foundations of mathematics (Y. Bar-Hillel et al., editors), Magnes Press, Jerusalem, 1961, pp. 132–139.

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Robert M. Solovay [Sol66] On the cardinality of Σ12 set of reals, Foundations of Mathematics: Symposium papers commemorating the 60th birthday of Kurt G¨odel (Jack J. Bulloff, Thomas C. Holyoke, and S. W. Hahn, editors), Springer-Verlag, 1966, pp. 58–73. [Sol78A] A ∆13 coding of the subsets of ù ù, In Kechris and Moschovakis [Cabal i], pp. 133–150. e John R. Steel [SteA] The derived model theorem, Available at http://www.math.berkeley.edu/∼ steel. [Ste83A] Scales in L(R), this volume, originally published in Kechris et al. [Cabal iii], pp. 107– 156. [Ste83B] Scales on Σ11 -sets, this volume, originally published in Kechris et al. [Cabal iii], pp. 72– 76. [Ste88] Long games, this volume, originally published in Kechris et al. [Cabal iv], pp. 56–97. [Ste95A] HODL(R) is a core model below Θ, The Bulletin of Symbolic Logic, vol. 1 (1995), pp. 75– 84. [Ste95B] Projectively wellordered inner models, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 77–104. [Ste96] The core model iterability problem, Lecture Notes in Logic, no. 8, Springer-Verlag, Berlin, 1996. [Ste05] PFA implies ADL(R) , The Journal of Symbolic Logic, vol. 70 (2005), no. 4, pp. 1255–1296. [Ste07A] The length-ù1 open game quantifier propagates scales, this volume, 2007. [Ste07B] An outline of inner model theory, In Kanamori and Foreman [KF07]. [Ste07C] Scales in K(R), this volume, 2007. M. Y. Suslin [Sus17] Sur une d´efinition des ensembles mesurables B sans nombres transfinis, Comptes Rendus Hebdomadaires des S´eances de l’Acad´emie des Sciences, vol. 164 (1917), pp. 88–91. John von Neumann [vN49] On rings of operators, reduction theory, Annals of Mathematics, vol. 50 (1949), pp. 448– 451. W. Hugh Woodin [Woo99] The axiom of determinacy, forcing axioms, and the nonstationary ideal, De Gruyter Series in Logic and its Applications, Walter de Gruyter, Berlin, 1999. A. Stuart Zoble [Zob00] Stationary reflection and the determinacy of inductive games, Ph.D. thesis, U.C. Berkeley, 2000. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720, USA

E-mail: [email protected]

NOTES ON THE THEORY OF SCALES

ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

Editorial Note. These informal notes were written in the summer of 1971 and were distributed fairly widely, but with the intention of publishing the results separately. Despite this intention, they were included in [Cabal i] because “it seemed like a good idea to include them . . . particularly since there are many references to them in the literature.” The notes are reproduced here in their [Cabal i] form (including the increasingly inaccurate remark “This paper is not meant for publication”), which added a few comments regarding progress since 1971.

It is the purpose of these notes to give an informal exposition of several recent results in Descriptive Set Theory, all centering around the notion of a scale. This was first isolated explicitly in the generalization of the Uniformization Theorem on the hypothesis of projective determinacy [Mos71A], but is surely implicit in some of the classical proofs. It has turned out that scales have many applications beyond the Uniformization Theorem, both in producing new results and in providing more elegant proofs of known results. Among the new results the Kunen-Martin theorem on the length of wellfounded relations is perhaps the most important. As for new proofs, one can now establish the beautiful results of Solovay on the regularity of Σ12 sets (Lebesgue measurae bility, property of Baire, etc.) without any use of Cohen’s forcing—in fact the new arguments are very simple and much in the spirit of classical Descriptive Set Theory. (Some of these new arguments are due to Solovay again.) This paper is not meant for publication; Moschovakis wants to keep his results for his forthcoming book [Mos80], Kechris is holding his for his Ph.D. thesis and the theorems which belong to neither of us will be presumably written up by their authors. In many ways this can be considered a first draft of part of [Mos80]. The point of making it available now is that sometimes books remain “forthcoming” for a long time, despite the best of authors’ intentions. We believe that these results are interesting enough to deserve early—if incomplete—dissemination. This draft should be comprehensible to one with some knowledge of classical Descriptive Set Theory, recursive functions with real arguments and at During the preparation of this manuscript, both authors were partially supported by NSF Grant #GP-27964, and Moschovakis was the recipient of a Sloan Foundation Fellowship. The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

28

29

NOTES ON THE THEORY OF SCALES

least the basic definitions of games and determinacy. Some parts depend on a knowledge of the theory of indiscernibles for L, but they are independent of the main results. Except for a few inessential changes (explained in §1), we shall follow the notation and terminology of [Mos70A, §1, §2]; it will be convenient and space-saving to assume that the reader is familiar with this material, although the rest of [Mos70A] is not relevant to this work, except for one result which will be identified when used. §1. Preliminaries. As usual, ù = {0, 1, 2, . . . } and R = reals. We study subsets of the product spaces

ù

ù = the set of

X = X1 × · · · × Xk (Xi = ù or Xi = R) which we call pointsets. Sometimes we think of these as relations and write interchangeably, x ∈ A ⇐⇒ A(x). A pointclass is a class of pointsets, not necessarily all in the same product space. Thus Σ11 consists of all relations expressible in the form (∃α)(∀n)R(α(n), ¯ x) (R recursive) and similarly for Π11 , etc. If A ⊆ X × ù, put ∃ùA = {x : ∃nA(x, n)}, ∀ùA = {x : ∀nA(x, n)} and if A ⊆ X × R, put ∃RA = {x : ∃αA(x, α)}, ∀RA = {x : ∀αA(x, α)}. If Γ is a pointclass, define Γ˘ = {X \ A : A ⊆ X, A ∈ Γ} and call it the dual class of Γ, define Γ = {A : for some B ⊆ R × X and some α0 ∈ R, x ∈ A ⇐⇒ hα0 , xi ∈ B}, e and for any operation Φ on pointsets, define ΦΓ = {ΦA : A ∈ Γ}. A pointclass Γ is adequate if it contains all recursive sets and is closed under disjunction, conjunction, bounded number quantification of both kinds and substitution of recursive functions. All the usual arithmetical, analytical and projective classes are adequate. If κ is an ordinal, then κ + is the least cardinal greater than κ. By “κ ë ” we always mean ordinal exponentiation.

30

ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

We work here entirely within ZF+DC, Zermelo-Fraenkel set theory with Dependent Choices, (∀u ∈ x)(∃v)hu, vi ∈ r =⇒ (∃f)(∀n)hf(n), f(n + 1)i ∈ r.

(DC)

We always state all additional hypotheses, including the full Axiom of Choice, AC, when we need them. §2. Norms and the Prewellordering property. The Prewellordering property on a pointclass Γ was formulated in order to extend elegantly to Σ12 some of the basic results about Π11 . It was later shown that if Projective Determinacy (PD) holds, then the Prewellordering property can be established for all Π1n (for odd n) and Σ1k (for even k), so that the same results could be extended to all analytical classes of the right kind and index. Our main purpose in this section is to establish the elementary facts about the Prewellordering property and prove this theorem. 2.1. Definition and elementary properties. A norm on a pointset A is any function ϕ : A ։ κ from A onto some ordinal κ, the length of ϕ. Each norm ϕ determines uniquely a prewellordering (reflexive, transitive, connected, wellfounded relation) ≤ϕ on A given by x ≤ϕ y ⇐⇒ ϕ(x) ≤ ϕ(y); conversely each prewellordering 4 on A determines a unique norm ϕ such that 4 = ≤ϕ . There are, of course, many trivial norms on a pointset, for example the constant 0 function. The concept becomes nontrivial when we place definability conditions on a norm in the following way: Let Γ be a pointclass, ϕ : A ։ κ a norm on some pointset. We call ϕ a ϕ ϕ Γ-norm if there exist relations ≤Γ , ≤Γ˘ in Γ and Γ˘ respectively such that for every y, y ∈ A =⇒ ∀x{(x ∈ A ∧ ϕ(x) ≤ ϕ(y)) ⇐⇒ x ≤ϕΓ y ⇐⇒ x ≤ϕΓ˘ y}.

(1)

Notice that if Γ is adequate and ϕ is a Γ-norm on A we can also define relations <ϕΓ , <ϕΓ˘ in Γ, Γ˘ respectively such that for every y, y ∈ A =⇒ ∀x{(x ∈ A ∧ ϕ(x) < ϕ(y)) ⇐⇒ x <ϕΓ y ⇐⇒ x <ϕΓ˘ y}.

(2)

In fact we put ϕ

ϕ

ϕ

x <Γ y ⇐⇒ x ≤Γ y ∧ ¬(x ≤Γ˘ y) x <ϕΓ˘ y ⇐⇒ x ≤ϕΓ˘ y ∧ ¬(x ≤ϕΓ y). It is quite important for the applications that the definition of a Γ-norm be precisely that given by (1). Notice that for Γ adequate and A ∈ Γ, this is stronger than simply requiring that ≤ϕ ∈ Γ, but weaker than insisting that ˘ ≤ϕ ∈ Γ ∩ Γ˘ (which implies A ∈ Γ ∩ Γ).

NOTES ON THE THEORY OF SCALES

31

Finally put: PWO(Γ) ⇐⇒ Every pointset A in Γ admits a Γ-norm. This notion is not interesting unless Γ is at least adequate. For adequate, parametrized Γ it is equivalent to that defined in [Mos70A, §2], where there is also a discussion of some properties that it implies (see also [AM68]). We only give here a short proof of the Reduction property from this form of the Prewellordering property. Theorem 2.1. Assume Γ is adequate and PWO(Γ); then Reduction(Γ), i.e., if A, B ∈ Γ, A ⊆ X, B ⊆ X, then there exists A1 ⊆ A, B1 ⊆ B, A1 ∈ Γ, B1 ∈ Γ such that A1 ∩ B1 = ∅, A1 ∪ B1 = A ∪ B. Proof. Let A, B ⊆ X, A, B ∈ Γ be given. Define C = (A×{0})∪(B ×{1}) (notice that C ⊆ X × ù). Since Γ is adequate, C ∈ Γ. Let ϕ be a Γ-norm on C , and put x ∈ A1 ⇐⇒ x ∈ A ∧ ¬(hx, 1i ≤ϕΓ˘ (x, 0)) x ∈ B1 ⇐⇒ x ∈ B ∧ ¬(hx, 0i <ϕΓ˘ (x, 1)). An easy checking shows that A1 , B1 have all the required properties. ⊣ The following trivial observation will save us having to deal separately, with the “lightface” and “boldface” cases: Proposition 2.2. If Γ is adequate and PWO(Γ), then PWO(Γ). e Proof. Let A ∈ Γ, A ⊆ X. Then for some α0 and some B ⊆ R × X, B ∈ Γ we have A = {α :e hα0 , αi ∈ B}. Find a Γ-norm ϕ¯ for B and define the following ordinal map on A ø(α) = ϕ(α ¯ 0 , α). Then let α 4 â ⇐⇒ ø(α) ≤ ø(â) and let ϕ be the norm on A such that 4 = ≤ϕ . (We define the norm in this roundabout way because ø need not be onto an ordinal.) Clearly ϕ is a Γ-norm on A. ⊣ e 2.2. Establishing the Prewellordering property. We first prove Theorem 2.3. PWO(Π11 ). Proof. Let A ∈ Π11 , A ⊆ X . For α ∈

ù

ù, let

≤α := {hm, ni : α(pm, nq) = 0}. Now put LOR = {α : ≤α is a linear ordering} WO = {α : ≤α is a wellordering}

32

ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

Then for some recursive function f : X → R, f(x) ∈ LOR, for all x ∈ X and x ∈ A ⇐⇒ f(x) ∈ WO. Define on A ø(x) = |f(x)| where for α ∈ WO, |α| is the length of ≤α . Then let 4 be defined by x 4 y ⇐⇒ ø(x) ≤ ø(y) and let ϕ be the norm of A such that 4 = ≤ϕ . To see that ϕ is actually a Π11 -norm, we use the well-known fact that there exist relations QΠ11 , QΣ11 in Π11 , Σ11 respectively, such that for â ∈ WO, (α ∈ WO ∧ |α| ≤ |â|) ⇐⇒ QΠ11 (α, â) ⇐⇒ QΣ11 (α, â).

⊣

We now give two theorems which establish the prewellordering property for some pointclasses closed under ∃R . Theorem 2.4. (Moschovakis, see [Mos70A]) Assume Γ is adequate, A ∈ Γ and A admits a Γ-norm. Then ∃R A admits an ∃R ∀R Γ-norm. Corollary 2.5. If Γ is adequate, PWO(Γ), and ∀R Γ ⊆ Γ imply PWO(∃R Γ). Corollary 2.6. PWO(Σ12 ). Proof of Theorem 2.4. Let A ⊆ X × R, A ∈ Γ, B = ∃R A = {x : ∃α(x, α) ∈ A}. Let ϕ be a Γ-norm on A, define 4 on B by x 4 y ⇐⇒ min{ϕ(x, α) : hx, αi ∈ A} ≤ min{ϕ(y, â) : hy, âi ∈ A} and let ø: B ։ ë be the unique norm such that ≤ = 4. ϕ ϕ ϕ ϕ To check that this is actually a ∃R ∀R Γ-norm, let ≤Γ , ≤Γ˘ , <Γ , <Γ˘ be the relations associated with ϕ and notice that for y ∈ B, ø

ϕ

(x ∈ B ∧ ø(x) ≤ ø(y)) ⇐⇒ (∃α)(hx, αi ∈ A ∧ ∀â(¬hy, âi <Γ˘ hx, αi)) ⇐⇒ (∀â)(hy, âi ∈ A =⇒ ∃α(hx, αi ≤ϕΓ˘ hy, âi)). ⊣ (Theorem 2.4) Let ≤ be a wellordering of R of order-type ℵ1 ; put In.Segment≤ (ã, α) ⇐⇒ {â : â ≤ α} = {(ã)n : n = 0, 1, 2, . . . }. We call ≤ Γ-good if both the relation ≤ and the relation In.Segment≤ are in ˘ Γ ∩ Γ. Theorem 2.7 (Essentially Addison [Add59]). Assume Γ is adequate, closed under both ∃ù , ∀ù and Γ˘ ⊆ ∃R Γ and there is some wellordering ≤ of R which is Γ-good; then PWO(∃R Γ).

NOTES ON THE THEORY OF SCALES

33

Proof. If A = ∃R B with B ∈ Γ, define on A x 4 y ⇐⇒ min{α : hx, αi ∈ B} ≤ min{â : hy, âi ∈ B}, where the minima are taken in the good wellordering ≤ and let ø be the associated norm. The computation is easy. ⊣ Corollary 2.8. V=L implies that for every k ≥ 2, PWO(Σ1k ). Proof. V=L implies there exists a wellordering of R which is ∆12 -good. ⊣ Corollary 2.9 (Silver, [Sil71]). If ì is a normal κ-additive measure on some cardinal, then V=L[ì] implies that for all k ≥ 2, PWO(Σ1k ). Proof. V=L[ì] =⇒ there exists a wellordering of R which is ∆13 -good. 2.3. The First Periodicity Theorem.

⊣

Theorem 2.10 (Martin, Moschovakis, [Mar68, AM68]). Assume that Γ is adequate and that Det(Γ ∩ Γ˘ ) holds. Then if A ∈ Γ and A admits a Γ-norm, e e ∀R A admits a ∀R ∃R Γ-norm. Corollary 2.11. If Γ is adequate with PWO(Γ), Det(Γ ∩ Γ˘ ) and ∃R Γ ⊆ Γ, e e then PWO(∀R Γ).

Proof of Theorem 2.10. Assume the hypotheses hold and let ø be a Γnorm on A ⊆ X × R, A ∈ Γ. Let B = ∀R A = {x : ∀α(hx, αi ∈ A)}. Instead of giving directly a norm ϕ on A we shall define the associated prewellordering ≤ϕ , for simplicity 4. Given x, y ∈ X consider the game G(x, y) I II α(0) â(0) α(1) â(1) α(2) â(2) .. .. . . α â Player I plays α, player II plays â and player II wins if hy, âi 6∈ A ∨ (hy, âi ∈ A ∧ hx, αi ∈ A ∧ ø(x, α) ≤ ø(y, â)). Now for x, y ∈ B put x 4 y ⇐⇒ player II has a winning strategy in G(x, y). The idea is that x 4 y if there is some procedure (strategy) ô which to each α assigns bit-by-bit some [α] ∗ ô so that ø(x, α) ≤ ø(y, [α] ∗ ô); in some sense, supα {ø(x, α)} ≤ supâ {ø(y, â)} effectively. Claim 2.12. ∀x ∈ B(x 4 x).

34

ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

Proof of Claim 2.12. In the game G(x, x), where x ∈ B, player II has always a trivial winning strategy, namely copying down the moves of player I. ⊣ (Claim 2.12) Claim 2.13. ∀x, y, z ∈ B(x 4 y ∧ y 4 z =⇒ x 4 z). Proof of Claim 2.13. Let x, y, z ∈ B ∧ x 4 y ∧ y 4 z. Then player II has winning strategies in both G(x, y) and G(y, z). Fix one in each game and consider the diagram: G(x, y) I II a0 Oo O O' b0 a1 Oo O O' b1 .. .. . . α â

G(y, z) I II / b0

/ c0

/ b1 .. . â

/ c1 .. . ã

G(x, z) I II a0 / c0 a1 / c1 .. .. . . ã α

We describe a strategy for player II in G(x, z) as follows: Player I plays in G(x, z) a0 . Then player I copies in G(x, y) a0 and player II answers in G(x, y) by his winning strategy to give b0 . Player I plays in G(y, z) this b0 and player II answers in G(y, z) by his winning strategy to give c0 . Player II’s answer in G(x, z) is this c0 , and so on as in the diagram. After the game is over, reals α, â, ã result, and, since player II played with his winning strategy in G(x, y) and G(y, z), it follows that ø(x, α) ≤ ø(y, â) ≤ ø(z, ã) (recall that x, y, z ∈ B). Thus ø(x, α) ≤ ø(z, ã) and player II wins G(x, z). We described a winning strategy for player II in G(x, z), so x 4 z. ⊣ (Claim 2.13) Claim 2.14. For any x, y, G(x, y) is determined. If x, y ∈ B, then x ≺ y ⇐⇒ player I has a winning strategy in G(y, x), where x ≺ y ⇐⇒ x 4 y ∧ ¬(y 4 x). Thus, ∀x, y ∈ B(x 4 y or y 4 x). Proof of Claim 2.14. Let x, y ∈ X be given. If y 6∈ B, pick â0 such that hy, â0 i 6∈ A; player II then wins by playing â0 . If y ∈ B then we have player II wins G(x, y) ⇐⇒ (hx, αi ∈ A ∧ ø(x, α) ≤ ø(y, â)) ⇐⇒ hx, αi ≤ø Γ hy, âi ⇐⇒ hx, αi ≤ø hy, âi. Γ˘ Thus the game is in Γ ∩ Γ˘ and is determined. Now assume x, y e ∈ B.e If x ≺ y, then ¬(y 4 x), so player II does not have a winning strategy in G(y, x) and player I has a winning strategy in G(y, x).

35

NOTES ON THE THEORY OF SCALES

Conversely assume that player I has a winning strategy in G(y, x). We shall show that x ≺ y, i.e., player II has a winning strategy in G(x, y), but player II has no winning strategy in G(y, x). The last statement is obvious so we proceed to invent a winning strategy for player II in G(x, y). Fix a strategy for player I in G(y, x). Consider the diagram G(y, x) G(x, y) I II I II a0 4 b0 b 0 o a0 o woo a1 4 b1 b 1 o a1 o woo 4 b2 b2 .. .. .. .. . . . . α α â â We describe a strategy for player II in G(x, y) as follows: Player I plays in G(x, y) a0 . Then player I plays b0 by his strategy in G(y, x) and player II plays this b0 in G(x, y) answering a0 . Then player I plays a1 in G(x, y). Player II copies a0 in G(y, x) and player I answers by his strategy in G(y, x) to give b1 . Player II plays this b1 in G(x, y) answering to a1 . Then player I gives a2 in G(x, y), player II copies a1 in G(y, x) and player I answers in G(y, x) by his strategy to give b2 . Player II then copies b2 in G(x, y), etc. After the end of the game reals α, â result as in the picture. Since player I played by his winning strategy in G(y, x), we have ø(y, â) > ø(x, α); thus player II wins G(x, y). Thus the above described strategy is a winning one for player II in G(x, y) and we are done. ⊣ (Claim 2.14) Claim 2.15. The relation 4 is wellfounded. Proof of Claim 2.15. We have to show that there is no infinite descending chain x0 ≻ x1 ≻ x2 ≻ · · · . Assume not, towards a contradiction. If x0 ≻ x1 ≻ x2 ≻ · · · , then player I wins G(xi , xi+1 ) for each i ≥ 0. Fix winning strategies for player I in each one of these games. Consider the diagram: G(x0 , x1 ) I II α0 (0) α1 (0) o xx {xxx α0 (1) α1 (1) o xx {xxx α0 (2) α1 (2) o .. .. . . α0 α1

G(x1 , x2 ) I II α1 (0) α2 (0) o xx {xxx

α2 (1) o x x {xxx α1 (2) α2 (2) o .. .. . . α1 α2 α1 (1)

G(x2 , x3 ) I II α2 (0) α3 (0) o xx {xxx α2 (1)

α3 (1) o x x {xxx α2 (2) α3 (2) o .. .. . . α2 α3

G(x3 , x4 ) I II α3 (0) α4 (0) . . . xx {xxx α3 (1)

α4 (1) . . . x x x {xx α3 (2) α4 (2) . . . .. .. . . α3 α4 . . .

36

ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

Let first player I play α0 (0), α1 (0), α2 (0), . . . in G(x0 , x1 ), G(x1 , x2 ), G(x2 , x3 ), . . ., respectively by following his winning strategies. Then let player II play α1 (0), α2 (0), α3 (0), . . . in G(x0 , x1 ), G(x1 , x2 ), G(x2 , x3 ), . . ., respectively. Player I answers by his winning strategies to give α0 (1), α1 (1), α2 (1), . . . in G(x0 , x1 ), G(x1 , x2 ), G(x2 , x3 ), . . ., respectively. Then player II plays α1 (1), α2 (1), α3 (1), . . . in G(x0 , x1 ), G(x1 , x2 ), G(x2 , x3 ), . . ., respectively. Player I answers by his strategies to give α0 (2),α1 (2),α2 (2), . . . in G(x0 , x1 ), G(x1 , x2 ), G(x2 , x3 ), . . ., respectively, etc. After all these moves have been played, reals α0 , α1 , α2 , . . . are created as in the picture and since player I wins all the games G(xi , xi+1 ), i ≥ 0, we have ø(x0 , α0 ) > ø(x1 , α1 ) > ø(x2 , α2 ) > · · · , which is a contradiction. ⊣ (Claim 2.15) Claim 2.16. The norm associated with 4 is a ∀R ∃R Γ-norm on B. Proof of Claim 2.16. Notice first that for y ∈ B, x ∈ B∧x 4 y ⇐⇒ player II has a winning strategy in G(x, y) ⇐⇒ ∃ô∀α(hx, αi ≤ø ˘ hy, [α] ∗ ôi). But Γ since G(x, y) is determined, player II has a winning strategy in G(x, y) ⇐⇒ player I has no winning strategy in G(x, y) ⇐⇒ ∀ó∃â(hx, ó ∗ [â]i ≤ø Γ hy, âi). Thus for y ∈ B, x ∈ B ∧ x 4 y ⇐⇒ ∃ô∀α(hx, αi ≤ø ˘ hy, [α] ∗ ôi) Γ ⇐⇒ ∀ó∃â(hx, ó ∗ [â]i ≤ø Γ hy, âi), and we are done.

⊣ (Claim 2.16) ⊣ (Theorem 2.10)

2.4. The zig-zag picture. It is not hard to verify that the Reduction property cannot hold both for Σ1n and Π1n for any n, hence the same is true for the Prewellordering property. If we make a diagram of the classes Σ1n , Π1n and circle those which have the Prewellordering property, we get the following two pictures, under the hypotheses V = L (or V = L[ì]) and PD:

V=L or V = L[ì]

Σ11

89:; ?>=< Σ12     89:; ?>=< Π11 Π12

?>=< 89:; Σ13

?>=< 89:; Σ14

?>=< 89:; Σ15

...

Π13

Π14

Π15

...

Σ11

PD

?>=< 89:; 89:; ?>=< Σ12 Σ13 Σ14 Σ15  ? ? ? ?  ?? ??      ?? ??      1 1 1 1 89:; ?>=< 89:; ?>=< 89:; ?>=< Π Π Π1 Π Π 1

2

3

4

5

NOTES ON THE THEORY OF SCALES

37

This second picture is the motivation for the name “Periodicity Theorem.” We will see later that (assuming PD) we can construct models in which this picture has any finite predetermined number of “teeth.” §3. Scales. In this section we define scales and the property Scale(Γ), we prove that Scale(Γ) =⇒ Unif(Γ) for suitable Γ (in particular Π11 , Π13 , Π15 , . . . ) and we establish Scale(Γ) for Γ = Π11 , Σ12 , Π13 , Σ14 , . . . under the hypothesis PD. The elementary theory of scales is quite similar to that of norms, so the structure of this section is parallel to that of §2. 3.1. Definitions and basic properties. A scale on a pointset A is a sequence of norms hϕn : n ∈ ùi on A with the following limit property: If x0 , x1 , x2 , · · · ∈ A, if limi→∞ xi = x, if for each n and all large i, ϕn (xi ) = ën , then x ∈ A and for each n, ϕn (x) ≤ ën . It is easy to see that assuming AC every pointset admits a scale (take every ϕn to be equal to a fixed 1-1 mapping from the pointset onto an ordinal). In order to get something interesting we place definability conditions on a scale as follows: Let Γ be a pointclass, hϕn : n ∈ ùi a scale on a pointset A. We call hϕn : n ∈ ùi a Γ-scale if there exist relations SΓ , SΓ˘ in Γ and Γ˘ respectively such that, for every y, y ∈ A =⇒ ∀x{[x ∈ A ∧ ϕn (x) ≤ ϕn (y)] ⇐⇒ SΓ (n, x, y) ⇐⇒ SΓ˘ (n, x, y)]. (3) Finally we put: Scale(Γ) ⇐⇒ Every pointset A in Γ admits a Γ-scale. The notion of a scale and the associated scale property were introduced in connection with the uniformization problem in [Mos71A]. If Γ, Γ∗ are pointclasses, put: Unif(Γ, Γ∗ ) ⇐⇒ For every P ∈ Γ, P ⊆ X × Y we can find P ∗ ∈ Γ∗ , P ∗ ⊆ X × Y such that P ∗ ⊆ P and ∀x(∃yP(x, y) ⇐⇒ ∃!yP ∗ (x, y)). (In this case we say that P ∗ uniformizes P.) In this definition we allow P ⊆ Y in which case P ∗ ⊆ P and ∃yP(y) ⇐⇒ ∃!yP ∗ (y), i.e., P ∗ is a singleton contained in P, if P 6= ∅. We abbreviate Unif(Γ) ⇐⇒ Unif(Γ, Γ). Theorem 3.1. Assume Γ is adequate and closed under ∀R . Then, Scale(Γ) =⇒ Unif(Γ).

38

ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

Proof. Let P ∈ Γ, P ⊆ X × Y, where for simplicity and without loss of generality we can assume Y = R. Let hϕn : n ∈ ùi be a Γ-scale for P with associated relations SΓ , SΓ˘ . Fix x ∈ X and define inductively the following (where we agree that min(∅) = 0):  x  P0 = {α : P(x, α)} ëx0 = min{ϕ0 (x, α) : α ∈ P0x }   x k0 = min{α(0) : α ∈ P0x ∧ ϕ0 (x, α) = ëx0 },  x x x x  Pn+1 = {α : α ∈ Pn ∧ α(n) = kn ∧ ϕn (x, α) = ën } x ëxn+1 = min{ϕn+1 (x, α) : α ∈ Pn+1 }   x x kn+1 = min{α(n + 1) : α ∈ Pn+1 ∧ ϕn+1 (x, α) = ëxn+1 }. Finally put x = P∞

\

Pnx .

n∈ù

We have now the following: (1) P0x ⊇ P1x ⊇ P2x ⊇ · · · . (2) ∃αP(x, α) =⇒ ∀n(Pnx 6= ∅). x = {α x }. (3) Assume ∃αP(x, α). Let α x = hk0x , k1x , k2x , . . .i. Then P∞ x Proof of (3). If α ∈ P∞ , then for each n, α(n) = knx , i.e., α = α x . Conversely, if ∃αP(x, α) pick reals αi ∈ Pix , i = 0, 1, . . . . Then for i > n, αi (n) = knx , thus αi → α x and hx, αi i → hx, α x i. Also for i > n, ϕn (x, αi ) = ëxn , hence by the limit property of scales, hx, α x i ∈ P and ϕn (x, α x ) ≤ ëxn . Then certainly α x ∈ P0x . But also α x (0) = k0x and ϕ0 (x, α x ) ≤ ëx0 , i.e., ϕ0 (x, α x ) = ëx0 . Thus α x ∈ P1x . A similar argument shows inductively that for all n, α x ∈ Pnx . ⊣ (3)

Put now P ∗ (x, α) ⇐⇒ ∃αP(x, α) ∧ α = α x . Clearly P ∗ ⊆ P and ∃αP(x, α) =⇒ ∃!αP ∗ (x, α). To complete the proof it will be enough to show that P ∗ ∈ Γ. It is eas˘ And this follows from the ier to show that the complement of P ∗ is in Γ. computation

NOTES ON THE THEORY OF SCALES

39

¬P ∗ (x, α) ⇐⇒ n n ¬P(x, α) ∨ P(x, α) ∧ ∃n∃â P(x, â)  ∧ (∀i < n) α(i) = â(i)
∧ ϕi (x, α) = ϕi (x, â) ∧ ϕn (x, â) < ϕn (x, α) ∨ (ϕn (x, â) = ϕn (x, oo α)
∧ â(n) < α(n)) ⇐⇒ n  ¬P(x, α) ∨ ∃n∃â ((∀i < n)(SΓ˘ (i, x, â, x, α) ∧ SΓ˘ (i, x, α, x, â) ∧ α(i) = â(i)) ∧ (SΓ˘ (n, x, â, x, α) ∧ ¬SΓ (n, x, α, x, â) ∨ (â(n) < α(n) ∧ SΓ˘ (n, x, â, x, α)

o ∧ SΓ˘ (n, x, α, x, â))) .

⊣ Again we should mention the trivial observation that if Γ is adequate, then Scale(Γ) implies Scale(Γ). e 3.2. Establishing the Scale property. Theorem 3.2. Scale(Π11 ). ˆ Corollary 3.3 (The classical Novikoff-Kondo-Addison Theorem). Unif(Π11 ). Proof of Theorem 3.2. Let A ⊆ X, A ∈ Π11 . Then for some recursive function f : X → R x ∈ A ⇐⇒ f(x) ∈ WO. For α ∈ WO put α↾n = {m : m <α n} (where m <α n ⇐⇒ m ≤α n ∧ m 6= n) and |α↾n| = the length of α↾n (α↾n is an initial segment of ≤α ). If n 6∈ Field(≤α ), i.e., if α(pn, nq) 6= 0, then of course |α↾n| = 0. Define now the following prewellorderings on A x ≤n y ⇐⇒|f(x)| < |f(y)| ∨ (|f(x)| = |f(y)| ∧ |f(x)↾n| ≤ |f(y)↾n|). Let ϕn be the norm on A such that ≤ϕn = ≤n . We will show that hϕn : n ∈ ùi is a Π11 -scale on A.

40

ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

Claim 3.4. hϕn : n ∈ ùi is a scale. Proof of Claim 3.4. Assume xi ∈ A for every i, xi → x and for some hën : n ∈ ùi, ϕn (xi ) = ën for all sufficiently large i. This implies that |f(xi )| = ë′ for some ë′ and all sufficiently large i and also that for some hë′n : n ∈ ùi, |f(xi )↾n| = ë′n for all sufficiently large i. We show first that x ∈ A, i.e., that f(x) ∈ WO, by proving that the mapping n 7→ ë′n is order preserving on the field of ≤f(x) . In fact, let n
⊣ (Claim 3.4)

Claim 3.5. hϕn : n ∈ ùi is a Π11 -scale. This is proved by a computation similar to that in the proof of Theorem 2.3. ⊣ (Theorem 3.2) It should be pointed out here that the proofs of Theorems 3.2 and 3.1 ˆ taken together constitute the classical proof of the Novikoff-Kondo-Addison theorem. We proceed now to prove the analogue of Theorem 2.4. Theorem 3.6 (Moschovakis, [Mos71A]). Assume Γ is adequate, A ∈ Γ and A admits a Γ-scale; then ∃R A admits an ∃R ∀R Γ-scale. Corollary 3.7. Scale(∃R Γ).

If Γ adequate and ∀R Γ ⊆ Γ, then Scale(Γ) implies

Corollary 3.8. Scale(Σ12 ). Proof of Theorem 3.6. If ë is any ordinal and n ∈ ù, consider the lexicographical wellordering of n ë, hî1 , . . . , în i ≤ hç1 , . . . , çn i ⇐⇒ î1 < ç1 ∨ (î1 = ç1 ∧ î2 < ç2 ) ∨ . . . ∨ (î1 = ç1 ∧ . . . ∧ în−1 = çn−1 ∧ în ≤ çn ).

41

NOTES ON THE THEORY OF SCALES

This wellorders n ë with ordinal ën and we let pî1 , . . . , înq = ordinal of hî1 , . . . , în i in the lexicographical wellordering. Each ϑ < ën can be written uniquely as ϑ = pî1 , . . . , înq for îi < ë. Now let A ⊆ X × R, A ∈ Γ, and assume hϕn : n ∈ ùi is a Γ-scale on A. If B = ∃R A = {x : ∃α(hx, αi ∈ A)}, define for x ∈ B øn′ (x) = min{pϕ0 (x, α), α(0), ϕ1 (x, α), α(1), . . . , ϕn (x, α), α(n)q : hx, αi ∈ A}. Let ≤n be the prewellordering (on B) x ≤n y ⇐⇒ øn′ (x) ≤ øn′ (y) and let øn be the associated norm. We will prove that høn : n ∈ ùi is a ∃R ∀R Γ-scale on A. Claim 3.9. høn : n ∈ ùi is a scale. Proof of Claim 3.9. Assume xi ∈ B, and xi → x and for some hën : n ∈ ùi we have for each n, øn (xi ) = ën , for all large enough i. In fact we may assume without loss of generality that for i ≥ n, øn (xi ) = ën . This implies that for some hën : n ∈ ùi and all i ≥ n, øn′ (xi ) = ë′n . Each ë′n can be written uniquely as ë′n = pën0 , k0n , . . . , ënn , knnq. We claim that for i ≥ n, ëin = ënn and kni = knn . Because if i ≥ n, øn′ (xi ) = ë′n = min{hϕ0 (xi , α), α(0), . . . , ϕn (xi , α), α(n)i : hxi , αi ∈ A} and øi′ (xi ) = ë′i =  min pϕ0 (xi , α), α(0), . . . , ϕn (xi , α), α(n), . . . , ϕi (xi , α), α(i)q :

hxi , αi ∈ A .

Thus if αi is such that hxi , αi i ∈ A and øi′ (xi ) = ë′i = pϕ0 (xi , αi ), αi (0), . . . , ϕi (xi , αi ), αi (i)q, then we must have pϕ0 (xi , αi ), αi (0), . . . , ϕn (xi , αi ), αi (n)q = ë′n (since to minimize a sequence lexicographically one has to minimize first all its initial segments). Thus ëin = ϕn (xi , αi ) = ënn and kni = αi (n) = knn , and the claim is proved. Now find αi such that hxi , αi i ∈ A and øi′ (xi ) = ë′i = pϕ0 (xi , αi ), αi (0), . . . , ϕi (xi , αi ), αi (i)q. As shown above, for i ≥ n, αi (n) = knn and therefore αi → α = hk00 , k11 , k22 , . . .i. Thus hxi , αi i → hx, αi. Moreover, for i ≥ n, ϕn (xi , αi ) = ënn , so that by the limit property of scales hx, αi ∈ A ∧ ϕn (x, α) ≤ ënn .

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ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

But then x ∈ B and øn′ (x) = min{pϕ0 (x, α), α(0), . . . , ϕn (x, α), α(n)q : hx, αi ∈ A} ≤ pë00 , α(0), ë11 , α(1), . . . , ënn , α(n)q = pën0 , k0n , ën1 , k1n , . . . , ënn , knnq = ë′n . Thus x ∈ B and øn (x) ≤ ën and we are done.

⊣ (Claim 3.9)

Claim 3.10. høn : n ∈ ùi is a ∃R ∀R Γ-scale. This is very similar to the computation done in the proof of Theorem 2.4 and we omit the details. ⊣ (Theorem 3.6) The direct analogue of Theorem 2.7 is true, but we have no use for it. The useful analogue of Theorem 2.7 gives uniformization directly. Theorem 3.11 (Essentially Addison, [Add59]). Assume Γ is adequate, ˘ = Γ and there is a wellordering closed under both ∃ù , ∀ù , Γ˘ ⊆ ∃R Γ, ∃R (Γ∩ Γ) ≤ of R which is Γ-good. Then Unif(∃R Γ, ∃R Γ). Proof. Since for Γ, Γ′ adequate Unif(Γ, Γ′ ) =⇒ Unif(∃R Γ, ∃R Γ′ ) ˘ Γ ∩ Γ). ˘ But if P ⊆ X × Y is in Γ ∩ Γ, ˘ it will be enough to prove Unif(Γ ∩ Γ, where we can assume without loss of generality Y = R, let P ∗ (x, α) ⇐⇒ P(x, α) ∧ (∀â < α)¬P(x, â). ˘ Then P ∗ uniformizes P and clearly P ∗ ∈ Γ ∩ Γ.

⊣

Corollary 3.12 (Addison, [Add59]). V = L =⇒ Unif(Σ1k , Σ1k ) (k ≥ 2). Corollary 3.13 (Silver, [Sil71]). V = L[ì] =⇒ Unif(Σ1k , Σ1k ) (k ≥ 2). 3.3. The Second Periodicity Theorem. Theorem 3.14 (Moschovakis, [Mos71A]). Assume Γ is adequate and Det(Γ ∩ Γ˘ ). Then if A ∈ Γ and A admits a Γ-scale, ∀R A admits a ∀R ∃R Γ-scale. e e Corollary 3.15. Suppose that Γ is adequate, and that Scale(Γ), Det(Γ∩Γ˘ ), e e R ∃ Γ ⊆ Γ hold. Then Scale(∀R Γ) holds. Corollary 3.16. PD implies Unif(Π12n+1 ), n ≥ 1. Proof of Theorem 3.14. Enumerate in a 1-1 recursive way all the finite sequences of integers, say u0 , u1 , u2 , . . . such that u0 = h i (the empty sequence) and if ui is a proper initial segment of uj then i < j. Put B = ∀R A ⊆ X and define x ∈ Bn ⇐⇒ (∀α ⊇ un )A(x, α)

NOTES ON THE THEORY OF SCALES

43

(where α ⊇ un ⇐⇒ α extends un ). Notice that B0 = B and for every n, B ⊆ Bn . We shall define norms on each Bn by considering games as follows: For each n ∈ ù and x, y ∈ X let Gn (x, y) be the following game un I α′

un II â′

Player I plays α ′ , player II plays â ′ and if we call α = un aα ′ , â = un aâ ′ then player II wins if and only if A(y, â) doesn’t hold, or (A(y, â) ∧ A(x, α) ∧ pϕ0 (x, α), . . . , ϕn (x, α)q ≤ pϕ0 (y, â), . . . , ϕn (y, â)q) holds, where hϕn : n ∈ ùi is a Γ-scale on A. Finally put for x, y ∈ Bn x ≤n y ⇐⇒ player II has a winning strategy in Gn (x, y). Proofs similar to those in 2.3 show that each ≤n is a prewellordering on Bn and that for some S, S ′ in ∀R ∃R Γ, ∃R ∀R Γ˘ respectively we have y ∈ Bn =⇒ ∀x((x ∈ Bn ∧ x ≤n y) ⇐⇒ S(n, x, y) ⇐⇒ S ′ (n, x, y)). Call øn the norm associated with ≤n . Suppose we can prove that if xi ∈ B, xi → x and øn (xi ) = ën for all large enough i, then x ∈ B and øn (x) ≤ ën . Then if we put for x ∈ B øn′ (x) = pø0 (x), øn (x)q and let øn′′ be the norm on B such that øn′′ (x) ≤ øn′′ (y) ⇐⇒ øn′ (x) ≤ øn′ (y) it is easy to check that høn′′ : n ∈ ùi is a ∀R ∃R Γ-scale on B. Thus, to complete the proof, we assume xi ∈ B, xi → x and øn (xi ) = ën for all i ≥ n, and try to show that x ∈ B and øn (x) ≤ ën . Claim 3.17. x ∈ B. Proof of Claim 3.17. We have to show that for every α, hx, αi ∈ A. Fix α. ¯ Notice that n0 = 0∧n0 < n1 < n2 · · · . Consider Let ni be such that uni = α(i). the subsequence hxn0 , xn1 , xn2 , . . .i. Then øni (xni ) = øni (xni+1 ). Thus player II has a winning strategy in all the games Gni (xni+1 , xni ) (since xni ≥ni xni+1 ). Fix strategies for player II in each one of the games Gni (xni+1 , xni ) and consider the diagram below:

44

ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

Gn0 (xn1 , xn0 )

I α(0) v

α1 (1) α1 (2) .. . α1

v

Gn1 (xn2 , xn1 )

II / α0 (0)

α(0) I α(1)

/ α0 (1)

α2 (2)

/ α0 (2) .. . α0

α2 (3) .. . α2

v

v

Gn2 (xn3 , xn2 )

α(0) II / α1 (1)

α(0) α(1) I α(2)

/ α1 (2)

α3 (3)

/ α1 (3) .. . α1

α3 (4) .. . α3

v

α(0) α(1) II / α2 (2)

v

/ α2 (3) / α2 (4) .. . α2

Gn3 (xn4 , xn3 ) α(0) α(0) α(1) α(1) α(2) α(2) I II / α(3) α3 (3)

... ... ...

... e eeeeee ree/ eeee . .. α4 (4) α3 (4) e eeeeee ree/ eeee . .. α4 (5) α3 (5) .. .. . . α3 α4

Player I plays α(0) in Gn0 (xn1 , xn0 ) and player II answers with his strategy to give α0 (0). Then player I plays α(1) in Gn1 (xn2 , xn1 ) and player II answers with his strategy to give α1 (1), etc. After these moves have been played, player I plays α1 (1) in Gn0 (xn1 , xn0 ) and player II answers with his strategy to give α0 (1). Then player I plays α2 (2) in Gn1 (xn2 , xn1 ) and player II answers by his strategy to give α1 (2), etc. Finally reals α0 , α1 , α2 , . . . are formed (where, e.g., α4 = hα(0), α(1), α(2), α(3), α4 (4), . . .i). Clearly αi → α. Moreover since player II always wins we have ϕ0 (xn1 , α1 ) ≤ ϕ0 (xn0 , α0 ) ϕ0 (xn2 , α2 ) ≤ ϕ0 (xn1 , α1 ) ϕ0 (xn3 , α3 ) ≤ ϕ0 (xn2 , α2 ) .. . Thus for all large enough i, ϕ0 (xni , αi ) is constant. Looking at such i’s and since again player II always wins we have ϕ1 (xni+1 , αi+1 ) ≤ ϕ1 (xni , αi ) ϕ1 (xni+2 , αi+2 ) ≤ ϕ1 (xni+1 , αi+1 ) .. . Thus again ϕ1 (xni , αi ) becomes eventually constant. Then we look at ϕ2 etc. Thus for the sequence hhxni , αi i : i ∈ ùi we have hxni , αi i ∈ A, hxni , αi i → hx, αi and for each n, ϕn (xni , αi ) becomes eventually constant. So hx, αi ∈ A. ⊣ (Claim 3.17) Claim 3.18. For each n, øn (x) ≤ ën .

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Proof of Claim 3.18. Since for i ≥ n, øn (xi ) = ën , it will be enough to prove that for each n, x ≤n xn , i.e., that player II has a winning strategy in Gn (x, xn ). Since øk (xk ) = øk (xm ), for k ≤ m, we have xm ≤k xk for m ≥ k, thus player II has a winning strategy in each one of the games Gk (xm , xk ) for m ≥ k. Fix strategies for player II in each one of these games. Fix n. In order to invent a strategy for player II in Gn (x, xn ) consider the diagram Gn (xn1 , xn ) un un

Gn1 (xn2 , xn1 ) u n1 u n1

I

II

I

a0 u

/ α(0)

a1

/ α(1)

α2 (2) .. . .. . .. . α2

v

α1 (1) α1 (2) .. . α1

v

/ α(2) .. . α

II v v

/ α1 (1) / α1 (2) .. . .. . .. . α1

Gn2 (xn3 , xn2 ) u n2 u n2 I

II

...

Gn (x, xn ) un un I

a2 gOO / α2 (2) a0 OOO .. . OO.O . . OOO .. .. OOO a1 OOO . . OOO .. .. OO . . .. .. a2 . . .. .. .. . . . . . . α3 α2 α′

II 3 α(0) 3 α(1) 3 α(2) .. . α

Let player I play a0 in Gn (x, xn ). Let un1 = un aa0 . Then n1 > n. Consider the game Gn (xn1 , xn ) and let player I play in Gn (xn1 , xn ) a0 and player II answer in this game by his winning strategy to give α(0). This α(0) is player II’s answer to a0 in Gn (x, xn ). Then player I plays a1 in Gn (x, xn ). Let un2 = un aa0 aa1 . Then n2 > n1 . Consider the game Gn1 (xn2 , xn1 ) and let player I play a1 in Gn1 (xn2 , xn1 ) and player II answer in this game by his winning strategy to give α1 (1). Then player I plays in Gn (xn1 , xn ) this α1 (1) and player II answers by his strategy in Gn (xn1 , xn ) to give α(1) which is player II’s next move in Gn (x, xn ) etc. After all these moves have been played α ′ , α, α1 , α2 , . . . are created (e.g., α2 = un1 aha1 , α2 (2), α2 (3), . . .i = un aha0 , a1 , α2 (2), α2 (3), . . .i). Clearly αn → α ′ = un aha0 , a1 , a2 , a3 , . . .i and, since player II always wins, we have ϕ0 (xn , α) ≥ ϕ0 (xn1 , α1 ) ≥ ϕ0 (xn2 , α2 ) ≥ · · · . Thus after a while ϕ0 (xni , αi ) becomes constant. Then we look at ϕ1 (xni , αi ) for such i’s; it is nonincreasing with i, thus becomes eventually constant, etc. Thus for each n, ϕn (xni , αi ) becomes eventually constant. But also hxni , αi i → hx, α ′ i, so hx, α ′ i ∈ A and ϕn (x, α ′ ) ≤ limi ϕn (xni , αi ). Then, as we saw above, ϕ0 (xn , α) ≥ limi ϕ0 (xni , αi ) ≥ ϕ0 (x, α ′ ). If ϕ0 (xn , α) > ϕ0 (x, α ′ ) clearly player II wins the game Gn (x, xn ) and we are done. If ϕ0 (xn , α) = ϕ0 (x, α ′ ), then for all i, ϕ0 (xni , αi ) = ϕ0 (xn , α) = ϕ0 (x, α ′ ). But then if n ≥ 1, ϕ1 (xn , α) ≥ ϕ1 (xn1 , α1 ) ≥ ϕ1 (xn2 , α2 ) ≥ · · · , thus ϕ1 (xn , α) ≥ ϕ1 (x, α ′ ). If again ϕ1 (xn , α) > ϕ1 (x, α ′ ) we are done, otherwise ϕ1 (xn , α) = ϕ1 (xni , αi ) =

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ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

ϕ1 (x, α ′ ) and then we look (if n ≥ 2) at ϕ2 , etc. In any case this shows that pϕ0 (xn , α),ϕ1 (xn , α), . . . , ϕn (xn , α)q ≥ pϕ0 (x, α ′ ), . . . , ϕn (x, α ′ )q, i.e., player II wins. Thus we have described a winning strategy for player II in Gn (x, xn ), so x ≤ n xn . ⊣ (Claim 3.18) ⊣ (Theorem 3.14) 3.4. The zig-zag picture. It follows from the results of this section that the pictures given in Section 2.4 for the Prewellordering property hold also for the Scale and Uniformization properties, i.e., under the stated hypotheses these properties hold for the circled pontclasses. That, assuming PD, they hold only for the circled pointclasses we will prove in the next section. 3.5. The Martin-Solovay Uniformization Theorem. From the results of this section it is obvious that Det(∆12 ) =⇒ Unif(Π12 , Π13 ). e However Martin and Solovay had obtained a similar theorem from weaker hypotheses before these results were proved, namely ∀α(α # exists) =⇒ Unif(Π12 , ∆14 ), see [MS69]; this in turn was strengthened by [Man71] to ∀α(α # exists) =⇒ Unif(Π12 , Π13 ). These proofs (and in fact the statements of the theorems) involve the theory of indiscernibles with which we are not concerned here. We will state one by-product of this work which will be useful later. Theorem 3.19. Assume that there exists a measurable cardinal or (the weaker hypothesis) that for each α, α # exists, let u1 = ℵ1 , u2 , u3 , . . . , uù be the first ù + 1 uniform indiscernibles. Then: 1. un ≤ ℵn , uù ≤ ℵù and cf(un+1 ) = cf(u2 ). 2. If AC holds, then uù < ℵ3 . 3. Every Π12 set A admits a Π13 scale on uù , i.e., a scale hϕn : n ∈ ùi with each ϕn : A → uù . ((1) and (2) are due to Solovay, (3) is implicit in [MS69, Man71]) §4. Bases. One of the most interesting corollaries of uniformization results is the computation of bases for pointclasses. If Γ is a pointclass and C a set of reals, put Basis(Γ, C ) ⇐⇒ for each A ∈ Γ, A ⊆ R, A 6= ∅ =⇒ A ∩ C 6= ∅. If Λ is a pointclass, we often abbreviate Basis(Γ, Λ) ⇐⇒ Basis(Γ, {α : the set {hn, mi : α(n) = m} ∈ Λ}).

NOTES ON THE THEORY OF SCALES

47

4.1. Computation of bases. It is immediately obvious that Unif(Γ, Γ′ ) implies Basis(Γ, {α : {α} ∈ Γ′ })

(4.1.1)

and it is easy to see that if Γ is adequate and Basis(Γ, C ) holds, then Basis(∃R Γ, {α : (∃â)(â ∈ C ∧ α is recursive in â)}). (4.1.2) From this and the results in §3 it is clear that

and

Det(∆12n ) implies Basis(Σ12n+2 , ∆12n+2 ) e

(4.1.3)

PD implies Basis(Σ12n , ∆12n ), n ≥ 2. On the other hand we have Theorem 4.4. Det(∆12n ) implies ¬ Basis(Σ12n+1 , ∆12n+1 ). e Proof. Since Det(∆12n ), we have PWO(Π12n+1 ). But this has a consequence that {α : α ∈ ∆12n+1 e } ∈ Π12n+1 . (This is announced in [AM68].) From this the result follows immediately. (For another proof see [MS69].) ⊣ The periodicity phenomenon is again clear in (4.1.3) and Theorem 4.4. Theorem 4.5 (Martin, Solovay, Mansfield, [MS69, Man71]). If α # exists for all reals α, then there exists a fixed Π13 singleton α0 (i.e., {α0 } ∈ Π13 ) such that Basis(Σ13 , {â : â is recursive in α0 }). Theorem 4.6 (Moschovakis, [Mos71A]). If Det(∆12n ) holds, then there exe ists a fixed Π12n+1 singleton α0 such that Basis(Σ12n+1 , {â : â is recursive in α0 }).

Proof. By (4.1.2) it will be enough to find a Π12n+1 singleton α0 such that every Π12n set contains a real recursive in α0 . Let B ⊆ ù × R be a universal Π12n set. Uniformize B by some B ∗ ∈ Π12n+1 . Then B ∗ ⊆ B and ∃αB(n, α) ⇐⇒ ∃!αB ∗ (n, α). Define B ∗∗ (n, α) ⇐⇒ B ∗ (n, α)∨(∀â(¬B(n, â))∧α = ët0}. Then B ∗∗ ∈ Π12n+1 and ∀n∃!αB ∗∗ (n, α). Put α ∈ C ⇐⇒ ∀nB ∗∗ (n, (α)n ). For this proof choose (α)n so that α is completely determined by {(α)n : n ∈ ù}. Thus C is a singleton and C ∈ Π12n+1 . If C = {α0 } we show that every Π12n set A contains a real recursive in α0 . In fact if A ∈ Π12n we have a ∈ A ⇐⇒ hn0 , αi ∈ B, for some n0 . Then A 6= ∅ =⇒ ∃α(hn0 , αi ∈ B), so hn0 , (α)n0 i ∈ B i.e., (α)n0 ∈ A. ⊣

48

ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

A well known basis theorem says that for some fixed Σ11 subset of ù, say A0 , we have Basis(Σ11 , {α : α is recursive in A0 }). The following question is open: Does this generalize (under any reasonable hypothesis) to Σ12n+1 , n ≥ 1?1 4.2. Independence results. It is clear that the weakest basis result one can expect for a (“lightface”) pointclass Γ is Basis(Γ, {α : α is ordinal definable}). But even such a weak result is not provable in ZFC for Γ beyond Σ12 as the next theorem shows. Theorem 4.7 (L´evy, [L´ev66]). In ZFC alone we cannot prove Basis(Π12 , {α : α is ordinal definable}). Proof. It is enough to show that if M is a countable model of ZF+V=L and α is a real Cohen generic over M , then in N = M [α] there is a Π12 set containing no ordinal definable real. In fact, in N , consider the set A = {â : â 6∈ L}. Then A = Π12 and A 6= ∅. But A cannot contain an ordinal definable real, since all such reals belong already to M = LN (because the notion of forcing is homogeneous). ⊣ 1 Of course we have a basis theorem for Σ3 assuming, for example, that there exists a measurable cardinal (Theorem 4.6). Unfortunately we cannot go further even with this stronger hypothesis. Theorem 4.8. (L´evy’s method for Theorem 4.7 using a key result of Silver [Sil71].) In ZFC + “there exists a measurable cardinal”, we cannot prove Basis(Π13 , {α : α is ordinal definable}). Proof. Repeat the proof of 4.7, but now start with an M which is a countable model of ZF+V=L[ì], where ì is a normal measure on a cardinal κ. ⊣ §5. Partially playful universes. We outline here a construction which (granting PD) yields for each n ≥ 3 a model M n of ZF+AC such that M n |= Det(∆1n−1 ), e M n |= R admits a Σ1n+1 -good wellordering. In particular the zig-zag picture of §2.4 for the scale property in M n has only finitely many teeth, i.e., the scale property settles on the Σ side for k ≥ n + 1. The results are due to Moschovakis. 1 Martin and Solovay have shown in 1972 that this generalization is false for n ≥ 1, granting Det(∆12n ). They also show that in Theorem 4.6, α0 can be any Π12n+1 singleton which is not ∆12n+1 . e This turns out to be the correct generalization of the Kleene Basis Theorem for Σ11 . See [MS].

NOTES ON THE THEORY OF SCALES

49

Fix n ≥ 3, let k be the largest even integer less than n such that k = n − 1 or k = n − 2 and assume Det(∆1k ). By the Second Periodicity Theorem we have e Unif(Π1n−1 , Π1n ) whether n is odd or even, so let Pn−1 (m, α, â) be the standard Π1n−1 universal ∗ relation and let Pn−1 (m, α, â) be the Π1n relation that comes out of the proof of the Second Periodicity Theorem such that ∗ Pn−1 (m, α, â) =⇒ Pn−1 (m, α, â), ∗ (∃â)Pn−1 (m, α, â) =⇒ (∃!â)Pn−1 (m, α, â).

(∗) (∗∗)

Finally define Fn∗ (m, α)

=

  the unique â  

ët0

∗ such that Pn−1 (m, α, â), if (∃â)Pn−1 (m, α, â), if ∀â¬Pn−1 (m, α, â).

Clearly Fn∗ is a function whose graph is Π1n . Let M be a model of ZF, transitive and containing all ordinals (for brevity, standard model). We call M Σ1n -correct if for every Σ1n formula ϑ(α1 , . . . , αℓ ), α1 , . . . , αℓ ∈ M =⇒ (ϑ(α1 , . . . αℓ ) ⇐⇒ M |= ϑ(α1 , . . . , αℓ )). Lemma 5.1. Assume Det(∆1k ). A standard model M of ZF+DC is Σ1n -correct e F ∗ , i.e., if and only if M is closed under n α ∈ M =⇒ Fn∗ (m, α) ∈ M.

Proof. Assume first that M is Σ1n -correct. Notice that if for some α ∈ M and some m1 , m2 , ∀â(Pn−1 (m1 , α, â) ⇐⇒ ¬Pn−1 (m2 , α, â)), then the same equivalence holds in M (it is expressible by a Π1n formula); thus hm1 , m2 , αi codes a ∆1n−1 set in M if and only if it does in the world. This e k ≤ n − 1, and it is now easy to verify that applies to ∆1k sets, since e M |= Det(∆1k ). e Hence the Second Periodicity Theorem holds in M , so that (∗) and (∗∗) hold. Now if for some m, α ∈ M, (∃â)Pn−1 (m, α, â), then M |= (∃â)Pn−1 (m, α, â), ∗ ∗ hence for some â ∈ M, M |= Pn−1 (m, α, â), hence Pn−1 (m, α, â) in the world ∗ and â = Fn (m, α) ∈ M . To prove the converse, assume that M is closed under Fn∗ and then show by induction on i ≤ n that M is Σ1i -correct. This part of the proof does not need the assumption that M |= DC. We omit the details. ⊣

50

ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

(Actually neither direction of the equivalence needs the assumption M |= DC, but the proof is a bit more complicated.) Define by induction on the ordinal î, M0n = ∅, n Mî+1 = Mîn ∪ {x ⊆ Mîn : x is definable in hMîn , ∈↾Mîn i}

∪ {Fn∗ (m, α) : α ∈ Mîn }, [ [ Mçn = Mîn if ç = ç > 0, î<ç

and put Mn =

[

Mîn .

î

Theorem 5.2. Let n ≥ 3, assume Det(∆1k ), with k = largest even integer e < n, let Mîn , M n be defined as above.

1. M n is a standard model of ZF, it is closed under Fn∗ and it is Σ1n -correct. 2. The relation “x ∈ Mîn ” is definable by a formula which is absolute for all Σ1n -correct models of ZF that are closed under Fn∗ . 3. M n |= ∀x∃î(x ∈ Mîn ) 4. M n |= AC. 5. M n is the smallest standard model of ZF+DC which is Σ1n -correct. 6. M n |= Det(∆1k ) and if Det(∆1n−1 ) holds in the world it also holds in M n . e 1 ) for i odd,ei ≤ n, M n |= Scale(Σ1 ) for i even, i ≤ n + 1. 7. M n |= Scale(Π i i 8. M n |= Generalized Continuum Hypothesis. 9. M n |= R admits a Σ1n+1 -good wellordering. 10. M n |= Scale(Σ1i ) for i ≥ n + 1.

Outline of proof. 1-5 are easy by standard methods, 6 follows by the remarks on absoluteness made in the proof of Lemma 5.1 and 7 follows from ¨ this. The key to 8 and 9 is a version of the Godel Condensation Lemma that is apropriate to Σ1n -correct models. First notice that there is a finite subset Φ0 of the axioms in ZF+DC+Det(∆1k ) such that the function î 7→ Mîn is absolute e for transitive sets which are models of Φ0 and closed under Fn∗ and hence the transitive models of Φ0 + ∀x∃î(x ∈ Mîn ) which are closed under Fn∗ are precisely of the form Mîn . Now get the Condensation Lemma as usually, except that in taking elementary submodels close under Fn∗ . We omit the details. ⊣ We draw the zig-zag picture for the scale property for the models M 3 , M 4 :

51

NOTES ON THE THEORY OF SCALES

V=M 3 or V=M 4

89:; ?>=< 89:; ?>=< Σ13 Σ12 Σ14 ??   ??   ??     89:; ?>=< 89:; ?>=< Π1 Π1 Π1 Π1 Σ11

1

2

3

4

?>=< 89:; Σ15

...

Π15

Another interesting model, M ù , can be obtained by closing under all = 3, 4, 5, . . . . This satisfies PD and has the same zig-zag picture as V (assuming PD of course), but in M ù , R admits a very simple (hyperanalytic) good wellordering. Kechris has shown by indiscernibility considerations that Fn∗ , n

M3 $ M4 $ M5 $ · · · and in fact for each n ≥ 3, there is an α ∈ M n+1 \M n . §6. Trees. We show here that the existence of a scale on a set A yields a representation for A in terms of a tree on ordinals which is very similar to the classical representation for Σ11 sets. This is the key to the applications of scales described in the remainder e of this paper. 6.1. Notation for trees. A tree on some set C is a set T of finite sequences from C such that if hc0 , c1 , . . . , ck i ∈ T ∧ i ≤ k, then hc0 , . . . , ci i ∈ T ; in particular every non-empty tree contains the empty sequence h i. A branch through (or of) a tree T on C is any function f ∈ ù C such that for all n, (hf(0), . . . , f(n)i ∈ T ). Put [T ] = the set of all branches through T and call T wellfounded if [T ] = ∅, i.e., if T has no infinite branches. The idea here is a bit clearer if we consider the relation ≻ of proper extension on finite sequences. hc0 , . . . , ck i ≻ hd0 , . . . , dℓ i ⇐⇒ k < ℓ ∧ c0 = d0 ∧ . . . ∧ ck = dk ; T is wellfounded if and only if ≺ ↾T has no infinite descending chains, i.e., if and only if ≺ ↾T is wellfounded. We can now assign an ordinal rank to every sequence of a wellfounded tree in the canonical way we do this for any wellfounded relation, |u|T = sup{|v|T + 1 : v ∈ T, u ≻ v} (where sup(∅) = 0) and define the rank of T , |T | = sup{|u|T : u ∈ T } = |h i|T .

52

ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

By convention let also |u|T = −1, if u 6∈ T . We shall often look at the subtree of T starting from some sequence, Tu = {v : u av ∈ T }, where u av is concatenation of sequences. Most useful for us will be trees of pairs, i.e., trees on sets C = A × B— usually C = ù × κ for some ordinal κ. A typical member of a tree T on A × B is a sequence hha0 , b0 i, ha1 , b1 i, . . . , han , bn ii and a branch through T is a function f ∈ ù (A × B). It will be convenient to represent each branch f by the pair hg, hi, g ∈ ù A, h ∈ ù B which determines it, f(n) = hg(n), h(n)i. For each fixed g ∈

ù

A now, we can define a new tree T (g) on B by

T (g) = {hb0 , . . . , bn i : hhg(0), b0 i, . . . , hg(n), bn ii ∈ T }. In the typical case when T is a tree on ù × κ, for each α ∈ R we will have a tree on κ T (α) = {hî0 , . . . , în i : hhα(0), î0 i, . . . , hα(n), în ii ∈ T }; notice that the function α 7→ T (α) is continuous in a strong sense, i.e., ¯ + 1) =⇒ hî0 , . . . , în i ∈ T (â). hî0 , . . . , în i ∈ T (α) ∧ α(n ¯ + 1) = â(n 6.2. κ-scales and their trees. Let hϕn : n ∈ ùi be a scale on A; we call hϕn : n ∈ ùi a κ-scale, if every ϕn is a function on A into κ, i.e., if the length of each prewellordering ≤ϕn is ≤ κ. With each κ-scale hϕn : n ∈ ùi on A we define the associated tree T on ù × κ by T = {hhα(0), ϕ0 (α)i, hα(1), ϕ1 (α)i, . . . , hα(n), ϕn (α)ii : α ∈ A}. Theorem 6.1. Let A be a pointset, A ⊆ R, hϕn : n ∈ ùi a κ-scale on A, T the associated tree. Then α ∈ A ⇐⇒ T (α) is not wellfounded ⇐⇒ (∃f)(α, f) ∈ [T ]. (this is an idea implicit in many of the classical proofs.)

NOTES ON THE THEORY OF SCALES

53

Proof. If α ∈ A, then hϕ0 (α), ϕ1 (α), ϕ2 (α), . . .i is a branch through T (α). Conversely, suppose hî0 , î1 , î2 . . .i is a branch through T (α), i.e., for each n, hhα(0), î0 i, . . . , hα(n), în ii ∈ T ; by the definition of T , there must exist reals α0 , α1 , . . . in A, so that for each n, hhαn (0), ϕ0 (αn )i,hαn (1), ϕ1 (αn )i, . . . , hαn (n), ϕn (αn )ii = hhα(0), î0 i, hα(1), î1 i, . . . , hα(n), în ii. This implies immediately that limn αn = α and for m ≤ n, ϕm (αn ) = îm , so that by the basic property of scales α ∈ A. ⊣ Kechris has shown that a converse to Theorem 6.1 is true, namely: if α ∈ A ⇐⇒ T (α) is not wellfounded, where T is a tree on ù × κ, then A admits a κ ù -scale. This shows a connection between the notion of scale and some ideas of Mansfield in [Man70]. 6.3. Computing lengths of scales. Theorem 6.2. If A ⊆ X × R admits a κ-scale, κ ≥ ù, then ∃R A admits a κ ù -scale. Proof. See the proof of Theorem 3.6. Σ11

⊣

set admits an ù -scale. e Proof. Every closed set admits an ù-scale. Now define Theorem 6.3. Every

ù

⊣

ä 1n = sup{î : î is the length of a ∆1n prewellordering of R}. e e Classically it is known that ä 11 = ℵ1 . Clearly every Π12n+1 -norme on a set has length ≤ ä 12n+1 . Thus: e e Theorem 6.4. Assume Det(∆12n ). Then every Π12n+1 set admits a ä 12n+1 -scale e e e (by the Periodicity Theorem 3.2).

Corollary 6.5. (a) Every Π11 set admits a ℵ1 -scale. e (b) Every Σ12 set admits a ℵù 1 -scale. e Corollary 6.6. Assume Det(∆12n ). Then every Σ12n+2 set admits a (ä 12n+1 )ù e e e scale. From §3.5 we also have

Theorem 6.7 (Martin, Solovay, [MS69]). If α # exists for all reals α, then every Π12 set admits a uù -scale. e Corollary 6.8 (Martin, [Mar70B]). If α # exists for all reals α, then every Σ13 set admits a (uù )ù -scale. If we also assume AC, then every Σ13 set admits a e e κ-scale, with κ < ℵ3 .

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ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

The reader should have noticed that in this section a considerable change in our attitude towards scales has happened. We started worrying not only about definability of a scale but also about its length. Later sections will show why. §7. Computing lengths of wellfounded relations. 7.1. The Kunen-Martin theorem. Recall that for a wellfounded relation < we put for x ∈ Field(<), |x|< = sup{|y|< + 1 : y < x} and we define the length of < by |<| = sup{|x|< : x ∈ Field(<)}. Theorem 7.1. Let <⊆ R × R be a wellfounded relation and assume < (as a pointset) admits a κ-scale. Then |<| < κ + . (Kunen, Martin, independently, unpublished; the proof below is Kunen’s) Proof. To the wellfounded relation < associate a tree T of reals as follows: T = {hα0 , α1 , . . . , αn i : α0 , α1 , . . . , αn ∈ Field(<) ∧ α0 > α1 > · · · > αn }. Notice that T is also wellfounded and in fact | < | ≤ |T |. To prove the last statement one can show by <-induction that for any α ∈ Field(<) and any α0 , . . . , αn such that α0 > α1 · · · > αn > α, we have |α|< = |hα0 , α1 , . . . , αn , αi|T . We shall define a mapping f : T ։ S, where S is a set of finite sequences from an ordinal ë < κ + such that hα0 , α1 , . . . , αn i ≻ hâ0 , . . . , âm i =⇒ f(hα0 , . . . , αn i) ≻ f(hâ0 , . . . , âm i) at least when m ≥ 1. Of course u ≻ v means u is a proper initial segment of v. Then we will show that ≺↾S is wellfounded, therefore |T | ≤ |≺↾S| + 1; but |≺↾S| < κ + and the proof will be complete. Let hϕn : n ∈ ùi be a κ-scale on >. To simplify the definition of f (though it is not essential) we put for α > â øn (α, â) = pα(0), â(0), ϕ0 (α, â), . . . , α(n), â(n), ϕn (α, â)q. Then notice the following limit property of høn : n ∈ ùi: If αi > âi for all i, and for each n, øn (αi , âi ) is eventually constant, then limi (αi , âi ) = (α, â) exists and α > â.

55

NOTES ON THE THEORY OF SCALES

Define now f by induction on the length of sequences: f(h i) = h i f(hα0 i) = h i f(hα0 , α1 i) = hø0 (α0 , α1 )i f(hα0 , α1 , α2 i) = hø0 (α0 , α1 ), ø1 (α0 , α1 ), ø1 (α1 , α2 ), ø0 (α1 , α2 )i, and in general f(hα0 , . . . , αn−1 , αn i) = f(hα0 , . . . , αn−1 i) a

høn−1 (α0 , α1 ), øn−1 (α1 , α2 ), . . . , øn−1 (αn−1 , αn ), øn−2 (αn−1 , αn ), . . . , ø0 (αn−1 , αn )i.

The idea is to include in f(hα0 , . . . , αn i) all øj (αi , αi+1 ) for i ≤ n − 1, j ≤ n − 1. The diagram below explains the way we have done it: ø0

ø1

ø2

ø3

...

hα0 , α1 i ◦

◦

◦

◦

...

hα1 , α2 i ◦

◦

◦

◦

...

hα2 , α3 i ◦

◦

◦

◦

...

hα3 , α4 i ◦

◦

◦

◦

...

.. .

.. .

.. .

.. .

.. .

Clearly f is an ≺-preserving (on T − {h i}) map from T onto a set of finite sequences S on κ ù = ë. Thus it will be enough to show that ≺ ↾S is wellfounded. Assume not, towards a contradiction. Then we have f(hα00 i) ≻ f(hα01 , α11 i) ≻ f(hα02 , α12 , α22 i) ≻ · · · for some α00 , α01 , α11 , . . . , such that α01 > α11 , α02 > α12 > α22 etc. Then in the diagram ø0 (α01 , α11 ) ø0 (α02 , α12 ) ø1 (α02 , α12 )

ø1 (α12 , α22 ) ø0 (α12 , α22 )

ø0 (α03 , α13 ) ø1 (α03 , α13 )

ø1 (α13 , α23 ) ø0 (α13 , α23 )

.. .

.. .

.. .

.. .

ø2 (α03 , α13 ) . . . .. .

56

ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

each column consists of identical ordinals, thus for each n and for each i j, øn (αji , αj+1 ) becomes constant for large enough i. Thus for each j, i i hαj , αj+1 i → hαj , αj+1 i and αj > αj+1 , i.e., α0 > α1 > α2 > · · · , a contradiction. ⊣ Corollary 7.2. Every Σ11 wellfounded relation has length < ℵ1 (classical e result).

Corollary 7.3. Every Σ12 wellfounded relation has length < ℵ2 . Thus if e of R, the Continuum Hypothesis holds. (Martin; there exists a Σ12 wellordering e by an unpublished forcing argument before scales were introduced.)

Corollary 7.4. Assume ∀α(α # exists). Then every Σ13 wellfounded relae tion has length < (uù )+ . If we also assume AC, then every Σ13 wellfounded e by Martin, 1 + relation has length < ℵ3 . (That ä 3 ≤ (uù ) was already shown e [Mar70B].)

7.2. Projective ordinals. We introduced in §6 the projective ordinals ä 1n and e first we mentioned that ä 11 = ℵ1 (this follows also independently from our e corollary in 7.1). By the results in § 7.1, it is then clear that

ä 12 ≤ ℵ2 (Martin) e ∀α(α # exists) + AC =⇒ ä 13 ≤ ℵ3 (Martin) e Det(∆12n ) =⇒ ä 12n+2 ≤ (ä 12n+1 )+ (Kunen, Martin) e e e (To prove 7.2.3 recall Theorem 6.4.)

(7.2.1)

(7.2.2) (7.2.3)

Det(∆12 ) + AC =⇒ ä 14 ≤ ℵ4 (Kunen, Martin) (7.2.4) e e Open Problem 7.5. Is it true that assuming AC (and any other reasonable hypotheses), ä 1n ≤ ℵn , n ≥ 5? e We shall mention some other known results about the projective ordinals in the last section. §8. Construction principles. A construction principle for a pointclass Γ asserts, roughly speaking, that every set in Γ can be expressed, in some canonical way, in terms of sets in a simpler pointclass Γ′ . A classical example is the result that every analytic (Σ11 ) set can be expressed both as a union and an e intersection of ℵ1 Borel sets. 8.1. Inductive analysis of projection of trees. Let T be a tree on ù × κ. We write A = p[T ] iff α ∈ A ⇐⇒ ∃f(hα, fi ∈ [T ]) ⇐⇒ T (α) is not wellfounded.

NOTES ON THE THEORY OF SCALES

57

Theorem 8.1. Let T be a tree on ù × κ and A = p[T ]. Put for 1 ≤ î < κ + and u a finite sequence from κ, Aîu = {α : |T (α)u | < î}, where for any tree J we abbreviate |J | < î ⇐⇒ J is wellfounded and |J | < î. Then, if lh(u) = n we have A0u = {α : hhα(0), u0 i, . . . , hα(n), un−1 ii 6∈ T } \ î Aî+1 = Aîu ∪ Auaç u ç<κ

Aëu =

[

Aîu , if ë =

[

ë>0

î<ë

and [

R\A=

Aîh i .

î<κ +

(For κ = ù this is apparently due to Sierpinski, see [Kur66, p. 32]; Martin [Mar70B] first applied these methods to κ > ù.) Proof. Notice that α 6∈ A ⇐⇒ T (α) is wellfounded ⇐⇒ (∃î < κ + )(|T (α)| < î) ⇐⇒ (∃î < κ + )(|T (α)h i | < î).

⊣

If ë is an ordinal put B(ë) = the smallest Boolean algebra containing all closed sets and closed under unions of length < ë. Let also Bn = B(ä 1n ). e Theorem 8.2. Let T be a tree on ù × κ and A = p[T ]. Then A is both the union and the intersection of κ + sets in B(κ + ). (Sierpinski for κ = ù.) Proof. In the notation of 8.1, we clearly have A1u clopen, for all u. Thus Aîu ∈ B(κ + ), for any î and u. So A is the intersection of κ + sets in B(κ + ). Now put Bî = {α : |T (α)| < î} ∪ {α : (∃u)(|T (α)u | = î)}. Since Bî = Aîh i ∪

[ (Aî+1 \ Aîu ), u u

58

ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

clearly each Bî is in B(κ + ). It is then easy to check that \ R\A= Bî ; î<κ +

thus A is the union of κ + sets in B(κ + ).

⊣

Corollary 8.3. (a) Each Σ11 set is both the union and the intersection of ℵ1 Borel sets. (Classical) e (b) Each Σ12 set is both the union and the intersection of ℵ2 sets in B(ℵ2 ). e (c) If α # exists for all reals α, each Σ13 set is both the union and the intersection of uù+ sets in B(uù+ ). Thus,eif also AC holds, each Σ13 set is both the union and the intersection of ℵ3 sets in B(ℵ3 ). (Martine[Mar70B]) (d) Det(∆12n ) implies that every Σ12n+2 set is both the union and the intersece (ä 1 )+ sets in B((ä 1e )+ ). Thus Det(∆1 )+AC =⇒ Every Σ1 set tion of 2 4 n+1 2n+1 e e in B(ℵ ). e e union and the intersection is both the of ℵ4 sets 4 In §§ 8.2 and 8.3 will see how most of the results of this corollary can be improved. 8.2. An extension of Suslin’s Theorem. Theorem 8.4. (Martin, [Mar70B] for n = 1; Moschovakis [Mos71A] in general.) Assume Det(∆12n ). Then e ∆12n+1 ⊆ B2n+1 . e Proof. (In this proof we use essentially [Mos70A, Lemmas 9 & 10].) Let A ⊆ R, A ∈ ∆12n+1 . Find a Π12n+1 -scale on R \ A, say hϕm : m ∈ ùi. Since R \ A ∈e ∆12n+1 , each prewellordering ≤ϕm is actually a ∆12n+1 e) > prewellordering, thus ite has length îm < ä 12n+1 . Since obviously cf(ä 12n+1 e e ù, supm îm < ä 12n+1 , say îm < κ < ä 12n+1 , for all m, and hϕm : m ∈ ùi is a e e κ-scale. For each α put J (α) = {hî0 , . . . , îk i : îi < κ ∧ ((∃α0 . . . αk ∈ R \ A)(∀i ≤ k(α¯ i (i) = α(i)) ¯ ∧ (∀i ≤ k∀j)(i ≤ j ≤ k =⇒ ϕi (αj ) = îi )))}. The mapping α → 7 J (α), from R into trees on κ, is continuous, i.e., for each hî0 , . . . , îk i, {α : hî0 , . . . , îk i ∈ J (α)} is open, and α 6∈ A ⇐⇒ J (α) is not wellfounded; thus α ∈ A ⇐⇒ J (α) is wellfounded. Put ë = sup{J (α) + 1 : α ∈ A}. Suppose we can prove ë < ä 12n+1 . Then as in 8.1 we can define Aîu = {α : |J (α)u | < î} for 1 ≤ î < ëe and u a finite sequence from κ and prove that S Aîu ∈ B2n+1 (since κ, ë < ä 12n+1 ). Since A = î<ë Aîh i we have A ∈ B2n+1 . e

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59

We prove now ë < ä 12n+1 . Define the tree e J ∗ (α) = {hα0 , . . . , αk i : hα0 , . . . , αk i ∈ R \ A ∧ ∀i ≤ k(α¯ i (i) = α(i)) ¯ ∧ ∀i ≤ k∀j∀ℓ(i ≤ j ≤ ℓ ≤ k =⇒ (ϕi (αj ) = ϕi (αℓ ))). Then there is an obvious surjective map f from J ∗ (α) onto J (α), namely f(hα0 , . . . , αk i) = hϕ0 (α0 ), . . . , ϕk (αk )i, which is clearly ≺-preserving. Thus J ∗ (α) is wellfounded for any α ∈ A and it is easy to check that for α ∈ A |hα0 , . . . , αk i|J ∗ (α) = |f(hα0 , . . . , αk i)|J (α) . Thus |J (α)| = |J ∗ (α)|, for every α ∈ A. It will then be enough to show sup{|J ∗ (α)| + 1 : α ∈ A} < ä 12n+1 . To prove this define e hα,hα0 , . . . , αk ii > hâ, hâ0 , . . . , αm ii ⇐⇒ α = â ∈ A ∧ hα0 , . . . , αk i, hâ0 , . . . , âm i ∈ J ∗ (α) ∧ hα0 , . . . , αk i ≻ hâ0 , . . . , âm i. Then < is a ∆12n+1 wellfounded relation, if we code the finite sequences by single reals. e But a simple variation of Lemma 10 in [Mos70A] shows that every ∆12n+1 wellfounded relation has length < ä 12n+1 , thus | < | < ä 12n+1 and e we areedone, since |J ∗ (α)| ≤ | < |, for each α ∈eA. ⊣ Corollary 8.5 (Moschovakis, [Mos71A]). Assume Det(∆12n ). Then every e set is the union of ä 12n+1 sets in B2n+1 . e e Proof. It follows from PWO(Π12n+1 ) that every Π12n+1 set is the union of e e 6.4). It also follows 1 from PWO(Π12n+1 ) ä 2n+1 ∆12n+1 sets (see remarks before e then e e 1 1 that a 1-1 continuous image of a ∆2n+1 set is also a ∆2n+1 set. We can get e e the result easily by applying the uniformization theorem and 8.4. ⊣ Let AD be the statement “Every set is determined”. Martin [Mar70B] has shown that AD implies that ∆12n+1 ⊇ B2n+1 ; thus AD implies that ∆12n+1 = e every e B2n+1 . Moschovakis has shown in [Mos70A], that AD implies that 1 1 1 union of ä 2n+1 sets in ∆2n+1 is in Σ2n+2 ; thus AD implies e e e A ∈ Σ12n+2 ⇐⇒ A is the union of ä 12n+1 sets in B2n+1 . e e 8.3. Unions of Borel sets.

Σ12n+2

Theorem 8.6 (Martin, [Mar70B]). Assume AC. If a pointset A admits a κ-scale with κ < ℵn+1 , 0 < n ∈ ù, then A is the union of ℵn Borel sets. Corollary 8.7. (a) Assume AC + ∀α(α # exists). Then every Σ13 set is the e union of ℵ2 Borel sets. 1 1 (b) Assume AC+Det(∆2 ). Then every Σ4 set is the union of ℵ3 Borel sets. e e

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Proof of the theorem. Assume A ⊆ R admits a κ-scale with κ < ℵn+1 . Then for some tree T, α ∈ A ⇐⇒ T (α) is not wellfounded, where we may assume that T is actually a tree on ù × ℵn (we replace if necessary the tree coming from the scale on A by an isomorphic tree, noticing that |κ| ≤ ℵn ). Since n > 0, cf(ℵn ) > ù, thus if T (α) is not wellfounded there is a î < ℵn such that T î (α) is not wellfounded, where T î = {hhk0 , î0 i, . . . , hkm , îm ii ∈ T : î0 , . . . , îm ≤ î}. Thus α ∈ A ⇐⇒ ∃în < ℵn [T în (α) is not wellfounded]. If în < ℵn we can replace T în by an isomorphic tree on ù × ℵn−1 , say T1în . Then α ∈ A ⇐⇒ (∃în < ℵn )(T1în (α) is not wellfounded). If n − 1 > 0 we have again T1în (α) is not wellfounded ⇐⇒ (∃în−1 < ℵn−1 )((T1în )în−1 is not wellfounded) and we proceed similarly. After at most n steps we get α ∈ A ⇐⇒∃în < ℵn ∃în−1 < ℵn−1 . . . ∃î1 < ℵ1 (T în ,...,î1 (α) is not wellfounded) where T în ,...,î1 is a tree on ù×ù. But then {α : T în ,...,î1 (α) is not wellfounded} is a Σ11 set, thus it is the union of ℵ1 Borel sets and the proof is complete. ⊣ e Open Problem 8.8. Prove assuming AC (and any other reasonable hypotheses) that every Σ1n+1 set is the union of ℵn Borel sets, n ≥ 4. Notice that a e solution to the problem at the end of §7 solves this problem too. §9. Constructibility in the tree associated with a scale. In §6 we associated with each κ-scale hϕn : n ∈ ùi on a set A ⊆ R a tree T on ù × κ. We introduce and study here the models L[T ], where T comes from a complete Π12n+1 set, granting Det(∆12n )—these are the basic tools for the results in the e theorem in the present section is that the tree that next two sections. The key comes from a complete Π11 set is in fact constructible. 9.1. The models L[T 2n+1 ]. Theorem 9.1 (Folk-type result). Let T be the tree associated with a κ-scale on some set A, let Q ⊆ R × R and assume that for some recursive f : R × R → R, Q(α, â) ⇐⇒ f(α, â) ∈ A.

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Then for some tree S ∈ L[T ] on ù × κ ∃âQ(α, â) ⇐⇒ S(α) is not wellfounded. ⇐⇒ (∃â ∈ L[T, α])Q(α, â)

(1) (2)

Proof. We have ∃âQ(α, â) ⇐⇒ ∃â(f(α, â) ∈ A) ⇐⇒ ∃â∃ã(f(α, â) = ã ∧ ã ∈ A) ¯ ⇐⇒ ∃â∃ã(∀nR(α(n), ¯ â(n), ã(n)) ¯ ∧ T (ã) is not wellfounded) where R is recursive. Put S ′ = {hha0 , hb0 , c0 , î0 ii, . . . , hak , hbk , ck , îk iii : hhc0 , î0 i, . . . , hck , îk ii ∈ T ∧ R(pa0 , . . . , akq, pb0 , . . . , bkq, pc0 , . . . , ckq)}. Then S ′ is a tree on ù × (ù × ù × κ), S ′ ∈ L[T ] and ∃âQ(α, â) ⇐⇒ S ′ (α) is not wellfounded ⇐⇒ ∃â ∈ L[T, α]Q(α, â) where the last equivalence follows from the usual absoluteness of wellfoundedness. To get instead of S ′ a tree on ù × κ fix a 1-1 mapping from ù × ù × κ onto κ in L[T ] and replace S ′ by an isomorphic tree on ù × κ, call it S. ⊣ Fix now a complete Π12n+1 set P2n+1 ⊆ R. (P2n+1 is such that for every A ⊆ X, A ∈ Π12n+1 we can find f : X → R recursive such that x ∈ A ⇐⇒ f(x) ∈ P2n+1 .) Assuming Det(∆12n ), let hϕn : n ∈ ùi be a fixed for the e let T 2n+1 be the associated tree. T 2n+1 is discussion Π12n+1 scale on P2n+1 and 1 a tree on ù × ä 2n+1 . e Theorem 9.2. Assume Det(∆12n ). Then for every Σ12n+2 set A we can find a tree S ∈ L[T 2n+1 ] such that e α ∈ A ⇐⇒ S(α) is not wellfounded.

Theorem 9.3. Assume Det(∆12n ). Then Σ12n+2 formulas are absolute for e L[T 2n+1 ].

Proof. We show successively that for 2n + 2 ≥ k ≥ 2, Σ1k formulas are absolute for L[T 2n+1 ]. For k = 2 this is Shoenfield’s theorem, while for k ≥ 3 we proceed using (3) and (4) of Theorem 9.1. ⊣ Unfortunately, except for the case n = 0, which we shall study in the rest of this section, there is practically nothing known about the internal structure of L[T 2n+1 ].2

2 It has been recently shown by Harrington and Kechris that for all n ≥ 0, R ∩ L[T 2n+1 ] is the largest countable Σ12n+2 set of reals (as conjectured by Moschovakis) and that additionally,

L[T 2n+1 ] |= “R has a ∆12n+2 -good wellordering”.

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9.2. Absoluteness of closed games. Let S be a set of even finite sequences from a set A. We define the game GS as follows: I a0 a1 .. .

II b0 b1 .. .

I plays a0 , a1 , . . . and II plays b0 , b1 . . . , ai , bi ∈ A. Then I wins iff for some n, ha0 , b0 , . . . , an , bn i ∈ S. Clearly the game is open in I.

The following is a folk-type result. Theorem 9.4. Let M |= ZF+DC and M ⊇ Ord. Let A, S ∈ M , and assume A is wellorderable in M . Then player I has a winning strategy in GS iff M |= player I has a winning strategy in GS and similarly for player II. Moreover the player who has a winning strategy has a winning strategy (for the game in the world) which lies in M . Proof. For each ha0 , b0 , . . . , an , bn i consider the subgame GS (a0 , b0 , . . . , an , bn ) defined by; I α

II â

I plays α, II plays â and I wins iff for some m ha0 , b0 , . . . , an , bn iahα(0), â(0), . . . , α(m), â(m)i ∈ S.

Then define S0 = S S î = {ha0 , b0 , . . . , an , bn i : ∃an+1 ∈ A∀bn+1 ∈ A ∃ç < î(ha0 , b0 , . . . , an+1 , bn+1 i ∈ S ç ). Then for each î, ha0 , b0 , . . . , an , bn i ∈ S î =⇒ player I has a winning strategy in GS (a0 , b0 , . . . , an , bn ). Using this we show: Claim 9.5. Player II has a winning strategy in G if and only if ∀î(h i 6∈ S î ). Proof of Claim 9.5. If player II has a winning strategy in GS = GS (h i), then player I has no winning strategy in GS (h i), thus for all î, h i 6∈ S î . Conversely assume that for each î, h i 6∈ S î . We describe a winning strategy for player II in GS as follows: If player I plays a0 , player II plays the least b0 (in a fixed wellordering of A) such that ∀î(ha0 , b0 i 6∈ S î ). Such a b0 exists, because otherwise for all b, there exists a î such that ha0 , bi ∈ S î . Let g(b) = least such î and find î0 > all g(b), b ∈ A. Then ∀b∃î < î0 (a0 , b) ∈ S î , thus h i ∈ S 0 , a contradiction. Similarly if player I plays a1 , player II picks the least b1 such that ∀î(ha0 , b0 , a1 , b1 i 6∈ S î ), etc. ⊣ (Claim 9.5) These results suggest that L[T 2n+1 ] is a correct higher level analogue of L. Their proof uses determinacy of all hyperprojective sets. See [HK77].

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Since the above equivalence was proved under the assumption “ZF+DC+A is wellorderable” and since î 7→ S î is clearly an absolute map and M ⊇ Ord, it is immediate the “player II has a winning strategy” is absolute for M , thus the same is true for “player I has a winning strategy.” Moreover the argument above clearly provides a winning strategy for player II which lies in M and wins in the world, thus it will be enough in order to complete the proof to show that when player I has a winning strategy we can find one (who wins in the world also) in M . Notice that Player I has a winning strategy ⇐⇒ ∃î(h i ∈ g S ) and check that the following is a winning strategy for player I which lies in M . Put î0 = least î such that h i ∈ S î . If î0 = 0, player I has already won. If î0 > 0, let player I play the least a0 such that for every b, ∃î < î0 (ha0 , bi ∈ S î ). If now player II plays b0 , let î1 = least î < î0 such that ha0 , b0 i ∈ S î . If î1 = 0, player I has already won, otherwise let player I play the least a1 such that for all b, ∃î < î1 (ha0 , b0 , a1 , bi ∈ S î ) etc. (Notice that î0 > î1 > · · · , so this cannot go on.) ⊣ (Theorem 9.4) 9.3. Proof that T 1 ∈ L. Suppose A ⊆ ℵ1 . We let Code(A) = {α ∈ WO : |α| ∈ A}. Similarly, if T is a tree on ù × ℵ1 , we let Code(T ) = {hk0 , α0 , . . . , kn , αn i : hhk0 , |α0 |i, . . . , hkn , |αn |ii ∈ T }. We say that Code(T ) is in Γ iff {pk0 , α0 , . . . , kn , αnq : hk0 , α0 , . . . , αn i ∈ Code(T )} ∈ Γ where pk0 , α0 , . . . , kn , αnq = hn, k0 , . . . , kn , α0 (0), . . . , αn (0), α0 (1), . . . , αn (1), . . .i ∈ R. Lemma 9.6 (Kechris). Let A ⊆ R, A ∈ Π11 and assume hϕn : n ∈ ùi is a on A. Then the tree T associated with hϕn : n ∈ ùi is Σ12 in the codes.

Π11 -scale

Proof. We have hk0 , α0 , . . . , kn , αn i ∈ Code(T ) ⇐⇒ α0 , . . . , αn ∈ WO ∧(∃α)(α ∈ A ∧ ϕ0 (α) = |α0 | ∧ α(0) = k0 ∧ . . . ∧ ϕn (α) = |αn | ∧ α(n) = kn ). The result follows immediately if we can show that for each n, α ∈ A ∧ â ∈ WO ∧ ϕn (α) = |â|

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is a Σ12 relation in α, â, n. But each ϕn is a Π11 -norm, thus every initial segment of ≤ϕn =≤n will have countable length (since ä 11 = ℵ1 ). From this we have e α ∈ A ∧ â ∈ WO ∧ ϕn (α) = |â| ⇐⇒ α ∈ A ∧ â ∈ WO ∧ ∃ä((∀m)(m ≤â m =⇒ (ä)m ≤n α) ∧ (∀ã)(ã ≤n α =⇒ (∃m)(m ≤â m ∧ (ä)m ≤n α ∧ α ≤n (ä)m ) ∧ ∀m, ℓ(m <â ℓ ⇐⇒ (ä)m
⊣

Theorem 9.7. Suppose A ⊆ ℵ1 and Code(A) is Σ12 . Then A is in L. Similarly if T is a tree on ù × ℵ1 and Code(A) is Σ12 , then T ∈ L. (For A ⊆ ℵ1 with Code(A) ∈ Π11 the result is implicit in methods (using forcing) of Solovay. The game method used below as a substitute of forcing was used by Moschovakis to prove a version of the corollary below and traces to [Mos70B]. The present version of the theorem is due to Kechris.) Corollary 9.8 (Moschovakis). T 1 ∈ L. Proof of Theorem 9.7. We give the proof for A ⊆ ℵ1 , the case of a tree being similar. Thus let A ⊆ ℵ1 and Code(A) = P ∈ Σ12 . Then î ∈ A ⇐⇒ (∃α)(α ∈ P ∧ |α| = î). Let α ∈ P ⇐⇒ ∃â Q(α, â) ⇐⇒ ∃â(f(α, â) ∈ WO), where Q ∈ Π11 and f : R × R → R is recursive and for all α, â, f(α, â) ∈ LOR. Then î ∈ A ⇐⇒ ∃α∃â(f(α, â) ∈ WO ∧ |α| = î). Consider the following game Gî : I î0 î1 .. .

a0 a1 .. .

b0 b1 .. .

α

â

II ç0 ç1 .. .

ϑ0 ϑ1 .. .

k0 k1 .. .

Player I and player II play as in the diagram natural numbers and ordinals < ℵ1 and player II wins iff for every n, either for some i ≤ n, îi ≥ î or all the following are true: (a) The mapping i 7→ çi (i ≤ n) is order preserving on the part of ≤f(α,â) already determined by hha0 , . . . , an i, hb0 , . . . , bn ii (notice that f is continuous). (b) The mapping i 7→ ϑi (i ≤ n) is order preserving on the part of ≤α already determined by ha0 , . . . , an i and ϑi < î, for each i ≤ n.

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If ki ≤ n, then ϑki = îi and if pki , kiq = j ≤ n, then aj = 0. Notice now the following: Claim 9.9. For î < ℵ1 , î ∈ A if and only if player II has a winning strategy in Gî . Proof of Claim 9.9. Assume î ∈ A; let α, â be such that f(α, â) ∈ WO and |α| = î and let i → çi be an order preserving map on ≤f(α,â) into ℵ1 and i → ϑi a mapping from ù into î such that its restriction to Field (≤α ) is an order preserving bijection onto î, with inverse g. Consider the following strategy for player II in Gî and verify easily that it is winning: If player I plays î0 , player II plays α(0), â(0), ç0 , ϑ0 , g(î0 ) (unless î0 ≥ î in which case player II plays anything). If player I plays î1 , player II gives α(1), â(1), ç1 , ϑ1 , g(î1 ) etc. Conversely assume player II has a winning strategy. Let player I play î0 , î1 , . . . enumerating without repetitions î, i.e., î = {î0 , î1 , î2 , . . . }. Then player II plays by his winning strategy and produces α, â, hç0 , ç1 , . . .i, hϑ0 , ϑ1 , . . .i, hk0 , k1 , . . .i, such that i → çi is order preserving on ≤f(α,â) , thus f(α, â) ∈ WO, i → ϑi is order preserving on ≤α into î, thus α ∈ WO, and finally îi → ki is an inverse to i → ϑi on Field(≤α ), thus |α| = î and the proof is complete. ⊣ (Claim 9.9) It is clear now that Gî = GSî where Sî is a set of finite sequences and moreover the map î 7→ Sî is absolute for L. Thus for î < ℵ1 , î ∈ A ⇐⇒ player II has a winning strategy in GSî ⇐⇒ L |= player II has a winning strategy in GSî . So A is definable in L, therefore A ∈ L. Notice that the definition of A involves as the only parameter ℵ1 , thus A = ô L (ℵ1 ) for some term ô. ⊣ The following converse to Theorem 9.7 was proved by Kechris: ∀α(α # exists) =⇒ Every A ∈ L, A ⊆ ℵ1 has Code(A) ∈ Σ12 . e The proof (as also the proof of Theorem 9.7) relativizes to any real and thus gives the elegant characterization: S If L˜ = α∈R L[α] and ∀α(α # exists), then A ⊆ ℵ1 =⇒ (A ∈ L˜ ⇐⇒ Code(A) ∈ Σ12 ). e Since T 2n+1 is a tree definable by some formula of set theory, it makes sense to talk of the tree (T 2n+1 )M for any model M of ZF+DC+Det(∆12n ). In particular, for the models M n introduced in §5, we have (by Theoremse5.2 and 9.3) that M = L[(T )M ]

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if M = M 2n+1 or M = M 2n+2 and T = T 2n+1 . Thus the models of §5 are ordinary relative constructibility models, but we do not have an independent characterization of the trees (T 2n+1 )M for these M ’s. Finally, we should mention that in contrast to L = L[T 1 ], Kechris has noticed that M 2n $ L[T 2n−1 ], n ≥ 2; this is because M 2n ⊆ L[α] for some α. §10. Lebesgue measurability and the property of Baire. We prove here Solovay’s results that Σ12 sets are Lebesgue measurable and have the property of e Baire, if ∀α(ℵL[α] < ℵ1 ). These will appear as corollaries of more general 1 results about approximations of sets which admit definable scales, but we do not know any other applications of these general theorems. The forcing-free proofs given here are an adaptation by Moschovakis of the original Solovay proofs which used the forcing method. 10.1. Sets which are ∞-Boolean over a model of ZFC. We define a set of codes C(κ) and for each a ∈ C(κ) a set Ba in the Boolean algebra B(κ), by the induction: 1. h1, pn0 , . . . , nkqi ∈ C(κ) and Bh1,pn0 ,...,nkqi = {α : α(k ¯ + 1) = pn0 , . . . , nkq}, 2. If a ∈ C(κ), then h2, ai ∈ C(κ) and Bh2,ai = R \ Ba . 3. If f : î → C(κ) is a function with domain some î < κ and values in C(κ), then h3, fi ∈ C(κ) and [ Bh3,fi = Bf(ç) . ç<î

Clearly, using AC, B(κ) = {Ba : a ∈ C(κ)}. Moreover the relation “a ∈ C(κ)” is definable by a formula which is absolute for models of ZFC. In particular the set C(ù + 1) gives canonical codes to the algebra B(ù + 1) of Borel sets. Suppose M is a (transitive, containing all ordinals) model of ZFC. A set of reals A is κ-Boolean over M, if A = Ba for some a ∈ C(κ) ∩ M.

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A is ∞-Boolean over M if it is κ-Boolean over M for some κ. A is ù + 1Boolean over M if it is “Borel rational over M” in Solovay’s terminology, [Sol70]. We would expect that sets which are ∞-Boolean over “thin”, definable models have some regularity properties. But first let us show how we can get such sets. Theorem 10.1. Let T be a tree on ù × κ and A = p[T ] = {α : T (α) is not wellfounded}. Then A is κ + + 1-Boolean over L[T ]. Proof. Immediate from Theorem 8.1. Corollary 10.2. If A ∈

Σ12 ,

⊣

then A is ℵ1 + 1-Boolean over L.

Remark. We know already that a Σ12 set is the union of ℵ1 Borel sets, a seemingly better representation than the above. Its only disadvantage is that S we cannot have A = î<ℵ1 Bf(î) , where f ∈ L, i.e., we cannot control this representation from inside L, unless ℵL1 = ℵ1 . 10.2. Ideals of Borel sets which are suitable over a model of ZFC. We isolate in a definition the key properties of the ideals of sets of measure 0 and sets of the first category that we need. Let M be a fixed model of ZFC, J a collection of Borel sets. We call J suitable over M if the following conditions hold: 1. J = 6 ∅ and J is closed under subsets and countable unions (i.e., J is a ó-ideal in the Boolean algebra of Borel sets). 2. There is a definable functor G: M → M (i.e., an operation on M definable by a formula, perhaps with parameters from M) such that if f : î → C (ù + 1), f ∈ M is a function in M which maps some ordinal î in the Borel codes of M, then G(f) = hçi : i ∈ ùi gives a countable sequence of ordinals less than î such that [ for all ç < î, Bf(ç) \ Bf(çi ) ∈ J . i

This last condition says in effect that the Boolean algebra B(ù + 1)/J has the countable chain condition for wellordered sequences of Borel sets in M. It is the key property that we need. If J is the ideal of Borel sets of measure 0 (where we can take the measure on R as coming from the measure on the true reals via the standard topological

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identification of R with the irrationals or we can take only the subsets of ù 2 with the product measure) and if J is the ideal of Borel sets of the first category, then J is suitable over every M which admits a definable (with parameters from M) wellordering, in particular any L[X ]. This is not hard to prove from standard analytic and topological facts about these ideals, e.g., see [Sol70, Kur66]. Let J be suitable over M and put Alg(M, J ) = {α : for some a ∈ C (ù + 1) ∩ M, Ba ∈ J and α ∈ Ba }. The reals in Alg(M, J ) are called J - algebraic over M. If we think of sets in J as small sets, they are reals which can be approximated in M, in the sense that they belong to a small set with code in M. Let Trans(M, J ) = {α : α 6∈ Alg(M, J )} be the set of reals J -transcendental over M. In the case of measure 0 or of the first category these are Solovay’s random and Cohen generic (over M) reals, respectively. This is the key notion needed for the statement and proof of the approximation theorem we want. Theorem 10.3. Let M be a model of ZFC. Let J be a ó-ideal of Borel sets which is suitable over M. Then for every set A which is ∞-Boolean over M, there is a Borel set A∗ , rational over M, such that A∆A∗ = (A \ A∗ ) ∪ (A∗ \ A) ⊆ Alg(M, J ). If in addition |M ∩ R| < ℵ1 , then A∆A∗ ∈ J . Proof. Let A be κ-Boolean over M. We assign by induction in M on the codes C(κ) ∩ M, to each a ∈ C(κ) ∩ M a code a ∗ ∈ C(ù + 1) ∩ M such that Ba ∆Ba ∗ ⊆ Alg(M, J ), i.e., for any α ∈ Trans(M, J ) α ∈ Ba ⇐⇒ α ∈ Ba ∗

(∗)

∗

Case 1. a = h1, pn0 , . . . , nkqi. Put a = a. Case 2. a = h2, bi. Put a ∗ = h2, b ∗ i. Case 3. a = h3, fi. Put f ∗ (î) = (f(î))∗ . Then let hçi : i ∈ ùi = G(f ∗ ) and f1 (i) = f ∗ (çi ). Clearly f1 : ù → C(ù + 1) ∩ M and we define a ∗ = h3, f1 i. The verification that (∗) holds is trivial, except for the third case. Thus let a = h3, fi ∈ M, where f : î → C(κ) ∩ M, and by induction hypothesis assume that for α ∈ Trans(M, J ) α ∈ Bf(ç) ⇐⇒ α ∈ B(f(ç))∗ ⇐⇒ α ∈ Bf ∗ (ç) .

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We have to show that for α ∈ Trans(M, J ), α ∈ Ba ⇐⇒ α ∈ Bh3,f1 i . Let α ∈ Trans(M, J ) and α ∈ Ba . Then αS∈ Bf(ç) for some S ç > î. Thus α ∈ Bf ∗ (ç) . Then by 2 above α 6∈ Bf ∗ (ç) \ i Bf ∗ (çi ) , so α ∈ i Bf ∗ (çi ) = Bh3,f1 i . S Conversely assume α ∈ Trans(M, J ) and α ∈ Bh3,f S 1 i . Then α ∈ i Bf ∗ (çi ) , ⊣ so for some i, α ∈ Bf ∗ (çi ) , thus α ∈ Bf(çi ) and α ∈ ç<î Bf(î) = Ba . Corollary 10.4. (a) Assume ∀α(ℵ1L[α] < ℵ1 ). Then every Σ12 set is Lebesgue Measurable and has the property of Baire. (Solovay,e unpublished.) (b) Assume Det(∆12n ) and that for each α, |R ∩ L[T 2n+1 , α]| = ℵ0 . Then e is Lebesgue Measurable and has the property of Baire. every Σ12n+2 set e (Solovay, unpublished.) It should be pointed out that it has been known for some time (see [MS64]) that PD implies that all projective sets are Lebesgue measurable and have the property of Baire. The proof of (b) above seems to use “less determinacy” (none if n = 0) and has a different flavor. §11. Perfect subsets of pointsets. The main results here are that if PD holds and for each n, |L[T 2n+1 ] ∩ R| = ℵ0 , then every uncountable projective set has a perfect subset and there exist largest countable Σ12n+2 sets. The first result does not need the hypothesis |L[T 2n+1 ]| ∩ R| = ℵ0 , see [Dav64], [Myc64], but the proof given here (due to Solovay and Mansfield) uses “less determinacy”, none if n = 0. 11.1. The theorem on perfect sets. Solovay in [Sol66] proved that, if ∀α [ℵ1L[α] < ℵ1 ], then every uncountable Σ12 set contains a perfect subset. His e method was one of the earliest applications of forcing to the proof of positive results. A few months later Mansfield obtained a similar theorem (see [Man69]) and in [Man70] he generalized the result to Theorem 11.1 below. His proof also used forcing. Finally Solovay obtained a new forcing-free proof of Mansfield’s result. This is essentially the proof reproduced below with one alteration: Solovay’s “inductive analysis” was replaced by the notion of “derivation on a tree”; as a result the proof becomes astonishingly similar to Cantor’s proof of the Cantor-Bendixson theorem. Theorem 11.1 (Mansfield, [Man70]). Assume T is a tree on ù × κ and A = p[T ]. Then if A contains an element not in L[T ], A contains a perfect set. Proof. For any tree T on ù × κ we define the derivative T ′ of T as follows: hhk0 , î0 i, . . . , hkn , în ii ∈ T ′ ⇐⇒ There are two, incompatible in

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the first coordinate, extensions of hhk0 , î0 i, . . . , hkn , în ii, both in T , i.e., ′ we can find hhk0′ , î0′ i, . . . , hkm′ , îm ii, hhk0′′ , î0′′ i, . . . , hkℓ′′ , îℓ′′ ii ∈ T extending hhk0 , î0 i, . . . , hkn , în ii, such that hk0′ , . . . , km′ i is incompatible with hk0′′ , . . . , kℓ′′ i. Notice that T ′ is a tree and T ′ ⊆ T . Then we define a´ la Cantor the îth-derivative of T by T0 = T T î+1 = (T î )′ \ [ Të = T î , if ë = ë > 0. î<ë

It is then clear that î → T is a function absolute for any model containing T , in particular for L[T ]. Moreover T 0 ⊇ T 1 ⊇ T 2 ⊇ · · · ⊇ T î ⊇ T î+1 ⊃ · · · ; thus let îT be the least î such that T î = T î+1 . Case 1. T îT = ∅. Then consider α ∈ A. Since A = p[T ], we can find f such that hα, fi ∈ [T ]. Since hα, fi 6∈ [T îT ] = ∅, let î < îT be such that hα, fi ∈ [T î ] \ [T î+1 ]. Let n be the least integer such that hhα(0), f(0)i, . . . , hα(n), f(n)ii 6∈ T î+1 . Since hhα(0), f(0)i, . . . , hα(n), f(n)ii ∈ T î , it is clear that all branches of T î extending hhα(0), f(0)i, . . . , hα(n), f(n)ii have î the same real part, namely α, therefore p[Thhα(0),f(0)i,...,hα(n),f(n)ii ] = {â}, î

where α = hα(0), . . . , α(n)iaâ. But then clearly â ∈ L[T ], since â is definable absolutely from elements of L[T ]; thus α ∈ L[T ]. So in this case A ⊆ L[T ]. Case 2. T îT 6= ∅. Then T îT = (T îT )′ 6= ∅, i.e., every sequence in T îT has two extensions in T îT which are incompatible in the first coordinate. Then it ⊣ is easy to show that p[T îT ] (⊆ p[T ] = A) contains a perfect set. Corollary 11.2 (Solovay, [Sol66]). Every Σ12 set with an element not in L contains a perfect set; thus, ∀α(ℵ1L[α] < ℵ1 ) =⇒ Every uncountable Σ12 set e contains a perfect subset.

Corollary 11.3. Assume Det(∆12n ). Then every Σ12n+2 set with an element not in L[T 2n+1 ] contains a perfecteset. Thus if |L[T 2n+1 , α] ∩ R| = ℵ0 , for all α ∈ R, every uncountable Σ12n+2 set contains a perfect subset. e 11.2. Largest countable Σ12n sets. The following is also a corollary of Theorem 11.1. Theorem 11.4 (Solovay, [Sol66]). Assume ℵL1 < ℵ1 . Then there exists a largest countable Σ12 set of reals, namely {α : α ∈ L}. The next result extends Theorem 11.4 to higher levels. Theorem 11.5 (Kechris, Moschovakis.). Assume Det(∆12n ). If the set R ∩ e Σ1 L[T 2n+1 ] is countable, then there exists a largest countable 2n+2 set.

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Proof. Notice first that Unif(Π12n+1 ) implies that for every countable Σ12n+2 set A we can find a countable Π12n+1 set B, so that every real in A is recursive in some real in B. Thus it will be enough to find a countable Σ12n+2 set C which contains all countable Π12n+1 sets. Then C ∗ = {α : (∃â)(â ∈ C ∧ α is recursive in â)} is the largest countable Σ12n+2 set. It will be convenient for this proof to choose a particular Π12n+1 -complete set P2n+1 and a Π12n+1 -scale on it as follows: Let W2n+1 ⊆ ù × R be universal for Π12n+1 subsets of R and put α ∈ P2n+1 ⇐⇒ (α(0), α ′ ) ∈ W2n+1 , where α ′ = hα(1), α(2), . . .i. Let also hϕn : n ∈ ùi be a Π12n+1 -scale on P2n+1 . Let T 2n+1 be the tree associated with this scale. We define now C and then we show that it works: α ∈ C ⇐⇒ ∃m∃î(|{maâ ∈ P2n+1 : ϕ0 (maâ) ≤ î}| ≤ ℵ0 ∧ ϕ0 (maα) ≤ î). Claim 11.6. C ∈ Σ12n+2 . Proof of Claim 11.6. Notice that the statements “α ∈ C ” and “∃m∃â(â ∈ P2n+1 ∧ ϕ0 (maα) ≤ ϕ0 (â) ∧ ∃ã∀ä(ϕ0 (maä) ≤ ϕ0 (â) implies ∃k(ä = (ã)k )))” are equivalent. ⊣ (Claim 11.6) Claim 11.7. C contains every countable Π12n+1 set. Proof of Claim 11.7. Let B ∈ Π12n+1 , B ⊆ R, |B| ≤ ℵ0 . Find m such that â ∈ B ⇐⇒ hm, âi ∈ W2n+1 ⇐⇒ maâ ∈ P2n+1 . If B 6⊆ C , let â0 ∈ B \ C . Put î = ϕ0 (maâ). Then since â0 6∈ C, |{maâ ∈ P2n+1 : ϕ0 (maâ) ≤ î}| > ℵ0 ; but B ⊇ {maâ ∈ P2n+1 : ϕ0 (maâ) ≤ î}, a contradiction. ⊣ (Claim 11.7) Claim 11.8. C ⊆ L[T 2n+1 ]; thus |C | = ℵ0 . Proof of Claim 11.8. It is enough to show that if for some m, î, |{maâ ∈ P2n+1 : ϕ0 (maâ) ≤ î}| ≤ ℵ0 , then {maâ ∈ P2n+1 : ϕ0 (maâ) ≤ î} ⊆ L[T 2n+1 ]. (T 2n+1 )m,î = {hhk0 , î0 i, . . . , hkℓ , îℓ ii ∈ T 2n+1 : k0 = m ∧ î0 ≤ î}. Clearly (T 2n+1 )m,î ∈ L[T 2n+1 ] and the limit property of scales shows that α ∈ p[(T 2n+1 )m,î ] ⇐⇒ α ∈ P2n+1 ∧ ϕ0 (α) ≤ î ∧ α(0) = m. Thus {maâ ∈ P2n+1 : ϕ0 (maâ) ≤ î} = p[(T 2n+1 )m,î ], so by Theorem 11.1 |{maâ ∈P2n+1 : ϕ0 (maâ) ≤ î}| ≤ ℵ0 =⇒ {maâ ∈ P2n+1 : ϕ0 (maâ) ≤ î} ⊆ L[(T 2n+1 )m,î ] ⊆ L[T ]. ⊣ (Claim 11.8) ⊣

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Open problem4. It is a well known result, that every countable Σ11 set contains only ∆11 reals. Thus there is no largest countable Σ11 set (since {α : α ∈ ∆11 } ∈ Π11 \ Σ11 ). Does any of these results generalize to Σ12n+1 (n ≥ 1), under any reasonable hypotheses? §12. A summary of results about projective ordinals. We give here a list of theorems about the projective ordinals. Proofs are omitted but many results follow from what we have already done. 1. ä 11 = ℵ1 (classical) 2. e(a) ä 12 ≤ ℵ2 (Martin, unpublished), If α # exists for all reals α, then äe12 ≤ u2 (Martin, unpublished), e α # exists for all reals α, then ä 1 ≥ u (Kechris, Martin, unpub(b) If 2 2 e α, then ä 1 = u . lished). Thus if α # exists for all reals 2 2 (c) If α # exists for all reals α, then for n ≥ 3,eä 1n = uä 1n (Kechris), e where u1 , u2 , . . . , uî , . . . is the increasing enumeration ofe the uniform indiscernibles. 3. If α # exists for all reals α and AC holds, then ä 13 ≤ ℵ3 (Martin, e [Mar70B]). 1 1 + 4. PD =⇒ ä 2n+2 ≤ (ä 2n+1 ) (Kunen, Martin, unpublished). e =⇒ eä 14 ≤ ℵ4 (Kunen, Martin, unpublished). 5. Det(∆12 )+AC e e < ä1 6. (a) PD implies ä 12n+1 2n+2 (Moschovakis, [Mos70A]). e 1e 1 (b) PD implies ä 2n < ä 2n+1 (Kechris). 7. PD implies that eevery eΠ12n+1 -norm on a universal Π12n+1 -set has length ä 12n+1 (Moschovakis, [Mos70A]). e If we now assume full determinacy (AD), the results have a different flavor. Assume AD; then: 1′ . (a) ä 1n is a cardinal and for n odd regular (Moschovakis, [Mos70A]). (b) äe1n is regular, for n even (Kunen, unpublished). ′ 2 . ä 12 =eℵ2 (= u2 ) (Martin). 3′ . e(a) uù = ℵù ; cf(ℵn ) = ℵ2 (n ≥ 2) (Martin, [Mar70B]). (b) un = ℵn (Kunen, Solovay, unpublished). 4′ . ä 13 = ℵù+1 (Martin, [Mar70B]). 5′ . e(a) ä 12n+2 = (ä 12n+1 )+ (Kunen, Martin, unpublished). (b) äe14 = ℵù+2e (Kunen, Martin, unpublished). ′ 6 . ä 12n+1e = (ën )+ , where ën is a cardinal and cf(ën ) = ù; thus ä 12n+1 ≥ ℵùn+1 e e (Kechris). 4 Kechris [Kec75] has proved from PD that for each n ≥ 1 there is no largest countable Σ1 2n+1 set. Martin [Mar73] then showed from PD that every countable Σ12n+1 set of reals contains only ∆12n+1 reals (Moschovakis has earlier shown this result for countable ∆12n+1 sets).

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7′ . (a) ℵ1 (= ä 11 ), ℵ2 (= ä 12 ) are measurable (Solovay; for ℵ1 see [Sol67]), for e e ℵ2 unpublished). 1 (b) ä 2n+1 is measurable (Martin, unpublished). (c) äe12n is measurable (Kunen, unpublished). e

Postscript. While this paper was being typed, we received a preprint from Martin titled “Projective sets and cardinal numbers: some questions related to the continuum problem”. This appears to contain most of the results of Martin that we have listed as “unpublished” or credited to [Mar70B]. REFERENCES

John W. Addison [Add59] Some consequences of the axiom of constructibility, Fundamenta Mathematicae, vol. 46 (1959), pp. 123–135. John W. Addison and Yiannis N. Moschovakis [AM68] Some consequences of the axiom of definable determinateness, Proceedings of the National Academy of Sciences of the United States of America, no. 59, 1968, pp. 708–712. Morton Davis [Dav64] Infinite games of perfect information, Advances in game theory (Melvin Dresher, Lloyd S. Shapley, and Alan W. Tucker, editors), Annals of Mathematical Studies, vol. 52, 1964, pp. 85– 101. Leo A. Harrington and Alexander S. Kechris [HK77] Ordinal quantification and the models L[T 2n+1 ], Mimeographed note, January 1977. Alexander S. Kechris [Kec75] The theory of countable analytical sets, Transactions of the American Mathematical Society, vol. 202 (1975), pp. 259–297. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. Kazimierz Kuratowski [Kur66] Topology, vol. 1, Academic Press, New York and London, 1966. Azriel L´evy [L´ev66] Definability in axiomatic set theory, Logic, methodology and philosophy of science. Proceedings of the 1964 international congress. (Amsterdam) (Yehoshua Bar-Hillel, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, 1966, pp. 127–151. Richard Mansfield [Man69] The theory of Σ12 sets, Ph.D. thesis, Stanford University, 1969. [Man70] Perfect subsets of definable sets of real numbers, Pacific Journal of Mathematics, vol. 35 (1970), no. 2, pp. 451– 457. [Man71] A Souslin operation on Π12 , Israel Journal of Mathematics, vol. 9 (1971), no. 3, pp. 367– 379. Donald A. Martin [Mar68] The axiom of determinateness and reduction principles in the analytical hierarchy, Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 687–689.

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[Mar70B] Pleasant and unpleasant consequences of determinateness, March 1970, unpublished manuscript. [Mar73] Countable Σ12n+1 sets, 1973, circulated note. Donald A. Martin and Robert M. Solovay [MS] Basis theorems for Π12k sets of reals, unpublished. [MS69] A basis theorem for Σ13 sets of reals, Annals of Mathematics, vol. 89 (1969), pp. 138–160. Yiannis N. Moschovakis [Mos70A] Determinacy and prewellorderings of the continuum, Mathematical logic and foundations of set theory. Proceedings of an international colloquium held under the auspices of the Israel Academy of Sciences and Humanities, Jerusalem, 11–14 November 1968 (Y. Bar-Hillel, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam-London, 1970, pp. 24–62. [Mos70B] The Suslin-Kleene theorem for countable structures, Duke Mathematical Journal, vol. 37 (1970), no. 2, pp. 341–352. [Mos71A] Uniformization in a playful universe, Bulletin of the American Mathematical Society, vol. 77 (1971), pp. 731–736. [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. Jan Mycielski [Myc64] On the axiom of determinateness, Fundamenta Mathematicae, vol. 53 (1964), pp. 205– 224. Jan Mycielski and Stanislaw Swierczkowski [MS64] On the Lebesgue measurability and the axiom of determinateness, Fundamenta Mathematicae, vol. 54 (1964), pp. 67–71. Jack H. Silver [Sil71] Measurable cardinals and ∆13 wellorderings, Annals of Mathematics, vol. 94 (1971), no. 2, pp. 141– 446. Robert M. Solovay [Sol66] On the cardinality of Σ12 set of reals, Foundations of Mathematics: Symposium papers commemorating the 60th birthday of Kurt G¨odel (Jack J. Bulloff, Thomas C. Holyoke, and S. W. Hahn, editors), Springer-Verlag, 1966, pp. 58–73. [Sol67] Measurable cardinals and the axiom of determinateness, Lecture notes prepared in connection with the Summer Institute of Axiomatic Set Theory held at UCLA, Summer 1967. [Sol70] A model of set theory in which every set is Lebesgue measurable, Annals of Mathematics, vol. 92 (1970), pp. 1–56. DEPARTMENT OF MATHEMATICS CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CA 91125, USA

E-mail: [email protected] DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES, CA 90095, USA

E-mail: [email protected]

PROPAGATION OF THE SCALE PROPERTY USING GAMES

ITAY NEEMAN

The aim of this short paper is to introduce the reader to the notion of a scale and to some of the basic techniques involved in the propagation of the scale property through the use of infinite games. None of the results presented is due to the author. For a full history see Moschovakis [Mos80]. We work throughout the paper with the space ù ù. For s ∈ <ù ù we use Ns to denote the set {x ∈ ù ù : x extends s}. The sets Ns , s ∈ <ù ù, form the basic open subsets of ù ù. Following standard abuse we refer to the space ù ù, equipped with the topology generated by these basic open sets, as R. Given a set A ⊂ R let G(A) denote the following game: Players I and II alternate playing x(n) for n ∈ ù subject to the order displayed in Diagram 1, with x(n) ∈ ù for each n. If, after ù moves, the real x = hx(n) : n < ùi belongs to A then player I wins. Otherwise player II wins. G(A) is determined if one of the two players has a winning strategy in the game. I II

x(0)

x(2) x(1)

...... x(3)

......

Diagram 1. The game G(A). For B ⊂ R × R and x ∈ R let Bx = {y ∈ R : hx, yi ∈ B}. This is the x-section of B. Define aB to be the set {x ∈ R : player I has a winning strategy in G(Bx )}. We sometimes write (ay)B(x, y), or (ay)hx, yi ∈ B, for the statement x ∈ aB. This is deliberately meant to conjure up the notation used for statements involving the quantifiers (∀y) and (∃y). (ay) really is a quantifier, giving precise meaning to the chain (∃y(0))(∀y(1))(∃(y(2)) · · · · · · of quantifiers over ù. Let B ⊂ R × R be open. Note that for each hx, yi ∈ B there exists some n < ù so that Nx↾n × Ny↾n ⊂ B. Let n(x, y) denote the least such n. We refer to n(x, y) as the time of entry of hx, yi into B. For hx, yi 6∈ B we set n(x, y) = ù. Supported by the National Science Foundation under Grant No. DMS-0094174. The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

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ITAY NEEMAN

F y(0)

S y(1) S y ∗ (0)

F y ∗ (1)

F y(2)

··· S y ∗ (2)

···

Diagram 2. The game H(x ∗ , x). Let A = aB. For x, x ∗ ∈ R define H(x ∗ , x) to be the following game: Players “first” and “second” (denoted F and S respectively) alternate moves subject to the format in Diagram 2. The moves are played sequentially from left to right, and are presented in two separate lines only for future convenience. The letters F and S indicate which player is responsible for each move. Each of the moves is a natural number. An infinite run leading to reals y = hy(i) : i < ùi and y ∗ = hy ∗ (i) : i < ùi is won by S if hx, yi 6∈ B, or hx, yi ∈ B & hx ∗ , y ∗ i ∈ B & n(x ∗ , y ∗ ) ≤ n(x, y). Otherwise the run is won by F. With our convention that n(x, y) = ù for hx, yi 6∈ B, a run hy ∗ , yi of H(x ∗ , x) is won by S just in case that n(x ∗ , y ∗ ) ≤ n(x, y). The game H(x ∗ , x) thus involves a simultaneous play of both G(Bx ), taking place on the upper line in Diagram 2, and G(Bx ∗ ), taking place on the lower line. We think of the former as owned by F, and of the latter as owned by S. Each plays for I on the line she owns, and for II on the line owned by her opponent. To win, S must make sure that the play on her line (namely the lower line, where she plays for I) does not lag behind the play on her opponent’s line: if the play on her opponent’s line enters B then she has to make sure that the play on her line enters B, at the same time or earlier. Define a relation  on R by setting x ∗  x iff S has a winning strategy in H(x ∗ , x). This winning strategy should be viewed as a “translation mechanism.” It translates a strategy for I in G(Bx ) into a strategy for I in G(Bx ∗ ), making sure that the translated strategy never lags behind the original strategy. Claim 1. The relation  is reflexive. Proof. S can win H(x, x) simply by copying the moves played by F. More precisely, the strategy defined by the conditions y ∗ (n) = y(n) for even n and y(n) = y ∗ (n) for odd n is winning for S in H(x, x). ⊣ Claim 2. The relation  is transitive. Proof. Suppose that x ∗∗  x ∗  x. Let ô1 be a winning strategy for S in H(x ∗ , x) and let ô2 be a winning strategy for S in H(x ∗∗ , x ∗ ). Let ô be the strategy in H(x ∗∗ , x) obtained by composing ô1 and ô2 . A typical play according to ô is illustrated in Diagram 3. The play starts in the upper left corner with a move by F, and proceeds along the arrows obtaining additional

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77

moves through plays by F in H(x ∗∗ , x), uses of ô1 , and uses of ô2 , as indicated in the diagram. Note that a play hy ∗∗ , yi is according to ô iff there is a real y ∗ so that hy ∗∗ , y ∗ i is according to ô2 and hy ∗ , yi is according to ô1 . It is easy using this characterization to check that ô is winning for S in H(x ∗∗ , x). ⊣

(a) H(x ,x) ∗

(b) H(x

∗∗

,x ) ∗

F y(0)  ô1 y ∗ (0)  ô2 y ∗∗ (0)

ô1 y(1) O

/ F y(2)

ô2 y ∗ (1) O / F y ∗∗ (1)

H(x ∗∗ ,x) (c)


/

Diagram 3. Composing ô1 and ô2 . Lemma 3. For x, x ∗ ∈ R set x ≺ x ∗ iff x ∗ 6 x. Suppose that each of the games H(x ∗ , x), x, x ∗ ∈ R, is determined. Then the relation ≺ is wellfounded. Proof. Suppose for contradiction that hxi : i < ùi is a sequence of reals so that xi+1 ≺ xi , meaning xi 6 xi+1 , for each i. Using the assumption of determinacy it follows that F has a winning strategy in H(xi , xi+1 ). Let ôi be such a strategy. Construct reals yi , i < ù, following Diagram 4. The construction proceeds column by column from left to right, setting y0 (k) = 0 for each even k and using the strategies ôi , as indicated in the diagram, to produce all other objects. For each i < ù, the lines corresponding to yi and yi+1 together form a play of H(xi , xi+1 ), according to ôi . (For i = 1 the progress of this play is indicated in the diagram through squiggly arrows, and the play itself is indicated in boldface.) Using the fact that ôi is winning for F in H(xi , xi+1 ) it follows that hxi+1 , yi+1 i ∈ B and that n(xi+1 , yi+1 ) < n(xi , yi ) for each i < ù. But this gives an infinite descending sequence of natural numbers, and hence a contradiction. ⊣ Remark 4. It follows from Lemma 3 that  is total: if x 6 x ∗ and x ∗ 6 x then hx, x ∗ , x, x ∗ , . . . i is an infinite descending sequence in ≺, a contradiction. We know now that the relation  is a prewellorder, meaning that the relation x ∼ y iff x  y & y  x is an equivalence relation and that  induces a wellordering of the equivalence classes of ∼. Let ϕ : R → Ord be the rank function associated to . Precisely, ϕ is defined through the condition ϕ(x) = sup{ϕ(x) ¯ + 1 : x¯ ≺ x} (with the supremum of the empty set taken to be 0). We shall see that the relationship between A and x 7→ ϕ(x) is analogous to the relationship between the open set B and the function hx, yi 7→ n(x, y) defined earlier.

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ITAY NEEMAN

/ ô2 y3 (0)

ô3 y3 (1)

ô2 y3 (2)

H(x2 ,x3 )

ô2 ô1 y2 (0) O  ô0 y1 (0) /o /o o/ / ô1

H(x1 ,x2 )

y2 (1) /o /o o/ / ô1 O O ô0 y1 (1)

y2 (2) O  y1 (2) /o /o /o /

H(x0 ,x1 )

y0 (0) = 0

ô0 y0 (1)

y0 (2) = 0

Diagram 4. The wellfoundedness of ≺. Note that if hx, yi ∈ B and n(x ∗ , y ∗ ) ≤ n(x, y) then hx ∗ , y ∗ i ∈ B.

(1)

We now establish a similar relationship between the set A = aB and the function x 7→ ϕ(x): Lemma 5. Suppose that x ∈ A and ϕ(x ∗ ) ≤ ϕ(x). Then x ∗ ∈ A. Proof. Since x ∈ A = aB, player I has a winning strategy in G(Bx ). Let ó be such a strategy. Since ϕ(x ∗ ) ≤ ϕ(x), S has a winning strategy in H(x ∗ , x). Let ô be such a strategy. ó and ô combine naturally to give rise to a strategy ó ∗ for player I in G(Bx ∗ ). ó ∗ is characterized by the condition that y ∗ is according to ó ∗ iff there is a real y so that y is according to ó and hy ∗ , yi is according to ô. A typical run y ∗ of ó ∗ is presented on the lower line of Diagram 5, and the associated real y is presented on the upper line. Since ó is winning for I in G(Bx ) the real y must belong to Bx . In other words hx, yi ∈ B. Since ô is winning for S in H(x ∗ , x), n(x ∗ , y ∗ ) ≤ n(x, y). By (1) above hx ∗ , y ∗ i ∈ B. This shows that ó ∗ is winning for I in G(Bx ∗ ), and hence x ∗ ∈ A. ⊣ G(Bx )

G(Bx ∗ )

ó y(0)  ô y ∗ (0)

ô y(1) O

/ ó y(2)

/ II y ∗ (1)




H(x ∗ ,x)

/

Diagram 5. Composing ó and ô. The last lemma shows that A forms an initial segment of R in the prewellorder given by ϕ, just as B forms an initial segment of R×R in the prewellorder given by hx, yi 7→ n(x, y). Note that the complement of B is a single equivalence

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79

class at the very top of the prewellorder given by hx, yi 7→ n(x, y). Precisely, if hx, yi 6∈ B then n(x ∗ , y ∗ ) ≤ n(x, y) for all hx ∗ , y ∗ i ∈ R × R.

(2)

The next claim establishes the same property for ϕ and A. Claim 6. Suppose that each of the games G(Bx ), x ∈ R, is determined. Let x, x ∗ ∈ R, and suppose that x 6∈ A. Then x ∗  x. Proof. Since x 6∈ A = aB, and G(Bx ) is determined, player II must have a winning strategy in G(Bx ). Let ó be such a strategy. Let ô be the strategy for S in H(x ∗ , x) which follows ó on the upper line, and plays 0s on the lower line. Plays hy ∗ , yi according to ô are characterized by the condition that y is according to ó and y ∗ (k) = 0 for each even k. From the first clause and the fact that ó is winning for II in G(Bx ) is follows that hx, yi 6∈ B, so that hy ∗ , yi is won by S in H(x ∗ , x). ⊣ Recall that B ⊂ R2 is open, that is Σ01 , and n(x, y) is equal to the least n so that Nx↾n × Ny↾n ⊂ B if hx, yi ∈e B, and to ù if hx, yi ∈ R2 − B. Define ⊑ by setting hx ∗ , y ∗ i ⊑ hx, yi iff n(x ∗ , y ∗ ) ≤ n(x, y), and define ⊏ by setting hx ∗ , y ∗ i ⊏ hx, yi iff n(x ∗ , y ∗ ) < n(x, y). Both are relations on R2 , equivalently subsets of R2 × R2 . It is easy to see that both ⊏ and ⊑ ∩ (B × R2 ) are Σ01 . (3) e By ⊑ ∩ (B × R2 ) we mean {hx, ¯ y, ¯ x, yi : n(x, ¯ y) ¯ ≤ n(x, y) & hx, ¯ yi ¯ ∈ B}. The restriction to hx, ¯ yi ¯ ∈ B is important. The full relation ⊑ is not Σ01 . (We e make could, for symmetry, also restrict ⊏ to B × R2 in (3). But this would not any difference: hx, ¯ yi ¯ ⊏ hx, yi already implies that hx, ¯ yi ¯ ∈ B.) For a pointclass1 Γ let aΓ be the pointclass {aD : D ∈ Γ}. The set A = aB is aΣ01 . The next lemma establishes the parallel of property (3) for A e and ϕ.

Lemma 7. Suppose that all length ù games with Σ01 payoff are determined. e Then ≺ and  ∩ (A × R) are both aΣ01 . e Proof. Using determinacy, x¯ ≺ x iff F has a winning strategy in H(x, x). ¯ A run hy, yi ¯ of H(x, x) ¯ is won by F iff n(x, ¯ y) ¯ < n(x, y), and this is a Σ01 e condition by property (3). Hence the set {hx, ¯ xi : F has a winning strategy 0 in H(x, x)} ¯ is aΣ1 . It remains to e prove that  ∩ (A × R) is aΣ01 . ′ ∗ Define H (x , x) to be played according e to the rules of H(x ∗ , x) but with the modified payoff condition that S wins just in case that hx ∗ , y ∗ i ∈ B and n(x ∗ , y ∗ ) ≤ n(x, y). Note that this modified condition is Σ01 by property (3). e Hence:

1 By a pointclass we always mean a class of sets closed under recursive substitutions, conjunctions, disjunctions, and bounded number quantifications.

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(i) The set {hx ∗ , xi : S has a winning strategy in H′ (x ∗ , x)} is aΣ01 . e Note further that the modified condition is harder for S than the original condition. Hence: (ii) If S has a winning strategy in H′ (x ∗ , x) then she also has a winning strategy in H(x ∗ , x). We intend to show that for x ∗ ∈ A the games are in fact equivalent. Claim 8. Let x ∗ ∈ A. Then S has a winning strategy in H′ (x ∗ , x ∗ ). Proof. Suppose for contradiction that S does not have a winning strategy in H′ (x ∗ , x ∗ ). By determinacy then F has a winning strategy. Let ô be such a strategy. Let ó be a winning strategy for I in G(Bx ∗ ). Player I has a winning strategy in this game since x ∗ is assumed to be in A. Diagram 6 shows how to compose ó and infinitely many copies of ô to produce a sequence of reals yi , i < ù, with the property that y0 is according to ó, and for each i, hyi , yi+1 i is according to ô. (Each copy of H′ (x ∗ , x ∗ ) in the diagram is labelled by a roman letter, and the copies of ô have superscripts indicating which of the copies of H′ (x ∗ , x ∗ ) they belong to. The copy of H′ (x ∗ , x ∗ ) labelled (b) is highlighted in squiggly arrows in the diagram, and the moves by F in this game, made by the copy of ô labelled (b), are indicated in boldface.) Since ó is winning for I in G(Bx ∗ ), hx ∗ , y0 i ∈ B. Since ô is winning for F in H′ (x ∗ , x ∗ ), either hx ∗ , yi i 6∈ B or n(x ∗ , yi ) 6≤ n(x ∗ , yi+1 ), for each i < ù. Notice that the latter disjunct implies that hx ∗ , yi+1 i ∈ B, by property (2). It therefore follows by induction that hx ∗ , yi i ∈ B and n(x ∗ , yi+1 ) < n(x ∗ , yi ) for each i < ù. But this gives an infinite descending sequence of natural numbers, a contradiction. ⊣

/ ô (c) y3 (0)

ô (d) y3 (1)

ô (c) y3 (2)

(c) H′ (x ∗ ,x ∗ )

(b) H′ (x ∗ ,x ∗ )

ô (c) y2 (1) /o /o o/ / ô (b) y2 (2) ô (b) y2 (0) O O O O   ô (a) y1 (2) /o /o /o / ô (a) y1 (0) /o /o o/ / ô (b) y1 (1)

(a) H′ (x ∗ ,x ∗ )

ó y0 (0)

ô (a) y0 (1)

ó y0 (2)

Diagram 6. F cannot have a winning strategy in H′ (x ∗ , x ∗ ) for x ∗ ∈ A.

PROPAGATION OF THE SCALE PROPERTY USING GAMES

81

Claim 9. Suppose that S has a winning strategy in H(x ∗ , x) and that x ∗ ∈ A. Then S has a winning strategy in H′ (x ∗ , x). Proof. Let ô1 be a winning strategy for S in H(x ∗ , x), and using the previous claim let ô2 be a winning strategy for S in H′ (x ∗ , x ∗ ). Let ô be obtained by composing ô1 and ô2 in the manner of Diagram 3. (But here the games labelled (b) and (c) in the diagram are H′ (x ∗ , x ∗ ) and H′ (x ∗ , x) respectively.) Then ô is winning for S in H′ (x ∗ , x). ⊣ ∗ From Claim 9 and condition (ii) above it follows that for x ∈ A, S has a winning strategy in H(x ∗ , x) iff she has a winning strategy in H′ (x ∗ , x). From this and condition (i) it follows that the set {hx ∗ , xi : x ∗ ∈ A and S has a ⊣ (Lemma 7) winning strategy in H(x ∗ , x)} is aΣ01 . e List 10. The following list summarizes the properties of ϕ and A obtained so far (with Γ standing for aΣ01 ): e 1. If x ∈ A and ϕ(x) ¯ ≤ ϕ(x) then x¯ ∈ A. 2. If x 6∈ A and x¯ 6∈ A then ϕ(x) = ϕ(x). ¯ 3. Both the sets {hx, ¯ xi : ϕ(x) ¯ < ϕ(x)} and {hx, ¯ xi : x¯ ∈ A & ϕ(x) ¯ ≤ ϕ(x)} are in Γ. Conditions (1) and (2) merely note that R − A forms a single equivalence class of the prewellorder induced by ϕ, located above all elements of A in this prewellorder. Notice that any function ϕ : A → Ord can be extended to a function on R satisfying conditions (1) and (2) simply by setting ϕ(x) = sup{ϕ(x) ¯ + 1 : x¯ ∈ A} for x ∈ R − A. We use the same letter to refer both to the function defined on A and to an extension of the function to R subject to conditions (1) and (2). Condition (3) in List 10 is the crucial one, connecting ϕ to the pointclass Γ. The next definition abstracts an equivalent condition, that refers only to the restriction of ϕ to A. The equivalence is proved in Claim 12. Definition 11. A function ϕ : A → Ord is a Γ norm on A just in case that there are sets U and V in Γ and ¬Γ respectively, so that, for every x ∈ A, {x¯ : ϕ(x) ¯ ≤ ϕ(x)} = {x¯ : hx, ¯ xi ∈ U } = {x¯ : hx, ¯ xi ∈ V }. Claim 12. Let ϕ : A → Ord and extend ϕ to R in line with conditions (1) and (2) in List 10. Then ϕ is a Γ norm iff it satisfies condition (3) in the list. Proof. Assuming condition (3), let U = {hx, ¯ xi : x¯ ∈ A & ϕ(x) ¯ ≤ ϕ(x)}, and let V = {hx, ¯ xi : ϕ(x) 6< ϕ(x)}. ¯ Assuming the condition in Definition 11 note that ϕ(x) ¯ < ϕ(x) iff x¯ ∈ A & hx, xi ¯ 6∈ V , and that x¯ ∈ A & ϕ(x) ¯ ≤ ϕ(x) iff (x ∈ A & hx, ¯ xi ∈ U ) ∨ (x¯ ∈ A & hx, xi ¯ 6∈ V ). ⊣ The condition in Definition 11 states that for each x ∈ A, the initial segment {x¯ : ϕ(x) ¯ ≤ ϕ(x)} belongs to both Γ(x) and ¬Γ(x), and that this holds uniformly in x. Working with norms in this paper it is more convenient to use

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the equivalent characterization in List 10, and we shall do this without further comment throughout the paper. A pointclass Γ is said to have the prewellordering property if every set in Γ admits a Γ norm. The prewellordering property has various applications to questions in descriptive set theory, see for example the theorem on reduction in [KM78B, §2]. The sequence of results given above shows how to produce a aΣ01 norm on a e specific given aΣ01 set, starting from Σ01 norms on Σ01 sets. But there is nothing e e 0 e to Σ1 sets in any of the proofs. They generalize routinely to yield the following e theorem: Theorem 13. Let Γ be a pointclass. Suppose that every length ù game with payoff in Γ is determined. Suppose that Γ has the prewellordering property. Then aΓ has the prewellordering property. Proof. Let B ⊂ R2 belong to Γ and let ϑ : R → Ord be a Γ norm on B. Follow the sequence of definitions and claims above, only replacing the uses of n(x, y) by uses of ϑ(x, y). It is easy to check that the proofs adapt, showing that the resulting function ϕ is a aΓ norm on A = aB. Let us only note that the definability expressed by condition (3) in List 10, for the norm ϑ, is such that all the games that come up in the adapted proofs have payoff sets in Γ, and are therefore determined. (Their determinacy is needed in the proofs.) ⊣ There is one crucial property of the norm hx, yi 7→ n(x, y) that was not considered in the discussion so far. It is easy to check that this norm and the Σ01 set B satisfy e let hxi , yi i, i < ù, be elements of B. Suppose that limi −→ ∞ xi (4) and limi −→ ∞ yi exist and let x∞ and y∞ respectively denote the limits. Suppose that n(xi , yi ) is eventually constant as i −→ ∞. Then hx∞ , y∞ i ∈ B and n(x∞ , y∞ ) ≤ eventual value of n(xi , yi ). As stated this additional property is not true at the level of aΣ01 sets, but we e can obtain a parallel property at that level by using countably many norms. ∗ <ù Given x, x ∈ R, p ∈ ù of even length, say 2k, and h, h ∗ ∈ ù, define Hp (x ∗ , h ∗ , x, h) to be played as follows: Players F and S alternate moves subject to the format in Diagram 7. The moves are played sequentially from left to right, starting from the vertical line, and each of the moves is a natural number. At the end of an infinite run we set y = pahhiahy(i) : 2k < i < ùi and y ∗ = pahh ∗ iahy ∗ (i) : 2k < i < ùi. The run is won by S if hx, yi 6∈ B, or hx, yi ∈ B & hx ∗ , y ∗ i ∈ B & n(x ∗ , y ∗ ) ≤ n(x, y), and otherwise the run is won by F. Hp (x ∗ , h ∗ , x, h) may thus be viewed as a version of H(x ∗ , x) with y↾2k + 1 set equal to pahhi and y ∗ ↾2k + 1 set equal to pahh ∗ i. For reason of notational

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PROPAGATION OF THE SCALE PROPERTY USING GAMES

p(0) · · · p(2k − 1) h p(0) · · · p(2k − 1) h∗

S y(2k + 1) F y(2k + 2) F y ∗ (2k + 1)

···

Diagram 7. The game Hp (x ∗ , h ∗ , x, h). convenience we refer to hy ∗ , yi, rather than the sequence of actual moves, as a run of Hp (x ∗ , h ∗ , x, h). Define p by setting hx ∗ , h ∗ i p hx, hi iff S has a winning strategy in the game Hp (x ∗ , h ∗ , x, h). The previous proofs adapt to show that p is a prewellorder. Let ϕp : R × ù → Ord be the associated rank function, defined ¯ + 1 : hx, ¯ ≺p hx, hi}. ϕp is then a aΣ0 norm by ϕp (x, h) = sup{ϕp (x, ¯ h) ¯ hi 1 e on the set {hx, hi : pahhi is a winning position for I in G(Bx )}. For p ∈ <ù ù of even length and x ∈ R let øp (x) = min{ϕp (x, h) : h ∈ ù} and let hp (x) be the smallest number h realizing the minimum, that is hp (x) = min{h : ϕp (x, h) = øp (x)}. Exercise 14. Let x ∈ A. Let p ∈ <ù ù be a node of even length, say 2k. Suppose that p(2i) = hp↾2i (x) for each i < k. Show that p is a winning position for I in G(Bx ). Hint for the case k = 1. Let ó be a winning strategy for I in G(Bx ). Let h be the first move played by ó. By assumption p(0) = h∅ (x), and it follows from the definition of h∅ (x) that ϕ∅ (x, p(0)) ≤ ϕ(x, h). So S has a winning strategy, ô say, in the game H∅ (x, p(0), x, h). Diagram 8 shows how to win G(Bx ) from p, against an opponent who plays for II, using a composition of ó and ô. ⊣ G(Bx )

óh

ô y(1) O

G(Bx ) from p

p(0)

p(1)

/ ó y(2)  ô y ∗ (2)

ô y(3) O

/ H(x,p(0),x,h)

/ II y ∗ (3)

Diagram 8. Hint for Exercise 14. Claim 15. The norm ø∅ is equivalent to the earlier norm ϕ on A, in the sense that ø∅ (x ∗ ) ≤ ø∅ (x) iff ϕ(x ∗ ) ≤ ϕ(x). Proof. ø∅ (x ∗ ) ≤ ø∅ (x) iff (∀n) (∃n ∗ ) so that S has a winning strategy in H∅ (x ∗ , n ∗ , x, n). Prepending moves corresponding to the quantifier string (∀n)(∃n ∗ ) to the game H∅ (x ∗ , n ∗ , x, n) we obtain precisely the game H(x ∗ , x). So ø∅ (x ∗ ) ≤ ø∅ (x) iff S has a winning strategy in H(x ∗ , x) iff ϕ(x ∗ ) ≤ ϕ(x). ⊣

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Exercise 16. Let Ap = {x : p is a winning position for I in G(Bx )}. Show ¯ xi : øp (x) ¯ < that øp is a aΣ01 norm on Ap . In fact, show that the sets {hp, x, e x, øp (x)} and {hp, ¯ xi : x¯ ∈ Ap & øp (x) ¯ ≤ øp (x)} are in aΣ01 . e Hint. That {hp, x, ¯ xi : øp (x) ¯ < øp (x)} is aΣ01 follows directly from (determinacy and) the definitions. For {hp, x, ¯ xi : ex¯ ∈ Ap & øp (x) ¯ ≤ øp (x)} use an argument similar to that in the proof of Lemma 7. ⊣ We now approach the parallel of property (4): Theorem 17. Let hxi : i < ùi be a sequence of reals in the set A = aB. Suppose that limi −→ ∞ xi exists and let x∞ denote this limit. Suppose that for each p, both øp (xi ) and hp (xi ) are eventually constant as i −→ ∞. Let ëp and hp respectively be their eventual values. Then: 1. x∞ belongs to A. 2. For each p, høp (x∞ ), hp (x∞ )i ≤lex hëp , hp i. Proof. We show that ϕ∅ (x∞ , h∅ ) ≤ ë∅ . A similar proof establishes that ϕp (x∞ , hp ) ≤ ëp for each p. Condition (2) of the theorem follows directly from this. Condition (1) follows from the instance ø∅ (x∞ ) ≤ ø∅ (xi ) (for all sufficiently large i) using Lemma 5 and Claim 15. For each p let k(p) < ù be large enough that øp (xi ) = ëp and hp (xi ) = hp for all i ≥ k(p). Choose k(p) inductively so that k(p) > k(p) ¯ whenever p¯ is a strict initial segment of p. Let p0 = ∅ and let k0 = k(∅). Suppose for contradiction that ϕ∅ (x∞ , h∅ ) 6≤ ë∅ . Since ë∅ = ø∅ (xk0 ) = min{ϕ∅ (xk0 , h) : h ∈ ù} this means that there is some h ∈ ù so that ϕ∅ (x∞ , h∅ ) 6≤ ϕ∅ (xk0 , h). Fix such an h. Using determinacy the fact that ϕ∅ (x∞ , h∅ ) 6≤ ϕ∅ (xk0 , h) implies that F has a winning strategy in the game H∅ (x∞ , h∅ , xk0 , h) which we denote H (∞) . Fix such a winning strategy ó. Now construct sequences hkn i, hpn i, hô (n) i, and hyn i so that: S (a) p0 ⊂ p1 ⊂ p2 · · · and lh(pn ) = 2n for each n. Let y∞ = n<ù pn . (b) y0 (0) = h, y1 (0) = h∅ , and the pair hy∞ , y0 i is a run of H (∞) played according to ó. (c) pn+1 = pn ahhpn , yn+1 (2n + 1)i. (d) kn+1 = k(pn+1 ). (e) ô (n) is a winning strategy for S in the game Hpn (xkn+1 , hpn , xkn , yn (2n)) which we denote H (n) . (f) The pair hyn+1 , yn i is a run of H (n) played according to ô (n) . Diagram 9 illustrates the construction. The construction begins on the upper left corner, with the assignments y0 (0) = h and y1 (0) = h∅ . The construction continues following the arrows in the diagram, assigning to each entry a value either by setting it equal to hpn for some n or by using one of the

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PROPAGATION OF THE SCALE PROPERTY USING GAMES

strategies, as indicated. The symbol ′′ in an entry indicates copying the value of the entry above it. Note that the values of yn+1 (2n) (set equal to hpn ) and yn+1 (2n + 1) (determined using ó) can be determined before kn+1 is known. (They do not depend on ô (n) .) Once these assignments are made we set pn+1 = pn ahyn+1 (2n), yn+1 (2n + 1)i, and set kn+1 = k(pn+1 ). Since yn+1 (2n) = hpn , and since both kn and kn+1 are greater than or equal to k(pn ), ϕpn (xkn+1 , yn+1 (2n)) = ëpn = ϕpn (xkn , hpn ) ≤ ϕpn (xkn , yn (2n)). Thus S has a winning strategy in H (n) = Hpn (xkn+1 , yn+1 (2n), xkn , yn (2n)), allowing us to pick ô (n) subject to condition (e) above and continue with the construction, following the arrows. The sequences y∞ (whose entries are indicated in boldface in Diagram 9) and y0 together form a run of H∅ (x∞ , h∅ , xk0 , h), played according to ó. Since ó is winning for F in the game, (i) n(xk0 , y0 ) < n(x∞ , y∞ ). The sequences yn+1 and yn together form a run of H (n) according to ô (n) . Since ô (n) is winning for S in the game, (ii) n(xkn+1 , yn+1 ) ≤ n(xkn , yn ). It follows from this, and the wellfoundedness of ù, that n(xkn , yn ) is eventually constant as n −→ ∞. Now limn −→ ∞ xkn = x∞ by the assumptions of the theorem, and limn −→ ∞ yn = y∞ by construction. Using property (4) above it follows that (iii) n(x∞ , y∞ ) ≤ the eventual value of n(xkn , yn ). But from this and condition (ii) it follows that n(x∞ , y∞ ) ≤ n(xk0 , y0 ), and this contradicts condition (i). ⊣ k0 H (0)

k1

h

ô (0) y0 (1)

O



hp0

/ó

ó y0 (2)



y1 (1)

H (1)

k2

/

ô (0) y1 (2)



′′

′′

hp1

′′

′′

′′

ô (0) y0 (3)

O

ó y0 (4) ô (0) y1 (4)

/

ó y2 (3)

ô (1) y2 (4)

′′

hp2

O

y0



ô (1) y1 (3)

y1



H (2)

k3

/



y2

/ y∞

Diagram 9. The proof of Theorem 17.

H (∞)

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Definition 18. Let A ⊂ R. A countable sequence hϑℓ i of norms on A is called a scale on A if it satisfies the following limit condition: (∗) Let xi for i < ù belong to A. Suppose that limi −→ ∞ xi exists and let x∞ denote the limit. Suppose that for each ℓ, ϑℓ (xi ) is eventually constant as i −→ ∞, and let ëℓ denote the eventual value. Then: x∞ ∈ A; and for each ℓ, ϑℓ (x∞ ) ≤ ëℓ . A scale hϑℓ i is called a Γ scale if both the sets {hℓ, x, ¯ xi : ϑℓ (x) ¯ < ϑℓ (x)} and {hℓ, x, ¯ xi : x¯ ∈ A & ϑℓ (x) ¯ ≤ ϑℓ (x)} belong to Γ, or, equivalently, if there are sets U and V in Γ and ¬Γ respectively so that {x¯ : ϑℓ (x) ¯ ≤ ϑℓ (x)} = {x¯ : hx, ¯ x, ℓi ∈ U } = {x¯ : hx, ¯ x, ℓi ∈ V } for all ℓ < ù and all x ∈ A. Γ has the scale property if each set in Γ admits a Γ scale. Remark 19. Many applications involve scales hϑℓ i so that the sets {hx, ¯ xi : ϑℓ (x) ¯ < ϑℓ (x)} and {hx, ¯ xi : x¯ ∈ A & ϑℓ (x) ¯ ≤ ϑℓ (x)} belong to Γ for each individual ℓ, meaning that each ϑℓ is a Γ norm, but the joins {hℓ, x, ¯ xi : ϑℓ (x) ¯ < ϑℓ (x)} and {hℓ, x, ¯ xi : x¯ ∈ A & ϑℓ (x) ¯ ≤ ϑℓ (x)} do not belong to Γ. We call such scales weakly Γ, and say that Γ is weakly scaled if every set in Γ admits a weakly Γ scale. Theorem 17 shows that høp , hp i is a scale on A = aB. But it need not be a aΣ01 scale. The problem is with the shift from A to Ap in the definability of the enorms øp in Exercise 16. We now solve this problem by restricting to p which are winning for I in G(Bx ), through a use of Exercise 14. Call p ∈ <ù ù of even length 2k correct for x ∈ A just in case that (∀i < k) p(2i) = hp↾2i (x). Remark 20. The assumption in Theorem 17, that øp (xi ) and hp (xi ) are eventually constant as i −→ ∞ for each p, can be weakened to apply only to p which are correct for xi for almost all i (meaning all but finitely many i), as these are the only p which come up during the proof of the theorem. Note that for every r ∈ <ù ù of length k there is a unique p which is correct for x and so that (∀i < k)p(2i + 1) = r(i). Let α ~ r (x) = høp↾0 (x), hp↾0 (x), øp↾2 (x), hp↾2 (x), . . . , øp (x), hp (x)i for this unique p. Set x¯ Er x iff x 6∈ A or x¯ ∈ A & x ∈ A & α ~ r (x) ¯ ≤lex α ~r (x). It is clear that Er is a prewellorder on R. Let ør′ : R → Ord be its rank function. Claim 21. The relations {hr, x, ¯ xi : ør′ (x) ¯ < ør′ (x)} and {hr, x, ¯ xi : 0 ′ ′ x¯ ∈ A & ør (x) ¯ ≤ ør (x)} are both aΣ1 . e Proof. This is a simple calculation using Exercise 16, a parallel of the same exercise for the norm ϕp , and the fact that if p is correct for x then x ∈ Ap , given by Exercise 14. ⊣

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Claim 22. Let xi be a sequence of reals in A and suppose that for each r, ør′ (xi ) is eventually constant as i −→ ∞. Suppose that p is correct for xi for almost all i. Then øp (xi ) and hp (xi ) are eventually constant as i −→ ∞. Proof. Let k be such that lh(p) = 2k. Let r = hp(1), . . . , p(2k − 1)i. Let n be large enough that p is correct for xi for all i > n. Then α ~ r (xi ) = høp↾0 (xi ), hp↾0 (xi ), øp↾2 (xi ), hp↾2 (xi ), . . . , øp (xi ), hp (xi )i for all i > n, and the fact that øp (xi ) and hp (xi ) are eventually constant follows from the fact that α ~ r (xi ) is eventually constant. ⊣ ùi is a aΣ01 scale on A = aB. e Proof. The definability required by Definition 18 is given by Claim 21. The limit condition is given by Theorem 17 using Remark 20 and Claim 22. ⊣ 0 0 Our work so far produced a aΣ1 scale on a given aΣ1 set, starting from a e e But in fact the argument norm satisfying property (4) above. can be adapted to start with a scale, rather than a single norm, and to use the limit condition in Definition 18, rather than property (4). The result is the following theorem: Corollary 23. The sequence hør′ : r ∈

<ù

Theorem 24. Let Γ be a pointclass. Let B ⊂ R2 belong to Γ. Let hϑℓ : ℓ < ùi be a Γ scale (respectively weakly Γ scale) on B. Suppose that Γ determinacy holds. Define Hp , p , ϕp , hp , øp , and ør′ as above, but replacing the condition “n(x ∗ , y ∗ ) ≤ n(x, y)” with the condition hϑ0 (x ∗ ), . . . , ϑlh(p)/2 (x ∗ )i ≤lex hϑ0 (x), . . . , ϑlh(p)/2 (x)i throughout. Then hør′ : r ∈ set aB.

<ù

ùi is a aΓ scale (respectively weakly aΓ scale) on the

Proof. The proofs given above generalize routinely to these settings. Let us only make the following comments: First, note that all the games that come up during the proofs have payoff sets in Γ, since the norms ϑℓ are all Γ norms. The games are therefore determined. This is important since the proofs require their determinacy. Second, condition (i) in the proof of Theorem 17 is revised in the general settings to state that (i)′ ϑ0 (xk0 , y0 ) < ϑ0 (x∞ , y∞ ). Third, condition (ii) in the proof of Theorem 17 is revised in the general settings to state that (ii)′ hϑ0 (xkn+1 , yn+1 ), . . . , ϑn (xkn+1 , yn+1 )i ≤lex hϑ0 (xkn , yn ), . . . , ϑn (xkn , yn )i. It follows from the revised condition, and from the wellfoundedness of the ordinals, that ϑℓ (xkn , yn ) is eventually constant as n −→ ∞, for each ℓ < ù. Using the limit condition in Definition 18 then (iii)′ ϑℓ (x∞ , y∞ ) ≤ eventual value of ϑℓ (xkn , yn ) as n −→ ∞, for each ℓ.

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ITAY NEEMAN

From this and condition (ii)′ it certainly follows that ϑ0 (x∞ , y∞ ) ≤ ϑ0 (xk0 , y0 ), contradicting condition (i)′ . ⊣ Theorems 24 and 13 are propagation theorems. They show that some desirable properties, the prewellordering property in the case of Theorem 13 and scale properties in the case of Theorem 24, propagate from a pointclass Γ to the pointclass aΓ. Infinite games are central to the proofs of both theorems, and both theorems require determinacy. We noted above that the prewellordering property can be used to settle the classical problem of reduction. The scale property too has applications to classical problems, specifically to the problem of uniformization. Roughly speaking the problem involves definably selecting elements from non-empty sets of reals. The following claim is an indication of how scales connect with such selections. Claim 25. Let E ⊂ R be non-empty and let hϑℓ : ℓ < ùi be a scale on E. For x ∈ R set α(x) ~ = hϑ0 (x), x(0), ϑ1 (x), x(1), . . . i. Let hë0 , h0 , ë1 , h1 , . . . i be the lexicographic infimum of the set {α ~ (x) : x ∈ E}. (The infimum is characterized precisely by the condition that for each n, hë0 , h0 , . . . , ën−1 , hn−1 i is the lexicographically smallest element of {α(x)↾n ~ : x ∈ E}.) Then the real y = hhn : n < ùi belongs to E. Proof. For each n pick xn ∈ E so that α(x ~ n )↾n = hë0 , h0 , . . . , ën−1 , hn−1 i. Note that if n −→ ∞, then xn −→ y, and that ϑℓ (xn ) is equal to ëℓ for all n > ℓ. By the limit condition in Definition 18 it follows that y ∈ E. ⊣ Thus there is a canonical way to select an element of E given a scale on E. This canonical selection process can be turned into a solution for the problem of uniformization, cf. [KM78B, §3.1]. In a similar fashion, scales can be used to select canonical winning strategies, producing aΓ winning strategies in Γ games won by the player aiming to enter the Γ set: Exercise 26. Let E ⊂ R belong to a pointclass Γ and let hϑℓ : ℓ < ùi be a Γ scale on E. Suppose that every length ù game with payoff in Γ is determined. Suppose that player I wins the game Gù (E). Prove that player I has a winning strategy ó which belongs to the pointclass aΓ, meaning that the relation (p is winning for I) & ó(p) = h is aΓ. Hint. Given p ∈ <ù ù of even length, say 2k, and h, h ∗ ∈ ù, define Hp (h ∗ , h) to be played according to Diagram 10, with the moves played sequentially from left to right, starting from the vertical line. At the end of an infinite run set y = pahhiahy(i) : 2k < i < ùi and y ∗ = pahh ∗ iahy ∗ (i) : 2k < i < ùi. The run is won by S if y 6∈ E, or y ∈ E & y ∗ ∈ E & hϑ0 (y ∗ ), . . . , ϑlh(p)/2 (y ∗ )i ≤lex hϑ0 (y), . . . , ϑlh(p)/2 (y)i,

PROPAGATION OF THE SCALE PROPERTY USING GAMES

p(0) · · · p(2k − 1) h p(0) · · · p(2k − 1) h∗

89

S y(2k + 1) F y(2k + 2) F y ∗ (2k + 1)

···

Diagram 10. The game Hp (h ∗ , h). and otherwise the run is won by F. Let Q be the set of positions p ∈ <ù ù of even length from which player I can continue to win Gù (E). For p ∈ Q and h, h ∗ ∈ ù set h ∗ p h iff S has a winning strategy in the game Hp (h ∗ , h). Notice the similarity between the definitions here and the ones preceding Exercise 14 (with the modification indicated in Theorem 24). Adapting the proofs connected to the exercise show that: (i) p is a prewellorder on ù. Let ó(p) be the smallest number in the set {h : h is p minimal}. Show that: (ii) The relation p ∈ Q & ó(p) = h is aΓ. (iii) If p is consistent with ó then p is a winning position for I in Gù (E). (iv) ó is a winning strategy for I in Gù (E). Items (i) and (ii) involve adaptations of proofs preceding Exercise 14. Item (iii), which precisely parallels Exercise 14, is a warm-up for item (iv). ⊣ Claim 25 and Exercise 26 are examples of basic applications of norms and scales in descriptive set theory. The papers in Part II of this volume contain many more applications, demonstrating the fundamental importance of scales in the study of consequences of determinacy. The papers in Part I for the most part concentrate on establishing the scale property for various pointclasses. In many cases this is done through propagation, building on and expanding the introductory methods presented here. REFERENCES

Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. [KM78B] Notes on the theory of scales, this volume, originally published in Cabal Seminar 76–77 [Cabal i], pp. 1–53. Yiannis N. Moschovakis [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES LOS ANGELES, CA 90095, USA

E-mail: [email protected]

SCALES ON Σ11 -SETS

JOHN R. STEEL

Let Λ be the class of sets which are (ù · n)-Π11 for some n < ù. We shall show that every set in Λ admits a very good scale all of whose norms are in Λ. Even for Σ11 sets, this is the best possible definability bound on very good scales. Busch shows in [Bus76] that every Σ11 set admits a not very good scale all of whose norms are (ù + 3)-Π11 . Some terminology: if L is an ordered set, then “≤lex ” denotes the lexicographic ordering on <ù L ∪ ù L. A tree on L is a subset of <ù L closed under initial segment. [T ] is the set of infinite branches of the tree T . If T is a tree on L × M and x ∈ ù L, then T (x) = {ô ∈ <ù M : hx↾ lh(ô), ôi ∈ T }, and p[T ] = {x ∈ ù L : [T (x)] 6= ∅}. If M is wellordered and x ∈ p[T ], then fxT is the leftmost branch of T (x). That is, fxT ∈ [T (x)] and ∀g ∈ [T (x)](fxT ≤lex g). Finally, if hAα : α < âi is a sequence of sets, then Diff α<â Aα = {x : ∃α < â(α is odd ∧ x ∈ Aα ∧ ∀ã < α(x 6∈ Aã ))}. A set A ⊆ ù ù is â-Π11 iff A = Diff α<â Aα for some sequence hAα : α < âi of e of Σ1 indices for the A ’s can be taken to be recursive, Σ11 sets. If the sequence α 1 e then A is â-Π11 . Lemma 1.1. If A and B are (ù · n)-Π11 , where n < ù, then A ∩ B is (ù · (n 2 ))-Π11 . Proof. We claim first that there is a wellorder 4 of ù 2 × ù 2 of order type ù which extends the product partial order. (That is, if α ≤ â and ã ≤ ä, then hα, ãi 4 hâ, äi.) For simply let 2

hù · i0 + j0 , ù · k0 + ℓ0 i 4 hù · i1 + j1 , ù · k1 + ℓ1 i ⇐⇒ (2 · 3k0 < 2i1 · 3k1 ∨ (2i0 · 3k0 = 2i1 · 3k1 ∧ 2j0 · 3ℓ0 ≤ 2j1 · 3ℓ1 )). i0

Notice that 4↾(ù · n × ù · n) has order type ù · (n 2 ). Now let A = Diff α<ù·n Aα and B = Diff α<ù·n Bα , where the Aα ’s and Bα ’s are Σ11 . Fix an α < ù · (n 2 ), and let hâ, ãi be the αth element of ù · n × ù · n under 4. If â and ã are both odd, set C2α = ∅ and C2α+1 = Aâ ∩Bã ; otherwise set C2α = Aâ ∩ Bã and C2α+1 = ∅. We claim that A ∩ B = Diff α<ù·(n2 ) Cα . For example, let x ∈ A∩B. Let â be least so that x ∈ Aâ and ã be least so that The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

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x ∈ Bã . Then â and ã are odd, so x ∈ C2α+1 , where hα, âi is the αth element of ù · n × ù · n under 4. If x ∈ Cä for some ä < 2α + 1, then x ∈ Aâ ′ ∩ Bã ′ for some hâ ′ , ã ′ i 4 hâ, ãi. But â ≤ â ′ and ã ≤ ã ′ by the definitions of â and ã, so hâ, ãi 4 hâ ′ , ã ′ i, a contradiction. Thus x 6∈ Cä for ä < 2α + 1, so that x ∈ Diff α<ù·(n2 ) Cα . It is equally easy to check that Diff α<ù·(n2 ) Cα ⊆ A ∩ B. thus A ∩ B is (ù · (n 2 ))-Π11 . ⊣ With a little care, one can show that the intersection of two (ù · n)-Π11 sets is (ù · (2n − 1))-Π11 . S Since the complement of an α-Π11 set is (α + 2)-Π11 , the class Λ = n<ù ((ù · n)-Π11 ) is closed under complement, intersection and union. This means that “interweaving” Λ-norms produces a Λ-norm. In particular, interweaving the norms of the Busch scale on a Σ11 set produces a very good scale on that set with norms in Λ. We give now a direct construction of such a scale. Theorem 1.2. Every Σ11 set A admits a very good scale hϕn i such that for all n, ϕn : A → ù · (n + 1) and ϕn is an (ù · (n + 1))-Π11 norm. Proof. Let A = p[T ], where T is a recursive tree on ù ×ù. We may assume that if hó, ôi ∈ T , then ∀i < lh(ó)∃k(ô(i) = 2ó(i) · 3k ), so that any branch of T (x) codes x. Let hói : i ≤ ni be a recursive enumeration of <ù ù such that for all n, {ói : i ≤ n} is a tree. For n < ù, let En = {ô ∈

<ù

ù : ô 6∈ {ói : i ≤ n} ∧ ô↾(lh(ô) − 1) ∈ {ói : i ≤ n}}.

For x ∈ A and n < ù, let ônx = unique ô ∈ En (ô ⊆ fxT ) and ϕn (x) = the ordinal rank of ônx in ≤lex ↾En . Notice that ≤lex ↾En has order type ù · (n + 1), so that ϕn : A → ù · (n + 1). We now check that hϕn i is a very good scale on A. Clearly, if x ∈ A and y x x n ≤ m, then ônx ⊆ ôm . So if x, y ∈ A and ônx
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JOHN R. STEEL

hx, fi ∈ [T ], so x ∈ A; moreover fxT ≤lex f, so ônx ≤lex ôn for all n. Thus hϕn i is a very good scale on A. It is easy to compute that ϕn is an (ù · (n + 1))-Π11 norm. ⊣ The universal Σ11 set admits no very good scale whose norms are all (ù · n)for some fixed n, at least if x # exists for all reals x. For if it did, the e Third Periodicity Theorem [Mos80, p. 335] would give an x ∈ ù ù so that every Σ11 (x) game has a winning strategy which is a((ù · n)-Π11 (x)). But every a((ù · n)-Π11 (x)) real is recursive in the type of the first n + 1 indiscernibles for L[x], and hence a member of L[x], by Martin’s analysis of (ù · n)-Π11 games. This is impossible since there are Σ11 (x) games without winning strategies in L[x]. Π11

Theorem 1.3. Let Λ be the class of sets which are (ù·n)-Π11 for some n < ù. Then every set in Λ admits a very good scale whose norms are in Λ. Proof. Let A = Diff α<ù·n Aα where the Aα sets are Σ11 . For α < ù · n, let hϕiα i be a scale on Aα as given in Theorem 1.2 (or in [Bus76]), and let hϑiα i be a very good Π11 scale on {x : ∀â < α(x 6∈ Aâ )}. Let hFi : i < ùi be an increasing sequence of finite sets whose union is {α < ù · n : α is odd}. For x ∈ A, let αx = min{â : x ∈ Aâ }, and let øi (x) =

(

hαx i if αx ∈ 6 Fi , αx αx αx αx hαx , ϕ0 (x), ϑ0 (x), · · · , ϕi (x), ϑi (x)i if αx ∈ Fi .

Here, of course, øi (x) is identified with an ordinal using ≤lex ↾ù1≤2i+1 . It is routine to check that høi i is a very good scale on A. We shall check that øi is a Λ norm; for notational simplicity, we consider only i = 0. First, notice that there is a relation R ∈ Λ so that for x, y ∈ A, R(x, y) iff αx < αy . For simply let B2·α = {hx, yi : y ∈ Aα } and B2·α+1 = {hx, yi : x ∈ Aα } for α < ù · n, and then set R = Diff α<ù·n Bα . Let E(x, y) iff ¬(R(x, y) ∨ R(y, x)), so that E ∈ Λ and for x, y ∈ A, E(x, y) ⇐⇒ αx = αy . Now let x ≤0 y ⇐⇒ x ∈ A ∧ (y ∈ 6 A ∨ ø0 (x) ≤ ø0 (y)), x <0 y ⇐⇒ x ∈ A ∧ (y ∈ 6 A ∨ ø0 (x) < ø0 (y)).

SCALES ON Σ11 -SETS

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Then x ≤0 y ⇐⇒ (x ∈ A ∧ (y 6∈ A ∨ (y ∈ A ∧ (R(x, y) ∨ (E(x, y) ∧ [ ^ (x ∈ Aâ \ Aã =⇒ hø0â (x), ϑ0â (x)i ≤lex hø0â (y), ϑ0â (y)i)))))). ã<â

â∈F0

Our Lemma 1.1 implies that ≤0 ∈ Λ. Similarly <0 ∈ Λ, so that ø0 is a Λ norm. ⊣ According to the Third Periodicity Theorem, if every Γ set has a very good scale with norms in Γ, then every aΓ set has a (very good) scale with norms in Γ. So we have S Corollary 1.4. Let Λ = n (ù · n)-Π11 . Then for all k < ù, every ak Λ set admits a very good scale with norms in ak Λ. In particular, every Π12 (= aΣ11 ) set admits a scale with aΛ norms. The Martin-Solovay scale [Kec78A, Theorem 6.3] also has this property. In fact, if one propagates the scale of Theorem 1.2 to Π12 via the second periodicity construction, one obtains the Martin-Solovay scale. (Martin and the author proved this independently.) One can see this by looking at the auxiliary closed games associated to the (ù · n)-Π11 games involved in propagating the scale of Theorem 1.2. S Theorem 1.3 generalizes easily to the class Λα = n (ù 2 · α + (ù · n)-Π11 ), e where α < ù . 1

REFERENCES

Douglas R. Busch [Bus76] ë-Scales, κ-Souslin sets and a new definition of analytic sets, The Journal of Symbolic Logic, vol. 41 (1976), p. 373. Alexander S. Kechris [Kec78A] AD and projective ordinals, In Kechris and Moschovakis [Cabal i], pp. 91–132. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. Yiannis N. Moschovakis [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720, USA

E-mail: [email protected]

INDUCTIVE SCALES ON INDUCTIVE SETS

YIANNIS N. MOSCHOVAKIS

Let X = X1 × · · · × Xn be any product of copies of ù and R = ù ù. A pointset P ⊆ X is inductive if there is a projective set Q ⊆ X × R such that P(x) ⇐⇒ ((∀α1 )(∃α2 )(∀α3 )(∃α4 ) . . . )(∃n)Q(x, pα1 , . . . , αnq),

(1)

where pα1 , . . . , αnq is the usual (recursive) coding of tuples and the (open) game quantifier is interpreted in the obvious way. If Q is analytical in some â0 ∈ R, we call P inductive in â0 and if we can choose â0 recursive, we call P absolutely inductive. Sets which are both inductive and coinductive are hyperprojective; “hyperprojective in â0 ” and “absolutely hyperprojective” sets are defined in the obvious way. The theory of inductive sets (on arbitrary structures) is developed in some detail in [Mos74A]. Our purpose here is to outline a proof of the following: Main Theorem. If every hyperprojective game is determined, then every absolutely inductive pointset admits an absolutely inductive scale; it follows that inductive sets admit inductive scales and hyperprojective sets admit hyperprojective scales. Part of the interest in this result lies in the fact that the collection of inductive sets is the largest collection of pointsets for which we can presently establish the scale property, from any hypotheses. §1. Proof of the Main Theorem. We assume that the tupling function pα1 , . . . , αnq is defined for the empty tuple (n = 0), p q = t 7→ 1 and that concatenation is given by a recursive function aon the codes, pα1 , . . . , αnqapâ1 , . . . , âmq = pα1 , . . . , αn , â1 , . . . , âmq. We will prove the main theorem in a sequence of simple lemmas. Generalizing slightly the definition (1) above, for any given Q ⊆ X × R, put R(x, α) ⇐⇒ ((∀α1 )(∃α2 )(∀α3 )(∃α4 ) . . . )(∃n)Q(x, α apα1 , . . . , αnq) The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

(2)

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and for each ordinal î define by induction Rî (x, α) ⇐⇒ Q(x, α) ∨ (∀â)(∃ã)(∃ç < î)Rç (x, α apâ, ãq).

(3)

Lemma 1.1. R(x, α) ⇐⇒ (∃î)Rî (x, α). Proof. First check by a simple induction on î that Rî (x, α) =⇒ R(x, α). For the converse, assume (∀î)¬Rî (x, α) and show by applying (3) repeatedly that in that case ((∃α1 )(∀α2 )(∃α3 )(∀α4 )...)(∀n )¬Q(x, α apα1 , . . . , αnq) which is equivalent to ¬R(x, α).

⊣

It follows from results of [Mos74A] that if Q is analytical (or projective), then R(x, α) ⇐⇒ (∃î < κ)Rî (x, α), where κ is the closure ordinal for positive elementary inductive definitions on R, or alternatively, κ = sup{rank() : 4 is a hyperprojective prewellordering of R}. Suppose now that R and Rî are defined by (2) and (3) and suppose we are given a scale ϕ ~ 0 = hϕn0 : n ∈ ùi on Q = R0 . We will define by induction on î î a scale ϕ ~ on each Rî . Assuming that ϕ ~ ç has been defined for each ç < î, define first on [ R<î = Rç ç

the sequence of norms ø ~ = î

hønî

: n ∈ ùi by

ø0î (x, α) = min{ç : Rç (x, α)}, î øn+1 (x, α)

=

pæ, ϕnæ (x, α)q,

where æ =

(4) ø0î (x, α).

(5)

Lemma 1.2. If each ϕ ~ ç is a scale on Rç (ç < î), then (4) and (5) define a î <î scale ø ~ on R . Proof. If hxi , αi i → hx, αi and all norms ønî (xi , αi ) are ultimately fixed, then in particular æ = ø0î (xi , αi ) = min{ç : Rç (xi , αi )} is ultimately fixed; hence Ræ (x, α) holds and for all n we have ϕnæ (x, α) ≤ ϕnæ (xi , αi ) (for all large i).

(6)

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YIANNIS N. MOSCHOVAKIS

î Now if ø0î (x, α) = æ, then (6) implies directly that for all n, øn+1 (x, α) = î æ æ pæ, ϕn (x, α)q ≤ pæ, ϕn (xi , αi )q = øn+1 (xi , αi ) for all large i. If on the other hand ø0î (x, α) = æ ′ < æ, then the lexicographic ordering used in (5) implies that î î øn+1 (x, α) = pæ ′ , ϕnæ (x, α)q < pæ, ϕnæ (xi , αi )q = øn+1 (xi , αi ), ′

so that in either case, ø ~ î has the critical lower semicontinuity property of scales. ⊣ To extend the construction and define a scale on Rî from given scales on all the Rç , ç < î, we need the determinacy hypothesis. Lemma 1.3. Assume that every hyperprojective game is determined and that both R<î and the scale ø ~ î on R<î constructed above is hyperprojective; then we can construct a hyperprojective scale on Rî . Proof. This is immediate from [KM78B, 3.2.2] and [KM78B, 3.3.1], taking Γ = all hyperprojective sets and using the rather obvious fact that Γ is adequate and closed under ∃R , ∀R . ⊣ κ To complete the construction, we now put a scale ø ~ on R = R<κ by taking î = κ in Lemma 1.2. To prove the main theorem, it will be enough to show that if P is defined from an analytical pointset Q by (1) and R is defined by (2), then this construction actually gives an inductive scale ø ~ κ on R, since we can then obviously get an inductive scale on P by setting ϕn (x) = ønκ (x, p q). The proof naturally rests on some elementary properties of inductive and hyperprojective sets, all proved in [Mos74A]. To begin with, the collection of absolutely inductive sets is adequate in the sense of [KM78B] and it is closed under ∃ù , ∀ù , ∃R , ∀R —this is almost obvious. The hyperprojective sets are additionally closed under ¬ and continuous substitutions. It is a bit less obvious that the inductive sets can be parameterized in R by absolutely inductive universal sets; i.e., for each X, there is an absolutely inductive G X = G ⊆ R × X such that for P ⊆ X, P is inductive ⇐⇒ for some α ∈ R, P = Gα = {x : G(α, x)}. This can be proved by checking that we can always take Q to be Σ12 —see e [Mos74A, 5D.2]. The most important fact about inductive sets that we need is the Stage Comparison Theorem, [Mos74A, 2A.2]:

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Theorem 1.4 (Stage Comparison Theorem). If Q is analytical in (2), then the obvious norm on R given by ϕ(x, α) = min{î : Rî (x, α)} is an absolutely inductive norm. In particular, the pointclasses of absolutely inductive and coinductive sets have the prewellordering property; see [KM78B, 2A]. For each X, fix an absolutely inductive G X ⊆ R × X which parametrizes the inductive subsets of X as above and put pα, âq ∈ C ⇐⇒ (∀x ∈ X)(Gα (x) ⇐⇒ ¬Gâ (x)). Each ã = pα, âq ∈ C codes a hyperprojective subset of X, Hã = Gα = X \ Gâ and every hyperprojective subset of X receives at least one code in C . It is not very hard to check that if the system {G X } of universal sets is chosen carefully, then the hyperprojective sets are effectively closed in this coding under every operation which preserves both absolute inductiveness and absolute coinductiveness; for example, there is a recursive function f: R×R → R such that if ã, ä ∈ C , then å = f(ã, ä) ∈ C and Hå = Hã ∩ Hä . The key to this argument is to choose universal sets so that for suitable recursive functions S X,Y = S, G(α, x, y) ⇐⇒ G(S(α, x), y) and such that in addition, P is absolutely inductive =⇒ P = Gα with a recursive α. We get effective closure under ∧ (as an example) using such universal sets by setting P0 (ã, ä, x) ⇐⇒ G((ã)0 , x) ∧ G((ä)0 , x), P1 (ã, ä, x) ⇐⇒ G((ã)1 , x) ∨ G(ä)1 , x), choosing recursive å0 , å1 so that P0 (ã, ä, x) ⇐⇒ G(å0 , ã, ä, x) ⇐⇒ G(S(å0 , ã, ä), x) P1 (ã, ä, x) ⇐⇒ G(å1 , ã, ä, x) ⇐⇒ G(S(å1 , ã, ä), x)

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and taking f(ã, ä) = pS(å0 , ã, ä), S(å1 , ã, ä)q. It is simple to compute that if ã, ä ∈ C and å = f(ã, ä), then å ∈ C and Hå = Hã ∩ Hä . One can always find “good” universal sets with these properties—see [Mos74A, 9C.7]. After these preliminaries we can now describe the main construction for the proof. If ϕ ~ = hϕn : n ∈ ùi is a scale on a set A ⊆ X, then a (hyperprojective) code for ϕ ~ is any α ∈ C which codes a set Hα ⊆ ù × X × X such that y ∈ A =⇒ (∀x)((x ∈ A ∧ ϕn (x) ≤ ϕn (y)) ⇐⇒ (n, x, y) ∈ Hα ). Lemma 1.5. Assume that every hyperprojective game is determined and that R(x, α) ⇐⇒ ((∀α1 )(∃α2 )(∀α3 )(∃α4 ) . . . )(∃n )Q(x, α apα1 . . . αnq) with Q analytical, let Rî (x, α) ⇐⇒ (∀â)(∃ã)(Q(x, α) ∨ (∃ç < î)Rç (x, α apâ, ãq)) as above and for hx, αi ∈ R, put |x, α| = min{î : Rî (x, α)}. For each î, there is a scale ϕ ~ î = hϕnî : n ∈ ùi on Rî such that for some recursive function ð : X × R → R, if R(x, α) and |x, α| = î, then ð(x, α) is a code of ϕ ~ î. î In particular, each ϕ ~ is a hyperprojective scale on Rî . Proof. The scales ϕ ~ î will be exactly those defined in Lemmas 1.2 and 1.3—we will have to make sure that the hypothesis of Lemma 1.3 is satisfied. Since the definition of ϕ ~ î is by transfinite induction and we want ð to be recursive, we must apply Kleene’s well-known method of definition of effective transfinite induction, using the recursion theorem; i.e., we will set ð(x, α) = {å0 }(x, α) where å0 is a fixed recursive real satisfying {å0 }(x, α) = f(å0 , x, α) with a recursive (possibly partial) f : R × X × R → R. (The necessary coding machinery for recursive partial functions on the reals can be found in [Mos70A] or [Mos80].) Now the key to the construction is the fact that the transfer theorems for scales [KM78B, 3.2.2] and [KM78B, 3.3.1] have effective proofs. Combined with the remarks about effective closure of the hyperprojective sets above, this

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implies easily that there are recursive functions f1 (α, ã), f2 (α, ã) such that if α happens to code a scale on Hã (ã ∈ C ), then f1 (α, ã) codes a scale on ∃R Hã and f2 (α, ã) codes a scale on ∀R Hã . Considering the positive analytical definition of Rî from R<î , this in turn implies that there is a recursive function f3 (α, ã) such that if R<î = Hα and ã happens to code a scale on R<î , then f3 (α, ã) codes a scale on Rî . (Notice that the hypothesis here implies that both R<î and the scale on it are hyperprojective, so the appropriate games in Lemma 1.3 are determined.) At the same time, the Stage Comparison Theorem implies easily that for a suitable recursive f4 (x, α), if R(x, α) holds and |x, α| = î, then f4 (x, α) is a hyperprojective code of R<î . Finally, looking at (4) and (5), it is not hard to believe that there is a further recursive function f5 (å, x, α) with the following property: if there are scales ϕ ~ ç on Rç for each ç < î and if for each hx ′ , α ′ i ∈ R ′ ′ such that |x , α | = ç < î, {å}(x ′ , α ′ ) is defined and codes ϕ ~ ç , then f5 (å, x, α) î ç codes the scale ø ~ defined from the ϕ ~ by (4), (5). Now choose a recursive å ∗ ∈ R such that f5 (å, x, α) = {å ∗ }(å, x, α) = {S(å ∗ , å)}(x, α) using the appropriate recursive S for our coding of recursive partial functions and set f(å, x, α) = f3 (f4 (x, α), S(å ∗ , å)). If we pick å0 by the recursion theorem and set ð(x, α) = {å0 }(x, α) = f(å0 , x, α), it is then trivial to verify by induction on î that ð has the required property. ⊣ This lemma completes the proof of the main theorem, since in the scale ø ~κ <κ κ we put on R = R by (4) and (5) the first norm ø0 is absolutely inductive by the Stage Comparison Theorem and the later norms can be computed easily using the recursive function ð. §2. Corollaries and remarks. Of course the main corollary concerns uniformization. Corollary 2.1. If every hyperprojective game is determined, then the collection of absolutely inductive pointsets has the uniformization property. Proof. See [KM78B, 3.1.1]. ⊣ To look a bit closer at one of the more interesting consequences of this uniformization result, let us go back and reconsider the open games on R that we used to define inductive sets. In a definition of the form P(x) ⇐⇒ ((∀α1 )(∃α2 )(∀α3 )(∃α4 ) . . . )(∃n)Q(x, pα1 , . . . , αnq)

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we normally interpret the expression on the right as asserting the existence of a winning strategy for player II in the obvious game. This understanding of the definitions will force us to use the axiom of choice in the proof of Lemma 1.2 which is best avoided in the context of determinacy; instead we can reinterpret the infinite alternating string using multiple-valued strategies (quasistrategies in [Mos74A]) and if we do that, then the proof of the main theorem is easily given in ZF+DC. On the other hand, once we have the theorem, we should point out that if P(x) holds, then player II can win the game on the right using a (single-valued) strategy which is hyperprojective in x. Corollary 2.2. Assume that every hyperprojective game is determined and suppose that ((∀α1 )(∃α2 )(∀α3 )(∃α4 ) . . . )(∃n)Q(pα1 , . . . , αnq), where Q is hyperprojective in some â0 . Then there exists a function ó: R → R which is hyperprojective in â0 such that ((∀α1 )(∀α3 )(∀α5 ) . . . )(∃n)Q(pα1 , α2 , α3 , α4 , . . . , αnq),

(7)

where for even n, αn = ó(pα1 , . . . , αn−1q). Proof. Put P(α, â) ⇐⇒ ((∀αn+1 )(∃αn+2 ) . . . )(∃k)Q(α apâ, αn+1 , αn+2 , . . . , αkq), so that P is inductive in â0 . Let P ∗ uniformize P and set for even n ó(pα1 , . . . , αn−1q) = â ⇐⇒ P ∗ (pα1 , . . . , αn−1q, â)

⊣

A further consequence of this remark concerns the absoluteness properties of the partially playful universe associated with the inductive sets. Corollary 2.3. Assume that every hyperprojective game is determined and let T be the tree on ù × κ associated with some absolutely inductive scale on some universal inductive set, let L[T ] be the associated model as in [KM78B]. If Q is analytical and for x ∈ L[T ] ((∀α1 )(∃α2 )(∀α3 )(∃α4 ) . . . )(∃n)Q(x, pα1 , . . . , αnq) holds (in the world), then the relativization of this open game assertion to L[T ] also holds. Proof. L[T ] is easily closed under functions hyperprojective in x, with x ∈ L[T ]. ⊣

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The corollary is quite strong, as there are a lot of things that one can say with open game assertions—particularly if one assumes full determinacy and uses the coding Lemma 3 of [Mos70A]. The proof of the main theorem makes it clear that the result goes through for various pointclasses (included in the inductive sets) which enjoy some of the structural and closure properties of the inductive sets. In particular, the careful reader will notice that closure of the inductive sets under ∃R is not used in the proof —although we did use (effective) closure of the hyperprojective sets under ∃R . Using this observation, anyone who is somewhat familiar with recursion in higher types (as presented in [KM77], for example) will verify easily Corollary 2.4 (to the proof). If every game which is Kleene recursive in E and some â0 ∈ R is determined, then every pointset which is semirecursive in 3 E admits a scale which is semirecursive in 3 E. 3

In particular, the collection of pointsets semirecursive in 3 E has the uniformization property (under this determinacy hypothesis). In the other direction, it is hard to see how one could extend the class of sets now known to possess definable scales without using some entirely new principle of constructing scales. It appears that the question whether coinductive sets admit definable scales is one of the critical open problems in this theory. REFERENCES

Alexander S. Kechris and Yiannis N. Moschovakis [KM77] Recursion in higher types, Handbook of mathematical logic (K. J. Barwise, editor), NorthHolland, 1977, pp. 681–737. [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. [KM78B] Notes on the theory of scales, this volume, originally published in Cabal Seminar 76–77 [Cabal i], pp. 1–53. Yiannis N. Moschovakis [Mos70A] Determinacy and prewellorderings of the continuum, Mathematical logic and foundations of set theory. Proceedings of an international colloquium held under the auspices of the Israel Academy of Sciences and Humanities, Jerusalem, 11–14 November 1968 (Y. Bar-Hillel, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam-London, 1970, pp. 24–62. [Mos74A] Elementary induction on abstract structures, North-Holland, 1974. [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES, CA 90095, USA

E-mail: [email protected]

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YIANNIS N. MOSCHOVAKIS

The main purpose of this note is to prove (under appropriate determinacy hypotheses) that coinductive pointsets admit definable scales. This solves the second Victoria Delfino problem [Cabal i, p. 279-282] and was announced in [MMS82], together with many stronger related results of Martin and Steel. §1. Lemmas on preservation of scales. When this question was first posed in [Mos78], there was a feeling (certainly on my part) that its answer might require methods for constructing scales that would be radically different from the simple and well-understood ideas of the periodicity theorems. As it happens, this is not true: one can define a scale on a given coinductive set quite easily, by a judicious mix of the constructions in the second periodicity theorems [Mos80, 6C.1 & 6C.3]. In fact, the simplest way to explain the proof of this theorem is to reexamine the proofs of these two results and extract from them a bit more than was stated explicitly in [Mos80]. Suppose P is a pointset, i.e., P ⊆ X = X1 × · · · × Xn with each Xi = ù or Xi = ù ù. A putative scale on P is a sequence ϕ ~ = hϕi : i ∈ ùi of norms on P, each norm ϕi : P → Ord mapping P into the ordinals. A putative scale ϕ ~ on P defines in a natural way a notion of convergence for sequences of points in P: we say that xn converges to x modulo ϕ ~ and write xn → x

(mod ϕ) ~

if limn→∞ xn = x in the usual topology on X and if in addition, for each fixed i, the sequence of ordinals ϕi (x0 ), ϕi (x1 ), ϕi (x2 ), . . . is ultimately constant. We say that ϕ ~ is a semiscale on P if for every sequence hxn : n ∈ ùi in P, xn → x

(mod ϕ) ~ =⇒ x ∈ P.

During the preparation of this paper the author was partially supported by NSF Grant MCS78-02989. The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

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A norm ϕ : P → Ord is lower semicontinuous relative to the putative scale ϕ, ~ if for each sequence hxn : n ∈ ùi in P, xn → x

(mod ϕ) ~ =⇒ (x ∈ P ∧ ϕ(x) ≤ lim ϕ(xn )); n→∞

we call ϕ ~ a scale if each ϕi is lower semicontinuous relative to ϕ. ~ To give a topological interpretation of these definitions (for those familiar with uniform spaces), let ô(~ ϕ ) be the topology on P generated by the neighbourhoods N (x0 , k) = {x : x↾k + 1 = x0 ↾k + 1 ∧ ∀i ≤ k(ϕi (x) = ϕi (x0 ))} relative to a putative scale ϕ. ~ It is easy to check that ô(~ ϕ ) is a uniform topology in the sense of Kelley [Kel55] and that ϕ ~ is a semiscale ⇐⇒ P is complete, ϕ ~ is a scale ⇐⇒ for each i and ë, the “closed ball” {x : ϕi (x) ≤ ë} is complete. Of course, the theory of uniform spaces is tailored to the examples where the topology is determined by a sequence of pseudo-metrics with real values and is not useful in the present context of ordinal-valued norms. If ϕ ~ and ϕ ~ ′ are both putative scales on P, we say that ϕ ~ is finer than ϕ ~ ′ if for every x0 , x1 , . . . in P, xn → x

(mod ϕ) ~ =⇒ xn → x

(mod ϕ ~ ′ ).

For example, one can refine a putative scale ϕ ~ by adding some norms or interweaving, i.e., setting øn (x) = pϕ0 (x), ϕ1 (x), . . . , ϕn (x)q where hî0 , . . . , în i 7→ pî0 , . . . , înq is some order-preserving map (depending on n) of the (n+1)-tuples of ordinals (ordered lexicographically) into the ordinals. Suppose that Q(x) ⇐⇒ ∃α(P(x, α)) and that we are given norms ϕ0 , . . . , ϕn on P. We define the norm øn = “inf ”{ϕ0 , . . . , ϕn } on Q by øn (x) = inf{pϕ0 (x, α), α(0), ϕ1 (x, α), α(1), . . . , ϕn (x, α), α(n)q : P(x, α)}

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where p. . .q again stands for an order-preserving map of 2n-tuples into the ordinals. If ϕ ~ = hϕn : n ∈ ùi is a putative scale on P, then this definition associates with ϕ ~ the infimum putative scale ø ~ = “inf ” ϕ ~ = høn : n ∈ ùi on Q—but it is important that the definition of each øn in “inf ” ϕ ~ depends only on ϕ0 , . . . , ϕn and not on the whole sequence ϕ ~. Now the proof of [Mos80, 6C.1] (stated there for “very good scales”) actually establishes the following result. Lemma 1.1 (The Infimum Lemma). Suppose ϕ ~ is a putative scale on P ⊆ X × ù ù, Q(x) ⇐⇒ ∃α(P(x, α)) and ø ~ = “inf ” ϕ ~. ′ 1. If ø ~ is any putative scale finer than ø ~ and if hxn : n ∈ ùi is any sequence of points in Q such that xn → x

(mod ø ~ ′ ),

then we can find a subsequence xk∗ = xnk with (n0 < n1 < . . . ) and irrationals α, α0 , α1 , . . . , such that P(xk , αk ) holds for each k and hxk∗ , αk i → hx, αi

(mod ϕ ~ ).

2. If ϕ ~ is a semiscale on P and for i = 0, . . . , m each ϕi is lower semicontinuous relative to ϕ, ~ then øm is lower semicontinuous relative to ø. ~ In particular, if ϕ ~ is a scale on P, then ø ~ = “inf ” ϕ ~ is a scale on Q. Proof. For 1., notice that it is enough to consider the case ø ~ = “inf ” ϕ ~ and choose nk to be the least n such that for i = 0, . . . , k, and all m ≥ n, øi (xm ) = øi (xn ). By definition, øi (xk∗ ) = pϕ0 (xk∗ , αik ), αik (0), ϕ1 (xk∗ , αik ), αik (1), . . . , ϕi (xk∗ , αik ), αik (i)q for some αik and for k ≥ i, ℓ ≥ i, αik (0) = αi (0), . . . , αik (i) = αiℓ (i) so we can take αk = αkk and the result follows immediately. 2. is not hard to prove directly from the definition of the infimum norms and we will skip it. ⊣

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We will also need the analogous modification of the second periodicity theorem [Mos80, 6C.3] which applies to arbitrary putative scales that need not be semiscales or scales. Suppose then that Q(x) ⇐⇒ ∀αP(x, α) and ϕ ~ is a given putative scale on Q and let u(0), u(1), . . . be a canonical enumeration of all finite sequences of integers such that if u(i) is a proper initial segment of u(j), then i < j. As in [Mos80, 6C.3] (but interweaving the norms in ϕ ~ ), we define for each x, y ∈ X and each n a game Gn (x, y) where player I plays α ′ , player II plays â ′ , α = u(n)aα ′ , â = u(n)aâ ′ , and Player II wins ⇐⇒ pϕ0 (x, α), . . . , ϕn (x, α)q ≤ pϕ0 (y, â), . . . , ϕn (y, â)q. The relation x ≤n y ⇐⇒ x, y ∈ Q ∧ player II wins Gn (x, y) is a prewellordering (granting the determinacy of these games), so we can define on Q for each n the norm øn = “sup”{ϕ0 , . . . ϕn } by setting øn (x) to be the rank of x in the prewellordering ≤n . The putative scale ø ~ = høn : n ∈ ùi on Q is the (fake) supremum putative scale associated with ϕ ~ on P, in symbols ø ~ = “sup” ϕ. ~ Notice again that the definition of each øn in “sup” ϕ ~ depends only on ϕ0 , . . . , ϕn . Lemma 1.2 (The Fake Supremum Lemma). Suppose ϕ ~ is a putative scale on P ⊆ X × ù ù, Q(x) ⇐⇒ ∀αP(x, α), assume that each ϕi is a Γ-norm where Γ is an adequate pointclass satisfying Det(∆) and let ø ~ = “sup” ϕ. ~ e ′ 1. If ø ~ is any putative scale on Q which is finer than ø ~ and if hxn : n ∈ ùi is a sequence of points in Q such that xn → x

(mod ø ~ ′ ),

then for each α, there exists a subsequence xk∗ = xnk with (n0 < n1 < . . . ) and irrationals α0 , α1 , . . . , such that hxk∗ , αk i → hx, αi

(mod ϕ ~ ).

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2. If ϕ ~ is a semiscale on P and for i = 0, . . . , m each ϕi is lower semicontinuous relative to ϕ, ~ then øm is lower semicontinuous relative to ø. ~ In particular, if ϕ ~ is a scale on P, then ø ~ = “sup” ϕ ~ is a scale on Q. Proof. The proof of this is exactly the proof of [Mos80, 6C.3] which was stated with additional hypotheses there. ⊣ §2. The main result. A pointset Q ⊆ X is inductive if there is a projective P ⊆ X × ù ù such that Q(x) ⇐⇒ ((∀α0 )(∃α1 )(∀α2 ) . . . )(∃t)P(x, pα0 , . . . , αt−1q).

(1)

This can be taken as the definition of inductive sets or it can be proved easily from the more natural definition in terms of positive elementary induction on ù ù, see [Mos80, 7C]. (A quick summary of some of the basic properties of inductive sets can be found in [Mos78].) For the proof in this paper, we will only need the following very simple normal form for inductive sets. Lemma 2.1. A set Q ⊆ X is inductive if and only if there is a numbertheoretic relation R such that Q(x) ⇐⇒ ((∀α0 )(∃α1 )(∀α2 ) . . . )(∃t)R(x(t), pα 0 (t), . . . , α t−1q(t));

(2)

consequently, Q is coinductive if for some R Q(x) ⇐⇒ ((∃α0 )(∀α1 )(∃α2 ) . . . )(∀t)R(x(t), pα 0 (t), . . . , α t−1q(t)). Proof. Suppose, for example, that (1) above holds with some P(x, α) in Σ12 , so that e P(x, α) ⇐⇒ (∃â)(∀ã)(∃s)R(x(s), α(s), â(s), ã(s)) with a number-theoretic R. An easy game-argument shows that Q(x) ⇐⇒ ((∀α0 )(∃α1 )(∃u1 )(∀α2 )(∃α3 )(∃u3 ) . . . )(∃t) (ut = 1 ∧ (∀i < t, i odd )(ui = 0) ∧

(3)

∧ if s = αt+2 (0), then R(x(s), pα(s), . . . , α t−1 (s)q, α t (s), α t+1 (s))). (From a strategy that establishes Q(x) by (1), get one that establishes Q(x) by (3) and vice versa; notice that both (1) and (3) are interpreted via open games.) Now (3) gives a normal form for Q(x) as in (2) (with a different R) by trivial recursive manipulations of the infinite quantifier string and the matrix. ⊣

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Let Σ∗0 be the pointclass of all Boolean combinations of inductive and e coinductive pointsets, and for n ≥ 1, by induction,

Q ∈ Σ∗n ⇐⇒ for some P in Σ∗n−1 , and all x, Q(x) ⇐⇒ (∃α)¬P(x, α). e e S Theorem 2.2. If every game with payoff in n∈ù Σ∗n is determined, then every coinductive pointset Q admits a scale ø ~ = høi :ei ∈ ùi, such that each ∗ øi is a Σi -norm. e Proof. Suppose by Lemma 2.1 that Q(x) ⇐⇒ ((∃α0 )(∀α1 ) . . . )(∀t)R(x(t), pα 0 (t), . . . , α t−1q(t))

and for each even n define Qn (x, α0 , . . . , αn−1 ) ⇐⇒ ((∃αn )(∀αn+1 ) . . . )(∀t)R(x(t), pα 0 (t), . . . , α t−1 (t)q). For odd n, let Qn (x, α0 , . . . , αn−1 ) ⇐⇒ (∀αn )Qn+1 (x, α0 , . . . , αn−1 , αn ). We will define a putative scale ø ~ n = {øin : i ∈ ù} on each Qn and then show 0 that ø ~ on Q = Q0 is actually a scale. The definition of øin is by induction on i, simultaneously for all n. We will use “vector notation” for tuples, i.e., α ~ = hα0 , . . . , αn−1 i, â~ = hâ0 , . . . , ân−1 i. Case 1. n is even and i = 0. If Qn (x, α) ~ holds, put ø0n (x, α) ~ = 0. This norm just records (by being defined) that Qn (x, α) ~ holds. It is a Σ∗0 -norm e since all Qn are clearly coinductive and (in the notation of [Mos80, 4B]), ~ ⇐⇒ Qn (x, α), hx, αi ~ ≤∗ n hy, âi ~ ø0

hx, αi ~

<∗ø0n

~ ⇐⇒ Qn (x, α) ~ hy, âi ~ ∧ ¬Qn (y, â).

It is not hard to check that in general we cannot expect ø0n to be either inductive or coinductive. Case 2. n is odd. Now Qn (x, α) ~ ⇐⇒ (∀αn+1 )Qn+1 (x, α, ~ αn+1 ) and assuming that øjn+1 is defined for j ≤ i, we put øin = “sup”{ø0n+1 , . . . , øin+1 }. Case 3. n is even and i > 0. Now Qn (x, α) ~ ⇐⇒ (∃αn+1 )Qn+1 (x, α, ~ αn+1 )

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and assuming that øjn+1 is defined for all j < i, we put n+1 øin = “inf ”{ø0n+1 , . . . , øi−1 }.

This completes the definition of a putative scale ø ~ n on each Qn and the definability assertion in the theorem is immediate from the proofs of [Mos80, 6C.1 & 6C.3], by induction on i. Notice that by the construction, immediately, for each even n, ø ~ n is finer n+1 n n+1 than “inf ” ø ~ and for each odd n, ø ~ = “sup” ø ~ , so the basic lemmas apply. To check first that ø ~ 0 is a semiscale on Q = Q0 , suppose xm → x

(mod ø ~ 0 );

to prove that Q(x) holds, we must describe a strategy for ∃ to win the closed game which interprets ((∃α0 )(∀α1 )(∃α2 ) . . . )(∀t)R(x(t), pα 0 (t), . . . , α t−1 (t)q). By the Infimum Lemma, choose a subsequence hxm0 : m ∈ ùi of hxm : 0 0 m ∈ ùi and irrationals α0 , α0,1 , α0,2 , . . . such that 0 hxm0 , α0,m i → hx, α0 i

(mod ø ~ 1)

and play this α0 . If your opponent responds with some α1 , choose by the Fake 1 0 Supremum Lemma a subsequence hhxm1 , α0,m i : m ∈ ùi of hhxm0 , α0,m i : m∈ 1 ùi and irrationals hα1,m : m ∈ ùi such that 1 1 hxm1 , α0,m , α1,m i → hx, α0 , α1 i (mod ø ~ 2) 1 1 and then apply the Infimum Lemma again to hhxm1 , α0,m , α1,m i : m ∈ ùi 2 2 to get a subsequence hhxm2 , α0,m , α1,m i : m ∈ ùi and irrationals α2 and 2 hα2,m : m ∈ ùi such that 2 2 2 hxm2 , α0,m , α1,m , α2,m i → hx, α0 , α1 , α2 i (mod ø ~ 3)

and play this α2 . It is clear that ∃ can continue to play in this manner indefinitely, so we have defined a strategy for him. To see that he wins, notice that by the construction, for each n we know that for each m, n−1 n−1 n−1 , α1,m , . . . , αn−1,m ) Qn (xmn−1 , α0,m

so that by the definition of Qn , taking t = n in the matrix, we have for each m n−1 n−1 R(x n−1 m (n), pα 0,m (n), . . . , α n−1,m (n)q);

letting m → ∞, since n−1 n−1 hxmn−1 , α0,m , . . . , αn−1,m i → hx, α0 , . . . , αn−1 i,

we have for each n R(x(n), pα 0 (n), . . . , α n−1 (n)q)

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which is precisely the condition for ∃ to win. Thus ø ~ 0 is a semiscale on Q0 . n Now the same argument shows easily that each ø ~ is a semiscale in Qn , with just some added notation. To show that these are actually scales, notice first that for n even, ø0n is surely semicontinuous since it is constant; the Infimum Lemma and the Fake Supremum Lemma then imply immediately that all øin are lower semicontinuous relative to ø ~ n , by induction on i. ⊣ REFERENCES

Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. John Kelley [Kel55] General topology, The University series in higher mathematics, Van Nostrand, Princeton, NJ, 1955. Donald A. Martin, Yiannis N. Moschovakis, and John R. Steel [MMS82] The extent of definable scales, Bulletin of the American Mathematical Society, vol. 6 (1982), pp. 435– 440. Yiannis N. Moschovakis [Mos78] Inductive scales on inductive sets, this volume, originally published in Kechris and Moschovakis [Cabal i], pp. 185–192. [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES, CA 90095, USA

E-mail: [email protected]

THE EXTENT OF SCALES IN L(R)

DONALD A. MARTIN AND JOHN R. STEEL

We shall show that every set of reals which is Σ1 -definable over L(R) from parameters in R ∪ {R} admits a scale in L(R), and in fact that the class of sets so definable has the scale property. Kechris and Solovay observed some time ago that no other sets admit scales in L(R), so that our result determines precisely the extent of scales in L(R). Some preliminaries: We work in ZF + DC, and state our additional determinacy hypotheses as we need them. We let R = ù ù, the Baire space, and call its elements reals. Variables z, y, x, . . . range over R, while α, â, ã, . . . range over the class Ord of ordinals. Let ∗ be a recursive homeomorphism from ù R to R such that hxn : n < ùi∗ ↾i depends only on hxn : n < ii. We use hxn : n < ii∗ to denote the common value of hxn : n < ùi∗ ↾i for all extensions hxn : n < ùi of hxn : n < ii. A pointclass is a class of relations on R closed under recursive substitutions; a boldface pointclass is a pointclass closed under continuous substitutions. Let Σn (M, X ) be the class of relations on R which are Σn -definable over M from parameters in X . We are mainly interested in the pointclass Σ1 (L(R), {R}), and its boldface counterpart Σ1 (L(R), R ∪ {R}). It is easy to show that Σ1 (L(R), {R}) = (Σ21 )L(R) and Σ1 (L(R), R ∪ {R}) = (Σ21 )L(R) (cf. [Ste83B, e Lemma 1.12]). We shall have no use for this fact, however. For any X , <ù X is the set of finite sequences from X . A tree on X is a subset of <ù X closed under initial segment. If T is a tree on X , [T ] = {f ∈ ù X : ∀n(f↾n ∈ T )}; T is well founded just in case [T ] = ∅. We sometimes regard a sequence of tuples as a tuple of sequences (of the same length), so that a tree on X1 × · · · × Xk becomes a subset of <ù X1 × · · · × <ù Xk . If T is a tree on X × Y and f ∈ ù X , then T (f) = {u ∈ <ù Y : (f↾ lh(u), u) ∈ T }. Notice that T (f) is a tree on Y depending continuously on f. A quasi-strategy for player I in a game on X is a nonempty tree S on X such that if u ∈ S and lh(u) is even, then ∃a ∈ X ∀b ∈ X (u ahaiahbi ∈ S). We say S is a winning quasi-strategy for player I in the game with payoff A just in The preparation of the this paper was partially supported by NSF Grant Number MCS7802989 The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

110

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THE EXTENT OF SCALES IN L(R)

case [S] ⊆ A. If G is the game on R with payoff A, and Γ is a pointclass, we call G a Γ game iff {hxn : n < ùi∗ : hxn : n < ùi ∈ A} is in Γ. DetR (Γ) is the assertion that all Γ games on R are determined. Similarly, if B ⊆ R × R, let aR B = {x : player I has a winning quasi-strategy in the game with payoff {hyn : n < ùi : B(x, hyn : n < ùi∗ }}, = {x : ∃y0 ∀y1 ∃y2 · · · B(x, hyn : n < ùi∗ )}, and if Γ is a pointclass, let aR Γ = {aR B : B ∈ Γ}. For whatever else we use from Descriptive Set Theory, and in particular for the notions of a scale and of the scale property, we refer the reader to [Mos80]. Out main theorem builds directly on a scale construction due to [Mos83]. We shall describe briefly the slight generalization of this construction we need. Suppose that α ∈ Ord, and that for each x ∈ R we have a game Gx in which player I’s moves come from R × α while player II’s moves come from R. Thus a typical run of Gx has the form player I

x0 , â 0

x2 , â 1 ···

player II

x1

x3

where xi ∈ R and âi < α for all i. Suppose that Gx is closed and continuously associated to x in the strong sense that for some tree T on ù × ù × α player I wins Gx iff (x, hxn : n < ùi∗ , hân : n < ùi) ∈ [T ]. Let Pk (x, u) iff u is a position of length k from which player I has a winning quasi-strategy in Gx , and P(x) iff P0 (x, ∅). If sufficiently many games are determined, then Moschovakis’ construction yields a scale on P. [As in [Mos81], we extend the concept of a scale to relations with arguments from Ord as well as R by giving Ord the discrete topology. Following [Mos83] then, we define putative scales ϕ~k on Pk for all k simultaneously. If Pk (x, u), then ϕ0k (x, u) = 0. (Otherwise ϕ0k (x, u) is undefined.) Now either (Pk (x, u) ⇐⇒ ∀yPk+1 (x, u ahyi)) or (Pk (x, u) ⇐⇒ ∃yPk+1 (x, u ahyi)) or (Pk (x, u) ⇐⇒ ∃âPk+1 (x, u ahâi)), depending on whose k turn it is to play what at move k. In the first two cases, we define ϕi+1 from k+1 k+1 hϕ0 , . . . , ϕi i as in [Mos83]. In the third case, if Pk (x, u) then k ϕi+1 (x, u) = hâ, ϕ0k+1 (x, u ahâi), . . . , ϕik+1 (x, u ahâi)i, k where â is least so that Pk+1 (x, u ahâi). (More precisely, ϕi+1 (x, u) is the ordinal of this tuple in the lexicographic order.) As in [Mos83], one can show that each ϕ~k is in fact a scale on Pk , and thus ϕ~0 is a scale on P, as

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desired.] The prewellordering ≤i induced by the ith norm of this scale on P is definable from the Pk ’s for k ≤ i by means of recursive substitution, the Boolean operations, and quantification over R × α. In particular, if the map k 7→ Pk is Σ1 (L(R), {α, R}), then so is the map i 7→≤i . The determinacy required to construct this scale on P is closely related to the definability of the scale constructed. In particular, if the map k 7→ Pk is in L(R), then the determinacy of all games on ù in L(R) suffices. In the circumstances just described we call the map x 7→ Gx a closed game representation of P. One can show directly that every Σ1 (L(R), R ∪ {R}) set admits a closed game representation, and hence a scale, in L(R). This is done in [Ste83B]. We shall take a slightly more circuitous route here, our reward being some additional information concerning Π11 games on R. e Lemma 1. Every aR Π11 set admits a closed game representation in L(R). e Proof. Let P be aR Π11 , and let T be a tree on ù × ù × ù so that e P(x) iff player I has a winning quasi-strategy in Gx , where Gx is the game on R whose payoff for player I is {hxn : n < ùi : T (x, hxn : n < ùi∗ ) is wellfounded}. We shall associate to each Gx auxilliary closed games G∗x,α for α ∈ Ord as in Martin’s proof of Π11 determinacy. A e typical run of G∗x,α has the form player I

x0 , â 0

x2 , â 1 ···

player II

x1

x3

where Xi ∈ R and âi < α for all i. Let hói : i < ùi be an enumeration of <ù ù such that if ói ⊆ ój then i ≤ j. We say player I wins the run of G∗x,α just displayed just in case whenever ói , ój ∈ T (x, hxn : n < ùi∗ ) and ói ( ój , we have âi > âj . Claim. Player I has a winning quasi-strategy in Gx iff ∃α(player I has a winning quasi-strategy in G∗x,α ). Proof. If player I has a winning quasi-strategy in G∗x,α , then player I has a winning quasi-strategy in Gx , since Gx requires less of him. So suppose S is a winning quasi-strategy for player I in Gx . Let U = {hu, vi ∈

<ù

R×

<ù

ù : lh(u) = lh(v) and u ∈ S and hx↾ lh(u), u ∗ , vi ∈ T }.

(1)

Then U is a tree on R × ù, and since S is a winning quasi-strategy for player I, U is wellfounded. Let khu, vikU be the rank of hu, vi in U if hu, vi ∈ U ,

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113

and 0 otherwise. Let α = kh∅, ∅ikU + 1. Let S0∗ = {hx0 , â0 , x1 , . . . , x2k , âk , x2k+1 i : hx0 , x1 , . . . , x2k+1 i ∈ S and ∀i ≤ k(âi = khhx0 , . . . , xℓ i, ói ikU , where ℓ = lh(ói ) − 1}, ∗

and let S be the closure of S0∗ under initial segment. It is easy to verify that S ∗ is a winning quasi-strategy for player I in G∗x,α . This proves the claim. ⊣Claim If player I has a winning quasi-strategy in Gx , let αx be the least α such that player I has a winning quasi-strategy in G∗x,α . Let ë = sup{αx : player I has a winning quasi-strategy in Gx }. Since any winning quasi-strategy for player I in G∗x,α is a winning quasi-strategy for player I in all G∗x,â with â ≥ α, the map x 7→ G∗x,ë is a closed game representation of P. The map x 7→ G∗x,ë is clearly definable over L(R) (from T and ë). ⊣ The proof of Lemma 1 easily gives the following corollary. Corollary 2. If G is a Π11 game on R and player I has a winning quasistrategy for G, then player Iehas a winning quasi-strategy for G in L(R). Thus aR Π11 = (aR Π11 )L(R) . e e Corollary 2 is the best absoluteness result for L(R) of its kind, at least if DetR (Π11 ) holds. For in that case, Corollary 6 below implies that there is a Σ11 game on R for which player I has a winning quasi-strategy in V, but no winning quasi-strategy in L(R). In proofs of determinacy from large cardinal hypotheses one associates to a complicated game G a closed game G ∗ in which one or both players make additional ordinal moves. One then uses measures on the possible additional moves to “integrate” strategies for G ∗ , and thereby show that whoever wins G ∗ wins G (and vice-versa, since G ∗ is determined). We could have proved Lemma 1 this way, but would have required the existence of R# in order to do so. The proof of Lemma 1 we did give shows that whoever wins G wins G ∗ (and vice-versa, if G is determined). In the large cardinals/determinacy problem this simpler method for proving the equivalence of G with G ∗ may give some guidance in discovering G ∗ , although integration is unavoidable in actually proving the determinacy of G. Lemma 3. aR Π11 = Σ1 (L(R), {R}). Proof. Since quantification over R is bounded, (aR Π11 )L(R) ⊆Σ1 (L(R), {R}). By Corollary 2 then, aR Π11 ⊆ Σ1 (L(R), {R}). For the other inclusion, let ϕ(v0 , v1 ) be a Σ1 formula. We must associate recursively to each real x a Π11 (x) game Gx on R so that L(R) |= ϕ[x, R] iff player I has a winning quasistrategy in Gx . Our plan is to force player I in Gx to describe a countable, wellfounded model of ZF− + V=L(R) + ϕ[x, R], which contains all the reals played in the course of Gx , and minimally so. If L(R) |= ϕ[x, R], then player

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I will be able to win Gx by describing an elementary submodel of Lα R, where α is least so that Lα R |= ZF− + ϕ[x, R]. On the other hand, if player I has a winning quasi-strategy in Gx , we shall be able to piece together the models he describes in different runs of Gx according to his strategy, and thereby produce a model of the form Lα R satisfying ϕ[x, R]. By “V=L(R)” we mean some (easily constructed) sentence ϑ in the language of set theory such that if M is a transitive model of ZF− , then M |= ϑ iff M = Lα (RM ) for some α. Player I describes his model in the language L which has, in addition to ∈ and =, constant symbols xi for i < ù. He uses xi to denote the ith real played in the course of Gx . Let us fix recursive maps m, n : {ϑ : ϑ is an L-formula} → {2n : 1 ≤ n < ù}. which are one-one, have disjoint recursive ranges, and are such that whenever xi occurs in ϑ, i < min(m(ϑ), n(ϑ)). These maps give stages sufficiently late in Gx for player I to decide certain statements about his model. Player I’s description must extend the following L-theory T . The axioms of T include ZF− + DC + V=L(R)

(2) (3)i

xi ∈ R −

ϕ(x0 , R) ∧ ∀â(Lâ (R) 6|= ZF + ϕ[x0 , R])

(4)

Finally, T has axioms which guarantee that for any model U of T , the definable closure of {xU i : i < ù} is an elementary submodel of U. Let ϑ0 (v0 , v1 , v2 ) be a Σ1 formula such that whenever M is transitive and M |= ZF− +V=L(R), ϑ0M defines the graph of a map from OrdM ×RM onto M . It is easy to construct such a formula; cf. [Ste83B, Lemma 1.4]. Now, for any L-formula ϕ(v) of one free variable, T has axioms ∃vϕ(v) =⇒ ∃v∃α(ϕ(v) ∧ ϑ0 (α, xm(ϕ) , v)),

(5)ϕ

∃v(ϕ(v) ∧ v ∈ R) =⇒ ϕ(xn(ϕ) ).

(6)ϕ

This completes the list of axioms of T . A typical run of Gx has the form player I player II

i0 , x

i 1 , x2 x1

··· x3

where for all k, ik ∈ {0, 1} and xk ∈ R. If u = hhik , x2k , x2k+1 i : k < ni is a position in Gx , then T ∗ (u) = {ϑ : ϑ is a sentence of L and n(ϑ) < n and in(ϑ) = 0},

THE EXTENT OF SCALES IN L(R)

115

and if p is a full run of Gx , T ∗ (p) =

[

T ∗ (p↾n).

n<ù

Now let p = hhik , x2k , x2k+1 i : k < ùi be a run of Gx ; we say that p is a winning run for player I iff (a) x0 = x (b) T ∗ (p) is a complete, consistent extension T such that for all i, m, n “xi (n) = m” ∈ T ∗ (p) iff xi (n) = m, and (c) There is no sequence hϑi : i < ùi of L-formulae of one free variable such that for all i, “évϑi+1 (v) ∈ évϑi (v)” ∈ T ∗ (p). In condition (c) we have used the “unique v” operator as an abbreviation: ø(évϑ(v)) abbreviates “∃v(ø(v) ∧ ∀u(ϑ(u) ⇐⇒ u = v))”. It is clear that Gx is a Π11 (x) game on R, uniformly in x; its complexity comes from the wellfoundedness condition (c). So we need only prove the following claim. Claim. L(R) |= ϕ[x, R] iff player I has a winning quasi-strategy in Gx . Proof. (⇒) Let α be least so that Lα (R) |= ZF− + ϕ[x, R]. We call a position u = hhik , x2k , x2k+1 i : k < ni honest iff (i) n > 0 =⇒ x0 = x, (ii) if we let Iu (xi ) = xi for i < 2n, then all axioms of T ∪ T ∗ (u) thereby interpreted in hLα (R), Iu i are true in the structure. It is easy to check that if u is an honest position of length n, then ∃i, x∀y (u ahi, x, yi) is honest. [If n = n(ϑ) for some sentence ϑ of L, put i = 0 iff hLα (R), Iu i |= ϑ. Otherwise let i be random. If n = m(ϕ) for ϕ an L-formula of one free variable, choose x so that hLα (R), Iuahi,x,yi i |= (5)ϕ . If n = n(ϕ), choose x for the sake of axiom (6)ϕ , and otherwise let x be random.] Now S if p is a run of Gx such that p↾n is honest for all n, and Ip = n Ip↾n , then hLα (R), Ip i |= T ∪ T ∗ (p). It follows at once that p is a winning run for player I. Thus {u : u is honest} determines a winning quasi-strategy for player I in Gx . (⇐) Let S be a winning quasi-strategy for player I in Gx . If p is a run of Gx , let Rp be the set of reals played during p, that is, let Rp = ran(Ip ). If p ∈ [S] then T ∗ (p) is consistent, and by (4)ϕ has, up to isomorphism, a unique model Mp such that every element of Mp is L-definable over Mp . By player I’s payoff condition (c), Mp is wellfounded, so we may assume that Mp is transitive. By (5)ϕ and player I’s payoff condition (b), RMp = Rp . Thus Mp = hLαp (R), Ip i for some αp such that no reals beyond those in Rp appear in Lαp (Rp ). Notice that by (3)ϕ , if q ∈ [S] and Rq = Rp , then αq = αp .

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Claim (Subclaim). If p, q ∈ [S] and Ip (xi ) = Iq (xni ) for i ≤ k, then for any formula ϑ(v0 , . . . , vk ) of the language of set theory, ϑ(x0 , . . . , xk ) ∈ T ∗ (p) iff ϑ(xn0 , . . . , xnk ) ∈ T ∗ (q). Proof. Suppose not, and let u = p↾ℓ and v = q↾ℓ where ℓ is so large that (say) ϑ(x0 , . . . , xk ) ∈ T ∗ (p↾ℓ) and ¬ϑ(xn0 , . . . , xn ) ∈ T ∗ (q↾ℓ). Since player II is free in Gx to play whatever reals he pleases, we can find s, t ∈ [S] so that u ⊆ s, v ⊆ t, and Rs = Rt . But then αs = αt . Since ϑ(v0 , . . . , vk ) involves no constant symbols xi , and since Is (xi ) = It (xni ) for i ≤ k, we have Ms |= ϑ(x0 , . . . , xk ) iff Mt |= ϑ(xn0 , . . . , xnk ). But Ms |= T ∗ (u) and Mt |= T ∗ (v), a contradiction. By the subclaim, if p, q ∈ [S] and Rp ⊆ Rq , then there is a unique elementary embedding jp,q : Lαp (Rp ) → Lαq (Rq ). The uniqueness of the embeddings implies that if p, q, s ∈ [S] and Rp ⊆ Rq ⊆ Rs , then jp,s = jq,s ◦ jp,q . Thus we can form the direct limit M of the Lαp (Rp )’s under these embeddings. Now whenever {pi : i < ù} ⊆ [S] we can find a q ∈ [S] so that Rpi ⊆ Rq for all i, and therefore M is wellfounded. So we may assume M is transitive. S Since R ⊆ {Rp : p ∈ [S]}, R ⊆ M , and so M = Lα (R) for some α. But M |= ϕ[x, R]. This completes the proof of the claim, and thereby Lemma 3. ⊣ Our proof of Lemma 3 sharpens some earlier arguments of Solovay [Sol78B] which implies, assuming ADR , that every set of reals in L(R) is aR ∆13 . e Theorem 4. Assume all games in L(R) are determined. Then the pointclasses Σ1 (L(R), {R}) and Σ1 (L(R), R ∪ {R}) have the scale property. Proof. It is enough to show Σ1 (L(R), {R}) has the scale property. Let P be Σ1 (L(R), {R}). Lemmas 1 and 3 yield a map hx, αi 7→ G∗x,α which is ∆1 (L(R), {R}) such that P(x) iff ∃α(player I has a winning quasi-strategy in G∗x,α ). Let P α (x) iff player I has a winning quasi-strategy in G∗x,α . Our slight generalisation of [Mos83], and the uniformity in its proof, yield a ∆1 (L(R), {R}) map α 7→ ø~α such that ø~α is a scale on P α for all α. Now for x ∈ P let ϕ0 (x) = min{α : P α (x)}, and ϕ (x)

ϕi+1 (x) = hϕ0 (x), øi 0

(x)i,

where of course we use the lexicographic order to identify ϕi+1 (x) with an ordinal. It is easy to check that ϕ ~ is a Σ1 (L(R), {R}) scale on P. ⊣ Combining Theorem 4 with the result of Kechris and Solovay we mentioned earlier, we have

THE EXTENT OF SCALES IN L(R)

117

Corollary 5. Assume all games in L(R) are determined. Then P admits a scale in L(R) iff P is Σ1 (L(R), {R}). Proof. One direction is Theorem 4; we prove the other for the sake of completeness. For x, y ∈ R, let C (x, y) iff L(R) |= y is ordinal definable from x. By the reflection theorem, C is Σ1 (L(R), {R}). Now ¬C has no uniformization, hence no scale, in L(R). [If D uniformizes ¬C and D ∈ L(R), then we can find x0 ∈ R so that L(R) |= D is ordinal definable from x0 . Then D(x0 , y0 ) =⇒ C (x0 , y0 ), a contradiction since ∃y¬C (x0 , y).] Now the class Γ of sets admitting scales in L(R) is a boldface pointclass. Since ¬C 6∈ Γ and C is Σ1 (L(R), R ∪ {R}), Wadge’s lemma implies that Γ ⊆ Σ1 (L(R), R ∪ {R}). ⊣ As promised, our proof of Theorem 4 gives some information about Π11 games on R. Corollary 6. If DetR (Π11 ) holds, then aR Σ11 6= (aR Σ11 )L(R) . Thus DetR (Π11 ) is false in L(R). Proof. If DetR (Π11 ) holds, then aR Σ11 is the dual of aR Π11 , so that aR Σ11 = Π1 (L(R), {R}). But (aR Σ11 )L(R) ⊆ Σ1 (L(R), {R}) by a direct computation. ⊣ Corollary 7. Assume AD. Then the following are equivalent: (a) DetR (Π11 ) (b) R# exists Proof. If R# exists, then DetR (Π11 ) follows by the argument of [Mar70A]. (AD is not necessary here.) If DetR (Π11 ) holds, then ∃A ⊆ R(A 6∈ L(R)) by Corollary 6. But in the presence of AD this last statement implies that R# exists; cf. [SVW82]. ⊣ # 1 It seems likely that DetR (Π1 ) and the existence of R are provably equivalent in ZF + DC. We have not tried to show this. We have seen that the complete Π1 (L(R), {R}) set admits no scale in L(R). On the other hand, this set admits a scale just beyond L(R): Solovay has shown that if R# exists and L(R) |= AD, then every set of reals in L(R) admits a scale each of whose norms is in L(R). (By [SVW82, Theorem 1.3.3], one needs R# in order to construct definable scales beyond those in L(R).) This is a special case of a more general result of [Mar83B]: if Γ is a reasonably closed pointclass with the scale property, DetR (Γ) holds, and aR Γ games on ù are determined, then aR Γ has the scale property. (Solovay’s result follows by taking Γ to be the class of sets which are ù · n-Π11 for some n < ù.) We e a way which avoids the shall conclude by proving part of Martin’s theorem in determinacy of games on R. Our proof generalizes the proof of Lemma 1.

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Theorem 8. Assume AD. Then whenever A admits a scale, aR A admits a scale. Proof. Let ϕ ~ be a scale on A whose norms map into κ, where κ < ϑ. Let R be the tree of ϕ ~ , that is R = {hs, t, ui ∈

<ù

ù×

<ù

ù×

<ù

κ :

lh(s) = lh(t) = lh(u) ∧ ∃x, y(A(x, y) ∧ ∀i < lh(u)(u(i) = ϕi (x, y)))}. Thus A = {(x, y) : [R(x, y)] 6= ∅}. We now obtain a “homogeneous” tree representation of ¬A in the standard way (cf. [Kec81A]). For any hs, ti ∈ <ù ù × <ù ù such that lh(s) = lh(t), let W (s, t) be the Brouwer-Kleene order of <ù κ restricted to fld(W (s, t)) = {u ∈

<ù

κ : lh(u) ≤ lh(s) ∧ (s↾ lh(u), t↾ lh(u), u) ∈ R}.

W (s, t) is a wellorder because the sequences in its field have length ≤ lh(s). By AD we have a ë > κ such that ë → (ë)ë . For any C ⊆ ë, let [C ]W (s,t) be the set of order-preserving maps from fld(W (s, t)) into C . Now let T be the S tree on ù × ù × hs,ti [ë]W (s,t) given by   T = {hs, t, hf1 , . . . , fℓ ii : lh(s) = lh(t) = ℓ ∧ ∀i ≤ ℓ fi ∈ [ë]W (S↾i,t↾i) ∧ ∀i, j ≤ ℓ(i ≤ j =⇒ fi = fj ↾W (s↾i, t↾i))}. A path through T is equivalent to a pair hx, yi of reals together with a map of R(x, y) into ë preserving the Brouwer-Kleene order on R(x, y). Thus A = {hx, yi : [T (x, y)] = ∅}. For any hs, ti ∈

<ù

ù×

<ù

ù such that lh(s) = lh(t), let

T (s, t) = {u : hs, t, ui ∈ T }. Each T (s, t) carries a canonical measure ìs,t given by the strong partition property of ë. [Fix α ≤ ë. For C ⊆ α, let C ↑ = {h ∈ α [ë] : ∃g ∈ [C ]ùα ∀â < α(h(â) = sup{g(ùâ + n) : n ∈ ù})}. Then since ë → (ë)ë , the sets of the form C ↑ for C closed and unbounded in ë are the base for an ultrafilter ìα on α [ë]. If α is the order type of W (s, t), then ìα is isomorphic in an obvious way to an ultrafilter ìs,t on [ë]W (s,t) . But an inspection of the definition of T shows that we may identify T (s, t) with W (s,t) [ë]. (See [Kec81A] for further details on this construction.)]. The measures ìs,t are compatible, in the sense that if ìs,t (X ) = 1, and i < lh(s), then ìs↾i,t↾i ({u↾i : u ∈ X }) = 1. They also have a “homogeneity” property: if [T (x, y)] 6= ∅ and ìx↾i,y↾i (Yi ) = 1 for all i < ù, then ∃f ∈ [T (x, y)]∀i(f↾i ∈ Yi ). We shall now produce a closed game representation

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of aR A. For x ∈ R and α ∈ Ord, we define a closed game G∗x,α . A typical run of G∗x,α has the form player I

x0 , â 0

x2 , â 1 ···

player II

x1

x3

where xi ∈ R and âi < α for all i. Player I wins this run of G∗x,α just in case, if we set y = hxn : n < ùi∗ , and for all i < ù pick maps Fi : T (X ↾i, y↾i) → Ord such that âi = [Fi ]ìx↾i,y↾i , then whenever i < k, ìx↾k,y↾k ({u : Fk (u) < Fi (u↾i)} = 1. The compatibility of the measures implies that player I’s payoff condition does not depend on which map Fi is chosen to represent âi in the ultrapower by ìx↾i,y↾i . Claim. x ∈ aR A iff ∃α(player I has a winning quasi-strategy in G∗x,α ). Proof. Let S ∗ be a winning quasi-strategy for player I in G∗x,α . Let S result from S ∗ by omitting the ordinals from player I’s moves. Let hxn : n < ùi ∈ [S]. Then the homogeneity property of the ìs,t ’s guarantees that [T (x, hxn : n < ùi∗ )] = ∅. Thus (x, hxn : n < ùi∗ ) ∈ A, and S witnesses that x ∈ aR A. Conversely, let S be a winning quasi-strategy for player I in the game whose payoff is {hxn : n < ùi : A(x, hxn : n < ùi∗ )}. Let U be the tree on S R × hs,ti W (s,t) [ë] given by U = {hu, vi : lh(u) = lh(v) ∧ u ∈ S ∧ hx↾ lh(u), u ∗ , vi ∈ T }. Then U is wellfounded since S is a winning quasi-strategy for player I. For u ∈ S, let Fu : T (x↾ lh(u), u ∗ ) → Ord be defined by Fu (v) = rank of hu, vi in U We can now define a winning quasi-strategy S ∗ for player I as follows: let S0∗ = {hx0 , â0 , x1 , . . . , x2k , âk , x2k+1 i : hx0 , . . . , x2k+1 i ∈ S ∧ ∀i ≤ k(âi = [Fi ]ì where Fi = Fhxn : n
120

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Alexander S. Kechris [Kec81A] Homogeneous trees and projective scales, In Kechris et al. [Cabal ii], pp. 33–74. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer, 1981. [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. Donald A. Martin [Mar70A] Measurable cardinals and analytic games, Fundamenta Mathematicae, (1970), no. LXVI, pp. 287–291. [Mar83B] The real game quantifier propagates scales, this volume, originally published in Kechris et al. [Cabal iii], pp. 157–171. Yiannis N. Moschovakis [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. [Mos81] Ordinal games and playful models, In Kechris et al. [Cabal ii], pp. 169–201. [Mos83] Scales on coinductive sets, this volume, originally published in Kechris et al. [Cabal iii], pp. 77–85. Robert M. Solovay [Sol78B] The independence of DC from AD, In Kechris and Moschovakis [Cabal i], pp. 171–184. John R. Steel [Ste83B] Scales on Σ11 -sets, this volume, originally published in Kechris et al. [Cabal iii], pp. 72– 76. John R. Steel and Robert Van Wesep [SVW82] Two consequences of determinacy consistent with choice, Transactions of the American Mathematical Society, (1982), no. 272, pp. 67–85. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES, CA 90095, USA

E-mail: [email protected] DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720, USA

E-mail: [email protected]

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DONALD A. MARTIN

§1. Introduction. The set ù ù ∩ L has, if it is countable, an implicit descriptive set-theoretic characterization as the largest countable Σ12 set [Mos80, p. 538]. In this paper, we give an explicit descriptive set-theoretic characterization of this and (under determinacy hypotheses) other related sets. A subset A of k ù × ( ù ù)m is α-Π11 , where α is an ordinal, if there are Π11 e e sets Aâ , â < α such that x ∈ A ⇐⇒ min{â : â = α ∨ x ∈ / Aâ } is odd.

If α is small, say recursive, the lightface notion, α-Π11 , has an obvious definition. A is aΓ if there is a B ∈ Γ such that x ∈ A ⇐⇒ player I has a winning strategy for Gx (B), where Gx (B) is the game in which the players cooperatively produce a y ∈ ù ù and player I wins just in case hx, yi ∈ B. Let C2n be the largest countable Σ12n subset of ù ù, for n ≥ 1. See [Mos80, p. 346]. In §2 and §3, we prove the following theorem (where ai+1Γ = a(ai Γ)): Theorem 1.1 (Projective Determinacy, PD). [ C2k = a2k−1 (ù · n)-Π11 . n∈ù #

Corollary 1.2 (0 exists). L=

[

a(ù · n)-Π11 .

n∈ù

We also use our proof to characterize 0# as the recursive join of a sequence of complete aù · n-Π11 sets. Let Σ∗0 be the class of unions of inductive (see [Mos80, p. 410]) and coinductive sets. (We take “inductive” in the lightface sense.) Let Π∗n be the class of complements of Σ∗n sets and let Σ∗n+1 be the class of projections of Π∗n sets. In §4, we prove Research partially supported by NSF grant. The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

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DONALD A. MARTIN ∗

1.3 (Det(Σn ) for all n). S Theorem ∗ n∈ù Σn .

The largest countable inductive set is

[Mos83] shows that every coinductive set admits a scale whose complexity is the join of the Σ∗n . It follows from the theorem that there is a coinductive set which has no Σ∗n scale for any n. §2. Every a2k−1 (ù · n)-Π11 set belongs to C2k . We first fix a coding of (ù · n)-Π11 sets. Let each z ∈ ù ù code, in some effective manner, a sequence hAzâ : â < ù 2 i of Π11 sets. For each z let Az,n be the ù · n-Π11 set determined by the sequence hAezâ : â < ù · ni. For each z, let Gnz be theegame with player I’s winning set Az,n . Do this coding so that {z : Gnz is a win for player I} is a complete a(ù · n)-Π11 set, uniformly in n. Lemma 2.1. For each z and each sequence ã1 , . . . , ãn of ordinals, there is an open game G∗z,ã1 ,...,ãn , played on ordinals strictly less than sup{ã1 , . . . , ãn } such that 1. The winning conditions for G∗z,ã1 ,...,ãn are definable in L[z] from z and ã1 , . . . , ãn by a definition independent of z and ã1 , . . . , ãn . 2. If ã1 < · · · < ãn are indiscernibles for L[z], then whoever has a winning strategy for G∗z,ã1 ,...,ãn has a winning strategy for Gnz . 3. There is a Π11 relation R(n, z, y) such that, if y1 , . . . , yn are codes for countable ordinals ã1 , . . . , ãn respectively, then player I wins G∗z,ã1 ,...,ãn ⇐⇒ R(n, z, hy1 , . . . , yn i). Proof. For simplicity, we suppress z and often n from the notation. Let hAâ : â < ù ·ni be the Π11 sets given by z. We can associate with each x ∈ ù ù e and each â < ù · n a linear ordering Rxâ of ù in an effective manner, such â that Rx ↾k depends only on x↾k, such that 0 is maximum in Rxâ , and such that x ∈ Aâ if and only if Rxâ is a well-ordering. Let h : ù → ù · n × ù be an effective bijection such that (a) if h(k) = hâ, ti, then â is even if and only if k is even; (b) if h(k1 ) = hâ, t1 i, h(k2 ) = hâ, t2 i, and t1 < t2 , then k1 ≤ k2 ; (c) if h(k1 ) = hù · i + j1 , 0i, h(k2 ) = hù · i + j2 , 0i, and j1 < j2 , then k1 < k2 . We play G∗ã1 ,...,ãn as follows: Player I

Player II

m0 , ̺0 m1 , ̺1 m2 , ̺2 .. .

m3 , ̺3

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123

The rules of the game are: (i) mi ∈ ù. (ii) If h(k) = hù · i + j, ti, then ̺k < ãi+1 . (iii) Let Fâ (t) = ̺k , where h(k) = hâ, ti. Fâ must embed the ordering Rxâ into the ordinals, where x = hmi : i ∈ ùi. The first player to violate the rules loses. Otherwise, player II wins. The players are thus trying to verify that the Rxâ are well-orderings. Player I is responsible for even â, player II for odd â. Condition (b) ensures that the maximum value of Fâ is played before any other values, and that Rxâ ↾t + 1 is known before Fâ (t) is played. Condition (c) ensures that Fâ (0) is played before Fä (0) whenever ä > â and â and ä are in the same ù-block. Condition (1) in the statement of the lemma is clear. Condition (3) is also easily verified, since given codes for ã1 , . . . , ãn (i.e., well-orderings of those order types), we can construe G∗ã1 ,...,ãn as an ordinary open game. Since aΣ01 = Π11 , this yields (3). By (1), it suffices to verify (2) in the special case ãi = ùi , where we may assume z # exists. Assume for definiteness that player I has a winning strategy ó ∗ for G∗ = ∗ Gù1 ,...,ùn . Let us say that player II plays well if player II obeys all rules, Fù·i+2j+1 > ãi for all i ≥ 1 and j ∈ ù, Fù·i+2j+1 (t) > Fù·i+j ′ (0) for all j ′ ≤ 2j, and Fù·i+2j+1 (t) is an indiscernible for L[z] for all i ∈ ù and j ∈ ù. We define a strategy ó for player I for Gnz by assuming player II plays well and player I plays ó ∗ . Note that, in positions in G∗ such that player II has played well, player I’s moves mi and Fâ (t) given by ó ∗ are independent of player II’s moves Fâ ′ (t) for â ′ > â. Let x be any play of G (= Gnz ) consistent with ó. Let â0 be the least â such that â = ù · n or x ∈ / Aâ . By induction, we define Fâ for â < â0 . Assume Fâ ′ ′ is defined for â < â. Suppose â is even. Let Fâ be given by letting x be played, letting player II play Fâ ′ for odd â ′ < â, letting player II play well otherwise, and letting player I play according to ó ∗ . Suppose â = ù · i + j with j odd. Let ç be the supremum of {ãi ′ : ′ i < i} ∪ {Fù·i+j ′ (0) : j ′ < j}. Let Fâ : ù → ùi+1 embed Rxâ into the indiscernibles for L[z] between ç and ãi+1 . Suppose â0 = ù · n. Then the Fâ extend x to a play of G∗ according to ∗ ó with all rules obeyed. Since this play is a win for player II, we have a contradiction. Suppose â0 is even. Let Fâ0 be given by letting x be played, letting player II play Fâ for odd â < â0 , letting player II play well otherwise, and letting player

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I play ó ∗ . Eventually Fâ0 must violate rule (iii). Extending to a position where player II has played well, we get a contradiction. Thus â0 is odd, and so x is a win for player I. Hence ó is a winning strategy for player I for G. ⊣ Lemma 2.2 (0# exists). If S is a aù · n-Π11 set of integers, then S ∈ L = C2 . Proof. There is recursive f : ù →

ù

ù such that

m ∈ S ⇐⇒ player I wins Gnf(m) . Let B = {S ′ : ∃ã1 , . . . , ãn < ù1 ∀m(player I wins G∗f(m),ã1 ,...,ãn ⇐⇒ m ∈ S ′ )}. By part (2) of Lemma 2.1, S ∈ B. It is clear that S ′ ∈ B implies that S ′ ∈ L, but we prove directly that B is a countable Σ12 set. By part (3) of Lemma 2.1, S ′ ∈ B if and only if there are codes y1 , . . . , yn for countable ordinals such that for all m, we have R(n, f(m), y1 , . . . , yn ) ⇐⇒ m ∈ S ′ . Thus B ∈ Σ12 . Since B has a Σ12 well-ordering, B is countable.

⊣

Lemma 2.3 (PD). For i ∈ ù, every a2i+1 (ù · n)-Π11 set S ⊆ ù belongs to C2i+2 . Proof. There is a recursive function f such that m ∈ S if and only if player I wins the following game G(m) of length ù · 2i: x1 , . . . , x2i is a win for player I just in case player I has a winning strategy for Gnf(m,x1 ,...,x2i ) . Given degrees of unsolvability d1 , . . . , d2i , consider the game G(m, d1 , . . . , d2i ): Player I Player II ó1 x1 ó2 .. .

x2

ó2i x2i ój must be a strategy for player I for an ordinary game on ù and must satisfy deg(ój ) ≤ dj . xj must be a play consistent with ój such that deg(xj ) ≤ dj . Player I wins if and only if x1 , . . . , x2i is a win for player I in Gnf(m,x1 ,...,x2i ) . Sublemma 2.3.1. For almost every d1 , . . . , d2i (with respect to the iterated product of the usual measure on the degrees), for all m player I wins G(m) ⇐⇒ player I wins G(m, d1 , . . . , d2i ).

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Proof of Sublemma. Suppose that player I has a winning strategy ó for G(m). For 1 ≤ j ≤ 2i and degrees d1 , . . . , dj−1 , let dj ∈ A(d1 , . . . , dj−1 ) just in case, for all hx1 , . . . , xj−1 i consistent with ó such that deg(xj ′ ) ≤ dj ′ for all j ′ < j, the degree of ó restricted to the next ù moves from the position hx1 , . . . , xj−1 i is ≤ dj . Let d1 , . . . , d2i ∈ A just in case dj ∈ A(d1 , . . . , dj−1 ) for 1 ≤ j ≤ 2i. Clearly ó gives a winning strategy for G(m, d1 , . . . , d2i ) whenever hd1 , . . . , d2i i ∈ A. Since A has iterated product measure 1, we have proved the first half of the sublemma. A similar argument shows that if II has a winning strategy for G(m) then II has a winning strategy for G(m, d1 , . . . , d2i ) for almost every hd1 , . . . , d2i i. ⊣ For each d1 , . . . , d2i , and each ã1 , . . . , ãn < ù1 , let G(m, d1 , . . . , d2i , ã1 , . . . , ãn ) be the game played like G(m, d1 , . . . , d2i ) which player I wins just in case player I wins G∗f(m,x1 ,...,x2i ),ã1 ,...,ãn . Let B = {S ′ : for almost all d1 , . . . , d2i there exist ã1 , . . . , ãn < ù1 such that for all m player I wins G(m, d1 , . . . , d2i , ã1 , . . . , ãn ) ⇐⇒ m ∈ S ′ }. We first note that B is (projectively) well-orderable. For S ′ ∈ B, let φS ′ (d1 , . . . , d2i ) be the lexicographically least hã1 , . . . , ãn i such that for all m, player I wins G(m, d1 , . . . , d2i , ã1 , . . . , ãn ) if and only if m ∈ S ′ , if it is defined. It is defined for almost all d1 , . . . , d2i . If φS1′ (d1 , . . . , d2i ) = φS2′ (d1 , . . . , d2i ) almost everywhere, then S1′ = S2′ . Next we show that B is Σ12i+1 : S ′ ∈ B if and only if ∃d1 ∀d1′ ≥ d1 ∀d2 ∃d2′ ≥ d2 . . . ∀d2i ∃d2i′ ≥ d2i ∃y1 , . . . , ym y1 , . . . , ym are codes for countable ordinals and for all m  ∃ó1 with deg(ó1 ) ≤ d1′ ∀x1 with deg(x1 ) ≤ d1′ . . . ∃ó2i with deg(ó2i ) ≤ d2i′ ∀x2i with deg(x2i ) ≤ d2i′
 R(n, f(m, x1 , . . . , x2i ), y1 , . . . , yn ) ⇐⇒ m ∈ S ′ . Finally we show that S ∈ B. Let d1 , . . . , d2i be such that for all m, player I wins G(m) if and only if player I wins G(m, d1 , . . . , d2i ). By Sublemma 2.3.1, almost all hd1 , . . . , d2i i have this property. Now let ã1 < · · · < ãn be countable indiscernibles for L[w] with deg(w) ≥ dj for j = 1, . . . , 2i. By Lemma 2.1, if x1 , . . . , x2i is the result of a play of G(m, d1 , . . . , d2i ), then x1 , . . . , x2i is a win for player I in G(m, d1 , . . . , d2i ) ⇐⇒ x1 , . . . , x2i is a win for player I in G(m, d1 , . . . , d2i , ã1 , . . . , ãn ). Now for all m, we have m ∈ S if and only if player I wins G(m) if and only if player I wins G(m, d1 , . . . , d2i , ã1 , . . . , ãn ). ⊣

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§3. Characterization of C2i and of 0# . Lemma 3.1 (Moschovakis; Sufficient determinacy). Suppose every set in Γ admits a very good scale hφi : i ∈ ùi such that each φi is a Γi norm. 1. Every set in aΓ admits a very good scale høi : i ∈ ùi such that each øi is a aΓi norm. 2. Every Γ game won by player I has a winning strategy ó such that ó restricted to positions of length i belongs to aΓi . An inspection of the proof of [Mos80, Theorem 6E.1] and [Mos80, Theorem 6E.15] will reveal that they prove the lemma. Lemma 3.2 ([Ste83B]). Every Σ11 set admits a very good scale hφi : i ∈ ùi such that each φi is (ù · (i + 1))-Π11 . Lemma 3.3 (PD). Every Π12i+1 (resp. Σ12i+2 ) game won by player II has a winning strategy ô such that ô restricted to positions of length n is a2i+1 (ù · (n + 1))-Π11 (resp. a2i+2 (ù · (n + 1))-Π11 ). Proof. Lemma 3.1 and Lemma 3.2 allow one to prove this lemma by induction. ⊣ Lemma 3.4. For each i ≥ 1, every member of C2i is a2i−1 (ù · n)-Π11 for some n. Proof. Let A be any countable Σ12i subset of ù 2. Let A = {x : ∃y(hx, yi ∈ B)} for B ∈ Π12i−1 . Consider the following game: Player I Player II m0 , s0 å0 m1 , s1 å1 .. . Each mj must be a natural number. Each sj must be an element of <ù 2 with sj+1 ⊇ sj aåj . åj must be S 0 or 1. If all rules are obeyed, player I wins just in case hx, yi ∈ B where x = j∈ù sj and y = hmi : i ∈ ùi. If player I has a winning strategy, then A has a perfect subset, contradicting its countability. By Lemma 3.3, let ô be a winning strategy for player II such that ô restricted to positions of length n is a2n−1 (ù · (n + 1))-Π11 . Suppose x, y ∈ B. Then there is a position p in our game with last move åj−1 (we permit p to be the initial position, i.e., j = 0) such that 1. mj ′ = y(j ′ ) for all j ′ < j;

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2. sj−1 aåj−1 ⊆ x (if j > 0); 3. for any sj such that sj−1 aåj−1 ⊆ sj ⊆ x, if player I plays hy(j), sj i at p, then ô calls for player II to play åj = 1 − x(lh(sj )). If no such position existed, then there would be a play of the game consistent with ô with result hx, yi. For such a position p, x is clearly recursive in ô restricted to positions of length 2j + 1. Thus every x ∈ A belongs to a2i−1 (ù · n)-Π11 for some n. ⊣ S Theorem 3.5 (PD). C2i = n∈ù a2i−1 (ù · n)-Π11 . S Corollary 3.6 (0# exists). ù ù ∩ L = n∈ù a(ù · n)-Π11 . Theorem 3.7 (0# exists). Let Bn be a complete a(ù · n)-Π11 subset of ù, uniformly in n. 0# is recursively isomorphic with {hn, mi : m ∈ Bn } = B. Proof. There is a one-one recursive function f such that hn, mi ∈ B if and only if player I has a winning strategy for G∗f(n,m),ã1 ,...,ãn , for any indiscernibles ã1 < · · · < ãn for L. This follows from Lemma 2.1, part (2). By Lemma 2.1, part (1), B is one-one reducible to 0# . For the other direction, let φ(x1 , . . . , xn ) be any formula in the language of set theory. We play a game Gφ as follows: player I chooses (dividing up ù) xâ ∈ ù ù for each â < ù · (n + 1). Player II chooses yâ ∈ ù ù for each â < ù · (n + 1). If some xâ or yâ is not a code for a countable ordinal, then player I wins if, for the least such â, xâ is a code for a countable ordinal, and player II wins otherwise. If xâ (resp. yâ ) codes an ordinal, let |xâ | (resp. |yâ |) be the ordinal coded. If all |xâ | and |yâ | are defined, let ãi = sup{max{|xù·i+j |, |yù·i+j |} : j ∈ ù}. Player I wins just in case Lãn |= φ[ã1 , . . . , ãn−1 ]. Gφ is (ù · (n + 1))-Π11 , so it is enough to prove that player I wins Gφ if and only if pφq ∈ 0# . Assume for definiteness that player I has a winning strategy ó for Gφ . By boundedness, there is a closed, unbounded C ⊆ ù1 , such that, for all â < ù · (n + 1), all α ∈ C , and all plays consistent with ó, if for all â ′ < â, the ordinal |yâ | is defined and sup{|yâ ′ | : â ′ < â} < α, then |xâ | < α. Let ã1 < · · · < ãn be indiscernibles for L which belong to C . Let player II play yâ such that each h|yù·i+j | : j ∈ ùi is an increasing sequence with limit ãi . If player I plays ó, then ãi = ãi and so Lãn |= φ[ã1 , . . . , ãn−1 ] and so ⊣ L |= φ[ã1 , . . . , ãn−1 ], i.e., pϕq ∈ 0# . §4. The largest countable inductive set. Lemma 4.1 (Inductive Determinacy). If A ⊆ ù is Σ∗n , then A belongs to the largest countable inductive set.

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Proof. Let A be Π∗n if n is even and Σ∗n if n is odd. Then m ∈ A ⇐⇒ (Q1 x1 )(Q2 x2 ) . . . (∃xn )(hm, x1 , . . . , xn i ∈ B ∩ C ) where the quantifiers are alternating and B is inductive and C is coinductive. By [Mos80, p. 419], let ϕ be an inductive norm on B and let ø be an inductive norm on ¬C . For degrees d1 , . . . , dn , let Ed1 ,...,dn = {S ⊆ ù : (Q1 x1 of degree ≤ d1 )(Q2 x2 of degree ≤ d2 ) . . . (∃xn of degree ≤ dn ) ∃ã∀m(m ∈ S ⇐⇒ (ϕ(m, x1 , . . . , xn ) < ã ∧ ø(m, x1 , . . . , xn ) 6< ã))}. Since the inductive sets are closed under number quantification, Ed1 ,...,dn is inductive uniformly in reals x1 , . . . , xn of degrees d1 , . . . , dn respectively. Let E = {S : For almost all (d1 , . . . , dn ), S ∈ Ed1 ,...,dn }. E is inductive, since S ∈ E just in case ∃d1 ∀d1′ ≥ d1 ∀d2 ∃d2′ ≥ d2 . . . Qn∗ dn′ ≥ dn S ∈ Ed1′ ,...,dn′ , and the inductive sets are closed under quantification over ù ù. E is (inductively) well-orderable, since the function ϕS (d1 , . . . , dn ) = the least ã which witnesses S ∈ Ed1 ,...,dn embeds E in the ultrapower of the ordinals by the iterated product measure (where we only need the measure defined on inductive sets). Let us finally show that A ∈ E. For almost all (d1 , . . . , dn ), (Q1 x1 of degree ≤ d1 ) . . . (∃n xn of degree ≤ dn ) (∀m)((m, x1 , . . . , xn ) ∈ B ∩ C ⇐⇒ m ∈ A). Let ã(m, d1 , . . . , dn ) = sup{ϕ(m, x1 , . . . , xn ) : (m, x1 , . . . , xn ) ∈ A ∧ ∀i ≤ n deg(xi ) ≤ di }. Let ã(d1 , . . . , dn ) = sup{ã(m, d1 , . . . , dn ) : m ∈ ù}. ⊣ ã(d1 , . . . , dn ) witnesses that A ∈ Ed1 ,...,dn . Lemma 4.2 (Determinacy for all Σ∗n games). Every member of any countable inductive set is Σ∗n for some n. Proof. By [Mos83] every coinductive set admits a very good scale hϕi : i ∈ ùi such that ϕi is Σ∗1 . By Lemma 3.3, every inductive game won by player II has a winning strategy ô such that ô restricted to positions of length i is Σ∗j for some j. Using a game like that in the proof of Lemma 3.4, except that there are no mi , we see that every member of a countable inductive set is Σ∗n for some n. ⊣ S Theorem 4.3. The largest countable inductive set is n∈ù Σ∗n .

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Corollary 4.4 (Determinacy of all Σ∗n games). There is a coinductive set which does not admit a Σ∗n scale for any n. e Proof. For each x, let Cx be the largest countable set inductive in x. Let C = {hx, yi : y ∈ Cx }. ¬C is coinductive. Suppose ¬C admits a Σ∗n scale. This scale is Σ∗n in x for some x. Thus ¬Cx has a member ∆∗n in ex. This e relativization of the theorem. contradicts the ⊣ The results of this section hold for wider classes than the inductive sets. Suppose Γ has the scale property and is closed under trivial operations and integer and real quantification. With the obvious definitions, Lemma 4.1 holds for Γ (by the same proof). Hence Corollary 4.4 holds for Γ. Lemma 4.2 does not hold in general. We must replace Σ∗n by, roughly speaking, the e set. nth norm on the largest countable Γ REFERENCES

Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Yiannis N. Moschovakis [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. [Mos83] Scales on coinductive sets, this volume, originally published in Kechris et al. [Cabal iii], pp. 77–85. John R. Steel [Ste83B] Scales on Σ11 -sets, this volume, originally published in Kechris et al. [Cabal iii], pp. 72– 76. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES, CA 90095, USA

E-mail: [email protected]

SCALES IN L(R)

JOHN R. STEEL

§0. Introduction. We now know the extent of scales in L(R): (Σ21 )L(R) sets e admit (Σ21 )L(R) scales, while properly (Π21 )L(R) sets admit no scales whatsoever e e in L(R). It follows that (∆21 )L(R) sets admit (∆21 )L(R) scales, but this is by no e e means a local result, in that the simplest possible scale on a given (∆21 )L(R) e shall set may be substantially more complicated than the set itself. Here we consider the problem of finding scales of minimal complexity on sets in L(R), and obtain a fairly complete solution. Given a set A in L(R), we shall identify by means of reflection properties of the Levy hierarchy for L(R) the first level Σn (Lα (R)) of this hierarchy at which a scale on A is definable. This level e occurs very near the least α such that A ∈ Lα (R) and for some Σ1 formula ϕ(v) and real x, Lα+1 (R) |= ϕ [x] while Lα (R) |= ¬ϕ [x]. That is, in L(R) the construction of new scales is closely tied to the verification of new Σ1 statements about reals. Scales are important in Descriptive Set Theory because they provide the only known general method which will take arbitrary definitions in a given logical form of sets of reals, and produce definitions of members of those sets. This is something a descriptive set theorist will often want to do. It is a pleasing consequence of our work and the earlier work upon which it builds that there is no better general method in L(R). We shall see that there are no simpler uniformizations of arbitrary Σn (Lα (R)) relations on reals than those e given by scales. Our work knits together earlier work of Kechris and Solovay, Martin [Mar83], Martin-Steel [MS83], and Moschovakis [Mos83]. We shall credit this work in the appropriate places. What is new is our systematic use of the Levy hierarchy for L(R), especially its reflection properties and fine structure. The paper is organized as follows. In §1 we exposit rather carefully the basic fine structure theory of L(R). Although there is nothing really new here, we have included this section as a service to the scrupulous reader. In §2 we present the heart of our analysis of the complexity of scales in L(R). §3 is devoted to the one case in this analysis not covered by §2; we have isolated this case because of it is technically more involved than the others, and the The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

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casual reader might want to skip it. Finally, in §4 we refine the results of §§2 and 3 slightly, and use these results to prove some theorems concerning Suslin cardinals and the pointclass S(κ) of κ-Suslin sets. Some preliminaries and notation: Except in §4, we work in ZF + DC, and state our additional determinacy hypotheses as we need them. (This is done chiefly as a service to the readers and authors of [KW83], who must keep close watch on the determinacy we assume in Theorems 2.1 and 3.7.) We let R = ù ù, the Baire space, and call its elements reals. Variables z, y, x, w, . . . range over R, while α, â, ã, ä, . . . range over the class Ord of ordinals. If 0 ≤ k ≤ ù and 1 ≤ ℓ ≤ ù, then k ù × ℓ (ù ù) is recursively homeomorphic to R, and we sometimes tacitly identify the two. A pointclass is a class of subsets of R closed under recursive substitutions; a boldface pointclass is a pointclass closed under continuous substitutions. If Γ is a pointclass, then Γ˘ = {R \ A : A ∈ Γ} is the dual of Γ, and ∃R Γ = {∃R A : A ∈ Γ} where ∃R A = {x : ∃y(hx, yi ∈ A)} and ∀R Γ = (∃R Γ)˘. By “Det(Γ)” we mean the assertion that all games whose payoff set is in Γ are determined. For whatever else we use from Descriptive Set Theory, and in particular for the notions of a scale and of the scale property, we refer the reader to [Mos80]. Our general set theoretic notation is standard. The least ordinal satisfying P is denoted by ìα P(α). For any set X , X <ù is the set of finite sequences of elements of X , [S]<ù is the set of finite subsets of X , and ℘(X ) is the set of all subsets of X . Vα is the set of sets of rank less than α. If X ∪ {Vù+1 } ⊆ M , where M is any set, then Σn (M, X ) is the class of relations on M definable over (M, ∈) by a Σn formula from parameters in X ∪ {Vù+1 }. Thus we are always allowed Vù+1 itself S (not necessarily its elements) as a parameter in definitions. Σù (M, X ) = n<ù Σn (M, X ). We write “Σn (M )” for “Σn (M, ∅)”, and “Σn (M )” for “Σn (M, M )”. Similar conventions apply to the Πn and ∆n e e X notations. If X ∪ {V ù+1 } ⊆ M ⊆ N , then “M ≺n N ” means that for all <ù a~ ∈ (X ∪ {Vù+1 }) and all Σn formulae ϕ, M |= ϕ [~ a ] iff N |= ϕ [~ a ]. We write “M ≺n N ” for “M ≺M N ”. n In the sequel we shall sometimes refer to “the pointclass Σn (M, X )”, or assert “Σn (M, X ) has the scale property”. In such contexts we are actually referring to Σn (M, X ) ∩ ℘(R); the context should make this clear. §1. The fine structure of L(R). This section contains a straightforward generalization to L(R) of Jensen’s fine structure theory of L. Although the reader could no doubt supply the generalization himself, we have assumed he would rather not, and so have given a fairly complete exposition. ZF + DC will suffice for the results of this section; no determinacy is required. Because it is convenient, we shall use Jensen J-hierarchy for L(R). We presume the basic facts about rudimentary functions set forth in [Jen72]. Let

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rud(M ) be the closure of M ∪ {M } under the rudimentary functions. By [Jen72, Corollary 1.7], rud(M ) ∩ ℘(M ) = Σù (M ) for transitive sets M . Now e let J1 (R) = Vù+1 = {a : rank(a) ≤ ù}

Jα+1 (R) = rud(Jα (R)), for α > 0, and Jë (R) =

[

Jα (R), for ë limit.

α<ë

S Of course, L(R) = α∈Ord Jα (R). For all α ≥ 1, Jα (R) is transitive and Jα+1 (R) ∩ ℘(Jα (R)) = Σù (Jα (R)). Thus Jα (R) ∈ Jα+1 (R) for α ≥ 1, and Jα (R) ⊆ Jâ (R) for 1 ≤ αe ≤ â. If α > 1, rank(Jα (R)) = Ord ∩Jα (R) = ùα. It is useful to refine this hierarchy. Recall from [Jen72] the rudimentary functions F1 , . . . , F8 from which all rudimentary functions can be generated by composition. Let F9 (a, b) = ha, bi, F10 (a, b) = a[b] = {c : hb, ci ∈ a}, and F11 (a, b, c) = ha, b, ci. Define S(M ) = M ∪ {M } ∪

11 [

Fi [M ∪ {M }].

i=1

Then if M is transitive, so is S(M ) (which is why we included F9 , F10 , and F11 ); moreover, [ rud(M ) = S n (M ). n<ù

Now let Sù (R) = J1 (R), Sα+1 (R) = S(Sα (R)) for α ≥ ù, and Së (R) =

[

Sα (R) for ë limit.

α<ë

Clearly the Sα (R)’s are transitive and cumulative, Sα (R) ∈ Sα+1 (R), and Sùα (R) = Jα (R) for all α. Lemma 1.1. The sequences hSã (R) : ã < ùαi and hJâ (R) : â < αi are uniformly Σ1 (Jα (R)) for α > 1. Proof. The reason is that the two sequences are defined by local Σ0 recursions. Notice that
M = Sã (R) = ∃f Φ(f) & f(ã) = M ,

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where Φ(f) ⇐⇒



f is a function & dom f ∈ Ord & f(ù) = Vù+1 &


∀α ∈ dom f α + 1 ∈ dom f ⇒ f(α + 1) = S(f(α)) & [
 ∀ë ∈ dom f ë limit ⇒ f(ë) = f(α) . α<ë

The formula Φ is Σ0 since rudimentary functions have Σ0 graphs and we are always allowed Vα+1 as a parameter. So it is enough to show that for ã < ùα
M = Sã (R) ⇐⇒ ∃f ∈ Jα (R) Φ(f) & f(ã) = M . That is, we must show that hSä (R) : ä ≤ ãi ∈ Jα (R) for all ã and α such that ã < ùα. This can be proved by induction on ã. For limit ã, say ã = ùâ, we use the fact that hSä (R) : ä < ãi is Σ1 (Jâ (R)) by induction hypothesis. A similar argument shows that hùâ : â < αi is uniformly Σ1 (Jα (R)), and therefore hJâ (R) : â < αi is uniformly Σ1 (Jα (R)). ⊣ In a similar vein, Lemma 1.2. There is a Π2 sentence ϑ such that for all transitive sets M with Vù+1 ∈ M ,
M |= ϑ iff ∃α > 1 M = Jα (R) . Proof. Let Φ(f, u) be the Σ0 formula resulting from the Σ0 formula Φ(f) of Lemma 1.1 by replacing “Vù+1 ” with the variable u. Let
ø(u) ⇐⇒ ∀a∃f∃ã ≥ ù Φ(f, u) & a ∈ f(ã) & ∀ã ∈ Ord ∃â ∈ Ord(ã < â) &


∀ã ∈ Ord ∃f Φ(f, u) & ã ∈ dom f .

Then ø is Π2 , and it is easy to check that for transitive M with Vù+1 ∈ M ,
M |= ø [Vù+1 ] iff ∃α > 1 M = Jα (R) . The desired sentence ϑ is therefore
∀u ∀x(x ∈ u ⇐⇒ rank(x) ≤ ù) → ø(u) . ⊣ We shall call the Π2 sentence ϑ provided by Lemma 1.2 “V=L(R)”. Corollary 1.3. If M ≺1 Jα (R) and Vù+1 ∈ M , then M ∼ = Jâ (R) for some â ≤ α. Of course, if there is any point to this paper there is no definable wellorder of Jα (R). The next lemma is as much as we can get in the direction of such a wellorder.

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Lemma 1.4. There are uniformly Σ1 (Jα (R)) maps fα such that fα : [ùα]<ù × R ։ Jα (R). Proof. Fix a rudimentary function h such that for all finite F ⊆ Ord, h[{F } × R] = {hG1 , G2 , G3 i : Gi ⊆ F for 1 ≤ i ≤ 3}. It is easy to construct such an h. We now define by induction on ã ≥ ù functions gã : [ã]<ù × R ։ Sã (R) such that gã ⊆ gä if ã ≤ ä. We leave gù to the reader’s discretion. If ë is a limit, let [ gë = gâ . â<ë

Finally, suppose gã is given. Let F and x be given, and set gã+1 (F, x) = gã (F, x) if ã 6∈ F . If ã ∈ F , let h(F − {ã}, (x)0 ) = hG1 , G2 , G3 i, and for 1 ≤ i ≤ 3, let ( gã (Gi , (x)i ) if (x)4 (i) = 0 ai = Sã (R) if (x)4 (i) 6= 0. Let also j ∈ {1, . . . , 11} be such that j = (x)4 (4) (mod 11). Then we set ( Fj (a1 , a2 ) if j 6= 11 gã+1 (F, x) = Fj (a1 , a2 , a3 ) if j = 11. By an easy induction we have gã : [ã]<ù × R ։ Sã (R) for all ã. But now notice that there is a rudimentary function G such that gã+1 = G(gã , Sã (R)) for all ã. By the proof of Lemma 1.1, hgã : ã < ùαi is uniformly Σ1 (Jα (R)). Thus the desired fα is given by [ fα = gã . ã<ùα

[Jen72, Lemma 2.10] implies that for all α there is a Σ1 (Jα (R)) map of α e J (R). However, onto [ùα]<ù , and therefore a Σ1 (Jα (R)) map of ùα ×R onto α e the parameters involved in the Σ1 definitions of these map, and the consequent lack of uniformity in their definitions, make the maps described in Lemma 1.4 more useful. One can identify finite sets of ordinals with descending sequences of ordinals. The tree of all such descending sequences is wellfounded, and so its Brouwer-Kleene order is a wellorder. This gives us a wellorder of [Ord]<ù . More explicitly, for F, G ∈ [Ord]<ù let
F ≤BK G iff ∃α ∈ G (G = F \ α) ∨ max(G △ F ) ∈ G .

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Then ≤BK is a Σ0 wellorder of [Ord]<ù . ⊣ We begin now to make our way toward the Σn selection theorem. One central feature of fine structure theory is that it allows us to form Skolem hulls in the most effective way possible. This involves Σn selection (or uniformization). Now in L(R) we cannot hope to select from arbitrary set in a definable way; for example, the relation R(α, x) ⇐⇒ α < ù1 & x ∈ R & x codes α cannot be uniformized at all in L(R). However, we shall throw all of R into the hulls we build anyway, so that in view of Lemma 1.4 it suffices to select effectively from subsets of [Ord]<ù . Here we have a chance. Definition 1.5. Jα (R) satisfies Σn selection iff whenever R ⊆ Jα (R) × [ùα]<ù is Σn (Jα ), there is a Σn (Jα (R)) set S ⊆ R such that e e
∀a ∃F R(a, F ) → ∃!F S(a, F ) . Definition 1.6. Let h : Jα (R) × R → Jα (R) be a partial Σn (Jα (R)) map. e (R), {a}) and Then h is a Σn Skolem function for Jα (R) iff whenever S is Σn (J α S 6= 0, then ∃x ∈ R (h(a, x) ∈ S). Recall [Jen72, Corollary 1.13], according to which the satisfaction relation restricted to Σn formula is uniformly Σn (M ) for transitive, rudimentarily closed M . Lemma 1.7. If Jα (R) satisfies Σn selection, then there is a Σn Skolem function for Jα (R). Proof. Let fα be the map given by Lemma 1.4, and let hϕi : i < ùi be an enumeration of the Σn formulae of two free variables. Let R(a, x, F ) iff Jα (R) |= ϕx(0) [a, fα (F, ëi.x(i + 1))]. Since R is Σn (Jα (R)), we have a partial Σn (Jα (R)) map g uniformizing R. Let e
h(a, x) = fα g(a, x), ëi.x(i + 1) . It is easy to check that h is the desired Skolem function.

⊣

Definition 1.8. Let X ⊆ M and 1 ≤ n ≤ ù. Then HullM n (X ) = {a ∈ M : {a} is Σn (M, X )}. We write “Hullαn (X )” for “HullJnα (R) (X )”. The connection between these hulls and Σn Skolem functions is explained in the next lemma. Lemma 1.9. Let H = Hullαn (X ) where n < ù and R ⊆ X . Suppose that for some p ∈ H there is a Σn Skolem function for Jα (R) which is in fact Σn (Jα (R), {p}). Then (a) X ≺n Jα (R),

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(b) H = h[X <ù ] for some partial Σn (H ) map h, and e then H = h[R] for some partial Σ (H ) (c) if X = Y ∪ R where Y is finite, n e map h. Proof. (a) Notice that if {c} is Σn (Jα (R), H ), then c ∈ H . Now let a1 , . . . , an ∈ H and suppose Jα (R) |= ϕ [a1 , . . . , an , b] where ϕ is Σn . We want to see that Jα (R) |= ϕ [a1 , . . . , an , c] for some c ∈ H . Let h˜ be a Σn Skolem function for Jα (R) which is Σn (Jα (R), {p}) with p ∈ H . Then for some ˜ 1 , . . . , an i, x) then Jα (R) |= ϕ [a1 , . . . , an , c]. Since x ∈ R, if we set c = h(ha {c} is Σn (Jα (R), {a1 , . . . , an , x, p}), c ∈ H . (b) Let h˜ be as in (a), and set ˜ 1 , . . . , an−1 , pi, an ). h(ha1 , . . . , an i) = h(ha One can easily check that h[X <ù ] = H . (c) This follows easily from (b). Incidentally, we don’t need Σn selection to show that for R ⊆ X ,

⊣

Hullαn+1 (X ) ≺n Jα (R) for all n < ù, and therefore Hullαù (X ) ≺ù Jα (R). The crude Skolem functions obtained by selecting the BK-least F in the proof of Lemma 1.7 will suffice for this. Lemma 1.10. For all α > 1, Jα (R) satisfies Σ1 selection. Proof. Let R ⊆ Jα (R) × [ùα]<ù , and


R(a, F ) iff ∃b Jα (R) |= ϕ [a, F, b, p] ,

where ϕ is Σ0 and p ∈ Jα (R). Define  S(a, F ) iff ∃ã < ùα ∃b ∈ Sã (R) ϕ(a, F, b, p) & ∀ä < ã∀b ∈ Sä (R)¬ϕ(a, F, b, p) &
 ∀G ∈ Sã (R)∀b ∈ Sã (R) G ≤BK F → ¬ϕ(a, G, b, p) . By Lemma 1.1, S is Σ1 (Jα (R), {p}), and clearly S uniformizes R. ⊣ 1 Lemma 1.10 fails for α = 1 since there is a Σ1 subset of R × {0, 1} with no Σ11 uniformization. Before going on to Σn selection, we make some simple use of our ability to construct Σ1 Skolem hulls. Lemma 1.11. Suppose that α > 1, and for no â < α do we have Jâ (R) ≺R 1 Jα (R). Then (a) There is a Σ1 (Jα (R)) partial map h : R ։ Jα (R), (b) If â < α, then there is a total map h : R ։ Jâ (R) such that h ∈ Jα (R).

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Proof. (a) Let H = Hullα1 (R). From the proofs of Lemmas 1.7 and 1.10, we see that Jα (R) has a Σ1 Skolem function which is (lightface) Σ1 (Jα (R)). By Lemma 1.9 then, H ≺1 Jα (R) and H is the image of R under a Σ1 (H ) map ˜ Let ð : H ∼ h. = Jâ (R) be the collapse map. Since ð↾(R ∪ {R}) is the identity, ˜ Jâ (R) ≺R J 1 α (R), and therefore â = α. But then ð[h] is the desired h. (b) Suppose â < α. By increasing â if necessary we may assume that for some Σ1 formula ϕ and real x, Jâ+1 (R) |= ϕ [x] but Jâ (R) 6|= ϕ [x]. Let Sùâ+n |= ϕ [x]. Now rudimentary functions are simple [Jen72, Lemma 1.2], and so we can find a Σ0 formula ø such that for any transitive M and x ∈ M , ø(M, x) ⇐⇒ S n (M ) |= ϕ [x]. Clearly, there is a first order formula ϑ such that for any transitive M and x∈M
M |= ϑ [x] ⇐⇒ ø(M, x). Let ϑ be Σn . Let H := Hulln+1 (R). Then H ≺n Jâ (R). Let ð : H = Jã (R) be the collapse map. Since ð(x) = x and ð(R) = R, Jã (R) |= ϑ [x], and so ã = â. It is easy to get a Σù (H ) map of R onto H , which then collapses to e J (R). This map belongs to J (R). a Σù (Jâ (R)) map of R onto ⊣ α â e 2 Let Θ be the least ordinal not the surjective image of R, and ä 1 the least ordinal not the image of R under a surjection f such that {hx, yie : f(x) ≤ f(y)} is ∆21 . A Skolem hull argument like those just given shows that ΘL(R) = e ìα [℘(R) ∩ L(R) ⊆ Jα (R)]. For (ä 21 )L(R) we have e Lemma 1.12. Let ó be least such that Jó (R) ≺R 1 L(R). Then (Σ21 )L(R) = Σ1 (Jó (R)) ∩ ℘(R) e e (∆21 )L(R) = Jó (R) ∩ ℘(R) e2 L(R) = ó. (ä 1 ) e Proof. (a) Clearly, any (Σ21 )L(R) set of reals is Σ1 (L(R), R), and therefore e let A be a Σ (J (R)) set of reals. By Lemma Σ1 (Jó (R), R). For the converse, 1 ó e 1.11, A is Σ1 (Jó (R), R). Let (a) (b) (c)

x ∈ A ⇐⇒ Jó (R) |= ϕ [x, y] ⇐⇒ L(R) |= ϕ [x, y]

where ϕ is Σ1 . Since Jó (R) is the surjective image of R under a map in L(R), x ∈ A ⇐⇒ ∃E ∈ L(R) M = (R, E) is a wellfounded extensional model
of V = L(R) & RM collapses to R & M |= ϕ [x M , y M ] , where we use x M and y M for the inverse images of x and y under the collapse of M. Thus A is (Σ21 )L(R) . e

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(b) By Lemma 1.11, Jó (R) ≺1 L(R), and so Jó (R) is admissible. If A is (∆21 )L(R) , then A is ∆1 (Jó (R)) by (a), and so A ∈ Jó (R) by admissibility. e e (c) (ä 21 )L(R) ≤ ó by (b) and the fact that Jó (R) is admissible. But if e â < ó, then Lemma 1.11 (b) guarantees a map h : R ։ â in Jó (R). Since {hx, yi : h(x) ≤ h(y)} ∈ Jó (R), it is (∆21 )L(R) and therefore â < (ä 21 )L(R) . e e ⊣ Thus ó ≤ (ä 21 )L(R) . e 2 L(R) and the Lemma 1.2 indicates an analogy between the pointclass (Σ1 ) e by the result class Σ12 ∩ ℘(ù) = (Σ12 )L ∩ ℘(ù). This analogy is strengthened of [MS83] that (Σ21 )L(R) = aR Π11 , e e while of course Σ12 ∩ ℘(ù) = aΠ11 . One further evidence of the analogy is that if A ∈ (∆21 )L(R) , then A ≤W B for some B ⊆ R such that {B} is Π11 . (A ≤W B e where for e continuous & A = f −1 (B))). To see this, let A ∈ J (R) iff ∃f (f α some formula ϕ(v) and real x, Jα+1 (R) |= ϕ [x] but Jα (R) |= ¬ϕ [x]. Take B = {hϑ, yi : y ∈ R & Jα+1 (R) |= ϑ [y]}. It is easy to show that {B} is Π11 (x), and A ≤W B. We proceed to the Σn selection theorem. Our proof is a transcription of Jensen’s proof for L, as simplified by Sy Friedman (cf. [Sim78]). Definition 1.13. For any α ∈ Ord and n ∈ ù ñnα = the least â such that there is a Σn (Jα (R)) partial map e f : Jâ (R) ։ Jα (R) if n ≥ 1, and ñnα = α if n = 0. By Lemma 1.4, there is in fact a Σn (Jα (R)) partial map from ùñnα × R onto Jα (R). We shall write “ñnα = R” toemean ñnα = 1. Lemma 1.14. Suppose n ≥ 1 and Jα (R) satisfies Σn selection. Then ñnα = the least â such that some Σn (Jα (R)) subset of Jâ (R) e is not a member of Jα (R) Proof. Let â be as on the right hand side. If h : Jñnα (R) ։ Jα (R) is partial Σn , then {a ∈ dom h : a 6∈ h(a)} witnesses that â ≤ ñnα . Now let p ∈ Jα (R) be such that Jα (R) has a Σn Skolem function which is Σn (Jα (R), {p}) and there is a Σn (Jα (R), {p}) subset of Jâ (R) not in Jα (R). Let H = Hullαn (Jâ (R) ∪ {p}). Then H ≺n Jα (R) and there is a Σn (H ) partial map h of Jâ (R) onto H , by e the collapse. Then ä = α, as our new Lemma 1.9. Let ð : H ∼ = Jä (R) be subset of Jâ (R) at α is constructed at ä. Thus ð[h] witnesses that ñnα ≤ â. ⊣

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The next lemma gives the basic quantifier complexity reduction behind the proof of the Σn selection theorem. Lemma 1.15. Let n ≥ 1 and assume Jα (R) satisfies Σm selection for all m ≤ n. Let Q be a Σn (Jα (R)) relation and f a Σn (Jα (R)) partial function. e e Define S by
S(a, â) iff â < ñnα & ∀b ∈ Jâ (R) b ∈ dom f → Q(a, â, f(b)) . Then S is Σn+1 (Jα (R)). e Proof. We proceed by induction on n. Suppose Q(a, b, c) ⇐⇒ ∃dP(a, b, α c, d ), where P is Πn−1 (Jα (R)). Let ñn = pnα and ñn−1 = ñn−1 . Let e h : Jñn−1 (R) ։ Jα (R) be a partial Σn−1 (Jα (R)) map. e Σ (J (R)) map g : J (R) → ñ , the Case 1. For some ä < ñn and total n α n−1 ä e range of g is cofinal in ñn−1 .

Proof. Suppose S(a, â). Then if b ∈ Jâ (R) and b ∈ dom(f), we can find c ∈ Jñn−1 (R) so that P(a, â, f(b), h(c)). Moreover, c ∈ Jg(d ) (R) for some d ∈ Jä (R). Let H (b, d ) ⇐⇒ b ∈ Jâ (R) ∩ dom f & d ∈ Jä (R) & ∀e∀ç (e = f(b) & ç = g(d ))
→ ∃c ∈ Jç (R) P(a, â, e, h(c)) .

We have just seen that S(a, â) → ∀b ∃d H (b, d ). Our induction hypothesis implies (or if n = 1, it is trivial that) the consequent of the final implication defining H is Πn . Thus H is Σn & Πn , and being bounded in Jñn (R), e then S(a, â) ise equivalent e H ∈ Jα (R). But to the existence of such an H in Jα (R); that is, S(a, â) ⇐⇒ â < ñn & ∃H ∈ Jα (R) 
∀b ∈ Jâ (R) b ∈ dom f → ∃d H (b, d ) & ∀b ∈ Jâ (R) ∀d ∀ç ∀e H (b, d ) & ç = g(d ) & e = f(b) →
 ∃c ∈ Jç (R) P(a, â, e, h(c)) . Using the induction hypothesis again, we have that S is Σn+1 (Jα (R)). e Case 2. Otherwise. Proof. Suppose S(a, â). Let D = dom f ∩ Jâ (R); then D ∈ Jα (R) since â < ñn . Define R(b, ç) ⇐⇒ b ∈ D & ç < ñn−1 & ∃c ∈ Jç (R) P(a, â, f(b), h(c)),

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so that R is Σn (Jα (R)). By Σn selection for Jα (R), we have a total Σn (Jα (R)) e of g is map g : D →e ñn−1 which uniformizes R. By case hypothesis, the range bounded in ñn−1 . But then S(a, â) ⇐⇒ ∃ç < ñn−1 ∀b ∈ Jâ (R) ∀d


d = f(b) → ∃c ∈ Jç (R) P(a, â, d, h(c)) .

By induction hypothesis (or obviously in the case n = 1) S is Σn+1 (Jα (R)). e Theorem 1.16. Suppose α ∈ Ord, n ≥ 1 and satisfies Σn selection.

α ñn−1

⊣

6= 1. Then Jα (R)

Proof. Suppose R(a, F ) ⇐⇒ ∃b Q(a, F, b), where Q is Πn−1 (Jα (R)) and e Σ (J (R)) α R ⊆ Jα (R) × [ùα]<ù . Let f : Jñn−1 (R) ։ Jα (R) be a partial n−1 α e map. Define α S(a, F ) ⇐⇒ ∃â < ñn−1 
∃b ∈ Jâ+1 (R) b ∈ dom f & f(b)0 = F & Q(a, f(b)0 , f(b)1 )
& ∀b ∈ Jâ (R) b ∈ dom f → ¬Q(a, f(b)0 , f(b)1 )

∀b ∈ Jâ+1 (R) b ∈ dom f →


i (F ≤BK f(b)0 ∨ ¬Q(a, f(b)0 , f(b)1 )) .

Here, of course, f(b) = hf(b)0 , f(b)1 i. Lemma 1.15 easily implies that S is Σn (Jα (R)), and clearly S uniformizes R. (By Lemma 1.10, we may assume α is a limit ordinal, so that ne ≥ 2, and then Lemma 1.4 easily implies that ñn−1 α “∀b ∈ Jâ+1 (R)” is a bounded quantification over Jñn−1 (R).) ⊣ α 6= 1 of the Σn selection theorem We shall see in §4 that the hypothesis ñn−1 is essential. Of course, if L(R) = L we can weaken it to “α 6= 1 or n 6= 1”; that is, we need only rule out Σ1 (J1 (R)). On the other hand, if Det(L(R)) then J1 (R) satisfies Σn selection iff n is even, and a similar periodicity of order two takes over on arbitrary Jα (R) above the least n such that ñnα = 1.

§2. Scales on Σ1 (Jα (R)) sets. Our positive results on the existence of scales are refinements of [MS83, Theorem 1], which builds directly on a scale construction due to Moschovakis [Mos83]. We shall describe briefly the slight generalization of this construction we need. Suppose that α ∈ Ord, and that for each x ∈ R we have a game Gx in which player I’s moves come from R × α while player II’s moves come from R. Thus a typical run of Gx has the form I

x0 , â 0

x2 , â 1 ···

II

x1

x3

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where xi ∈ R and âi < α for all i. Suppose that Gx is closed and continuously associated to x in the strong sense that for some Q ⊆ (ù <ù )<ù × α <ù , player I wins Gx iff ∀n Q(hx↾n, x0 ↾n, . . . , xn ↾ni, hâ0 , . . . , ân i). Thus by Gale-Stewart, one of the players in Gx has a winning quasi-strategy. (We are not assuming the Axiom of Choice, so we don’t get full strategies.) Let Pk (x, u) iff u is a position of length k from which player I has a winning quasi-strategy and P(x) iff P0 (x, ∅). If sufficiently many games are determined , then Moschovakis’ construction yields a scale on P. The prewellordering ≤i induced by the ith norm in this scale is definable from the Pk ’s for k ≤ i by means of recursive substitution, the Boolean operations, and quantification over R × α. In particular, if α < ùã and hPk : k ≤ ii ∈ Jã (R), then ≤i ∈ Jã (R). The determinacy required to construct the Moschovakis scale is closely related to the definability of the scale constructed. In particular, if α < ùã and Pk ∈ Jã (R) for all k ≤ ù, then Det(Jã (R)) suffices. In the circumstances just described we call the map x 7→ Gx a closed game representation of P. Which sets P admit game representations, and therefore scales? According to [MS83], every (Σ21 )L(R) set admits a closed e game representation in L(R). Our first theorem refines that result.

Theorem 2.1. If α > 1 and Det(Jα (R)), then the pointclass Σ1 (Jα (R)) has the scale property. Proof. Let us first assume that α is a limit ordinal, and deal with the general case later. Let ϕ0 (v) be a Σ1 formula, and let P(x) iff Jα (R) |= ϕ0 [x], for x ∈ R, For â < α, let Thus P =

S

â<α

P â iff Jâ (R) |= ϕ0 [x]. P â . For each â < α we will construct a closed game repre-

sentation x 7→ Gâx of P â . Let Pkâ (x, u) ⇐⇒ u is a position of length k from which player I has a winning quasi-strategy in Gâx . We shall arrange that for each Pkâ ∈ Jα (R), and that the map (â, k) 7→ Pkâ is Σ1 (Jα (R)). This will suffice for Theorem 2.1. For let {ϕkâ } be the Moschovakis scale on P â , and let ≤âk be the prewellorder of R induced by

142 ϕkâ .

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≤âk ∈

Then Jα (R) by the remarks above. A simple inspection of [Mos83] shows that those remarks are true uniformly in â, so that the map (â, k) 7→≤âk is Σ1 (Jα (R)). We can then define a scale {øk } on P: ø0 (x) = ìâ P â (x), and øk+1 (x) = hø0 (x), ϕkø0 (x) (x)i. (In the definition of øk+1 , we use the lexicographic order to assign ordinals to pairs of ordinals.) It is easy to check that {øk } is a Σ1 (Jα (R)) scale on P. So let â and x be given; we want to define Gâx . Our plan is to force player I in â Gx to describe a countable model of V = L(R) + ϕ0 (x) + ∀ã (Jã (R) 6|= ϕ0 [x]) which contains all the reals played by player II, while using ordinals less than ùâ to prove that his model is wellfounded. If Jâ (R) |= ϕ0 [x], then player I will be able to win Gâx by describing an elementary submodel of Jã (R), where ã is least so that Jã (R) |= ϕ0 [x]. On the other hand, if player I has a winning quasi-strategy in Gâx , we will be able to piece together the models he describes in different runs of Gâx according to his strategy, and thereby produce a model of the form Jã (R) so that ã ≤ â and Jã (R) |= ϕ0 [x]. Player I describes his model in the language L which has, in addition to ∈ and =, constant symbols x i for i < ù. He uses x i to denote the ith real played in the course of Gâx . Let us fix recursive maps m, n : {ϑ : ϑ is an L-formula} → {2n : 1 ≤ n < ù}

which is one-one, have disjoint recursive ranges, and are such that whenever x i occurs in ϑ, then i < min(m(ϑ), n(ϑ)). These maps give stages sufficiently late in Gâx for player I to decide certain statements about his model. Player I’s description must extend the following L-theory T . The axioms of T include (1) Extensionality (2) V = L(R) (3)ϕ ∃vϕ(v) → ∃v (ϕ(v) & ∀u ∈ v ¬ϕ(u)) (4)i x i ∈ R. Of course, these axioms are true in any Jã (R) as long as the x i ’s are interpreted as reals. The next axiom restricts the models of this form to at most one possibility (5) ϕ0 (x 0 ) & ∀ä (Jä (R) 6|= ϕ [x 0 ]). Finally, T has axioms which guarantee that any model A can be regarded as a definable closure of {x A i : i < ù}. Recall from Lemma 1.4 the uniformly definable maps fã : [ùã]<ù × R ։ Jã (R); let ϑ0 (v0 , v1 , v2 ) be a Σ1 formula

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which for all ã defines the graph of fã over Jã (R). Now, for any L-formula ϕ(v) of one free variable, T has axioms (6)ϕ (7)ϕ

∃vϕ(v) → ∃v∃F (ϕ(v) & ϑ0 (F, x m(ϕ) , v)) ∃v (ϕ(v) & v ∈ R) → ϕ(x n(ϕ) ).

This completes the list of axioms of T . A typical run of Gâx has the form I

i 0 , x0 , ç 0

i 1 , x2 , ç 1 ···

II

x1

x3

where for all k, ik ∈ {0, 1}, xk ∈ R, and çk < ùâ. If u is a position of length n, say u = hhik , x2k , çk , x2k+1 i : k < ni, then we let T ∗ (u) = {ϑ : ϑ is a sentence of L & in(ϑ) = 0}, and if p is a full run of Gâx , T ∗ (p) =

[

T ∗ (p↾n).

n<ù

Now let p = hhik , x2k , çk , x2k+1 i : k < ùi be a run of Gx ; we say that p is a winning run for player I iff (a) x0 = x (b) T ∗ (p) is complete, consistent extension of T such that for all i, m, n, “x i (n) = m” ∈ T ∗ (p) iff xi (n) = m, and (c) If ϕ and ø are L-formulae of one free variable, and “év ϕ(v) ∈ Ord & év ø(v) ∈ Ord ” ∈ T ∗ (p), then “év ϕ(v) ≤ év ø(v)” ∈ T ∗ (p) iff çn(ϕ) ≤ çn(ø) . In condition (c) we have used the “unique v” operator as an abbreviation: if ϑ and ϕ are L-formulae, then ϑ(év ϕ(v)) abbreviates “∃v(ϑ(v) & ∀u (ϕ(u) ⇐⇒ u = v))”. It is clear that Gx is closed and continuously associated to x. In order to show that x 7→ Gx is the desired closed game representation of P â , we want to characterize the winning positions for player I in Gx as those in which player I has been “honest” about the minimal model of ϕ0 (x). More precisely, let us call a position u = hhik , x2k , çk , x2k+1 i : k < ni of length n (â, x)-honest iff Jâ (R) |= ϕ0 [x], and if ã ≤ â is least such that Jã (R) |= ϕ0 [x], then (i) n > 0 → x0 = x, (ii) if we let Iu (x i ) = xi for i < 2n, then all axioms of T ∗ (u) ∪ T thereby interpreted in (Jã (R), Iu ) are true in this structure, and (iii) If ϑ0 , . . . , ϑm enumerate those L-formulae ϑ of one free variable such that n(ϑ) < n and (Jã (R), Iu ) |= év ϑ(v) ∈ Ord

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and if äi < ùã is such that (Jã (R), Iu ) |= év ϑi (v) = äi , then the map äi 7→ çn(ϑi ) is well-defined and extendible to an order preserving map of ùã into ùâ. According to the following claim, honesty is the only rational policy for player I in Gâx . Claim. Let u be a position in Gâx . Then player I has a winning quasi-strategy in Gâx starting from u iff u is (â, x)-honest. Proof. (→) Let u be a (â, x)-honest position of length n; then it is easy to check that ∃i, x, ç ∀y (u ahi, x, ç, yi) is (â, x)-honest. [If n = n(ϑ) for some sentence ϑ of L, put i = 0 iff (Jã (R), Iu ) |= ϑ, otherwise i is random. If n = m(ϕ) for ϕ an L formula of one free variable, choose x so that (Jã (R), Iuahi,x,ç,yi ) |= axiom (6)ϕ of T . If n = n(ϕ), choose x for the sake of (7)ϕ , and otherwise let x be random. Finally, if n = ϕ(n) for ϕ of one free variable, and ä < ùã is such that (Jã (R), Iu ) |= év ϕ(v) = ä, then set ç = f(ä) where f witnesses part (iii) of honesty for u. Otherwise, choose ç randomly.] Moreover, no (â, x)-honest position is an immediate loss for player I in Gâx . Thus if u is (â, x)-honest , player I can win Gâx from u by keeping to (â, x)-honest positions. (⇐) The proof we give here is due to Hugh Woodin; our original proof followed more closely the corresponding proof in [MS83]. Let ϑ be the sentence we want to prove, that is, let ϑ = “For all â, x, and u, if player I has a winning quasi-strategy in Gâx starting from u, then u is (â, x)-honest.” We show that M |= ϑ whenever M is a countable, transitive model of (a sufficiently large fragment of) ZF + DC. But then ZF + DC ⊢ ϑ, and we’re done. So, let M be such a model, and let M |= S is a winning quasi-strategy for player I in Gâx starting from u. In M , define P = {v : v is a position in Gâx extending u and allowed by S}, S and let P = (P, ⊇). Let G be P-generic over M , and let p = G. We can think of p as a “generic run” of Gâx according to S. In particular, p satisfies the requirements for being a winning run for player I, since these requirements are closed.

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Thus T ∗ (p) is a complete, consistent extension of T . Let B |= T ∗ (p), and let A be the substructure of B whose universe is |A| = {b ∈ |B| : ∃i ∈ ù ∃ø ø is a formula of L with


no constant symbols but x i and b = év (B |= ø [v]) }.

Then by axioms (3) and (6) of T , A ≺ B. Let p = hhik , x2k , çk , x2k+1 i : k < ùi. If b ∈ OrdA , and b = év (B |= ø [v]), then let f(b) = çn(ø) . By requirement (c) on player I in Gâx , f is a well-defined order-preserving map of OrdA into ùâ. Thus we may assume that A = (Jã (RA ), I ) for some ã ≤ â, where by requirement (b) on player I in Gâx , I (x i ) = xi . By axioms (7) of T , RA = {xi : i < ù}; on the other hand, since p is a generic run and player II M is free to play whatever reals he pleases, S{xi : i < ù} = R . M Thus A = (Jã (R ), I ), where I = k Ip↾k . Moreover, by axiom (5) of T and requirement (a) on player I in Gâx , ã is least such that Jã (R) |= ϕ0 [x]. It is now easy to verify conditions (i)–(iii) of (â, x)-honesty for u in M [G]; the map f defined above witnesses (iii). Since honesty is absolute, u is (â, x)-honest in M . This proves the claim. ⊣ Notice that the empty position is (â, x)-honest iff Jâ (R) |= ϕ0 [x]. The claim therefore implies that player I has a winning quasi-strategy in Gâx iff Jâ (R) |= ϕ0 [x]. If we let Pnâ (x, u) ⇐⇒ player I has a winning quasi-strategy in Gâx from the length n position u ⇐⇒ u has length n and is (â, x)-honest, then it is easy to see that Pnâ ∈ Jâ+1 (R) for all â, n, and in fact that the map (â, n) 7→ Pnâ is Σ1 (Jα (R)). (To see that Pnâ ∈ Jâ+1 (R), notice that the sentences which come up in evaluating the honesty of positions of length n have quantifier complexity bounded by some recursive function on n. Also, notice that if there is any order preserving map as required by part (iii) of honesty, then there is a Σ1 (Jâ (R)) such map.) Thus the claim gives us Theorem 2.1 in the case that is ea limit. Minor modifications make this proof work for arbitrary α. Again, let P(x) ⇐⇒ Jα (R) |= ϕ0 [x], where ϕ0 is Σ1 . Using the simplicity of rudimentary function S as in the proof of Lemma 1.11(b), we can find first order formulae øn (v) for n < ù so that

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for all â ∈ Ord and x ∈ R, Jâ (R) |= øn [x] iff Sùâ+ç (R) |= ϕ0 [x]. For each â < α and n < ù we can define a game Gâ,n x so that player I wins, â,n Gx iff ∃ã ≤ â (Jã (R) |= øn [x]) iff Sùâ+n (R) |= ϕ0 [x]. The rules of Gâ,n x are exactly those of Gâx except that axiom (5) of T is replaced by
(5)’ øn (x 0 ) & ∀ä Jä (R) |= −øn (x 0 ) . We can define (â, n, x)-honesty and prove the analogue of our claim just as before. This analogous claim yields Theorem 2.1 in the general case at once. ⊣ Of course, the existence of a Σ1 (Jα (R)) set of reals universal for Σ1 (Jα (R), R) sets of reals immediately gives the boldface version of Theorem 2.1. Corollary 2.2. If α > 1 and Det(Jα (R)), then the boldface pointclass Σ1 (Jα (R), R) has the scale property. The hypothesis that α > 1 is necessary in Theorem 2.1 and Corollary 2.2, since Σ1 (J1 (R)) ∩ ℘(R) = Σ11 , and Σ11 doesn’t have the scale property. Theorem 2.1 immediately implies the special cases from which it was abstracted. Corollary 2.3. If Det(L(R)), then (a) The pointclass of inductive sets has the scale property ([Mos78]), (b) The pointclass Σ∗ù has the scale property; in particular, every coinductive set admits a Σ∗ù scale ([Mos83]), (c) The pointclass (Σ21 )L(R) has the scale property ([MS83]). Proof. Let κ R = ìα [Jα (R) is admissible]. Then inductive = Σ1 (JκR (R)) and Σ∗ù = Σ1 (JκR +1 (R)), so we have (a) and (b). Let ó = ìα [Jα (R) ≺1 L(R)]. (Σ21 )L(R)

Then by Lemma 1.12, = Σ1 (Jó (R)), so we have (c). ⊣ We shall use Theorem 2.1 together with some further work to give a complete description of those levels of the Levy hierarchy for L(R) (that is, those pointclasses of the form Σn (Jã (R))) which have the scale property. For this, e following definition. it is convenient to have the Definition 2.4. Let α, â ∈ Ord and α ≤ â. The interval [α, â] is a Σ1 -gap iff (i) Jα (R) ≺R 1 Jâ (R) (ii) ∀α ′ < α (Jα ′ (R) 6≺R 1 Jα (R))

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(iii) ∀â ′ > â (Jâ (R) 6≺R 1 Jâ ′ (R)). That is, a Σ1 -gap is a maximal interval of ordinals in which no new Σ1 facts about elements of R ∪ {Vù+1 } are verified. If [α, â] is a Σ1 -gap, we say α begins the gap and â ends it. Notice that we allow α = â. We shall also allow [(ä 21 )L(R) , ΘL(R) ] as a Σ1 -gap. e Lemma 2.5. The Σ1 -gaps partition ΘL(R) . Proof. Given ã < ΘL(R) , let α ≤ ã be least so that Jα (R) ≺R 1 Jã (R), and L(R) R let â = sup{ä < Θ : Jã (R) ≺1 Jä (R)}. Then [α, â] is a Σ1 -gap, and ã ∈ [α, â]. Any two distinct Σ1 -gaps are disjoint because we have restricted the parameters involved to R ∪ {Vù+1 }. ⊣ Let us work our way now through an arbitrary Σ1 -gap, looking for levels of the Levy hierarchy having the scale property. If α begins a Σ1 -gap, then by Lemma 1.11 there is a partial Σ1 (Jα (R)) map from R onto Jα (R), and therefore Σ1 (Jα (R)) = Σ1 (Jα (R), R). e Corollary 2.6. If α begins a Σ1 -gap, α > 1, and Det(Jα (R)), then Σ1 (Jα (R)) e has the scale property. The next classes to consider are the Σn (Jα (R)), n > 1, for α which begin e upon the admissibility of J (R). a Σ1 -gap. The scale property here depends α

Lemma 2.7. Suppose α begins a Σ1 -gap and Jα (R) is not admissible. Then for all n ≥ 1

and

Σn+1 (Jα (R)) = ∃R (Πn (Jα (R))) e e

Πn+1 (Jα (R)) = ∀R (Σn (Jα (R))). e e Proof. The two conclusions are of course equivalent. Let S be Σn+1 (Jα (R)); e say S(u) ⇐⇒ ∃v P(u, v)

where P is Σn (Jα (R)). Let e

f : R ։ Jα (R)

be a partial Σ1 (Jα (R)) map; There is such a map since α begins a Σ1 -gap. Then clearly
S(u) ⇐⇒ ∃x ∈ R x ∈ dom f & ∀v (v = f(x) → P(u, v)) . If n ≥ 2, this implies S ∈ ∃R (Πn (Jα (R))), as desired. For n = 1, we need to know that “x ∈ dom f” ∈ ∃Re(Π1 (Jα (R))). This is a direct consequence of e inadmissibility. For by inadmissibility we have a total Σ1 (Jα (R)) map e h : D → ùα

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such that D ∈ Jα (R) and h has range cofinal in ùα. Since α begins a Σ1 -gap, and easy Skolem hull argument gives a total g: R ։ D such that g ∈ Jα (R). Let k = h ◦ g. Let Q(u) ⇐⇒ Jα (R) |= ϕ [u, p], where ϕ is Σ1 , be any Σ1 (Jα (R)) set. Then e
Q(u) ⇐⇒ ∃x ∈ R Sk(x) (R) |= ϕ [u, p] 
 ∃x ∈ R ∀S ∀ã ã = k(x) & S = Sã (R) → S |= ϕ [u, p] so that Q ∈ ∃R (Π1 (Jα (R))). ⊣ The second periodicity theorem, Corollary 2.6 and Lemma 2.7 yield at once Corollary 2.8. Suppose α begins a Σ1 -gap, α > 1, Jα (R) is not admissible, and Det(Jα+1 (R)). Then for all n < ù, the classes Σ2n+1 (Jα (R)) and e Π2n+2 (Jα (R)) have the scale property. e Martin [Mar83] shows that at admissible α beginning a gap the scale property fails above Σ1 (Jα (R)) in a strong way. Theorem 2.9 (Martin). Suppose α begins a Σ1 -gap, Jα (R) is admissible, and Det(Jα+1 (R)). Then there is a Π1 (Jα (R)) subset of R × R with no uniformization in Jα+1 (R). Corollary 2.10 (Martin). If α begins a Σ1 -gap, Jα (R) is admissible, and Det(Jα+1 (R)), then none of the classes Σn (Jα (R)) or Πn (Jα (R)), for n > 1, e e have the scale property. We are ready to venture inside our Σ1 gaps. There we shall find no new scales. Theorem 2.11. Let [α, â] be a Σ1 -gap, and assume Det(Jα+1 (R)). Then there is a Π1 (Jα (R)) subset of R × R with no Σ1 (Jâ (R)) uniformization. e Proof. For x, y ∈ R, let
Cα (x, y) ⇐⇒ ∃ã < ùα ∃F ⊆ ã F is finite & {y} is Σ1 (Sã (R), {x, F }) . By Lemma 1.1, Cα is Σ1 (Jα (R)). Suppose for a contradiction that R is a Σ1 (Jâ (R)) relation uniformizing ¬Cα . By Lemma 1.4, we can fix an x ∈ R e Σ formula ϕ so that for some finite F ⊆ ùâ and 0
R(u, v) ⇐⇒ ∃w Jâ (R) |= ϕ [u, v, w, x, F ] . Now {y : Cα (x, y)} has a wellorder definable over Jα (R), and so by Det(Jα+1 (R)) is countable. Thus we have a unique y ∈ R so that R(x, y). Let

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ϑ(u, v) be the Σ1 formula ∃ã ∃F ⊆ ã ∃S F is finite & S = Sã (R)



& ∀z ∈ R z = v ⇐⇒ ∃w ∈ S ϕ(u, z, w, u, F ) .

Then Jâ (R) |= ϑ [x, y], so Jα (R) |= ϑ [x, y]. But inspecting ϑ, we see that this means Cα (x, y), a contradiction. ⊣ Corollary 2.12. If [α, â] is a Σ1 -gap and α < ã < â, then none of the classes Σn (Jã (R)) or Πn (Jã (R)), for n < ù, have the scale property. e e Theorem 2.11 extends in a simple way an example due to Solovay. For x, y ∈ R, let OD(x, y) iff y is ordinal definable from x. Solovay showed that if ∀x∃y ¬OD(x, y), then ¬OD has no uniformization ordinal definable from a real. [If R is such a uniformization, ordinal definable from x, and R(x, y), then OD(x, y), a contradiction.] Let ä = (ä 21 )L(R) and Θ = ΘL(R) . By the reflection theorem e and the stability of ä, ODL(R) is just the set Cä defined in Theorem 2.11. In L(R), every set is ordinal definable from a real, so Solovay’s example interpreted in L(R) gives Theorem 2.11 for the gap [ä, Θ] (as noticed by Kechris and Solovay). Theorem 2.11 comes from simply localizing this example. It is interesting that the non-uniformizable Π1 (Jα (R)) relation referred to in Martin’s Theorem 2.9 is just ¬Cα ; all counterexamples to uniformization in L(R) come from localizing a single example. The next easy proposition sheds more light on the Cα ’s. Proposition 2.13. Suppose Jα (R) is admissible and Det(Jα (R)). Let Cα be the set defined in the proof of Theorem 2.11. Then for all x, {y : Cα (x, y)} is the largest countable Σ1 (Jα (R), {x}) set of reals. Proof (Sketch). Let P be countable and Σ1 (Jα (R), {x}). Since Σ1 (Jα (R), {x}) has the scale property, P is Suslin via a tree on ù × α which is ∆1 (Jα (R), {x}) (here we use admissibility). Since P is countable, P is Suslin via T ↾ä for some ä < α. Taking the appropriate derivatives of T ↾ä, we eventually isolate any y ∈ P at some stage â < α. But then {y} is Σ1 (Jâ (R), {x, F }) for the appropriate F . ⊣ Let us return to the question of scales in Σ1 -gaps. We have only one question left, but its answer is sufficiently long-winded to warrant a new section. §3. Scales at the end of a Σ1 -gap. The results of §2 leave open the question whether any of the classes Σn (Jâ (R)) have scale property when â ends a Σ1 e gap. Notice that there are new pointclasses here, that is, ñnâ = R for some n < ù. This follows at once from part (b) of Lemma 1.11. It turns out that the scale property for these classes hinges on a reflection property of â

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somewhat subtler than the “Σ1 reflection on reals” involved in Theorem 2.1 and Corollary 2.6. For a ∈ Jα (R), let Σna,α be the Σn -type realized by a in Jα (R), that is Σna,α = {ϑ(v) : ϑ is either Σn or Πn & Jα (R) |= ϑ [a]}. Definition 3.1. An ordinal â is strongly Πn -reflecting iff every Σn type realized in Jâ (R) is realized in Jα (R) for some α < â; that is, iff ∀b ∈ Jâ (R) ∃α < â ∃a ∈ Jα (R) (Σnb,â = Σna,α ). Definition 3.2. Let [α, â] be a Σn -gap. We call [α, â] strong iff â is strongly Πn -reflecting, where n is least such that ñnâ = R. Otherwise, [α, â] is weak. Martin’s proof of Theorem 2.9 easily gives the following generalization. Theorem 3.3. Let [α, â] be a strong Σ1 -gap, and assume Det(Jα+1 (R)). Then there is a Π1 (Jα (R)) relation (namely, the ¬Cα of Theorem 2.11) which has no uniformization in Jâ+1 (R). Corollary 3.4. If [α, â] is a strong Σ1 -gap and Det(Jα+1 (R)), then none of the classes Σn (Jâ (R)) or Πn (Jâ (R)), for n < ù, have the scale property. e e Thus at strong gaps [α, â], the scale property first re-appears on Σ1 (Jâ+1 (R)). e At weak gaps it re-appears on Σn (Jâ (R)) where n is least so that ñnâ = R; that e is, as soon as possible given Theorem 2.11. Before we show this, let us consider examples of the two different kinds if gap. Example 3.5. Let [α, â] be the first Σ1 gap such that â ≥ α + ù1 . A Σ1 Skolem hull argument shows easily that â = α + ù1 and ñ1â = R. Since cf(â) > ù, any Σ1 -type realized in Jâ (R) is realized in some Jä (R) for ä < â. Thus â is strongly Π1 reflecting. Example 3.6. Let â be least such that ñ1â 6= R, and let α < â be least so that Jα (R) ≺R 1 Jâ (R). The minimality of â implies that [α, â] is a Σ1 gap, and â that ñ2 = R. Now for any n, let ϑn be a Σ2 sentence so that for all ã, Jã (R) |= ϑn iff ∃α0 . . . ∃αn (α0 < · · · < αn < ã and Jα0 (R) ≺1 · · · ≺1 Jαn (R) ≺1 Jã (R)). Every Σ2 -type realized in Jâ (R) contains all of the ϑn ’s, but no Σ2 -type realized in any Jä (R) for ä < â includes all of the ϑn ’s. Thus â is not strongly Π2 reflecting. There are simpler weak gaps than that given by Example 3.6. For example, the first gap of the form [α, α + 1] is weak. However, Example 3.6 seems to better illustrate the general situation. Indeed, the reader might gain some feeling for the proof of our next theorem by trying to show that for the â described in Example 3.6, Σ2 (Jâ (R)) has the scale property. e

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Theorem 3.7. Let [α, â] be a weak Σ1 gap, and suppose Det(Jα (R)). Then if n is least so that ñnâ = R, the class Σn (Jâ (R)) has the scale property. e Proof. Let α, â, and n satisfy the hypothesis. The case n = 1 is somewhat special, so we assume first that n > 1. We shall follow the proof of Theorem 2.1 in outline, but some new complications arise in this situation. In order to eventually obtain a Σn (Jâ (R)) e of Σ scale, we must regard Jâ (R) as the union of a canonical sequence n−1 substructures of Jâ (R), each of which collapse to some Jä (R) with ä < â. Our non-reflecting Σn type gives such a sequence. Let h : Jâ (R) × R → Jâ (R) be a Σn−1 Skolem function. We want to standardize the parameter from which such a function is Σn−1 definable. Fix ϑsk a Σn−1 formula, w1 a real, and F ′ a finite subset of ùâ such that h(u, x) = v iff Jâ (R) |= ϑsk [u, x, v, w1 , F ′ ]. Now fix F ≤BK F ′ to be the BK-least finite subset of ùâ so that {hu, x, vi : Jâ (R) |= ϑsk [u, x, v, w1 , F ]} is a Σn−1 Skolem function for Jâ (R). We next standardize the parameter satisfying a non-reflected Σn type. Let b ∈ Jâ (R) be such that Σnb,â is not realized in any Jã (R) for ã < â. Let b = f(G ′ , w2 ), where f is Σ1 (Jâ (R)), G ′ is a finite subset of ùâ, and w2 ∈ R. Set Σ = ΣnhG ′ ,w2 i,â . Clearly Σ is not realized in Jã (R) for any ã < â. Now fix G ≤BK G ′ to be BK-least so that Σ = ΣnhG,w2 i,â . We can now define our canonical sequence hHi : i < ùi of Σn−1 hulls. Simultaneously we shall define sequence hKi : i < ùi of finite subsets of ùâ and hϑi : i < ùi of Σn formulae in Σ. Let K0 = ∅. Given K0 , . . . , Ki , let
Hi = Hullân−1 {F, G, K0 . . . Ki } ∪ R 
Jâ (R) = a : {a} is Σn−1 {F, G, K0 . . . Ki } ∪ R . Because we have thrown F into Hi , Hi ≺n−1 Jâ (R) and Hi is the image of â R under a partial Σn−1 (Hi ) function. Since ñn−1 6= R, Hi must collapse to e some Jã (R) for ã < â. But then ð(hG, w2 i) does not realize Σ in Jã (R), where ð is the collapse map, and therefore hG, w2 i does not realize Σ in Hi . Since Hi ≺n−1 Jâ (R), there must be a Σn formula ϑ(v) in Σ so that Hi 6|= ϑ [hG, w2 i]. Let ϑi = the least Σn formula ϑ(v) ∈ Σ such that Hi 6|= ϑ [hG, w2 i],

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where by “least” we mean least in some fixed ordering <Σ of Σ of order type ù. Now every element of Jâ (R) is Σ1 definable from a real and a finite subset of ùâ, so that for some finite K ⊆ ùâ, Hullân−1 ({F, K} ∪ R) |= ϑi [hG, w2 i]. Let Ki+1 be the BK-least finite subset K of ùâ such that Hullân−1 ({F, G, K0 , . . . , Ki , K} ∪ R) |= ϑi [hG, w2 i]. This completes the definitions of the Hi ’s, Ki ’s, and ϑi ’s. We record some simple properties of these sequences in a claim. Claim 3.8. (a) ∀i (Hi ≺n−1 Hi+1 ≺n−1 Jâ (R)). (b) ∀i (Hi 6|= ϑi [hG, w2 i] & Hi+1 |= ϑi [hG, w2 i]). ∼ (c) ∀i S ∃ã < â (Hi = Jã (R)). (d) i<ù Hi = Jâ (R). Proof. (a) and (b) are obvious, and (c) was proved in the course of defining ϑi . For (d), notice that [ Hi |= ϑ [hG, w2 i], ∀ϑ ∈ Σ, i<ù

because the ϑi ’s were chosen least in an ordering of type ù. Thus if ð collapses S i<ù Hi to a trasnsitive set, then [ Hi ∼ ð: = Jâ (R), i<ù

as otherwise ð(hG, w2 i) would realize Σ in some Jã (R) for ã < â. We shall show ð(F ) = F , ð(G) = G, andSð(Ki ) = Ki for all i. Of course ð(x) = x
S i Hi for x ∈ R. Since i<ù Hi = Hulln−1 {F, G, K0 , K1 , . . . } ∪ R , it follows that Jâ (R) = Hullân−1 (ð[{F, G, . . . } ∪ R]), so Jâ (R) = Hullân−1 ({F, G, . . . } ∪ R), S that is, Jâ (R) = i<ù Hi . To see thatSð(F ) = F , notice that ϑsk defines a Σn−1 Skolem function from hF, w1 i over i Hi , and thus from hð(F ), w1 i over Jâ (R). But ð(F ) ≤BK F , while F is BK-least so that ϑsk defines a Σn−1 Skolem function from hF, w1 i over Jâ (R). Thus ð(F ) = F . Similarly, hð(G), w2 i realizes Σ in Jâ (R) and ð(G) ≤BK G, so the BK-minimality of G implies ð(G) = G. Finally, ð(Ki ) = Ki by induction on i. Certainly ð(K0 ) = K0 . If ð(Kj ) = Kj for j ≤ i, then ð[Hi+1 ] = Hullân−1 ({F, G, K0 . . . Ki , ð(Ki+1 )} ∪ R). Since ð[Hi+1 ] |= ϑi [hG, w2 i], ð(Ki+1 ) ≤BK Ki+1 , and Ki+1 is BK-least so that Hullân−1 ({F, G, K0 . . . Ki+1 } ∪ R) |= ϑi [hG, w2 i], we have ð(Ki+1 ) = Ki+1 . ⊣ (Claim 3.8)

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Let P be a Σn (Jâ (R)) set of reals. We want a closed game representation of e to yield a Σ (J (R)) scale on P. By part (d) of Claim 1, we P simple enough n â can fix e ∈ ù so that P is Σne(Jâ (R), {F, G, K0 . . . Ke } ∪ R). Let P(x) iff Jâ (R) |= ϕ0 [x, a] where ϕ0 is Σn , and a is a parameter of the form hF, G, K0 , . . . , Ke , w3 i for w3 ∈ R. For i ≥ e, let P i (x) iff Hi |= ϕ0 [x, a]. We shall construct closed game representations x 7→ Gix of the P i ’s in such a way that if Pki (x, u) iff u is a winning position for player I in Gix of length k, then Pki is Σù (Hmax(i,k) ). Suppose we have done this. Inspecting [Mos83] e that this gives a scale {ϕ i } i again, we see k k∈ù on P so that the prewellordering i i ≤k of R induced by ϕk is Σù (Hmax(i,k) ). But Hmax(i,k) collapses to some Jã (R) with ã < â, and ≤ik is fixedeby the collapse map, so ≤ik ∈ Jâ (R). Similarly, each P i ∈ Jâ (R). But now ñnâ = R, so any countable subset of Jâ (R) is Σn (Jâ (R)), e so the map (i, k) 7→ (P i , ≤ik ) is Σn (Jâ (R)). We then have the obvious scale e {øi }i<ù on P: ø0 (x) = ìi [P i (x)], and øk+1 (x) = hø0 (x), ϕkø0 (x) i, where again øk+1 is defined using the lexicographical order. Since (i, k) 7→ (P i , ≤ik ) is Σn (Jâ (R)), the scale {øi }i<ù is Σn (Jâ (R)). e e It suffices, then, to construct the representations x 7→ Gix in such a way that i Pk is first-order definable over Hmax(i,k) for all i, k. Our plan is to force player I in Gix to describe the truth in Jâ (R) about F , G, and the Kj ’s. Since â

Jâ (R) = Hulln−1 ({F, G} ∪ {Kj : j < ù} ∪ R) it is enough that player I describe only Σn−1 truths; moreover, this restriction is important if Pki is to be first-order definable over Hmax(i,k) . For this reason we also restrict player I to playing at move k only ordinals in Hk and formulae involving no Kj for j > k. Our main problem is how, within these restrictions, to prevent player I from describing the Σn−1 truths about some parameters different from F , G, or the Kj ’s. Player I’s description is given in the language L, which has ∈, =, constant symbols x i for i ∈ ù, and constant symbols F , G, and K i for i ∈ ù. If ϕ is an L-formula containing no constants K i for i > m, we say ϕ has support m.

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Let Bn−1 be the class of boolean combinations of Σn−1 formulae of L. I will actually describe the truth of Bn−1 formulae. We shall use the “unique v” operator applied to Σn−1 formulae to abbreviate formulae in Bn−1 . Let ó, ô, and ϕ be Σn−1 formulae, and let ø be Π1 . The reader can easily check that ϕ(év ó(v)), ø(év ó(v)), and ø(év ó(v, éu ô(u))) can be considered abbreviations of Bn−1 formulae. There are the sorts of abbreviations we shall use. For expository reasons, we allow player I to play finitely many sentences and finitely many reals in a single move of Gix . A typical run of Gix then has the form I T0 , s0 , ç0 , m0 T1 , s1 , ç1 , m1 ··· II y0 y1 where Tk is a finite set of sentences in Bn−1 , all of which have support k, sk ∈ R<ù , çk ∈ Ord ∩Hk , mk ∈ ù, and mk > k, and yk ∈ R. The roles of these objects have been explained, with the exception of the mk ’s. We shall explain their role shortly. Given the run of Gix displayed above, set hxi : i < ùi = s0 ahy0 ias1 ahy1 ia· · · , and T∗ =

[

Tk .

k<ù

We say this run is a win for player I just in case it meets the following requirements. Let n : Bn−1 ֒→ ù be such that any ϑ ∈ Bn−1 has support n(ϑ) and involves no x j for j ≥ n(ϑ). 1. s0 (0) = x, s0 (1) = w1 , s0 (2) = w2 , and s0 (3) = w3 . 2. (a) T ∗ is consistent, (b) if ϑ ∈ Bn−1 and ϑ is a sentence, then either ϑ ∈ Tn(ϑ) or ¬ϑ ∈ Tn(ϑ) , (c) if ϑ ∈ Tk , then ϑ has support k and involves no constant x j for j ≥ dom(s0 ahy0 ia· · · hyk−1 iask ), (d) The Axiom of Extensionality is in T0 . (e) if ô(v) is Σn−1 and ∃u (u = év ô(v)) ∈ Tk , then “∃α (év ô(v) ∈ Jα (R))” ∈ Tk+1 , (f) x k (n) = m ∈ T ∗ iff xk (n) = m. Requirement 2(e) could be replaced by “V=L(R) ∈ T ∗ ” in the case n > 2. Next we require player I to verify that ϕ0 (x, a) holds in Hi . Let ϕ0 = ∃vø0 where ø0 is Π1 . 3. For some Σn−1 formula ô(v) with support i, ø0 (x 0 , hF , G , K 0 . . . K e , x 3 i, év ô(v)) ∈ Ti .

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Player I must verify that any real he describes is one of the xk ’s: 4. If ô(v) is Σn−1 and “év ô(v) ∈ R” ∈ Tk , then for some j, “év ô(v) = x j ” ∈ Tk+1 . Player I must verify that the model he is describing is well-founded: 5. If ó(v) and ô(v) are Σn−1 , and “év ó(v) ∈ Ord & év ô(v) ∈ Ord ” ∈ T ∗ , then “év ó(v) ≤ év ô(v)” ∈ T ∗ iff çn(ó) ≤ çn(ô) . Finally, player I must verify that F denotes F , G denotes G, and K j denotes Kj for j < ù. In order to do so, player I must verify certain Σn and Πn sentences which come up as he plays. Now in order to verify ∀vϑ(v), where ϑ is Σn−1 , player I must simply refrain from putting ¬ϑ(év ó(v)) into T ∗ for any Σn−1 formula ó(v). This is a closed requirement on player I’s play. Dually, in order to verify ∃vϑ(v), where ϑ is Π1 , player I must put ϑ(év ó(v)) into T ∗ for some Σn−1 formula ó(v). Unfortunately, this is an open requirement on player I’s play. In simple cases, such as requirement 3, we can bound in advance the move at which player I is expected to verify his Σn sentence, so that the requirement becomes closed. In view of our restriction on sentences in Tk to those with support k, this amounts to bounding in advance the hull which is to satisfy player I’s Σn sentence. In some cases we can’t do this, and so we require player I himself to provide the bound. This is the role of mk ; it is player I’s prediction at move k of the later move at which he will discharge an accrued obligation to verify certain Σn sentences. (In 7(c)(ii) below, mk plays a similar but slightly different role.) One may object that this violates the spirit of our restrictions on the first k moves: since the mj for j < k can refer to arbitrarily large hulls, Pki won’t be first-order definable over Hmax(i,k) . The Coding Lemma (of all things!) will overcome this objection. Requirements 6(a) and (b) force player I to assert that ϑsk defines a Σn−1 Skolem function from F and x 1 , while 6(c) forces him to assert that nothing ≤BK F has this property of F . 6. (a) The sentence
  ∀v, w, u, x ϑsk (u, x, v, F , x 1 ) & ϑsk (u, x, w, F , x 1 ) → v = w is in T0 , (b) if ϑ(v0 , v1 ) and ô(u) are Σn−1 , and ∃v ϑ(v, éu ô(u)) ∈ Tk , then for some j
∃v ϑ(v, éu ô(u)) & ϑsk (éu ô(u), x j , v, F , x 1 ) is in Tk+1 , (c) if “év ó(v)



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is in Tmk , or for some Σn−1 formulae ϑ(v0 , v1 ) and ô(u), the sentence
∃v ϑ(v, éu ϑ(u)) & ¬∃x ∃v ϑsk (éu ô(u), x, v, évó(v), x 1 ) & ϑ(v, éu ô(u)) is in Tmk . A similar requirement will fix the meaning of G. Recall the Σn formula ϑj (v) in Σ used to define Kj+1 , and the order <Σ of Σ. 7. (a) If ∀u ϑ(u, v) ∈ Σ, where ϑ is Σn−1 , and if ô(u) is any Σn−1 formula, then ¬ϑ(éu ô(u), hG , x 2 i) 6∈ Tk (b) if ∃u ϑ(u, v) ∈ Σ, where ϑ is Π1 , and ∃u ϑ(u, v) <Σ ϑk , then for some Σn−1 formula ô(u) ϑ(éu ô(u), G , x 2 ) ∈ Tk , (c) if “év ó(v) 0, there is a Σn−1 ô(u) with support j such that ′ ϑj−1 (éu ô(u), hG , x 2 i) ∈ Tj ,

(c) For any j > 0, and any k, if “év ó(v)
This completes the description of the payoff for player I in Gix . We next want to characterize the winning positions for player I in Gix as the honest ones. Let u = hhTk , sk , çk , mk , yk i : k < si

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be a position of length s. Let hxj : j < l i = s0ahy0 ia· · ·ask ahyk i, and let Iu (x j ) = xj

for j < l,

Iu (F ) = F,

Iu (G) = G,

Iu (K j ) = Kj

for all j < ù.

and

Let us call u reasonable if it is not an immediate loss for player I because of hTk : k < si, in the sense that all assertions in 1– 4 and 6–8 about the Tk ’s hold so far for the Tk with k < s. [Take for example 7(c). If u is unreasonable on its account, there must be a k < s such that mk < s, and a formula “év ó(v) 0 → s0 (0) S = x & s0 (1) = w1 & s0 (2) = w2 & s0 (3) = w3 . 4. (Jâ (R), Iu ) |= k<s Tk . 5. if hó0 , . . . , óm i enumerates those Σn−1 formulae ó such that n(ó) < s and (Jâ (R), Iu ) |= év ó(v) ∈ Ord, and if ä1 < ùâ is such that (Jâ (R), Iu ) |= év ói (v) = äi then the map äi 7→ çn(ói ) is well-defined and extendible to an order-preserving map of Ord ∩Hs into Ord ∩Hs . The final two requirements for (i, x)-honesty guarantee that player I has made commitments mk , k < s, which can be kept. 6. If k < s and “év ó(v)
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7. If k < s and “év ó(v)
It suffices to show that A is Σù (Hs ); from this it easily follows that condition 7 e is Σù (Hmax(i,s) ). e Since there is a Σn−1 (Hs ) partial map of R onto Hs , there is a Σn (Hs ) total e e map f : R ։ {K ∈ Hs : K
For y, z ∈ R, let


y ≤∗ z iff f(y)0
Thus ≤∗ is a Σn (Hs ) prewellorder of R. Since Hs ∼ = Jã (R) for some ãù, e for m < ù, let ≤∗ ∈ Jâ (R). Now Am = {hK, ϑi : hK, ϑ, mi ∈ A}.

Now Am is Σù (Hmax(m,s) ), so f −1 (Am ) is Σù (Hmax(m,s) ), and so f −1 (Am ) ∈ e Jâ (R) for allem. By the Coding Lemma [Mos80, 7D.5], f −1 (Am ) is Σ11 (≤∗ ) for e ∗ −1 each m. [Since ≤ , f (Am ) ∈ Jâ (R), Det(Jâ (R)) suffices for this application R of the Coding Lemma. Now Jα (R) ≺1 Jâ (R), so if there is a non-determined game in Jâ (R), there is a non-determined game in Jα (R). However, we have assumed Det(Jα (R)).] Since Σ11 (≤∗ ) is closed under countable unions, we e have B ∈ Σ11 (≤∗ ), where e [ f −1 (Am ) × {m}). B= m

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But then B is Σù (Hs ). Since e
(K, ϑ, m) ∈ A iff ∃y hy, mi ∈ B & f(y)0 = K & f(y)1 = ϑ we have that A is Σù (Hs ), as desired. ⊣ (Claim 3.9) e Our original proof of Claim 3.9 required more that Det(Jα (R)), since we had applied the Coding Lemma directly to A. Kechris had the idea of applying the Coding Lemma to the Am ’s individually. This reduces the determinacy needed in the hypotheses of Theorem 3.7 to Det(Jα (R)), a reduction which is crucial for the use of Theorem 3.7 in Kechris-Woodin [KW83]. The next claim finishes our proof of Theorem 3.7 in the case that n > 1. Claim 3.10. For all i, x, and u, u is (i, x)-honest iff u is a winning position for player I in Gix . Proof (Sketch). (⇒) It is enough to show that whenever v is (i, x)-honest, then ∃T, s, ç, m ∀y (v ahT, s, ç, m, yi is (i, x)-honest). For if so, then player I can win Gix by keeping to (i, x)-honest positions. So let v be an (i, x)-honest position of length k. Because v satisfies 2 and 4 of honesty, player I has told as much of the truth about Jâ (R) in his first k moves as we required him to tell. Because v satisfies 6 and 7 of honesty, player I has made predictions mi for i < k which he can fulfill. Thus player I can choose T and s to insure continued satisfaction of 2 and 4 by the new position; that is, he can tell as much more of the truth as he must. (If k = i, we need that v satisfies 1 as well.) Continued satisfaction of 1 and 3 are trivial to insure, while continued satisfaction of 6 and 7 can be insured by choosing mk large enough. Finally, player I must choose ç so as to insure continued satisfaction of 5. Now let v determine the map g(äi ) = çn(ói ) ,

for 0 ≤ i ≤ m,

as in 5 of honesty for v. We may assume that i ≤ j → äi ≤ äj . It is enough to see that g can be extended to an order-preserving map of Ord ∩Hk+1 into Ord ∩Hk+1 . Now by 5 for v and the fact that Hk ≺1 Jâ (R), op

Hk |= ∃f : [äi , äi+1 ] −→ [g(ä1 ), g(äi+1 )] for all i < m. Since Hk ≺1 Hk+1 , op

Hk+1 |= ∃f : [äi , äi+1 ] −→ [g(ä1 ), g(äi+1 )] for all i < m. Similarly, op

Hk+1 |= ∃f : [0, ä0 ] −→ [0, g(ä0 )]. Thus in order to obtain the desired extensions of g, it is enough to show that there is an order-preserving f : (Ord ∩Hk+1 ) − äm → (Ord ∩Hk+1 ) − g(äm ).

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But such an f fails to exist just in case for some n < ù
Hk+1 |= g(äm ) − äm · n doesn’t exist, as the following diagram makes clear (see page 160). ∼

∼ .. .

äm

s     s s      s  1s      s

Ord ∩Hk+1

g(äm )

Ord ∩Hk+1

This is a Π1 property of Hk+1 , and so would pass downward to Hk , which is impossible since g can be extended over Hk . Thus there is such an f. (⇐) Let S be a winning strategy for player I starting from u. As in Theorem 2.1, let hhTk , sk , çk , mk , yk i : k < ùi be a “generic S play” according to S, and let hxk : k < ùi = s0 ahy0 ias1 ahy1 i · · · . Since k Tk is consistent, it has a model B. By payoff requirements 6(a) and (b), A ≺ B where B B B A = HullB ({x B j : j < ù} ∪ {F , G } ∪ {K j : j < ù}).

If A |= év ó(v) ∈ Ord, then set f(év ó(v)A ) = çn(ó) . By payoff requirement 5, f is well-defined and order-preserving. Thus A∼ = (Jã (RA ), I ) for some ã ≤ â and interpretation of constants I . By payoff requirements 7(a) and (b), A realizes Σ, so ã = â. By 2(f), 4, and genericity, RA = RM , where M is our ground model. Clearly I (x j ) = xj

for j < ù,

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while payoff requirements 6, 7, and 8 insure that I (F ) = F,

I (G ) = G,

and I (K j ) = Kj

for j < ù.

Therefore A = (Jâ (R), I ) where I is the natural interpretation. It is easy to verify conditions 1 through 4 and conditions 6 and 7 of (i, x)-honesty for u at this point. For condition 5, notice that the order-preserving map f defined above actually maps any Hs into Hs (since I is the natural interpretation). Thus f witnesses the satisfaction of condition 5. ⊣ (Claim 3.10) Of course, Claim 3.10 applied to the empty position implies that x 7→ Gix is a closed game representation of P i . Claims 3.9 and 3.10 imply that the sets Psi are simply definable enough to yield Σn (Jâ (R)) scale on P. e in the case n = 1. Since the basic We are left with the proof of the theorem plan in the case is the same as the plan in the case n > 1, we shall just sketch a few of the details. Assume first that â is a limit ordinal and n = 1. We shall need no analogue of F . Let G be a finite subset of â and w1 a real such that Σ, the Σ1 -type of hG, w1 i over Jâ (R), is not realized in Jã (R) for any ã < â. Further, let G be BK-least so that hG, w1 i realizes Σ. Define âi < â and ϑi (v) ∈ Σ by: â0 = 0, and ϑi = least Σ1 formula ϑ(v) ∈ Σ such that Jâi (R) 6|= ϑ [hG, w1 i] and âi+1 = least â > âi such that Jâ (R) |= ϑ [hG, w1 i]. The âi ’s are analogues of the Ki ’s. Let also âi+1 Hi = Hullù ({G, â0 . . . âi } ∪ R).

Of course, Hi ∼ = Jã (R) for some S ã < â, moreover, Hi is the image of R under a Σ1 (Hi+1 ) map. As before, i<ù Hi = Jâ (R). eLet P(x) ⇐⇒ Jâ (R) |= ϕ [x],

where ϕ is Σ1 and we have dropped the parameter for simplicity. Let P i (x) ⇐⇒ Hi |= ϕ [x]. Again, it is enough to construct representations x 7→ Gix of the P i in such a way that the corresponding Pki are first-order over Hmax(i,k) . For this, it is convenient to have in player I’s language L, besides the x i , constants â i

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and Jâi (R) for i < ù. In Gix , player I must play a Σ0 -complete. consistent set of Σ0 sentences of L, mentioning at move k no sentences involving â j or Jâi (R) for j > k. For each j, player I must play at move j the Σ0 sentence “Jâj (R) |= V = L(R)”. At move i + 1, player I must assert that some object definable over Jâi+1 (R) from G, â0 , . . . , âi , and some real (which he has played) witnesses that ϕ(x 0 ). Player I must play ordinals çk ∈ Hk to prove well-foundness. Finally, player I must prove that he is interpreting his constant symbols correctly; for G this involves commitments mk made at move k as in the case n > 1. The Coding Lemma argument of Claim 3.9 goes through because each Hk is the image of R under a map in Jâ (R). Claim 3.10 is proved as in the case n > 1. Finally, we have the case â = ã + 1 and n = 1. Let hG, wi realize over Jâ (R) a non-reflected Σ1 -type. Now G = h(p, Jã (R)) for some rud function h and p ∈ Jã (R). Using the simplicity of rud functions (as in Theorem 2.1), we can recursively associate to any Σ1 formula ϑ and any n ∈ ù a formula øϑ,n so that for all α and all a ∈ Jα (R), Jα (R) |= øϑ,n [a] iff Sùα+n (R) |= ϑ [h(a, Jα (R))]. It follows that if Σ is the full elementary type realized by hp, wi over Jã (R), then Σ is not realized in Jä (R) for any ä < ã. Again, we may replace p by some finite G ⊆ ùã (changing w) and take G to be BK-least so that hG, wi realizes Σ over Jã (R). Then Jã (R) = Hull({G} ∪ R). Let P(x) ⇐⇒ Jâ (R) |= ϕ [x] where ϕ is Σ1 and we have ignored the parameter. Let P i (x) ⇐⇒ Sùã+1 (R) |= ϕ [x]. As above, we can find first-order formulae øi such that P i (x) ⇐⇒ Jã (R) |= øi [x]. We construct the desired representation x 7→ Gix of P i as follows. Let L have ∈, =, x i for i < ù, and G. In Gix player I must produce a (fully) complete, consistent L theory containing øi (x 0 ), while using ordinals less than ùã to prove well-foundness. At move k he can put only Σk sentences into his hierarchy. His theory must be “Skolemized”, and whenever he describes a real he must play it bodily on the board, as before. Finally, player I must prove that he is interpreting G by G; this again involves commitments of the form mk . The Coding Lemma argument of Claim 3.9 goes through because for each k we have a map f : R → Jã (R) such that f ∈ Jâ (R) and G ∈ f[R] and f[R] ≺k Jã (R). Claim 3.10 is proved as in the case n > 1. This completes the proof of Theorem 3.7. ⊣

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In each case in the proof of Theorem 3.7, we expressed our arbitrary Σn (Jâ (R)) set of reals as a countable union of sets belonging to Jâ (R). Thus e have we â Corollary 3.11. If [α, â] is a weak SΣ1 -gap, and n is least such that ñn = R, then P ∈ Σn (Jâ (R)) ∩ ℘(R) iff P = i Pi , where Pi ∈ Jâ (R) for all i. e Corollary 3.12. If [α, â] is a weak Σ1 -gap, and n is least such that ñnâ = R, then each of the classes Σn+2k (Jâ (R)) and Πn+2k+1 (Jâ (R)), for k < ù, has e e the scale property.

Proof. By the second periodicity theorem, it is enough to show that Σn+k+1 (Jâ (R)) ∩ ℘(R) = ∃R (Πn+k (Jâ (R))) ∩ ℘(R). e e The proof of this is like the proof of Lemma 2.7, the key fact being that Σn (Jâ (R)) ∩ ℘(R) ⊆ ∃R (Πn (Jâ (R))). This fact follows trivially from Corollary 3.11.

⊣

Our description of the levels of the Levy hierarchy for L(R) having the scale property is now complete. It is interesting that the negative results on scales in L(R) (Corollaries 2.10, 2.12, and 3.4) all come from negative results on uniformization (Theorems 2.9, 2.11, and 3.3). In L(R), the best way to uniformize arbitrary relations in a given Levy class is by means of scales on those relations, as we claimed in the introduction. It follows easily from Wadge’s Lemma that if A ∈ Π1 (Jα (R)) − Σ1 (Jα (R)) e Π (J (R)) eset. This and A admits a Σn (Jâ (R)) scale, then so does every 1 α e observation and an inspection of our results above yield ethe promised characterization of the smallest Levy class Σn (Jâ (R)) at which a scale on a given set A e is definable. Let hâ, ni be lexicographically least so that A admits a Σn (Jâ (R)) e the scale scale; then A ∈ Σn (Jâ (R)) and either Σn (Jâ (R)) or Σn+1 (Jâ (R)) has e e e property. Our results on the scale property therefore characterize hâ, ni by means of reflection properties. §4. Suslin cardinals. We shall use results of §2 and §3 to extend some work of [Kec81] and [KSS81] (see also [Ste81]) in the global theory of boldface pointclasses. To simplify the exposition, we assume ZF+ AD+ DC throughout this section. The results of [Kec81] and [KSS81] are formulated in terms of a hierarchy slightly finer and considerably more general than the Levy hierarchy for L(R). We now define this hierarchy. Let Sep(Γ), PWO(Γ), and Scale(Γ) mean, respectively, that Γ has the separation, prewellordering, and scale properties. If Γ and Λ are nonselfdual boldface pointclasses, let e e ˘. {Γ, Γ} <W {Λ, Λ} iff Γ ⊆ Λ ∩ Λ e e e e e e e

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Wadge and Martin have shown that <W is a wellorder. The set of pairs {Γ, Γ˘ } e e such that ∃R Γ ⊆ Γ or ∃R Γ˘ ⊆ Γ˘ has order type Θ under ≤W ; for 0 ≤ α < Θ, e e e e let Pα be the αth such pair. Now let 
 Σ1α = unique Γ ∈ Pα ∃R Γ ⊂ Γ & ∀R Γ * Γ ∨ Sep(Γ) . e e e e e e e Since exactly one of Sep(Γ) and Sep(Γ˘ ) holds for every nonselfdual boldface e Γ, Σ1α is well defined. Lete e e Π1α = (Σ1α )˘, e1 e ∆α = Σ1α ∩ Π1α , e e e and ä 1α = sup{ë : ∃f f : R ։ ë & {hx, yi : f(x) ≤ f(y)} ∈ ∆1α )}. e e The sequence hΣ1α : α < ϑi is just the natural extension of the projective hierarchy cofinallye through all boldface pointclasses. In particular, Σ10 = Σ01 , e e and for 1 ≤ n < ù, Σ1n is just the class usually given that name. For α < ΘL(R) , Σ1α is essentially thee αth level of the restriction to sets of reals of the Levy e hierarchy for L(R), as we now show. First, more terminology. If Γ is a boldface pointclass, then e n[ o [ Γ= Aâ : ∀â < α (Aâ ∈ Γ) e α e â<α and

\ α

Γ= e

[  Γ ˘. α e

We can classify limit ordinals ë according to the closure properties of Σ1ë : e \ \ ë is type I ⇐⇒ Σ1ë ⊆ Σ1ë & Σ1ë * Σ1ë , e e e ù e 2 \ 1 1 ë is type II ⇐⇒ Σë 6⊆ Σë , e e 2 \ 1 ë is type III ⇐⇒ Σë ⊆ Σ1ë & ∀R Σ1ë * Σ1ë , e e e ù e

ë is type IV ⇐⇒ ∀R Σ1ë ⊆ Σ1ë . e e The type of ë is just the type, in the sense of [KSS81], of the projective-like hierarchy immediately above Σ1ë . Some facts from [KSS81] and [Ste81]: ë S S is of type I iff cf(ë) = ù iffeΣ1ë = ù ( α<ù Σ1α ). If ë is of type II,III, S e ë is of type I,ethen PWO(Σ1 ); otherwise or IV, then ∆1ë = α<ë Σ1α . If ë e e e L(R) (provided ë < Θ ) PWO(Π1ë ). (At this point we should warn the reader e

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that our definition of the extended projective hierarchy differ slightly from that of [Kec81]. The difference is that the classes Σ1ë for ë of type I or II are e omitted from the hierarchy of [Kec81].) L(R) Let häα : α < Θ i enumerate in increasing order those ordinals ä such that ñnä = R for some n, with ä0 = 1. Let nα be the least n such that ñnäα = R. Lemma 4.1. For all α < ΘL(R) , (a) PWO(Σnα (Jäα (R))), S (b) Σnα (Jäαe(R)) ∩ ℘(R) ⊆ äα (Jäα (R) ∩ ℘(R)). e ä Proof. Let ä = äα , n = nα , and ñ = ñn−1 . Fix a partial Σn−1 (Jä (R)) map f : Jñ (R) ։ Jä (R). Suppose x ∈ A ⇐⇒ Jä (R) |= ∃vϕ [x], where ϕ is Π1 , and for x ∈ A let 
 ø(x) = ìã ≤ ñ ∃u ∈ Jã (R) u ∈ dom f & Jä (R) |= ϕ [x, f(u)] . By Lemma 1.15, ø is a Σn (Jä (R)) norm, so we have (a). Fix ã < ñ. Then dom f ∩ Jã (R) ∈ Jä (R),efrom which it easily follows that {x : ø(x) ≤ ã} ∈ Jä (R). This proves (b). ⊣ Theorem 4.2. For all α < ΘL(R) , (a) if ùα = 0 or ùα is of type I, then for all k < ù Σ1ùα+k = Σnα +k (Jäα (R)) ; e e (b) if ùα is of type II or III, then for all k < ù Σ1ùα+k+1 = Σnα +k (Jäα (R)) ; e e (c) if ùα is of type IV, then Π1ùα = Σnα (Jäα (R)), e e and for k < ù such that k 6= 0, Σ1ùα+k+1 = Σnα +k (Jäα (R)). e e S Proof. By induction on α. The case α = 0 is clear. Set Λ = â<ùα Σ1â = Jäα (R) ∩ ℘(R). S (a) Since Σ1ùα = ù Λ, Σ1ùα ⊆ Σnα (Jäα (R)). Let A ∈ Σnα (Jäα (R)) ∩ ℘(R). e e thatecf(ùα) = ù e and the fact By Lemma 4.1(b) [ [ Anâ ), A= n<ù â<äα

where for some fixed ãn < ∈ Σ1ãn for all â < äα . By Kechris [Kec81], we S e may assume that Σ1ãn is closed under wellordered unions, so that A ∈ ù Λ = e Σ1ùα , as desired. Thus (a) is true for k = 0, and the case k > 0 follows easily. e ùα, Anâ

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(b) Σ1ùα 6= Σnα (Jäα (R) ∩ ℘(R)), by Lemma 4.1(a) and the fact that e Σ1 1 PWO(Πe1ùα ). Thus ùα+1 ⊆ Σnα (Jäα (R) ∩ ℘(R)). But Σùα+1 is closed under e e e wellordered unions by Kechris’ theorem, so Lemma 4.1(b) gives Σnα (Jäα (R))∩ e easily. ℘(R) ⊆ Σ1ùα+1 . Thus (b) is true for k = 0, and for k > 0 again follows S e 1 1 (c) The Coding lemma implies ùα Πa ⊆ Πa , which with the proof of Lemma 4.1(b) implies the first statement. The second statement is immediate S ⊣ once we notice that Σ1ùα+1 = ∃R 2 (Σ1ùα ∪ Π1ùα ). e e e If T is a tree on ù × κ, then [T ] is the set of infinite branches of T , and p([T ]) the projection of this set, that is, 
p([T ]) = x ∈ R : ∃f∀n (x↾n, f↾n) ∈ T . We let S(κ) = {p([T ]) : T is a tree on ù × κ}, and call the members of S(κ) κ-Suslin sets. If κ > ù, then the κ-Suslin sets are precisely those admitting scales all of whose norms map into κ. We say S κ is Suslin iff S(κ) \ α<κ S(α) 6= ∅. Suslin ordinals are cardinals. We shall locate the pointclasses S(κ) among the Σ1í ’s, and the Suslin cardinals among e the ä 1í ’s. e í be the αth ordinal í such that either Scale(Σ1 ) or Scale(Π1 ). By the Let α í í e ordinal second periodicity theorem, íα+1 = íα + 1 as long aseíα is not a limit of type IV. If íα is of type IV, then íα+1 is a limit ordinal of type I by the results of §§2 and 3. Theorem 4.3. Let ë < (ä 21 )L(R) be a limit ordinal, and let í = sup{íα : e α < ë}. Then (a) If í is type I, then for all n < ù

Scale(Σ1í+2n ), Scale(Π1í+2n+1 ), e e S(κë+n ) = Σ1í+n+1 , e κë+2n+1 = ä 1í+2n+1 = (κë+2n )+ , e cf(κë+2n ) = ù ; (b) if í is type II or III, then for all n < ù Scale(Π1í+2n ), Scale(Σ1í+2n+1 ), e e S(κë+n ) = Σ1í+n+1 , e κë+2n+2 = ä 1í+2n+2 = (κë+2n+1 )+ , e cf(κë+2n+1 ) = ù ;

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(c) if í is type IV, then Scale(Π1í ), S(κë ) = Π1í , e e and if ì = íë+1 , then for all n < ù

κë = ä 1í , e

Scale(Σ1ì+2n ), Scale(Π1ì+2n+1 ), e e S(κë+n+1 ) = Σ1ì+n+1 , e κë+2n+2 = ä 1ì+2n+1 = (κë+2n+1 )+ , e cf(κë+2n+1 ) = ù. Proof (Sketch). By induction on ë. (a) By Theorem 4.2 and the results of §§2 and 3, Σ1í = Σ1 (Jα (R)) for some α e κ e= sup{ä 1 : â < ë}. beginning a Σ1 gap. Thus Scale(Σ1í ), and by induction ë íâ e Thus κë < ä 1í+1 , and S(κë ) ⊆ Σ1í+1 . But Σ1í+1 ⊆ S(κë ) since Σ1í ⊆ S(κë ) e ) = Σ1 . e e and S(κ) is closed under ∃R andecountable intersection. Thus S(κ ë í+1 e The remaining assertions follow by arguments like those for the projective hierarchy (Σ1í being analogous to Σ10 ). e of §§2 and 3, Σ1 = Σ (J (R)) for some e (b) By Theorem 4.2 and the results 1 α í+1 α beginning a Σ1 gap. Thus Scale(Σ1í+1 ), and κë e= sup{äeí1â : â < ë} = ä 1í . e e We have S(κë ) = Σ1í+1 for the same reasons as in (a). The remaining assertions e follow by the arguments for the projective hierarchy, except for Scale(Π1í ). This can be proved using the ideas of the proof of PWO(Π1í ); cf. [Kec81, eTheorem e 3.1(iii)] and [Ste81, Theorem 3.1], . 1 (c) By Theorem 4.2 and §§2 and 3, Πí = Σ1 (Jα (R)), where α begins e second set follows from e a Σ1 gap. This proves the first set of assertions. The our analysis of the gaps in §§2 and 3, and the arguments for the projective ⊣ hierarchy (Σ1ì being analogous to Σ10 ). e e Corollary 4.4. (a) The sequences hκα : α ≤ (ä 21 )L(R) i and híα : α ≤ e 2 L(R) (ä 1 ) i are continuous at limits. e (b) For any κ ≤ (ä 21 )L(R) , either S(κ) or its dual has the scale property. e As a final application of §2 and §3, we prove Corollary 4.5. In L(R), the reliable cardinals are precisely the Suslin cardinals. This result should be stamped “Made in Los Angeles”. The reader who cares can untangle some of the credits for it from what follows. Recall that an ordinal ë is reliable iff there is a scale {ϕi } on a set A ⊆ R so that ϕi : A → ë

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for all i, and ë = {ϕ0 (x) : x ∈ A}. The interest of the notion of reliability stems from results of [Bec80] and [Mos81], which to date have been proved only for reliable ordinals. One direction of Corollary 4.5 is easy. Lemma 4.6. Every Suslin cardinal is reliable. Proof. Clearly ù is reliable. Let κ be an uncountable Suslin cardinal, let S A ∈ S(κ) − α<κ S(α), and let {ϕi } be a scale on A all of whose norms map onto (perhaps improper) initial segments of κ. Let B = {x ∈ R : ëi.x(i + 1) ∈ A}, and for x ∈ B ø0 (x) = ϕx(0) (ëi.x(i + 1)), øn+1 (x) = ϕn (ëi.x(i + 1)). Then {øi } is a scale on B all of whose norms map into κ. Since A 6∈ S S(α), κ = {ø0 (x) : x ∈ B}. ⊣ α<κ There are reliable ordinals which are not cardinals (cf. [Bec80]). The first step toward showing that in L(R) all reliable cardinals are Suslin, and the realization that it is a first step, are due to Kechris. Lemma 4.7 (Kechris). Let ë be reliable, and let κ = sup{ã ≤ ë : ã is Suslin}. Then there is a strictly increasing sequence of sets in S(κ) of length ë. Proof. Let {ϕi } be a scale on A witnessing the reliability of ë. For α < ë, let Aα = {x ∈ A : ϕ0 (x) ≤ α}. Then α < â → Aα ( Aâ ; moreover, each Aα is in S(κ) via a subtree of the tree of {ϕi }. But S(ë) = S(κ) by the definition of κ. ⊣ The next and most substantial step toward Corollary 4.5 is due to Martin (exploiting an idea of Jackson). Theorem 4.8. [JM83] For 1 ≤ n < ù, there is no strictly increasing sequence of Σ12n sets of length (ä 12n−1 )+ . e e By Lemma 4.7 and Theorem 4.8 and standard facts about the Suslin cardinals below ä 1ù , every reliable cardinal below ä 1ù is Suslin. But Martin’s e e more than Theorem 4.8, and in fact, argument gives with some care we can prove Theorem 4.9. Suppose α < ΘL(R) and Scale(Π1α ). Then there is no strictly increasing sequence of ∃R Π1α sets of length (ä 1α )+e. e e

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Proof. We shall assume that the reader is familiar with the proof of Theorem 4.8 given in [JM83]. Our only problem in extending that proof is to show that the coding of ordinals below (ä 12n−1 )+ which it employs generalizes e suitably. 1 R 1 Since Scale(Πα ), every ∃ Πα set has a scale whose norms map into ä 1α . Let e U ⊆ R3 be universal ∃R Π1α , eand {ϕi } a scale on U mapping into ä 1α . eDefine e by a tree T on ù × ù × ä 1α e e n T = (s, t, u) : lh(s) = lh(t) = lh(u) &  ^ ∃x ⊇ s ∃y ⊇ t U (x, (y)i , (y)i+1 ) & 1
^

o ϕ(i)0 (x, (y)(i)1 , (y)(i)1 +1 ) = U (i) .

1
Here we let h·, ·i be a bijection of ù × ù onto ù, and for any i, i = h(i)0 , (i)1 i, and for any y, i, n, (y)i (n) = y(hi, ni). For x ∈ R, let Tx = {(t, u) : (↾ lh(t), t, u) ∈ T }. Tx is the “Kunen tree” associated to x. If Ux is a wellfounded relation, then Tx is a wellfounded tree, and |Ux | ≤ |Tx | (where |R| denotes the rank of the relation R). In order to carry out the argument of [JM83] in the present situation, we need only to verify (a) Suppose that f : R → R is continuous and that the following holds for all x ∈ R and â < ä 1α : if for all ã < â, T(x)0 ↾ã is wellfounded, then Tf(x) ↾â is wellfounded. Then,eif there is a ä < (ä 1α )+ such that for all x ∈ R, T(x)0 is e wellfounded, then |Tf(x) | < ä; (b) Suppose that f : R → R is continuous and that the following holds for all x ∈ R and â < ä 1α : if for all ã < â, T(x)0 ↾ã is wellfounded, then Tf(x) ↾â e is wellfounded. Then there are unboundedly many ä < (ä 1α )+ such that for all e x ∈ R, we have |Tx | < ä → |Tf(x) | < ä ;

(c) For all sufficiently large ä < (ä 1α )+ , {x : |Tx | < ä} is not Π1α+1 . e (If R is a tree on ù k × α, then R↾ãe= {(ó1 . . . ók ) ∈ R : ran(ô) ⊆ ã}.) Now (c) can be proved exactly as in [JM83]. Since (a) and (b) have similar proofs, we shall just prove (b). Lemma 4.10. Suppose α < ΘL(R) , α is a successor, and Scale(Π1α ). Let R be a tree on ù × â, where â < ä 1α , and let f : R → R be continuous eand such that ∀x(Rx is wellfounded → Ref(x) is wellfounded). Then there is a club C ⊆ ä 1α e so that for ä ∈ C , ∀x(|Rx | < ä → |Rf (x)| < ä).

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S Proof. By a result of Martin, ä ∆1α ⊆ ∆1α for all ä < ä 1α (cf. [Kec78, e e Theorem e all u ∈ â <ù , Theorem 3.7]). It follows (as in [Kec78, 3.8]) that for 1 and ä < ä α , e {x : |Rxu | < ä} ∈ ∆1α , e where Rxu is the subtree of Rx below u; the proof is by induction on ä. It is enough to show that ∀ä < ä 1α ∃ç < ä 1α ∀x ∈ R (|Rx | < ä → |Rf(x) | < ç). So fix ä < ä 1α , and let A = {x :e |Rx | < e ä}. Now our hypotheses on α and e Theorem 4.3 together imply that Σ1α = S(ã) for some ã < ä 1α . Thus we can fix e a tree Q on ù × ã so that A = p([Q]). Define a tree P on eù × ã × ù × â by P = {(s, t, u, v) : lh(s) = lh(t) = lh(u) = lh(v) & (s, t) ∈ Q & (u, v) ∈ R & ∃x ⊇ s ∃y ⊇ u (f(x) = y)}. Then P is wellfounded, and if x ∈ A then Rf(x) can be embedded in P. Thus ç = |P| is as desired. ⊣ We can now complete the proof of (b) in the case α is S a successor as in [JM83]. Since α is a successor and Scale(Π1α ), we have ù Π1α ⊆ Π1α . e But then the ù-club subsets of ä 1α generate a normal ultrafilter ì e on ä 1α . eSet e e ã < κ, we can apply Kunen’s argument κ = ä 1α . Since Σ1α = S(ã) for some κ e e + ([Kec78, Theorem 14.3]) to show that κ /ì has order type κ . For â < κ let 
Câ = ä < κ : ∀x |Tx ↾â| < ä → |Tf(x) ↾â| < ä , so that Câ is club in κ by Lemma 4.10. Let D = {ä < κ : ∀â < ä (ä ∈ Câ )}, and let C = {[h]ì : h : κ → D & [h]ì ≥ κ}. Clearly C is unbounded in κ + . Suppose |Tx | < ä, where ä ∈ C . Let ä = [h], where h : κ → D. Then |Tx ↾â| < h(â)

(ì-a.e.),

so |Tf(x) ↾â| < h(â) (ì-a.e.), so |Tx | < [h]ì = ä, as desired.

⊣ Scale(Π1α ),

Theorems 4.3 and 4.2 tell us that Finally, let α be a limit. Since e cf(ñ) > ù. Let ë = cf(ñ), and let ∃R Π1α = Σ1 (Jñ (R)) for some ñ such that e e

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U be the supercompactness measure on Pù1 (ë) given by of [HK81, Theorem 6.2.1]. Let ì be the measure on ë defined by ì(A) = 1 iff {X : sup X ∈ A} ∈ U. Then ì is weakly normal (that is, if h(â) < â a.e., then ∃ã < ë (h(â) < ã a.e.)) and ì(A) = 1 for every ù-club A. Let cofinal

g : ë −−−→ ñ be strictly increasing, continuous, have range cofinal in ñ, and be such that for all â < ë, Jg(â) (R) ⊀R 1 Jg(â)+1 (R). For â < ë, let also h(â) = sup{|≤| : ≤ is a prewellorder of R in Jg(â) (R)}. Then h is strictly increasing, continuous, and has range cofinal in ä 1α . e case α is The following claim is the crucial new ingredient we need in the a limit ordinal. The proof of part (2), its non-trivial part, is due to Kechris. Claim. (1) [h]ì = ä 1α , (2) [ëâ.h(â)+ ] = (äe1α )+ . e S Proof. (1) ∆1α = â<α ∆1â = Jñ (R) ∩ ℘(R), and therefore every ∆1α set e e e is ã-Suslin for some ã < ä 1α . If ä 1α is regular (i.e. ë = ä 1α ), then the proof of S e e e [Ste81, Theorem 3.2] implies that ù Π1α ⊆ Π1α , so that the ù-club subsets e be ì. But then [h] = ä 1 . e must of ä 1α generate a normal ultrafilter, which ì α e e 1 So suppose ë < ä α . The proof of [HK81, Theorem 6.2.1] shows in this case e that ì ∈ Jñ (R). Thus if ã < ä 1α , then ëã /ì has order type less than ä 1α . By e e the weak normality of ì, [h]ì is just the supremum of these order types for 1 1 ã < ä α . Thus [h]ì = ä α . (2)e That [ëâ.h(â)e+ ]ì+ ≥ (ä 1α )+ is easy to see. For the other inequality, let e all â. We shall construct a wellfounded tree ℓ : ë → ä 1α and ℓ(â) < h(â)+ for e 1 W on ä α so that for ì-a.e. â < ë, ℓ(â) ≤ |W ↾h(â)|. This implies that e [ℓ]ì ≤ [ëâ.|W ↾h(â)|]ì = |[ëâ.W ↾h(â)]ì |. But by (a) of our claim, [ëâ.W ↾h(â)]ì is a wellfounded tree on ä 1α . Thus e [ℓ]ì < (ä 1α )+ , as desired. e In order to construct W , we construct first a tree R on ù × ù × ù × ä 1α so e that for all limit ordinals ä < ë, h(â)+ = sup{|Rxy ↾h(â)| : Rxy is wellfounded}.

For this, let V = {hx, yi : Jâ (R) |= óx(0) [ëi.x(i + 1), y]},

(∗)

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where ói is the ith Σ1 formula of two free variables, and let ϑ be a Σ1 formula so that (identifying R2 with R) V (x) iff Jñ (R) |= ϑ [x]. Let also L(x, y) iff ∃ä < ñ (Jä (R) |= ϑ [x] & −ϑ [y]). Since L is Σ1 (Jñ (R)), Theorem 2.1 gives us a scale {ϕi } on L. Notice that the scale {ϕi } we get from Theorem 2.1 has the property that if L(x, y), then  ϕ0 (x, y) = ìä [Jä (R) |= ϑ [x] , and for all i > 0. ϕ(x, y) is the order type of a prewellorder of R which is Σù (Jϕ0 (x,y)(R)). Let e M (x, y) iff ∀n < ù L((x)n , y), and let {øi } be the scale on M given by øhm,ni (x, y) = ϕm ((x)n , y). Finally, let  R = (s, t, u, v) : lh(s) = lh(t) = lh(u) = lh(v) & ^
∃x ⊇ s ∃y ⊇ t ∃z ⊇ u M (x, hy, zi) & v(i) = øi (x, hy, zi) . i
In order to show (∗), fix a limit ordinal â < ë, and fix an x ∈ R so that (x)n ∈ V for all n, and in fact 
 g(â) = ìã ∀n < ù Jã (R) |= ϑ [(x)n ] . We can find such an x by the universality of V and the fact that g grows as fast as it does. Then for all y, z, M (x, hy, zi) iff hy, zi ∈ p([Rx ]), iff hy, zi ∈ p([Rx ↾h(â)]), where the second equivalence follows from our observations on thee “local” nature of {ϕi }. Thus for any y, Rxy is wellfounded iff Rxy ↾h(â) is wellfounded. Now {hy, zi : M (x, hy, zi)} is clearly universal for the class Π1 (Jg(â) (R), R), and so if we let y ∈ N iff ∃z M (x, hy, zi), R

then N is in the class ∃ Π1 (Jg(â) (R), R), but not in its dual. From Theorem 4.2 we see easily that ∃R Π1 (Jg(â) (R), R) is a class of the form Σ1ã+1 , where e Scale(Π1ã+1 ) and, by Theorem 4.3, ä 1ã+1 = h(â)+ . Now for ç < ä 1ã+1 , {y : e e e |Rxy ↾h(â)| < ç} is ∆1ã+1 , by the argument in the proof of Lemma 4.10. Since e N = {y : Rxy is wellfounded} = {y : Rxy ↾h(â) is wellfounded}, and N is not ∆1ã+1 , we have (∗). e

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We can now complete the proof of (2) of the claim in the case ä 1α is singular, that is ë < ä 1α . Fix a norm ϕ : R ։ ë so that the induced eprewellordering ≤ϕ is in eJñ (R). By the Coding lemma there is a relation P such that ∀w∃x, y P(w, x, y), and for â < ë a limit

ϕ(w) = â & P(w, x, y) → Rxy is wellfounded & ℓ(â) < hRxy ↾h(â)i . and P ∈ Jñ (R). Thus P is ã-Suslin for some ã < ä 1α ; let P = p([S]), where S e is a tree on ù 3 × ã and ã < ä 1α . Finally, let e  W = (q, r, s, t, u) : lh(q) = lh(r) = · · · = lh(u) & (q, r, s, t) ∈ S & (r, s, u) ∈ R . Then W is wellfounded. If â < ë is a limit ordinal, ã < h(â), ϕ(w) = â, and P(w, x, y) then the tree Rxy ↾h(â) can be embedded into Wwxy ↾h(â). Since |Rxy ↾h(â)| > ℓ(â), |W ↾h(â)| > ℓ(â), as desired. We shall just outline the proof of Claim (b) in the case that ä 1α is regular. S Fix a complete Π1α set A, and a Π1α -scale {ôi } on A. Since eù Π1α ⊆ Π1α , e (by [Ste81]), anyeΣ1α subset of A isebounded in ô0 —here we use theeregularity e of ä 1α . Thus ì is just the ù-club measure on ë = ä 1α , and g(â) = h(â) = â foreì-a.e. â. Consider the following Solovay game.e I II

w z, x, y



Player II wins if ∃n (w)n 6∈ A & ∀m < n ((z)m ∈ A) or ∀n (w)n , (z)n ∈ A and, if â = sup{ô((w)n ), ô0 ((z)n ) : n < w}, then Rxy is wellfounded and l (â) < |Rxy ↾â|. By boundedness and the property (∗) of R, player II has a winning strategy ó. We can now use ó and the scale {ôi } to construct W just as we used the tree S to construct W in the case ä 1α is singular. Where we e ä 1 , we now use that for used before that E was a tree on an ordinal less that α ì-a.e. â, whenever ô0 ((w)n ) < â for all n, then ôi ((w)en ) < â for all i, n. This proves the claim. ⊣ Finally, we prove boundedness property (b) of the Kunen tree coding in the case α is a limit. Suppose f : R → R is continuous and ∀ç < ä 1α ∀x ∈ R (Tx ↾ç is wellfounded → Tf(x) is wellfounded). Now for ì-a.e. eâ < ë, h(â)+ = ä 1ã+1 for some ã of type I such that Scale(Π1ã+1 ). By Lemma 4.10, e the set e 
Câ = ä < h(â)+ : ∀x ∈ R |Tx ↾h(â)| < ä → Tf(x) ↾h(â) < ä is club in h(â)+ for such â. Let C ′ = [ëâ.Câ ]ì − ä 1α , e so that by our claim C ′ is club in (ä 1α )+ . Now there is easy to show that there is a function F : (ä 1α )α → (ä 1α )+ soethat whenever â < (ä 1α )+ and W is a tree e e e

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on ä 1α so that |W | = â, then |[ëã.W ↾h(ã)]ì | = F (â). (If ì is normal, then F e identity.) Let is the C = C ′ ∩ {â : F [â] ⊆ â}.

Suppose [p]ì ∈ C and |Tx | < [p]. Then F (|Tx |) < [p]ì , so |Tx ↾h(â)| < p(â) for ì-a.e. â, so |Tf(x) ↾f(â)| < p(â) for ì-a.e. â, so F (|Tf(x) |) < [p]. But it is easy to check that â ≤ F (â) for all â, so in fact |Tf(x) | < [p]. Thus C is as demanded by boundedness requirement (b), and we have (modulo [JM83], of course) proved Theorem 4.9. From Theorem 4.9 and Lemma 4.7 we have at once the following theorem. Theorem 4.11. Let ë < ΘL(R) be a reliable ordinal. There is a Suslin cardinal κ so that κ ≤ ë < κ + . Proof. Let κ = sup{ã ≤ ë : ã is Suslin}. By Corollary 4.4, κ is Suslin. If S(κ) = Π1α where α is type IV, then Scale(Π1α ) and κ = ä 1α by Theorem 4.3. e e 1 + 1 But ∃R Πe α = Πα , so Lemma 4.7 and Theorem 4.9 combined tell us that ë < κ . e e 1 R 1 Otherwise, inspection of Theorem 4.3 shows that S(κ) = Σα+1 = ∃ Πα for e 4.7 some α. If Scale(Π1α ), then κ = ä 1α by Theorem 4.3, and eagain Lemma e and Theorem 4.9 imply that ë < eκ + . If Scale(Π1α ) fails, then Scale(Π1α+1 ) e 1 of κ, ë < κ + . e ⊣ must hold, and ä α+1 = κ + is Suslin. By the definition e Theorem 4.11 clearly implies that all reliable cardinals below ΘL(R) are Suslin. This completes the proof of Corollary 4.5. REFERENCES

Howard S. Becker [Bec80] Thin collections of sets of projective ordinals and analogs of L, Annals of Mathematical Logic, vol. 19 (1980), pp. 205–241. Leo A. Harrington and Alexander S. Kechris [HK81] On the determinacy of games on ordinals, Annals of Mathematical Logic, vol. 20 (1981), pp. 109–154. Stephen Jackson and Donald A. Martin [JM83] Pointclasses and wellordered unions, In Kechris et al. [Cabal iii], pp. 55–66. Ronald B. Jensen [Jen72] The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308. Alexander S. Kechris [Kec78] AD and projective ordinals, In Kechris and Moschovakis [Cabal i], pp. 91–132. [Kec81] Suslin cardinals, κ-Suslin sets, and the scale property in the hyperprojective hierarchy, this volume, originally published in Kechris et al. [Cabal ii], pp. 127–146. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer, 1981.

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[Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. Alexander S. Kechris, Robert M. Solovay, and John R. Steel [KSS81] The axiom of determinacy and the prewellordering property, In Kechris et al. [Cabal ii], pp. 101–125. Alexander S. Kechris and W. Hugh Woodin [KW83] Equivalence of determinacy and partition properties, Proceedings of the National Academy of Sciences of the United States of America, vol. 80 (1983), no. 6 i., pp. 1783–1786. Donald A. Martin [Mar83] The largest countable this, that, and the other, this volume, originally published in Kechris et al. [Cabal iii], pp. 97–106. Donald A. Martin and John R. Steel [MS83] The extent of scales in L(R), this volume, originally published in Kechris et al. [Cabal iii], pp. 86–96. Yiannis N. Moschovakis [Mos78] Inductive scales on inductive sets, this volume, originally published in Kechris and Moschovakis [Cabal i], pp. 185–192. [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. [Mos81] Ordinal games and playful models, In Kechris et al. [Cabal ii], pp. 169–201. [Mos83] Scales on coinductive sets, this volume, originally published in Kechris et al. [Cabal iii], pp. 77–85. Stephen G. Simpson [Sim78] A short course in admissible recursion theory, Generalized recursion theory II, Studies in Logic, vol. 94, North Holland, Amsterdam, 1978. John R. Steel [Ste81] Closure properties of pointclasses, In Kechris et al. [Cabal ii], pp. 147–163. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720, USA

E-mail: [email protected]

SCALES IN K(R)

JOHN R. STEEL

§1. Introduction. In this paper, we shall extend the fine-structural analysis ~ R), of scales in L(R) [Ste83A] and L(ì, R) [Cun90] to models of the form L(E, ~ constructed over the reals from a coherent sequence E of extenders. We shall ~ R) show that in the natural hierarchy in an iterable model of the form L(E, satisfying AD, the appearance of scales on sets of reals not previously admitting ~ and individual a scale is tied to the verification of new Σ1 statements about E ~ reals in exactly the same way as it is in the special case E = ∅ of [Ste83A]. For example, we shall show: Theorem 1.1. Let M be a passive, countably iterable premouse over R, and suppose M |= AD; then the pointclass consisting of all ΣM 1 sets of reals has the scale property. A premouse is said to be countably iterable if all its countable elementary ¨ submodels are (ù1 + 1)-iterable. It is easy to show, using a simple LowenheimSkolem argument, that if M and N are ù-sound, countably iterable premice over R which project to R, then either M is an initial segment of N , or vice versa. We shall write K(R) for the “union” of all such premice over R, regarded as itself a premouse over R. This is a small abuse of notation, since our K(R) is determined by its sets of reals, but since we are concerned with the scale property, sets of reals are all that matter here. In fact, as in [Ste83A] and the work of [Mos83] and [MS83] on which it rests, our existence results for scales require determinacy hypotheses, and so we are really only concerned here with the longest initial segment of K(R) satisfying AD . Section 2 is devoted to preliminaries. In section 3 we show that for any R-mouse M satisfying “Θ exists”, HODM is a T -mouse, for some T ⊆ ΘM .1 We use this representation of HODM in the proof of Theorem 1.1, which is given in section 4. There we also extend the proof of Theorem 1.1 so as to obtain a complete description of those pointclasses which have the scale property and are definable over initial segments of K(R) satisfying AD. 1 What

we actually show is slightly weaker than this in some very technical respects.

The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

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§2. Preliminaries. 2.1. Potential R-premice. We shall be interested in premice built over R, which we take to be Vù+1 in this context, but nevertheless refer to as the set of all reals on occasion. In most respects, the basic theory of premice built over R is a completely routine generalization of the theory of ordinary premice (built over ∅); however, because R-premice do not in general satisfy the axiom of choice, one must be careful at a few points. Here are some details. Let M be a transitive, rud-closed set, and X ∈ M . Let E be an extender over M . We say that E is (M, X )-completeT iff whenever a is a finite subset of lh(E) and f : X → Ea and f ∈ M , then ran(f) ∈ Ea . In the contrapositive: whenever g : [crit(E)]<ù → ℘(X ) is in M , then the following implication holds: if for Ea -almost every u there is an i ∈ X such that i ∈ g(u), then there is an i ∈ X such that for Ea -almost every u, we have i ∈ g(u). It is clear that if E is (M, X )-complete and (M, Y )-complete, then E is (M, X × Y )-complete. Thus if E is (M, X )-complete and α < crit(E), then E is (M, X ×α)-complete. For any transitive set M , let o(M ) be the least ordinal not in M . Definition 2.1. Let M be transitive and X ∈ M ; we say M is wellordered mod X iff ∀Y ∈ M ∃α ∈ o(M )∃g ∈ M (g : X × α ։ Y ). Our R-premice will be wellordered mod R, moreover, if we take an ultrapower of such a premouse M by an extender E, then E will be (M, R)complete. In this context we have Proposition 2.2. Let M be transitive, rud-closed, and wellordered mod X , where X is transitive. Let E be an extender over M ; then the following are equivalent: 1. E is (M, X )-complete, 2. ult(M, E) satisfies the Ło´s theorem for Σ0 formulae, and the canonical embedding from M to ult(M, E) is the identity on X ∪ {X }. Proof. We shall just sketch (1) ⇒ (2), which is the direction we use anyway. The usual proof of Ło´s’s theorem works except at the point where one would invoke the axiom of choice in M . At this point we have assumed
for Ea -almost every u M |= ∃v ∈ g(u)ϕ[v, f1 (u), . . . , fk (u)] , where ϕ is Σ0 , and g, f1 , . . . , fk are in M , and we wish to find f in M such that
for Ea -almost every u M |= ϕ[f(u), f1 (u), . . . , fk (u)] . Now since M is wellordered mod X , we can fix h ∈ M so that h : X × α ։ S ran(g). For u ∈ dom(g) and â < α, set f ∗ (u, â) = {i ∈ X : M |= ϕ[h(i, â), f1 (u), . . . , fk (u)]}. For u ∈ dom(g), let t(u) = f ∗ (u, âu ), where âu is least s.t. f ∗ (u, âu ) 6= ∅,

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and let t(u) = ∅ if f ∗ (u, â) = ∅ for all â. Because M is rud-closed, the functions f ∗ and t are in M . But now for Ea -almost every u there is an i ∈ t(u), and so by (M, X )-completeness we can fix i0 such that for Ea -almost every u, we have i0 ∈ t(u). The desired function f is then given by f(u) = h(i0 , âu ). To see that the canonical embedding j is the identity on X ∪ {X }, suppose g ∈ M maps [crit(E)]|a| to X . We have that for Ea -almost every u there is an i ∈ X such that i = g(u), hence we can fix i ∈ X such that g(u) = i for Ea -almost every u. It follows that [g] = j(i). ⊣ ~

~ we define JE For X transitive and appropriate E, α (X ) by: ~

JE 0 (X ) = X, ~

~

~

E E JE α+1 (X ) = rud-closure of Jα (X ) ∪ {Jα (X ), Eα },

~ are those such that each and taking unions at limits. Here the appropriate E ~ ~ E Eα is either the emptyset or an extender over Jα (X ) which is (JE α (X ), X )complete. We write ~ ~ ~ JαE (X ) = (JE α (X ), ∈, E ↾ α, Eα , X )

for the structure for the language of set theory expanded by predicate symbols ~ F˙ for Eα , and a constant symbol R ˙ for X (chosen because X = E˙ for E, ~ E 2 R ∩ Jα (X ) is the case of greatest interest). This language of relativised premice we call L∗ . ~ is X -acceptable at α iff ∀â < α∀κ Definition 2.3. An appropriate E ~ ~ ~ ~ ~ ~ E E E E E (℘(Jκ (X ) ∩ (Jâ+1 (X ) \ Jâ (X )) 6= ∅) ⇒ (JE â+1 (X ) |= ∃f : Jκ (X ) ։ Jâ (X )). The following proposition is a uniform, local version of the fact that that ~ X ] is ordinal-definable from parameters in X ∪ {X }. every set in L[E, Proposition 2.4. 1. There is a fixed Σ1 formula ϕ0 of our expanded ~ is appropriate for X , and language such that whenever X is transitive, E ~ E ~ then ϕ defines over Jα (X ) a map h : (X <ù × [α]<ù ) ։ α < lh(E), ~ ~ E,X JE for the map h so defined. α (X ). We write hα ~ 2. We can (and do) take the maps hαE,R to have domain R×[α]<ù (replacing ϕ0 with ϕ1 , another Σ1 formula). ~ is appropriate for R, then for any α, there is a map from R × α onto 3. If E ~ ~ E Jα (R) which is Σ1 definable from parameters over JαE (R). 2 If

˙ to name 0. α = 0, take R

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The proof is a routine extension of Jensen’s [Jen72]. See [Ste83A] for the ~ = ∅. We shall need the uniformly Σ1 maps of assertion (2) later on. case E Using Proposition 2.4, we can reformulate R-acceptability with R × κ re~ placing JE κ (R), etc. Let us call ë an R-cardinal iff there is no map f : κ ×R ։ ë ~ ~ is R-acceptable at α, and JE with κ < ë. It is then easy to see that if E α (R) |= ë ~ ~ ~ E E is an R-cardinal, then for all κ < ë, ℘(JE κ (R)) ∩ Jα (R) ⊆ Jë (R). Definition 2.5. Θ is the least ordinal which is not the surjective image of R. Let M be an RM -premouse satisfying “Θ exists”, and let è = ΘM . It is easy to see, using Proposition 2.4, that è is regular in M. By acceptability, M|è satisfies “every set is the surjective image of R”. We can conclude then that the structure M|è is admissible. It is easy to show, without using the axiom of choice, that for ë > 1, ë is an ~ is R-acceptable at α, then R-cardinal iff ë is a cardinal and ë ≥ Θ. Thus if E ~ ~ ~ E E + Jα (R) satisfies: “whenever κ ≥ Θ and κ exists, then ℘(JE κ (R)) ⊆ Jκ + (R)”. Definition 2.6. Let X be transitive; then a fine extender sequence over X ~ such that for each α ∈ dom(E), ~ E ~ is X -acceptable at α, and is a sequence E ~ either Eα = ∅ or Eα is a (κ, α) pre-extender over JE α (X ) for some κ such that ~ ~ ~ E E JE α (X ) |= ℘(Jκ (X )) exists, and Eα is (Jα (X ), X )-complete, and: Eα satisfies clauses (1), (2), and (3) of [Ste07B, Definition 2.4]. Of course, [Ste07B, Definition 2.4] was formulated there for the case X = ∅, but it is now easy to see what its clauses should mean in the general case. Definition 2.7. A potential premouse over X (or X -ppm) is a structure of ~ ~ is a fine extender sequence over X . If M = JαE~ (X ) the form JαE (X ), where E ~ is an X -ppm, we write JâM , or simply M|â, for the structure JâE (X ). We say N is an initial segment of M, and write N E M, iff N = M|â for some â. Active potential premice are in general not amenable structures, but we can code an active X -ppm M by an amenable structure C0 (M). That involves replacing the last extender F = F˙ M with F ↾í(F ) in the case í(F ) (the sup of the generators of F ) is a limit ordinal, and with a certain predicate F ∗ coding the fragments of F in the case í(F ) is a successor ordinal. The details are given in [Ste07B, 2.11]. The “Σ0 -code” C0 (M) is also a structure for the language L∗ , and it is really this interpretation of L∗ which is of importance in what follows. Whenever we speak of definability over a ppm M, we shall in reality mean definabilty over C0 (M). 2.2. Cores, Projecta, Soundness. These notions carry over routinely to X ppm. One need only remember that X ∪ {X } is contained in all cores of

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X -ppm. For example, we define the first projectum, standard parameter, and core of an X -ppm M by:3 ñ1 (M) = least α such that for some boldface Σ1C0 (M) set A, e A ⊆ M|α and A 6∈ M, p1 (M) = least p ∈ <ù [o(M)] such that there is an A ⊆ M|ñ1 (M) s.t. A is Σ1C0 (M) in p but A 6∈ M, C1 (M) = Σ1 Skolem hull of M generated by M|ñ1 (M) ∪ p1 (M). Here we order finite sets of ordinals by listing their elements in decreasing order, and comparing the resulting finite sequences lexicographically. It is possible that M|ñ1 (M) = M; if this is not the case, then ñ1 (M) is an X cardinal of M. In the case ñ1 (M) = 1, we shall generally write ñ1 (M) = X instead. The core C1 (M) is taken to be transitive, and a structure for the language of X -ppm, and so taken, it is in fact an X -ppm. The definitions of solidity and universality for the standard parameter go over to X -ppm in the obvious way. We say p1 (M) is universal if ℘(M|ñ1 (M))∩ M ⊆ C1 (M). We say p = p1 (M) is solid if for each α ∈ p, letting b be the set of Σ1 sentences in our expanded language augmented further by names for all elements of (p \ (α + 1)) ∪ α which are true in M, we have b ∈ M. We call such b the solidity witnesses for p1 (M). If p1 (M) is solid and universal, we go on to define ñ2 (M), p2 (M), and C2 (M); the reader should consult [MS94] for all further details here regarding the ñn (M), pn (M), and Cn (M) for n > 1. In this paper, when we need to go into fine-structural details, we shall stick to the representative case n = 1. Definition 2.8. An X -ppm M is n-solid iff Cn (M) exists, and pn (Cn (M)) is solid and universal. M is n-sound iff M is n-solid, and M = Cn (M). M is an X -premouse iff every proper initial segment of M is ù-sound (i.e., n-sound for all n < ù). 2.3. Ultrapowers, Iteration Trees. It is easy to adapt the material of [MS94, §§ 4& 5], or [Ste07B, §§ 2&3], to X -premice. The definitions make sense, and the theorems continue to hold true, if one replaces “premouse” with “X premouse” everywhere. The following propositions summarize some of the basic facts: Proposition 2.9. Let M be an n-sound premouse over X , and let E be an extender over M which is (M, X )-complete; then 3 Here and elsewhere, we write M for the universe of the structure M, if no confusion can come from doing so.

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1. For any generalized rΣn formula ϕ, functions fi definable over M from parameters using Σn Skolem terms ôi ∈ Skn , and a ∈ [lh(E)]<ù such that dom(fi ) = [crit(E)]|a| for all i, the following are equivalent: (a) ultn (M, E) |= ϕ[[a, f0 ], . . . , [a, fk ]], and (b) for Ea -almost every u, M |= ϕ[f0 (u), . . . , fk (u)]. 2. The canonical embedding iEM from M to ultn (M, E) is an n-embedding. 3. Suppose also E is close to M, ñn+1 (M) ≤ crit(E), and pn+1 (M) is solid and universal; then ñn+1 (M) = ñn+1 (ultn (M, E)), and iEM (pn+1 (M)) = pn+1 (ultn (M, E)), and pn+1 (ultn (M, E)) is solid and universal. Here the ultrapower ult0 (M, E) is formed using the functions fi ∈ M. Thus we have already proved the n = 0 case of Proposition 2.9 (1). There are no new ideas in the rest of the proof. Proposition 2.10. Suppose T is an n-maximal iteration tree on the n-sound X -premouse M, and that αTâ and D T ∩ (α, â]T = ∅; then the canonical T embedding iα,â ◦ iα∗ from M∗α to MTâ is a k-embedding, where k = degT (â). T ◦ i ∗ (pk+1 (M∗α )) = pk+1 (MTâ ). Moreover, if crit(iα∗ ) ≥ ñk+1 (M∗α ), then iα,â The notions of [MS94] and [Ste07B] associated with iterability relativise to X -premice in an obvious way. Definition 2.11. Let M be a k-sound X -premouse; then we say M is countably k-iterable iff whenever M is a countable X -premouse, and there is a ð : M → M which is fully L∗ -elementary, then M is (k, ù1 + 1)-iterable. We say M is countably iterable just in case it is countably ù-iterable. The comparison process yields Theorem 2.12. Let M and N be X -premice. and suppose that M is m + 1sound and countably m-iterable, where ñm+1 (M) = X , and N is n + 1sound and countably n-iterable, where ñn+1 (N ) = X ; then either M E N or N E M. We can therefore define Definition 2.13. For any transitive set X , K(X ) is the unique X premouse M whose proper initial segments are precisely all those countably iterable X -premice N such that ñù (N ) = X .

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§3. Some local HOD’s. Our analysis of scales proceeds by getting optimal closed game representations. In the relevant closed game, player I attempts to verify that a given R-mouse M satisfies ϕ(x), where ϕ is Σ1 and x ∈ R. He ˙ N -premouse N satisfying ϕ(x); one can think of does so by describing an R him as claiming that his N is an elementary submodel of M. Player II helps keep player I honest about this by playing reals which player I must then put into N . In order to ensure that player I is indeed being honest about what is true in M, we must ask him in addition to verify that his N is iterable. Of course, the obvious way to verify this is to play an elementary ð : N → M, but this leads to a payoff condition for player I which is not closed in the appropriate topology.4 Our main new idea here is just that player I can verify iterability by elementarily embedding HODN into HODM . The key here is that HODM is definably wellordered,5 so that the embedding is essentially an ù-sequence of ordinals, and the elementarity condition is closed in the appropriate topology. We must consider here iterations of N involving Σn -ultrapowers of the form ultn . We shall reduce such ultrapowers to Σn -ultrapowers of HODN , and to do so we need a fine-structure theory for HODN . Fortunately, we can restrict ourselves to N satisfying “Θ exists”, and be content with a fine-structural analysis of HODN above ΘN : that is, a representation of HODN as an X mouse, for some X ⊆ ΘN . Now in the case N = L(R), the following theorem of Woodin does the job: Theorem 3.1 (Woodin). There is a partial order P on ΘL(R) such that HODL(R) = L(P), and moreover L(R) is an inner model of a P-generic extension of HODL(R) . Here P is a modification of the Vopˇenka partial order designed to add a generic enumeration of R. We shall extend Woodin’s argument so as to show that if M is an R-premouse satisfying “Θ exists”, then there is a P ⊆ ΘM such that HODM is the universe of a P-premouse H. The main new thing here is to show that the projecta and standard parameters of levels of H match those of the corresponding levels of M, and indeed establish level-by-level intertranslatability of the theories of initial segments of H and M respectively. This we get from the fact that 4 Player I may play reals as well as ordinals in our game, and the elements of ran(ð) can be coded by pairs hx, αi where x ∈ R and α ∈ Ord. However, the elementarity requirement on ð would not be closed in the appropriate topology on (Ord ×R)ù , which is the product of ù copies of the discrete topology on Ord and the Baire (not discrete) topology on R. 5 The definition which guarantees a set is in HODM must use the language L∗ of premice. E˙ and F˙ are allowed, but names for individual reals are not! Further, this definition must be interpreted over some proper initial segment of M; there may be sets of ordinals in M which are definable over M itself, yet not in HODM .

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M is a symmetric6 inner model of a P-extension of H, using the level-by-level definability of forcing. In turning to the details, it will be convenient to replace P with a superficially more powerful set. Let us fix for the remainder of this section an RM-premouse M such that • M |= “Θ exists”. We set è = ΘM , and we also fix an n0 < ù such that • M is n0 -sound and ñn0 (M) ≥ è. Finally, letting o(M) = ùã0 , we assume that • for all hî, ki
H1

(RM ) ∼ = M|ô

for some ô < è. In fact, if hî, ki ≤lex hã0 , n0 i and k ≥ 1, then for any finite F ⊆ ùî, M|î Hk (RM ∪ F ) ∈ M|è, since the theory of the hull is in M|è, and the latter M|î is an admissible structure. However, Hk (RM ∪ F ) may not be sound, and hence may not be of the form M|ô. ˙M

˙ M ↾ ã0 , ∅). Corollary 3.3. M|è is a Σ1 -elementary submodel of (JE ã0 , ∈, E Set T M = {hϕ, αi ~ : α ~∈

<ù

è and M|è |= ϕ[α]}. ~

¨ Since è is a cardinal in M, we can use Godel’s pairing function to identify T M with a subset of è. Letting ùç = o(M), we have that HODM|è ∩ Vè = Jç (T M ) ∩ Vè . Since M|è is a Σ1 -elementary submodel of M with its last extender removed, we have HODM ∩ Vè = Jç (T M ) ∩ Vè . We can construct a T M -premouse whose universe is the whole of HODM by simply constructing from T M together with the extenders from the Msequence having critical points above è. More precisely, letting M = JãE0 (RM ), ~

6 We

shall explain the meaning of this shortly. analysis of HODM we are developing will be used to show that M is countably n0 iterable, given that HODM is. 7 The

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we define an appropriate sequence F~ over è ∪ {T M } by setting Fα = Eè+α ∩ JFα ↾α (è ∪ {T M }) ~

for all α such that è +α ≤ ã0 . It is not hard to see that the sequence F~ is indeed appropriate for è ∪ {T M }; the main point is that crit(Eè+α ) > ΘM|(è+α) = è, which implies that Fα is sufficiently complete. We set Hα = JαF (è ∪ {T M }), ~

for all α such that è + α ≤ ã0 . It will be convenient to ignore the Hα for small α. Therefore, we add to our assumptions on M that there is some ë ≤ ã0 such that M|ë |= ZF, and let ë0 = least ë such that M|ë |= ZF. Notice that our new assumption on M holds if some Eè+α 6= ∅, and in this case crit(Eè+α ) > ë0 . Since in our closed game representation we only need the Hα to “verify” extenders from the M-sequence with index above è, we can afford to ignore Hα except when ë0 exists and ë0 ≤ è + α. Note that è + α = α for the α we do not ignore, so that we are already rewarded for our ignorance. We set H = Hã0 . We shall show that for α ≥ ë0 , Hα is a T M -premouse and M|α is an inner model of a generic extension of Hα .8 The relevant partial order is the same Vopˇenka-like partial order used by Woodin. Let us work in M for a while. Fix a bijection ð : è → O, where O is the collection of all subsets of Rn = {s : s : n → R} which are definable from ordinal parameters over M|è. We choose ð so that it is definable over M|è. We write A∗ for ð(A) henceforth. Let A ∈ Vopn ⇔ ∃n < ù(A∗ ⊆ Rn ∧ A∗ 6= ∅), and A ∈ Vopù ⇔ ∃n < ù(A ∈ Vopn ). For A in Vopù , we write s(A) for the unique n < ù such that A ∈ Vopn . For A, B ∈ Vopù , we put A ≤v B ⇔ s(B) ≤ s(A) ∧ ∀s ∈ A∗ (s↾s(B) ∈ B ∗ ). We also use Vopù to denote the partial order (Vopù , ≤v ). Clearly, Vopù is coded into T M in a simple way, and hence Vopù ∈ H|2. The standard Vopˇenka argument shows that for any n < ù, Vopn is a complete Boolean algebra in H, and each s ∈ Rn determines an H-generic filter Gs = {A ∈ Vopn : s ∈ A} on Vopn .9 It is easy to see that the inclusion 8 The

first assertion is true for smaller α as well. that any subset of R which is ODM is actually ODM|è , since M|è is a Σ1 elementary submodel of M. 9 Note

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map is a complete embedding of Vopn into Vopù . Motivated by this, we define for h : ù → R and A ∈ Vopù : A ∈ Gh ⇔ h↾s(A) ∈ A∗ . Lemma 3.4. If h is M-generic over Col(ù, R), then Gh is H-generic over Vopù . Proof. Let D be dense in Vopù , and D ∈ M. Let s ∈ Rn be a condition in Col(ù, R). It will be enough to find a t extending s in Col(ù, R) such that Gt ∩ D 6= ∅. Let X = {u ∈ Rn : ∃B ∈ D∃t(u ⊆ t ∧ t ∈ B ∗ )}. We want to see s ∈ X , so it will be enough to see X = Rn . Suppose not; then since X is clearly OD in M, there is an A ∈ Vopn such that A∗ = Rn \ X. Since D is dense, we can find B ∈ D such that B ≤ A. But now pick any t ∈ B, and it is clear that t↾n ∈ X , a contradiction. ⊣ We can recover h from Gh in a simple way. For b ∈ Vù and n < ù, let Ab,n ∈ Vopn+1 be such that A∗b,n = {s ∈ Rn+1 : b ∈ s(n)}. We assume that the map hb, ni 7→ Ab,n is definable over Mè , and hence in H, as any natural such map will be.10 Then clearly, b ∈ h(n) ⇔ Ab,n ∈ Gh . We define Vopù -terms for the h(n) and ran(h) by ˇ : A ≤v Ab,n }, ón = {hA, bi and

R˙ = {hA, ón i : A ∈ Vopù ∧ n < ù}. These terms are of course in H. It is easy to see Lemma 3.5. 1. For any h : ù → R, ónGh = h(n) for all n, and R˙ Gh = ran(h). 2. If h is M-generic for Col(ù, RM ), then R˙ Gh = RM . 3. For any condition A ∈ Vopù , there is an H-generic filter G on Vopù such that A ∈ G and R˙ G = RM . By Lemma 3.5, truth in H(RM ) can be reduced to truth in H via the forcing relation for Vopù . In order to see that H(RM ) determines M we need to know ~ For this, that the extenders on F~ generate the corresponding extenders on E. we need that the forcing relation for Vopù is locally definable. We also need

10 The ∗ map is one-one on the separative quotient of Vop , so the question as to what to ù choose for Ab,n disappears if we replace Vopù with its separative quotient.

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this local definability to show that the reduction of M-truth to H-truth is local, and thereby that H is a T M -premouse. We shall use the usual Shoenfield terms for our forcing language. Besides these terms, the language of forcing over an amenable structure (JA î (X ), ∈, A, X, B) has ∈, =, a constant symbol R˙ for X , and predicate symbols E˙ and F˙ for A and B. A filter G over a poset P ∈ JA î (X ) is generic over this structure just in case A G it meets all P-dense sets D ∈ JA : ô ∈ JA î (X ). We let Jî (X )[G] = {ô î (X )} be the set of G-interpretations of terms, and say A p ϕ ⇔ ∀G(G is generic over JA î (X ) ⇒ (Jî (X )[G], ∈, A, X, B) |= ϕ).

We use SαA (X ) for the αth level of Jensen’s S-hierarchy on JA î (X ). Let Σ0,n be the collection of Σ0 sentences of the forcing language containing at most n bounded quantifiers. Lemma 3.6. Let (JA î (X ), ∈, B) be amenable, and let P be a poset, with P ∈ JA (X ), where í < î and JA í í (X ) |= ZFC. For í ≤ α < ùî, let Fα,n = {hp, ϕi : p ∈ P ∧ ϕ ∈ (Σ0,n ∩ SαA (X )) ∧ p ϕ}. Then 1. ∀α < ùî∀n < ù(Fα,n ∈ JA î (X )); moreover the function hα, ni 7→ Fα,n is (JA (X ),∈,A,X,B)

Σ1 î in the parameter P (uniformly in î). A 2. If (Jî (X )[G], ∈, A, X, B) |= ϕ, where ϕ is Σ0 and G is generic over JA î (X ), then ∃p ∈ G(p ϕ). ∗ A proof of Lemma 3.6 can be organized as follows. Let Fα,n = {hp, ϕi ∈ ∗ Fα,n : F˙ does not occur in ϕ}. One first proves the lemma with Fα,n replacing Fα,n . This amounts to observing that the standard inductive definition of forcing for Σ0 sentences is a “local Σ0 -recursion” of the same sort that defines the function α 7→ SαA (X ) itself in a Σ1 way over (JA î (X ), ∈, A, X ). Of course, one needs that everything true is forced to verify that the inductive definition works. The reason for restricting ourselves to α ≥ í is that we need a starting point for the induction, and when α = í, Lemma 3.6 literally is a standard basic forcing lemma.11 Finally, one can show that Fα,0 is uniformly ∗ rudimentary in hFα,0 , B ∩ SαA (X )i (since p F˙ (ô) ⇔ ∀q ≤ p∃r ≤ q∃x ∈ B(r ô = x)). ˇ Also, Fα,n+1 is uniformly rudimentary in Fα,n . This completes our pseudo-proof of Lemma 3.6. As a consequence of Lemma 3.6, we get the level-by-level adequacy of the Shoenfield terms: 11 For small α, we have the problem that there may be sentences in S A (X ) involving terms of α rank greater than α.

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Lemma 3.7. Let G be H-generic over Vopù ; then for all î such that ë0 ≤ ~ ~ î ≤ ã0 , JFî (T M )[G] = JFî (hT M , Gi). If G is a Vopˇenka-generic over H such that R˙ G = RM , then H[G] can recover M: Lemma 3.8. If G is Vopù -generic over H, R˙ M = RM , and ë0 ≤ î ≤ ã0 , then M|î is ∆1 -definable over Hî [G] from the parameter G; moreover, this definition is uniform in such G and î. Proof (Sketch.) For î = ë0 this is clear. In general, what we need to see is that if Fî 6= ∅, then Fî determines the corresponding extender Eî H [G] on the M-sequence in a ∆1 î way. We may assume by induction that ~

M JE î (R ) ⊆ Hî [G]. Since Vopù has cardinality strictly less than crit(Fî ) in ~

M <ù Hî , Fî lifts to an extender F ∗ over JE î (R ) defined by: for a ∈ [lh(Fî )]

and Z ⊆ [crit(Fî )]|a| such that Z ∈

~ M JE î (R ),

Z ∈ Fa∗ ⇔ ∃Y (Y ∈ (Fî )a ∧ Y ⊆ Z). ~

M Clearly, any extender over JE î (R ) whose restriction to sets in Hî is Fî must ∗ then be equal to F . Thus Eî = F ∗ , and hence Eî is ∆1 over Hî [G] in the parameter G. The uniformity in î and G is obvious upon inspection of the definition we have given. (The uniformity is needed to pass through limit stages.) ⊣

Theorem 3.9. H is a T M -premouse; moreover for all k ≤ n0 , H is k-sound, ñk (H) = ñk (M), and pk (H) = pk (M) \ {è}. Proof of Theorem 3.9. We show by induction on î such that ë0 ≤ î ≤ ã0 , that Hî is a T M -premouse, and if 0 ≤ k ≤ ù, and k ≤ n0 if î = ã0 , then Hî is k-sound, ñk (Hî ) = ñk (M|î), and pk (Hî ) = pk (M|î). This is clear for î = ë0 . Now let î > ë0 . We first show that Hî is a premouse. Since all proper initial segments of Hî are ù-sound T M -premice by our induction hypotheses, it suffices to show that Hî is a T M -ppm. If Fî = ∅, this is trivial, so assume Fî = Eî ∩ Hî where Eî is an extender over M|î. We must verify that F~ has the properties of a fine extender sequence at î, that is, that it satisfies clauses (1)-(3) of [Ste07B, Definition 2.4]. Let us write F = Fî and E = Eî . Notice that î = lh(F ) = lh(E) = o(M|î) = o(Hî ). Set κ = crit(F ) = crit(E). Claim 3.10. If a ∈ [î]<ù and f ∈ M|î and f : [κ]|a| → Hî , then there is a Z ∈ Fa such that f↾Z ∈ Hî . Proof. We need to take a little care with the standard argument because Fa 6∈ M is possible. Note that by Proposition 2.4, f is definable over some M|ã, where ã < î, from ordinals and a real x0 . We can therefore fix a term f˙

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in Hî such that whenever h is M-generic over Col(ù, R), and G = Gh is the associated Vopˇenka generic, then f˙G = f. We claim there is an A ∈ Vopù such that ˙ for Fa -almost every u, ∃ç(A f(u) = ç), and ∃s ∈ A∗ (s(0) = x0 ). If not, then for each A ∈ Vopù such that ∃s ∈ A∗ (s(0) = x0 ), the set ZA of all ˙ u ∈ [κ]|a| such that A forces no value for f(u) is in Fa . The local definability of Vopˇenka forcing implies that the function A 7→ ZA is in M (in fact, in Hî ). But F is (M, R)-complete, and hence we have a u such that u ∈ ZA whenever ZA is defined. Now let h be M-generic over Col(ù, R) with h(0) = x0 . Then ˙ ∃A ∈ Gh ∃ç(A f(u) = ç), so ZA is defined and u 6∈ ZA , a contradiction. Now let A be as in our claim, and let Z be the set of all u such that ˙ ∃çA f(u) = ç. It is clear that f↾Z can be computed inside Hî from ˙ A, f, and the forcing relation. (Note that there is an M-generic h such that h(0) = x0 and A ∈ Gh .) ⊣ (Claim 3.10) Claim 3.11. The extenders F and E have the same generators. Proof. If ç < lh(F ) is not a generator of F , then there is an f ∈ Hî and finite a ⊆ ç such that f(u) = v for (F )a∪{ç} -almost every u ∪ {v}. Since f ∈ M|î and F ⊆ E, this means that ç is not a generator of E. Conversely, if ç is not a generator of E, as witnessed by f ∈ Mî and a ⊆ ç finite, then we can apply Claim 3.10 to see that ç is not a generator of F . ⊣ (Claim 3.11) Claim 3.12. For all è ≤ ç < î, ç is a cardinal of Hî iff ç is a cardinal of M|î. Proof. If ç is a cardinal of M|î, then ç is a cardinal of the smaller model Hî . If ç > è is a cardinal of Hî and G is Vopù -generic over Hî , then ç is a cardinal of Hî [G]. Choosing G so that R˙ G = RM , we see that ç is a cardinal of M|î. For ç = è, we have that ç is a cardinal of both models. ⊣ (Claim 3.12) We can now verify the first clause in the definition of fine extender sequences, that î = í(F )+ in ult(Hî , F ). We have í(F ) = í(E) by Claim 3.11, ~ is a fine extender sequence. Letting and î = í(E)+ in ult(M|î, E) because E iE : M|î → ult(M|î, E) be the canonical embedding, we have ult(Hî , F ) = iE (Hî ) by our first claim.12 By Claim 3.12 and the elementarity of iE , ult(M|î, E) has the same cardinals as iE (Hî ), so î = í(E)+ in iE (Hî ). Thus î = í(F )+ in ult(Hî , F ), as desired. 12 Where

iE (Hî ) is the “union” of the iE (Hç ) for ç < î.

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To verify clause 2, coherence, notice that iE (F~ )↾î = F ↾î by coherence for ~ sequence and the fact that the Fα are uniformly locally definable from the E the Eα . Since by Claim 3.10, iF (F~ ) = iE (F~ ), we are done. The initial segment condition for E easily implies the initial segment condition for F ; we leave the details to the reader. We have therefore shown that Hî is a T M -premouse. We now show by induction on k such that k ≤ n0 if î = ã0 that ñk (Hî ) = ñk (M|î), pk (Hî ) = pk (M|î) \ {è}, and Hî is k-sound. This is trivial if k = 0. Let us first consider the case k = 1. The key is that the Σ1 theories (in the language L∗ ) of Hî and M|î are intertranslatable. First, let us translate the Σ1 theory of Hî into that of M|î. Here we shall expand the latter theory by allowing a name for è; notice that T M is Σ1 -definable over M|î from the parameter è. We can then see that the universe of Hî , together with the interpretations of E˙ and F˙ in Hî , are ∆1 -definable over M|î from è. Clearly ì˙ Hî = ì˙ M|î = κ, and í˙ Hî = í˙ M|î by Claim 3.11 above. We leave it to the reader to show that ã˙ Hî is Σ1 -definable over M|î from è; this is a bit of a mess because of the “one ultrapower away” case in the initial segment condition, but otherwise routine.13 These calculations constitute a proof of: Claim 3.13. There is a recursive map ϕ(v1 , . . . , vn ) 7→ ϕ ∗ (v0 , v1 , . . . , vn ) associating to each Σ1 formula of L∗ a Σ1 formula of L∗ with one additional free variable, such that for all ϕ(v1 , . . . , vn ) and a1 , . . . , an , Hî |= ϕ[a1 , . . . , an ] ⇔ M|î |= ϕ ∗ [è, a1 , . . . , an ]. We translate in the other direction using the strong forcing relation for Σ1 formulae and Lemma 3.6. Let L∗∗ be the sublanguage of L∗ with symbols ˙ F˙ . If ∈, =, E, ϕ(v1 , . . . , vn ) = ∃u1 . . . ∃uk ø(u1 , .., uk , v1 , . . . , vn ) where ø is a Σ0 formula of L∗∗ , then for p ∈ Vopù and ô1 , . . . , ôn Shoenfield terms, we put p s ϕ(ô1 , . . . , ôn ) ⇔ ∃r1 . . . ∃rk (p ø(r1 , . . . , rk , ô1 , . . . , ôn )). Now if G is generic over Hî for Vopù and R˙ G = RM , then the universe of M|î is ∆1 definable over Hî [G] from R˙ G ; moreover, the interpretations in M|î of the symbols of L∗ are ∆1 definable over Hî [G] from their interpretations in Hî .14 Since strong forcing equals truth, we get 13 The extenders E and F fall under the same case in the initial segment condition, and ã˙ Hî = ã˙ M|î unless E and F are type II, and their last initial segments are an ultrapower away from the corresponding sequence. 14 Our previous comments regarding ã˙ apply here too.

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Claim 3.14. There is a recursive map ϕ(v1 , . . . , vn ) 7→ ϕ † (w, x, y, z, v1 , . . . , vn ) associating to each Σ1 formula of L∗ a Σ1 formula of L∗∗ with four additional free variables, such that whenever G is generic over Hî for Vopù and R˙ G = RM , and ô1 , . . . , ôn are Shoenfield terms, then M|î |= ϕ[ô1G , . . . , ônG ] if and only if ˙ x, ∃p∃w, x, y(p ∈ G∧hx, y, zi = hã˙ Hî , ì˙ Hî , í˙Hî i∧p s ϕ † (R, ˇ y, ˇ z, ˇ ô1 , . . . , ôn )). These translations give Claim 3.15. ñ1 (Hî ) = ñ1 (M|î). Proof. We first show ñ1 (Hî ) ≥ ñ1 (M|î). This follows at once from M|î

Subclaim 3.15A. Let S ⊆ Hî be Σ1 and suppose S ∈ M|î; then S ∈ Hî .

-definable from parameters in Hî ,

Proof of Subclaim 3.15A. We may as well assume S is a set of ordinals; say S ⊆ ñ < ùî. By Proposition 2.4 we can fix a real x0 and an ordinal ä such ~ M that S = hîE,R (ä, x0 ). Now for y a real, let ~

M

y ∈ Ord ↔ ∃ç ∈ S∃Z(Z = hîE,R (ä, y) ∧ ç 6∈ Z). M|î

M|î

Since S is Σ1 in parameters from Hî , O is Σ1 in ordinal parameters, so by Lemma 3.2, O is ordinal definable over M|è. Hence there is a condition r ∈ Vop1 such that r ∗ = RM \ O. Notice that x0 ∈ r ∗ . ~ M Let ϕh be a Σ1 formula defining hîE,R over M|î. Let α < ùî be large enough that ~ ↾ α ∗ ) |= ϕh [ä, x0 , S], (SαE (RM ), ∈, E ~

where α ∗ is largest such that ùα ∗ ≤ α. By the proof of Lemma 3.8, we can fix a term ô ∈ Hî such that whenever G is Vopù -generic over Hî and R˙ G = RM , then ~ ~ ↾ α ∗ ). ô G = (SαE (RM ), ∈, E Subclaim 3.15B. For any ç < ñ ˇ ó0 , Z) ∧ çˇ 6∈ Z)]). ç 6∈ S ⇔ ∃p ≤v r(p [ô |= (ϕh (ä, Proof of Subclaim 3.15B. Assume ç 6∈ S. Let f : ù ։ RM be M-generic ~ over Col(ù, RM ), with f(0) = x0 . It follows that r ∈ Gf , ô Gf = (SαE (RM ), ∈, G G ~ ↾ α ∗ ), and ó f = x0 . But then ô Gf |= ∃Z(ϕh [ä, ó f , Z] ∧ ç 6∈ Z), since S E 0 0 is in fact the unique such Z. Hence we have some p ∈ Gf forcing this fact, and we may as well take p ≤v r, so that p witnesses the right hand side of our equivalence.

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Conversely, let p be as on the right hand side of our equivalence. By Lemma 3.5, we can find and f : ù ։ RM which is M-generic over Col(ù, RM ) such ~ that p ∈ Gf . Then ô Gf = (SαE (RM )), and what’s forced by p is true in the G ~ generic extension, so (SαE (RM )) |= ∃Z(ϕh [ä, ó0 f , Z] ∧ ç 6∈ Z). If ç ∈ S, this G G implies ó0 f ∈ O by the definition of O; however, ó0 f = f(0) ∈ r ∗ = ¬O because r ∈ Gf . Thus ç 6∈ S, as desired. ⊣ (Subclaim 3.15B) From Subclaim 3.15B and the definability of the forcing relation for Σ0 sentences given by Lemma 3.6, we get that that S ∈ Hî . ⊣ (Subclaim 3.15A) H

Now let S ⊆ ñ1 (Hî ) be boldface Σ1 î but not in Hî . By Claim 3.13 and e Subclaim 3.15A, S 6∈ M|î, and this implies ñ1 (M|î) ≤ ñ1 (Hî ). M|î We now show ñ1 (Hî ) ≤ ñ1 (M|î). Let S ⊆ (RM × ñ) be boldface Σ1 e but not a member of M|î, where ñ = ñ1 (M|î). Let hx, çi ∈ S ⇔ M|î |= ϕ[x, ç, y, â],

M

where y ∈ R and â < ùî are fixed parameters. Now consider the strong forcing relation: ˇ hp, çi ∈ F ⇔ (p ∈ Vopù ∧ p s ϕ(ó0 , ç, ˇ ó1 , â)). H

F is Σ1 î by 3.6, and a subset of è × ñ. Since è ≤ ñ, we will be done if we show F 6∈ Hî . In fact, F 6∈ M|î, for hx, çi ∈ S ⇔ ∃p[hp, çi ∈ F ∧ (∃s ∈ p ∗ (s(0) = x ∧ s(1) = y)], so that if F ∈ M|î, then S ∈ M|î. Both directions of the equivalence displayed are proved by considering Vopˇenka generics of the form Gf , where f is Col(ù, RM ) generic over M|î and f(0) = x and f(1) = y. We leave the rest to the reader. ⊣ (Claim 3.15) Claim 3.16. p1 (Hî ) = p1 (M|î) \ {è}. Proof. The proof of Claim 3.15 actually shows that for any finite F ⊆ ùî and any α such that è < α, H

M|î

Th1 î (α ∪ F ) ∈ Hî ⇔ Th1

(RM ∪ α ∪ F ) ∈ M|î,

where ThP 1 (X ) denotes the Σ1 -theory in P of parameters in X . Letting p1 (Hî ) = hα0 , . . . , αn i, we have by the solidity of p1 (Hî ) that for i ≤ n, H

αi = least â such that Th1 î (â ∪ {α0 , . . . , αi−1 }) 6∈ Hî . Using the equivalence displayed above and the solidity of p1 (M|î), we then get by induction on i that αi is the ith member of p1 (M|î). Thus p1 (Hî ) ⊆ p1 (M|î) \ {è}. (Note that è 6∈ p1 (Hî ), since è is easily definable over Hî .) A similar argument shows p1 (Mî ) \ {è} ⊆ p1 (Hî ). ⊣ (Claim 3.16)

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Claim 3.17. Hî is 1-sound. H

Proof. Let ç < ùî; we must show ç is Σ1 î -definable, as a point, from parameters in ñ ∪ p, where ñ = ñ1 (Hî ) and p = p1 (Hî ). Since M|î is 1sound, we can find a finite F ⊆ ñ ∪p and a real x and a Σ1 formula ϕ(t, u, v, w) such that ç = unique â such that M|î |= ϕ[x, F, è, â]. We may assume that ϕ has been “uniformised”, so that over any premouse it defines the graph of a partial function of its first three variables. Now, letting f be Col(ù, RM )-generic over M|î with f(0) = x, we can find a A ∈ Gf such that ˇ ç). A s ϕ(ó0 , Fˇ , è, ˇ H

Since ϕ has been uniformised, this gives us a Σ1 î definition of ç from A, ó0 , F, and è. But A < è, and ó0 ⊆ è is coded into T M in a simple way. Thus ç is H ⊣ (Claim 3.17) Σ1 î definable from F , as desired. This finishes the k = 1 case in our induction on hî, ki. The case k > 1 can be handled quite similarly, using the master code structures. For example, let ~ ~ M ~ P = (JFñ (T M ), ∈, F~ ↾ñ, A) and Q = (JE ñ (R ), ∈, E↾ñ, B) be the first master code structures of Hî and M|î respectively. The arguments above show that α 7→ A ∩ α is ∆Q 1 and total on Q, and this can be used to show as above that ñ1 (Q) ≤ ñ1 (P), that is, ñ2 (M|î) ≤ ñ2 (Hî ). In the other direction, one can show that Q is ∆P[G] , uniformly in all Vopˇenka-generic G such that 1 R˙ G = RM (as in Lemma 3.8), and using the definability of forcing over P given by Lemma 3.6, this implies ñ1 (P) ≤ ñ1 (Q), that is ñ2 (Hî ) ≤ ñ2 (M|î). We leave the remaining details to the reader. ⊣ (Theorem 3.9) If M is a model of ZFC minus the powerset axiom, and H is the T M premouse we have defined above, then it is easy to see that the universe of H is just HODM . Indeed, H ⊆ HODM is clear, and HODM ⊆ H follows at once from Subclaim 3.15A. In general, for arbitrary M satisfying the assumptions behind Theorem 3.9, x ∈ H ⇔ ∃α(ùα < o(M) ∧ ∀y ∈ tcl(x) ∪ {x}(y ∈ ODM|α ). It is therefore tempting to write H = HODM in general, as this would be a reasonable general meaning for HODM . However, we shall stick to H = H(M) = HM . Finally, we come to the main reason we have isolated H. Theorem 3.18. Let M be n0 -sound RM -premouse, and satisfy “Θ exists”. Suppose ñn0 (M) ≥ ΘM , and M|ë |= ZF, for some ë ≥ ΘM . Finally, suppose

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M|î is countably k-iterable, for all hî, ki
T T • iα,â = iα,â ↾ MTα . ∗

∗

Given that T has limit length < ô, and we have a T ∗ as above, we simply define Γ(T ) = Σ(T ∗ ). Setting b = Γ(T ), it is routine to verify that the bextensions of T and T ∗ satisfy the conditions above. The main point is that for α ∈ b sufficiently large, T T MTb = iα,b (MTα ) = iα,b (H(MTα )) = H(MTb ), ∗

∗

∗

where we have applied the embeddings to classes of their domain models in the usual way. It is also clear that the existence of T ∗ as above propagates through successor steps in the construction of T . This completes our sketch. ⊣ §4. The scale property in K(R). Using the local HOD’s of the last section to verify iterability, in the same way that the ordinals were used to verify wellfoundedness in [Ste83A], we shall construct closed game representations of minimal complexity for sets in K(R). As explained in [Ste83A], an argument due to Moschovakis [Mos83] converts these closed game representations to scales of minimal complexity. Modulo the use of the local HOD’s to verify iterability, everything goes pretty much as it did in [Ste83A]. We shall therefore keep our notation as close as possible to that of [Ste83A], and omit many of the details treated more carefully there. 4.1. Scales on ΣM 1 sets, for M passive. In this subsection we prove Theorem 1.1. In fact, we prove the slightly stronger Theorem 4.1. Let M be a passive, countably iterable premouse over RM , and suppose M |= AD; then M |= “the pointclass ΣM 1 has the scale property.”

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It should be clear what it means for M to believe that a pointclass of M-definable sets of reals has the scale property: the norms of the putative scale, which are M-definable, must have the limit and lower semi-continuity properties of a scale with respect to all sequences of reals hxi : i < ùi ∈ M. If RM = R and M believes that the pointclass ΣM 1 has the scale property, then indeed ΣM 1 does have the scale property, as every ù-sequence of reals is in M. Thus Theorem 4.1 implies Theorem 1.1. Proof. Let us fix a passive, countably 0-iterable RM -premouse M such that M |= AD. We want to show that M satisfies a certain sentence, so by taking a Skolem hull we may assume that M is countable. For x ∈ RM , let P(x) ⇔ M |= ϕ0 [x], where ϕ0 is a Σ1 formula of L∗ . Since M is passive, we may (and do) assume ˙ We that ϕ0 does not contain F˙ , ì, ˙ í, ˙ or ã; ˙ that is, it contains only ∈ and E. want to show that M believes that there is ΣM scale on P. 1 Let us first assume that o(M) = ùα, where α is a limit ordinal, and deal with the general case later. If M satisfies “Θ exists”, then set α ∗ = ΘM , and otherwise set α ∗ = α. For â < α ∗ and x ∈ RM , let P â (x) ⇔ M|â |= ϕ0 [x], S

so that P = â<α∗ P â . Here we use Lemma 3.2 to see that the union out to α ∗ suffices.15 For each â < α ∗ , we will construct a closed game representation x 7→ Gxâ of P â . Letting â

Pk (x, u) ⇔ u is a position of length k from which player I has a winning quasi-strategy in Gxâ , we shall arrange that each Pkâ ∈ M, and that the map hâ, ki 7→ Pkâ is ΣM 1 . As explained in [Ste83A], Moschovakis’ argument then gives that M believes there is a scale on P of the desired complexity.16 So let â and x be given; we want to define Gxâ . Our plan is to force player ~ I to describe a model of V = K(R) + ϕ0 (x) + ∀ã(JãE (R) 6|= ϕ[x]) which includes all the reals played by player II. Player I will verify that his model is wellfounded by embedding its ordinals into ùâ, and verify his model is iterable by embedding its local HOD’s into the local HOD’s of M|â corresponding to them under his embedding of the ordinals. 15 We have restricted ourselves to â < α ∗ for a minor technical reason connected to the definability of “honesty”. 16 Moschovakis uses the “second periodicity” method to construct scales on the P â . It is here k that one needs M |= AD .

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Player I describes his model in the language L, which is L∗ together with new constant symbols x˙ i for i < ù. He uses x˙ i to denote the ith real played in the course of Gxâ . Let us fix recursive maps m, n : {ó : ó is an L-formula } → {2n : 1 ≤ n < ù} which are one-one, have disjoint recursive ranges, and are such that whenever x˙ i occurs in ó, then i < min (m(ó), n(ó)). These maps give stages sufficiently late in Gxâ for player I to decide certain statements about his model. Let us call an ordinal î of an RP -premouse P relevant iff P|î satisfies “Θ ˙ exists, and there is a ë > Θ such that JE ë (R) |= ZF.” That is, the relevant î are just those for which, under the additional assumption that P is countably iterable, we have defined and proved the existence of HîP . Also, “v is relevant” is the L-formula which expresses that v is relevant viv-a-vis the universe as P. Similarly for the L-formula “Hv exists”. Player I’s description must extend the following L-theory T . The axioms of T include (1) Extensionality plus V = K(R) (2) ∀v(v is relevant ⇒ Hv exists ) (3)ϕ ∃vϕ(v) ⇒ ∃v(ϕ(v) ∧ ∀u ∈ v¬ϕ(u)) (4)i x˙ i ∈ R. ˙ (5) ϕ0 (x˙ 0 ) ∧ ∀ä(JäE (R) 6|= ϕ0 [x˙ 0 ]) Finally, T has axioms which guarantee that in any model, the definable closure of the interpretations of the x˙ i constitute an elementary submodel. Recall from Proposition 2.4 the uniformly definable maps hã : [ùã]<ù ։ M|ã; let ó0 (v0 , v1 , v2 ) be a Σ1 formula which for all ã defines the graph of hã over M|ã. Now, for any L-formula ϕ(v) of one free variable, T has axioms (6) ∀v0 ∀v1 ∀y∀z(ó0 (v0 , v1 , y) ∧ ó0 (v0 , v1 , z) ⇒ y = z) (7)ϕ ∃vϕ(v) ⇒ ∃v∃F ∈ [Ord]<ù (ϕ(v) ∧ ó0 (F, x˙ m(ϕ) , v)) (8)ϕ ∃v(ϕ(v) ∧ v ∈ R) ⇒ ϕ(x˙n(ϕ) ). This completes the axioms of T . A typical run of Gxâ has the form I

i 0 , x0 , ç 0

i 1 , x2 , ç 1 ...

II

x1

x3

where for all k, ik ∈ {0, 1}, xk ∈ R, and çk < ùâ. If u = h(ik , x2k , çk , x2k+1 ) : k < ni is a position of length n, then we set T ∗ (u) = {ó : ó is a sentence of L ∧ in(ó) = 0}, and if p is a full run of Gxâ , T ∗ (p) =

[ n<ù

T ∗ (p ↾ n).

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Now let p = h(ik , x2k , çk , x2k+1 ) : k < ùi be a run of Gxâ ; we say that p is winning for player I iff (a) x0 = x, (b) T ∗ (p) is a complete, consistent extension of T such that for all i, m, n, “x˙ i (n) = m” ∈ T ∗ (p) iff xi (n) = m, (c) if ϕ and ø are L-formulae of one free variable, and “évϕ(v) ∈ Ord ∧ évø(v) ∈ Ord” ∈ T ∗ (p), then “évϕ(v) ≤ évø(v)” ∈ T ∗ (p) iff çn(ϕ) ≤ çn(ø) , and (d) if ø is an L-formula of one free variable, and “évø(v) ∈ Ord ∧évø(v) is relevant” ∈ T ∗ (p), then çn(ø) is relevant vis-a-vis M; moreover if ó1 , . . . , ón are L formulae of one free variable such that for all k, “évók (v) < (évø(v))” ∈ T ∗ (p), then for any L∗ formula è(v1 , . . . , vn ), “Hévø(v) |= è[évó1 (v), . . . , évón (v)]” ∈ T ∗ (p) if and only if HçM |= è[çn(ó1 ) , . . . , çn(ón ) ]. n(ø) Clearly, Gxâ is a game on R × ùâ whose payoff is continuously associated to x. It remains to show that the winning positions for player I in Gxâ are those in which he has been honest. More precisely, let us call a position u = h(ik , x2k , çk , x2k+1 ) : k < ni (â, x)-honest iff M|â |= ϕ0 [x], and letting ã ≤ â be least such that M|ã |= ϕ0 [x], we have (i) n > 0 ⇒ x0 = x, (ii) if we let Iu (x˙ i ) = xi for i < 2n, then all axioms of T ∗ (u) ∪ T thereby interpreted in (M|ã, Iu ) are true in this structure, and (iii) if ó0 , . . . , óm enumerates those L-formulae ó of one free variable such that n(ó) < n and (M|ã, Iu ) |= évó(v) ∈ Ord, and if äi < ùã is such that (M|ã, Iu ) |= évói (v) = äi , then the map äi 7→ çn(ói ) is well-defined and extendible to an order preserving map ð : ùã → ùâ with the additional property that whenever äi is relevant vis-a-vis M, so that çn(ói ) is as well, then ð↾ùäi is extendible to an elementary embedding .17 to HçM from HäM i n(ó ) i

It is not immediately clear that the set of (â, x)-honest positions even belongs to M, because of condition (iii). 17 Notice

ordinal.

that this extension is determined by ð, since every point in Häi is definable from an

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Qkâ (x, u)

Claim 4.2. Letting iff u is a (â, x)-honest position of length k, we have that Qkâ ∈ M for all â, and the map (â, k) 7→ Qkâ is ΣM 1 . Proof (Sketch). It is enough to see that the truth of clause (iii) can be determined within M. But note that (iii) is equivalent to the existence of a winning strategy for the closed player in a certain “embedding game” on ùâ. It is enough then to see that if the closed player wins the embedding game in V, then he wins it in M. (The converse is obvious.) So suppose the closed player wins the embedding game in V. Let A ∈ M be a set of ordinals which codes up this game; since â < ΘM , and M |= AD, we can find a model N of ZFC such that A ∈ N , and (N, ∈) is coded by a set of reals BN ∈ M. (E.g., let N = Lα [A], where α is the supremum of the order types of the M|ã ∆n prewellorders of RM , for an appropriate n and ã.) Since N |= ZFC, the closed player wins the embedding game via a strategy Σ ∈ N . For ã < ùâ, let f(ã) = {z ∈ RM : z codes ã via BN }. We can arrange that f ∈ M, and use f to show that Σ ∈ M. ⊣ (Claim 4.2) We now show that the winning positions are the honest ones. Claim 4.3. For any position u in Gxâ , player I has a winning quasi-strategy starting from u iff u is (â, x)-honest; that is, Pkâ (x, u) ⇔ Qkâ (x, u) for all k. Proof. It is easy to see that player I can win from honest positions u by continuing to tell the truth, while continuing to play his ç’s according to some map ð satisfying (iii) in the definition of honesty for u. Conversely, let Σ be a winning quasi-strategy for player I in Gxâ from u. Since RM is countable, we can easily construct a complete run p = h(ik , x2k , çk , x2k+1 ) : k < ùi â

of Gx according to Σ such that RM = {xi : i < ù}. Since T ∗ (p) is consistent, it has a model A, and by the axioms in groups (3), (6), (7), and (8), the L-definable points of A constitute an elementary submodel N ≺ A. Let ð(évø(v)N ) = çn(ø) whenever “évø(v) ∈ Ord” ∈ T ∗ (p). Then ð witnesses that N is wellfounded, and so we may assume N is transitive. By axiom (1) of T , N is a premouse. Note that N is a premouse over {x˙ iA : i < ù}, and this set is RM since x˙ iA = xi by (ii). The main thing we need to show is that N is an initial segment of M|â. Since ð guarantees o(N ) ≤ ùâ, it suffices to show N is an initial segment of M. We show by induction on ã that if ñù (N |ã) = 1, then N |ã = M|ã. This is clear if ã = 1.

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Suppose first that N |ã 6|= Θ exists . Note that then ã is not active in N . If ã is a limit ordinal, then there are arbitrarily large î < ã such that ñù (N |î) = 1, and by induction N |î = M|î for such î. This implies ã is not active in M, and N |ã = M|ã, as desired. If ã = î + 1, then ñù (N |î) = 1, so N |î = M|î. Since successor ordinals are not active, we get N |ã = M|ã, as desired. (Note that if ùã = o(N ), then it falls under this last case by axiom (5) of T , so we may assume ùã < o(N ) henceforth.) Next, suppose N |ã |= Θ exists, and let è = ΘN |ã . Note N |è = M|è by the argument of the last paragraph. If there is no î ∈ (è, ã) such that N |î |= ZF, then N |ã is just the constructible closure of N |è through ã steps. Also, no î ∈ (è, ã] can be active in M, and so N |ã = M|ã. So we may assume that there is a î ∈ (è, ã) such that N |î |= ZF; that is, ã is relevant in N . Let n0 be the largest n < ù such that ñn (N |ã) ≥ è. By rule (iii), ð determines an elementary embedding from H(N |î) to H(M|ð(î)), for each relevant î of N . Using Theorem 3.18 and a simple induction, we can then see that N |ã is countably n0 -iterable. Now â < α∗, and hence we can find a î ≥ â such that ñù (M|î) = 1. Applying the comparison theorem 2.12 to N |ã and M|î, we get that N |ã = M|ã, as desired. Thus N is an initial segment of M|â. Clearly, N = M|ã, where ã is least such that M|ã |= ϕ0 [x]. The theory T ∗ (u) is true in N = M|ã. The remainder of (â, x) honesty for u is witnessed by ð. ⊣ (Claim 4.3) Claims 4.2 and 4.3 yield the desired scale, as we have explained. This completes the proof of Theorem 4.1 in the case that o(M) = ùα for α a limit. The case that α is a successor ordinal can be handled similarly, using Jensen’s S-hierarchy. See [Ste83A]. ⊣ 4.2. Σ1 gaps. Definition 4.4. Let M and N be X -premice; then we write M ≺1 N iff M is an initial segment of N , and whenever ϕ(v1 , . . . , vn ) is a Σ1 formula of the language L∗ in which F˙ , ì, ˙ í, ˙ and ã˙ do not occur, then for any a1 , . . . , an ∈ X ∪ {X }, N |= ϕ[a1 , . . . , an ] ⇒ M |= ϕ[a1 , . . . , an ]. Notice here that such Σ1 formulae go up from M to N simply because M is an initial segment of N . This uses our restriction that the symbols of L∗ which have to do with the last extender of a premouse do not occur in ϕ. Definition 4.5. Let M be an X -premouse, and suppose ùα ≤ ùâ ≤ o(M); then we call the interval [α, â] a Σ1 -gap of M iff 1. M|α ≺1 M|â, 2. ∀ã < α(M|ã 6≺1 M|α), and 3. ∀ã > â(M|â 6≺1 M|ã).

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That is, a Σ1 -gap is a maximal interval of ordinals in which no new Σ1 facts ~ are verified. If [α, â] about members of X ∪ {X } and the extender sequence E is a Σ1 -gap, we say α begins the gap and â ends it. Notice that we allow α = â. It is easy to see Lemma 4.6. Let o(M) = ùα; then the Σ1 -gaps of M partition α + 1. We shall use the Σ1 -gaps of K(R) to characterize the levels of the Levy hierarchy in the initial segment of K(R) satisfying AD which have the Scale Property. Until we get to the end-of-gap case, the proofs are quite easy, and completely parallel to those of [Ste83A], so we shall omit them. If M is an RM -premouse, then by the pointclass ΣM n we mean the collection of all A ⊆ RM such that A is Σn definable over Cn−1 (M) from arbitrary parameters in Cn−1 (M), using the language L∗ . Theorem 4.7. Let M be a countably 0-iterable RM -premouse, and suppose α begins a Σ1 gap of M, and that M|α |= AD; then M believes that the M|α pointclass Σ1 has the Scale Property. Theorem 4.8. Let M be a countably 0-iterable RM -premouse, and suppose α begins a Σ1 -gap of M, and that M|(α + 1) |= AD, and that M|α is not an admissible structure. Then for all n < ù, M|α

M|α

(a) Σn+2 = ∃R (Πn+1 ) M|α

M|α

Πn+2 = ∀R (Σn+1 ), and M|α M|α (b) M believes that the pointclasses Σ2n+1 and Π2n+2 have the Scale Property. As in L(R), our negative results on the Scale Property are localizations of the fact that the relation “x is not ordinal definable from y” has no ordinal definable uniformization. Definition 4.9. If M is an RM -premouse and o(M) = ùα, then for x, y ∈ R , we put M

C M (x, y) ⇔ ∃ã < α({y} is M|ã-definable from parameters in {x} ∪ ùã). We also set ¬C M = (RM × RM ) \ C M . It is clear that C M is ΣM 1 , and indeed, it is so via a formula which does not refer to the last extender F˙ M . Theorem 4.10 (Martin, [Mar83A]). Let M be a countably 0-iterable RM premouse, and suppose that α begins a Σ1 -gap of M, that M|(α + 1) |= AD, and that the structure M|α is admissible. Then M|α

(a) there is a Π1 relation on RM , namely ¬C M|α , which has no uniformization in M|(α + 1), and hence

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JOHN R. STEEL M|α

(b) M believes that none of the pointclasses Σn the Scale Property.

M|α

or Πn

, for n > 1, have

In the interior of a Σ1 -gap, we find no new scales. Theorem 4.11 (Kechris, Solovay). Let M be a countably 0-iterable RM premouse, and suppose [α, â] is a Σ1 -gap of M, and that M|α |= AD. Then (a) the relation ¬C M|α has no uniformizing function f such that f ∈ M|â, and hence M|ã (b) if α < ã < â, then M believes that none of the pointclasses Σn or M|ã Πn , for n < ù, have the Scale Property. 4.3. Scales at the end of a gap. We are left with the question as to which, M|â M|â have the Scale Property in the case if any, of the pointclasses Σn and Πn that â ends a Σ1 gap [α, â] of M, and α < â. As in [Ste83A], the answer turns on the following reflection property of â. Definition 4.12. For M a relativised premouse and 1 ≤ n < ù and a ∈ M, we let Σn,M be the Σn -type realized by a in M; that is a Σn,M = {è(v) : è is either Σn or Πn and Cn−1 (M) |= è[a]}. a We are allowing formulae of the full language L∗ of relativised premice, so that the last extender F˙ M is (partially) described in Σn,M . a Definition 4.13. An ordinal â is strongly Πn -reflecting in M iff every Σn type realized in Cn−1 (M|â) is realized in Cn−1 (M|î) for some î < â; that is n,M|â n,M|î ). ∀a ∈ Cn−1 (M|â)∃î < â∃b ∈ Cn−1 (M|î)(Σa = Σb Definition 4.14. Let [α, â] be a Σ1 gap of M, with α < â; then we call [α, â] strong iff â is strongly Πn -reflecting in M, where n is least such that ñn (M|â) = RM . Otherwise, [α, â] is weak. Martin’s reflection argument of [Mar83A] yields Theorem 4.15 (Martin). Let M be a countably 0-iterable RM -mouse which satisfies AD , and let [α, â] be a strong Σ1 -gap of M such that ùâ < o(M); then M|α (a) there is a Π1 relation on RM which has no uniformization which is definable over M|â, and hence M|â M|â (b) M believes that none of the pointclasses Σn or Πn have the Scale Property. Thus at the end of strong gaps [α, â], the Scale Property first re-appears M|(â+1) with the pointclass Σ1 . The weak gap case is settled, under stronger determinacy hypotheses than should be necessary, by

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Theorem 4.16. Let M be a countably ù-iterable RM -mouse which satisfies AD, and [α, â] a weak gap of M and ùâ < o(M); then letting n be least M|â such that ñn (M|â) = RM , we have that M believes that Σn has the Scale Property. Remark. The hypothesis that ùâ < o(M) should not be necessary. The proof below needs it because at a certain point we apply the Coding Lemma to a bounded subset of ΘM|â such that A is merely definable over M|â, and we need enough determinacy to do this. (See below.) Unfortunately, adding this determinacy as a hypothesis in Theorem 4.16 makes the theorem significantly less useful in core model induction arguments than it would be otherwise. We can eliminate the additional determinacy hypothesis in one case: Theorem 4.17. Let M be a countably ù-iterable RM -mouse which satisfies AD, and [α, â] a weak gap of M. Suppose that either ΘM does not exist, or there are no extenders on the M-sequence with index above ΘM ; then letting M|â has n be least such that ñn (M|â) = RM , we have that M believes that Σn the Scale Property. One can combine Theorem 4.17 with the work of [SteB], and thereby obtain a construction of scales at the end of a weak gap in K(R) which is more useful in a core model induction. Proof of Theorem 4.16. (Sketch) One gets a proof by integrating our use of the local HODM ’s into the proof of the corresponding result (theorem 3.7) of [Ste83A]. This is fairly routine, yet involves many details. We shall therefore just sketch one case in which some care with the details is needed. We want also to point out the place where the additional determinacy hypothesis is used.18 The case we consider is n = 1 and M|â is active of type II. Let us make these assumptions. Let F ∗ be the amenable-to-M predicate coding F˙ M which is described in [Ste07B]. For ã < â, let ˙M

˙M

M||ã = (JE , ∈, E˙ M|ã , F ∗ ∩ JE ). ã ã The M||ã are just the initial segments of the Σ0 -code C0 (M). They are structures for the language L∗ of C0 (M), and for ϕ a Σ1 formula of L∗ and x ∈ M, we have C0 (M) |= ϕ[x] ⇔ ∃ã < â(M||ã |= ϕ[x]). Further, M||ã ∈ M for all ã < â. Let Σ = Σ1,M be our nonreflecting Σ1 -type. We may assume a = hG, w1 i, a where G is a finite subset of â and w1 ∈ R, and that G is Brouwer-Kleene 18 This

hypothesis is needed in the other cases not covered by Theorem 4.17 as well.

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minimal, in the sense that whenever H ∈ [â]<ù and H
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We shall construct closed game representations i 7→ Gxi of the P i in such a way that if Pki (x, u) ⇔ u is a winning position for player I in Gxi of length k , then Pki is first order definable over M||âsup(i,k) . Such a closed game representation yields scales on each P i each of whose norms belongs to M, and hence a ΣM 1 scale on P. In Gxi , player I describes C0 (M) as the union of the M||âk . The language ˙ and M ˙ k , â˙k , and L in which he does this has ∈, =, and constant symbols G, ˙ x˙k for all k < ù. If ϕ is an L-formula involving no constants Mk or â˙k for k ≥ m, then we say ϕ has support m. Player I will produce a Σ0 -complete theory in L, restricting himself at move m to Σ0 sentences with support m. Let B0 be the collection of Σ0 formulae of L, and let n : B0 ֒→ ù be such that any è ∈ B0 has support n(è) and involves no x˙k for k ≥ n(è). A typical run of Gxi has the form I

T0 , s0 , ç0 , m0

T1 , s2 , ç1 , m1

... II s1 s3 where for all k, Tk is a finite set of sentences in B0 , all of which have support k, sk ∈ R<ù , çk < ùâ, and mk ∈ ù. Given such a run of Gxi , let hxk : k < ùi = concatenation of hsk : k < ùi, and T∗ =

[

Tk .

k

Let S0 be the set of sentences in B0 which involve no constants of the form x˙i for i 6∈ {1, 2}, and are true in the interpretation under which x˙1 denotes ˙ k denote âk and M||âk for all k < ù. S0 w1 , x˙2 denotes w2 , and â˙k and M will enter as a real parameter in the payoff condition for Gxi , and hence in the definition of our scale on P. We could avoid this by replacing S0 with an appropriate finitely axiomatized subtheory, but since real parameters will enter elsewhere, there is no point in doing so. Notice that it is part of S0 that ˙ k is an L∗ -structure. For è any formula of L∗ , let è M˙ k be the natural each M ˙ k |= è. B0 -formula expressing that M We say that the run of Gxi displayed above is a win for player I iff the following conditions hold: (1) x0 = x, x1 = w1 , and x2 = w2 . (2) T ∗ is a consistent extension of S0 such that for all k, m, n, “x˙ k (n) = m” ∈ T ∗ iff xk (n) = m. (3) If è ∈ B0 is a sentence, then either è ∈ Tn(è) or (¬è) ∈ Tn(è) .

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(4) If ∃v(v ∈ R ∧ ó) ∈ Tk , then for some j, ó(x˙j ) ∈ Tk+1 . ˙ (5) (ϕ0 (x˙0 , hG˙ , â˙0 , . . . , â˙e , x˙2 i))Mi ∈ Ti+1 . ∗ (6) If è(v1 , . . . , vn+2 ) is an L -formula, and ó1 , . . . , ón are B0 formulae of one free variable with support k, and “évóm (v) ∈ Ord”∈ T ∗ for all m ≤ n, then ˙

è Mk (évó1 (v), . . . , évón (v), x˙ 1 , x˙ 2 ) ∈ T ∗ ⇔M||âk |= è[çn(ó1 ) , . . . , çn(ón ) , w1 , w2 ]. ˙ ∈ Tk , then either (8) If (évó(v)
(¬ϕ(hévó(v), w˙1 i))Mmk ∈ Tmk +1 , or (b) there is a Σ1 formula ϕ which is one of the first mk elements of Σ such that for all j and ℓ, ˙

(ϕ(hévó(v), w˙1 i))Mℓ 6∈ Tj . This completes the description of the payoff set for player I in Gxi . We now show that player I wins Gxi iff M||âi |= ϕ0 [x, b], and that the Pki are appropriately definable. Both claims follow from the fact that Pki (x, u) iff u is honest. Honesty is defined as follows: let Iu be the interpretation of L under which x˙j denotes xj whenever xj is the jth real determined by u, and G˙ , â˙k ˙ k denote G, âk , and M||âk for all k. For u a position in Gxi , we say u and M is x-honest iff (i) T ∗ (u) is true in (|M|, ∈, Iu ), (ii) M||âi |= ϕ0 [x, b], (iii) x0 = x, x1 = w1 ,and x2 = w2 , if u determines x0 , x1 , and x2 . (iv) the commitments represented by the mk can be kept, (v) if ó0 , . . . , ón enumerates those B0 -formulae ó of one free variable such that n(ó) ∈ dom(u) and (|M|, ∈, Iu ) |= évó(v) ∈ Ord, and if äm < o(M) is such that (|M|, ∈, Iu ) |= évóm (v) = äm , for all m ≤ n, then the map äm 7→ çn(óm ) is well-defined and extendible to an order preserving map ð : o(M) → o(M) such that for all k, all formulae è of L∗ , and all tuples ã of ordinals from M||âk , ð ↾ M||âk ⊆ M||âk and M||âk |= è[ã, w1 , w2 ] ⇔ M||âk |= è[ð(ã), w1 , w2 ].

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Claim 4.19. For any position u of Gxi , I wins Gxi from u iff u is x-honest. Proof. If u is x-honest, then I can win Gxi from u by continuing to tell the truth, while using the map ð given by condition (v) to play further ç’s. Now suppose I wins Gxi from u. Let p be a run of Gxi by such a strategy, with u ⊆ p, such that the associated sequence of reals hxk : k < ùi enumerates RM . Let T ∗ = T ∗ (p) be the B0 -theory played by I. Let A be the unique model of T ∗ which is pointwise definable from parameters in RM . (There is such a model by rule (4).) By rule (6) of Gxi , A is wellfounded, and so we assume it is transitive. Let ∗ ˙A M k = (Nk , Fk ), âk∗ = â˙kA , and G ∗ = G˙ A . Since S0 ⊆ T , the Nk are premice, and Nk  Nk+1 for all k. Let N be the union of the Nk , and F ∗ the union of the Fk∗ . We can define ð : o(N ) → o(N ) by ð(évó(v)A ) = çn(ó) , ∗

for all B0 -formulae ó such that évó(v) ∈ Ord is in T ∗ . Clearly, ð is welldefined, and for any tuple ã of ordinals from Nk and any formula è of L∗ , (Nk , Fk∗ ) |= è[ã, w1 , w2 ] ⇔ M||âk |= è[ð(ã), w1 , w2 ]. As in the proof of Theorem 4.1, this implies that N is countably iterable, and that ð extends to an embedding, which we also call ð, such that ð : HN → HM is Σ1 -elementary (in L∗ ). ∗ Because S0 is true in A, Fk+1 measures all subsets of its critical point in (N ,F ∗ )

∗ H1 k k (RM ∪{â0∗ , . . . , âk−1 , G ∗ }). But the union of these hulls has the same ∗ universe as N , and thus F is an extender over N . Similarly, we get

Fk∗ ∩ HNk ∈ HNk+1 for all k, and because ð is sufficiently elementary, ð(Fk∗ ∩ HNk ) = F ∩ HM|âk . Letting E ∗ = F ∗ ∩ HN and E = F ∩ HM , we then have that Σ

ð : (HN , E ∗ ) →1 HM is Σ1 elementary. It follows that (HN , E ∗ ) is a countably iterable premouse, and hence that (N , F ∗ ) is a countably iterable premouse. It is part of S0 that our non-reflecting type Σ is realized for the first time, and thus we have (N , F ∗ ) = M. Because player I has kept the commitments

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he made according to rule (7) of Gxi , we have G ∗ = G, and âk∗ = âk and (Nk , Fk∗ ) = M||âk for all k. It is now easy to verify that u was x-honest; the map ð witnesses that condition (v) of x-honesty is met. ⊣ (Claim 4.19) Claim 4.20. Let k ≤ i; then {u : u is an x-honest position of length k} is a member of M|â. Proof. It is clear that the set of u satisfying conditions (i)-(iii) of x-honesty is definable over M||âk , and hence in M|â. The Coding Lemma argument of [Ste83A] shows that the set of u satisfying condition (iv) is also in M|â. Here, as in [Ste83A], we can apply the Coding Lemma to sets belonging to M|â, so we don’t actually need determinacy beyond the sets in M|â. For (v), let s = (évó(v)M||âk 7→ çn(ó) ) be a finite map coded into a position u of length k satisfying (i)-(iv). Note that dom(s) ⊆ Yk , so that if s can be extended to a ð as demanded in (v), then as Yk is Σ1 -definable over M||âk+1 from â0 , . . . , âk−1 , G, ran(s) ⊆ Yk as well. (Note here that by the proof of Claim 4.2, ð must fix G and the âm for m < ù.) So if we let Zk = {t : Yk → Yk : |t| < ù ∧ ∃ð ⊇ t(ð is as in (v) }, then it suffices to show that Zk is definable over M||âsup(i,k) . We proceed as in the proof of Claim 4.2. M. Note that t ∈ Zk iff the closed player has a winning strategy in a certain “embedding game” on è M|â . We claim that if the closed player wins the embedding game in V, then he wins it in M. (The converse is obvious.) So suppose the closed player wins the embedding game in V. Let A ∈ M be a set of ordinals which codes up the payoff of game; since è M|â < è M , and M |= AD, we can find a model N of ZFC such that A ∈ N , and (N, ∈) is coded by a set of reals BN ∈ M. (E.g., let N = Lα [A], where α is the supremum of the order types of the M|â ∆n prewellorders of RM , for an appropriate n.) Since N |= ZFC, the closed player wins the embedding game via a strategy Σ ∈ N . For ã < è M|â , let f(ã) = {z ∈ RM : z codes ã via BN }. We can arrange that f ∈ M, and use f to show that Σ ∈ M. The argument of the last paragraph actually shows that there is a fixed M|â n < ù such that for all t, t ∈ Zk iff player II has a ∆n winning strategy in the embedding game associated to t. It follows that Zk ∈ M. But M |= AD, and Zk can be identified with a bounded subset of è M|â , since Yk ∈ M|â and is the surjective image of R by a map in M|â. It follows

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from the Coding Lemma that Zk ∈ M|â, and in fact Zk is definable over M||âsup(i,k) .19 ⊣ (Claim 4.20) ⊣ (Theorem 4.16) As the reader can see, we use the determinacy of sets definable over M|â in the proof of Claim 4.20 above. The determinacy of sets belonging to M|â is not enough for the proof because the payoff set A for the embedding game may not be a member of M|â. One can get by with the determinacy of sets in M|â in the proof of Theorem 4.17 because in that case, the only “global” role of the ordinals played by player I in Gxi is to verify that the model he is playing is wellfounded. This aspect of honesty can be explicitly defined; player I needs only to have spaced his ordinals adequately. See [Ste83A] for the details. It is still true that player I will have to verify that the HíA for í < o(A) are iterable, M|â by embedding them into a corresponding Hì , but these embeddings no longer need to fit together into a single embedding, and thus this aspect of honesty does not lead out of M|â. We leave the further details of the proof of Theorem 4.17 to the reader. REFERENCES

Daniel Cunningham [Cun90] The real core model, Ph.D. thesis, UCLA, 1990. Ronald B. Jensen [Jen72] The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308. Akihiro Kanamori and Matthew Foreman [KF07] Handbook set theory into the 21st century, Springer, 2007. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Donald A. Martin [Mar83A] The largest countable this, that, and the other, this volume, originally published in Kechris et al. [Cabal iii], pp. 97–106. Donald A. Martin and John R. Steel [MS83] The extent of scales in L(R), this volume, originally published in Kechris et al. [Cabal iii], pp. 86–96. William J. Mitchell and John R. Steel [MS94] Fine structure and iteration trees, Lecture Notes in Logic, vol. 3, Springer-Verlag, Berlin, 1994. Yiannis N. Moschovakis [Mos83] Scales on coinductive sets, this volume, originally published in Kechris et al. [Cabal iii], pp. 77–85. 19 All

we really need to get the desired scale is that Zk ∈ M|â.

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John R. Steel [SteB] Scales in K(R) at the end of a weak gap, Preprint, available at http://www.math. berkeley.edu/∼ steel. [Ste83A] Scales in L(R), this volume, originally published in Kechris et al. [Cabal iii], pp. 107– 156. [Ste07B] An outline of inner model theory, In Kanamori and Foreman [KF07]. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720, USA

E-mail: [email protected]

THE REAL GAME QUANTIFIER PROPAGATES SCALES

DONALD A. MARTIN

§1. Introduction. Moschovakis [Mos80, 6E] shows, assuming sufficient determinacy, that the game quantifier (for games on the integers) propagates scales. In [Mos83] he shows how to put scales on sets defined by applying the real game quantifier to a closed matrix. In this paper we show, from appropriate determinacy hypotheses, that the real game quantifier, and the real or integer game quantifier of any fixed countable length, propagate scales. Throughout the paper, we work in ZF+DC. If α is an ordinal, a game type of length α is a function g : α → {0, 1}. If g is a game type of length α and Y ⊆ α ù, the game GY g is played as follows: White and Black produce, in α moves, an element y of α ù. White chooses y(â) if g(â) = 0 and Black chooses y(â) if g(â) = 1. White wins just in case y ∈ Y. If Y ⊆ (ã+α) ù and g is a game type of length α, x g(Y ) = {x ∈ ã ù : White has a winning strategy for GY g }

where Yx = {z : x az ∈ Y }. (Note that g(Y ) depends in general on ã, though our notation does not indicate this dependence.) The notion of a scale on a subset of α ù is defined just like that of a scale on a subset of ù ù, using the product topology on α ù. The purpose of this paper is to show that, if Y ⊆ (ã+α) ù, α is countable, Y admits a scale, and g is a game type of length α, then g(Y ) admits a scale. We shall need to assume determinacy for certain games of length ≤ 4α (= α x if α is a limit ordinal), the games GY g and certain other games whose winning conditions are related to the norms on Y . Our definability results for the scale on g(Y ) will be natural generalizations of those of Moschovakis, the principal difference being that we need to use a well-ordering of a subset of ù of order type α. Our plan is to break down Moschovakis’ scale-propagation technique into three basic pieces and then to assemble the pieces in a routine fashion. The three pieces are: The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

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1. Infimum norms. Here we show how to go from a sequence of norms on Y to a sequence of norms on g(Y ) when α = 1 and g(0) = 0 (i.e., for one-move games where White moves). If the original sequence is a scale, the resulting sequence will be a scale, but our main result is similar to Moschovakis’ Infimum Lemma for existential real quantification. 2. Supremum norms. This is like (1) except that g(0) = 1 (Black moves). Our method is similar to that of Moschovakis’ Supremum Lemma. 3. Game norms. If g is any game type and ϕ is any norm on Y , a natural generalization of a result of Moschovakis [Mos80] gives a norm g(ϕ) on G(Y ), granted appropriate determinacy. In §2 we discuss infimum norms, in §3 we discuss supremum norms, in §4 we discuss game norms, and in §5 we assemble a scale on g(Y ) from a scale on Y , infimum norms, supremum norms, game norms, and a well-ordering of a subset of ù of order type α. In §6 we define canonical winning strategies. In §7 we prove results about propagation of the scale property. §2. Infimum norms. Let Y ⊆ (ã+1) ù. Suppose ϕ ~ = hϕi : i ∈ ùi is a putative scale on Y , i.e. ϕi : Y → Ord for each i and ϕi+1 (y1 ) ≤ ϕi+1 (y2 ) implies ϕi (y1 ) ≤ ϕi (y2 ) for all y1 , y2 ∈ Y and every i ∈ ù. (The last clause is merely for convenience, and differs from Moschovakis’ definition [Mos80].) Let k ∈ ù. We define a putative scale ø ~ = høi : i ∈ ùi = Inf k ϕ ~ on g(Y ), where g : 1 → {0, 1} and g(0) = 0. Our definition will have the property that each øi is defined solely from ϕ0 , . . . , ϕi . For i ≤ k, let øi (x) = inf{ϕi (x an) : n ∈ ù ∧ x an ∈ Y }. For i > k let øi (x) = inf{pϕk (x an), n, ϕi (x an)q : n ∈ ù ∧ x an ∈ Y } where p·, ·, ·q embeds the lexicographic order on an appropriate κ × ù × κ into the ordinals. For each i ∈ ù and x1 , x2 ∈ ã ù we now define the game for verifying x1 ∈ g(Y ) and either x2 6∈ g(Y ) or øi (x1 ) ≤ øi (x2 ). i ≤ k. Player II plays n1 and player I plays n2 . Player II wins just in case x1 an1 ∈ Y and either x2an2 6∈ Y or ϕi (x1 an1 ) ≤ ϕi (x2 an2 ). i > k. Player II plays n1 , player I plays n2 , player I plays j1 < 3, and player II plays j2 ≤ j1 . If j1 = j2 = 2, player II wins if x1 an1 ∈ Y and either x2an2 6∈ Y or ϕi (x1 an1 ) ≤ ϕi (x2 an2 ). If j1 = j2 = 1, player II wins if n1 ≤ n2 . If j1 = j2 = 0, player II wins if x1 an1 ∈ Y and either x2 an2 6∈ Y or ϕk (x1 an1 ) ≤ ϕk (x2 an2 ). If 1 = j2 < j1 , player II wins if n1 < n2 . If 0 = j2 < j1 , player II wins if x1 an1 ∈ Y and either x2 an2 6∈ Y or ϕk (x1 an1 ) < ϕk (x2 an2 ). It is easily checked that player II has a winning strategy for this game just in case x1 ∈ g(Y ) and either x2 6∈ g(Y ) or øi (x1 ) ≤ øi (x2 ).

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There is a similar game for verifying x1 ∈ g(Y ) and either x2 6∈ g(Y ) or øi (x1 ) < øi (x2 ). We omit the definition. A standard play of the game for verifying x1 ∈ g(Y ) and either x2 6∈ g(Y ) or øi (x1 ) ≤ øi (x2 ) is a play of the game in which either i ≤ k or i > k and j1 = j2 6= 1. The terminal condition of a standard play is hx1 an1 , x2 an2 , ki if i > k and j1 = 0 and hx1 an1 , x2 an2 , ii if i ≤ k or j1 = 2. Notice that player II wins a standard play just in case x1 an1 ∈ Y and either x2 an2 6∈ Y or ϕj (x1 an1 ) ≤ ϕj (x2 an2 ) where j is the third component of the terminal condition. Lemma 2.1. Let ϕ ~ be a putative scale on Y . Let ø ~ = Inf k ϕ ~ . Let f : ù → ù be such that a. f(i) ≤ f(i + 1) for all i; b. f(i) ≤ i for all i; c. ran(f) is unbounded. Let xi ∈ g(Y ) for each i ∈ ù, where g : 1 → {0, 1} and g(0) = 0. Assume that øf(i) (xi ) ≥ øf(i) (xi+1 ) for all i. Let ói be a winning strategy for player II for the game for verifying xi+1 ∈ g(Y ) and either xi 6∈ g(Y ) or øf(i) (xi+1 ) ≤ øf(i) (xi ), for each i ∈ ù. There are numbers n, n1 , n2 , . . . such that limi ni = n and, for every n0 with x0 an0 ∈ Y , there is an f ∗ : ù → ù satisfying (a), (b) and (c), and d. f ∗ (i) ≤ f(i) for all i, e. f ∗ (i) ≥ min{f(i), k}, and such that hxi+1 ani+1 , xi ani , f ∗ (i)i is the terminal condition of a standard play consistent with ói for each i. Proof. Let ni+1 be player II’s first move as given by ói , for i = 0, 1, 2, . . .. Let n0 be such that x0 an0 ∈ Y . Assume inductively that xi ani ∈ Y . Since hni+1 , ni i is consistent with ói for f(i) ≤ k and hni+1 , ni , 0, 0i is consistent with ói for i > k we have xi+1 ani+1 ∈ Y . For f(i) ≤ k we have also ϕf(i) (xi+1 ani+1 ) ≤ ϕf(i) (xi ani ). For f(i) > k, we have ϕk (xi+1 ani+1 ) ≤ ϕk (xi ani ). Since the ϕk (xi ani ) are non-increasing for f(i) > k, there must be an i0 with f(i0 ) > k such that i ≥ i0 implies ϕk (xi ani ) = ϕk (xi0 ani0 ). Let i ≥ i0 . The play hni+1 , ni , 1, 0i is inconsistent with ói , since it is a win for player I. Thus hni+1 , ni , 1, 1i is consistent with ói . This means ni+1 ≤ ni . Let then i1 be large enough that i1 ≥ i0 and ni = ni1 for all i ≥ i1 . Let n = ni1 . Let f ∗ (i) = f(i) if f(i) ≤ k or i ≥ i1 . Let f ∗ (i) = k otherwise. We have already seen that hni+1 , ni , f ∗ (i)i is the terminal condition of a standard play consistent with ói when i < i1 . Suppose i ≥ i1 . Consider the play hni+1 , ni , 2, j2 i consistent with ó1 . j2 = 0 or j2 = 1 would lose for player II, so j2 = 2. Hence hni+1 , ni , f(i)i is the terminal condition of a standard play consistent with ó. ⊣

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Lemma 2.1 is similar to Moschovakis’ [Mos83] Infimum Lemma. Our situation is simpler than his in that our moves are integers instead of reals, but our situation is complicated by the fact that we are going to use our lemma in studying long games, so we need the ói and the functions f and f ∗ for bookkeeping and avoiding the axiom of choice. Let us say that a putative scale ϕ ~ = hϕi : i ∈ ùi on Y is i-lsc if whenever hyj : j ∈ ùi converges to y, each yj ∈ Y , and the norms ϕi (yj ) are all eventually constant as j increases, then y ∈ Y and ϕi (y) ≤ limj ϕi (yj ). Note that ϕ ~ is a scale just in case ϕ ~ is i-lsc for every i ∈ ù. Lemma 2.2. Let ϕ ~ be a putative scale on Y , let ø ~ = Inf k ϕ ~ , and let i0 ∈ ù. If ϕ ~ is i0 -lsc so is ø. ~ Proof. Let xj ∈ g(Y ) for j ∈ ù and suppose hxj : j ∈ ùi converges to x and the norms øi (xj ) are eventually constant. By thinning the sequence if necessary, we may assume that øi (xi ) = limj øi (xj ) for each i. Let ni be such that xi ani ∈ Y and ϕi (xi ani ), for i ≤ k, or pϕk (xi ani ), ni , ϕi (xi ani )q, for i > k, is as small as possible. If i ≤ k, we have that ϕi (xi ani ) = ϕi (xj anj ) for all j ≥ i, since the minimality of ϕj (xj anj ) implies that of ϕi (xj anj ). If i > k, we have then that ϕk (xi ani ) = ϕk (xk ank ). Hence, for all j > k, nj = nk+1 = n. Hence hxi ani : i ∈ ùi converges to (x an). If j > i > k, we must have ϕi (xi ani ) = ϕi (xj anj ). Thus all norms ϕi (xj anj ) are eventually constant. It follows that x an ∈ Y and ϕi0 (x an) ≤ limj ϕi0 (xj anj ). Thus x ∈ g(Y ). If i0 ≤ k, then øi0 (x) ≤ ϕi0 (x an) ≤ limj ϕi0 (xj anj ) = limj øi0 (xj ). If i > k, it suffices to note that inf{ϕk (x am) : m ∈ ù} ≤ ϕk (x an) ≤ limj ϕk (xj anj ) and so that øi0 (x) = inf{pϕk (x am), m, ϕi (x am)q : m ∈ ù} ≤ pϕk (x an), n, ϕi0 (x an)q ≤ limpϕk (xj anj ), nj , ϕi0 (xj anj )q.

⊣

j

§3. Supremum norms. Let Y ⊆ (ã+1) ù. Suppose ϕ ~ = hϕi : i ∈ ùi is a putative scale on Y . Let k ∈ ù. We define a putative scale ø ~ = høk : k ∈ ùi = Supk ϕ ~ on g(Y ), where g : 1 → {0, 1} and g(0) = 1. If i ≤ k, let øi (x) = sup{ϕi (x an) : n ∈ ù}. If i > k, let øi (x) = pøk (x), ϕp0 (x am0 ), ϕp1 (x am1 ), . . . , ϕpi−k−1 (x ami−k−1 )q, where n 7→ (pn , mn ) is a bijection between ù and ù × ù with pn ≤ n for all n, with p , . . . , q an appropriate embedding of the lexicographic ordering into the ordinals. ø ~ is clearly a putative scale.

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For each i and x1 , x2 , we define the game for verifying that x1 ∈ g(Y ) and either x2 6∈ g(Y ) or øi (x1 ) ≤ øi (x2 ). For i ≤ k, player I plays n1 and player II plays n2 . Player II wins if x1 an1 ∈ Y and either x2an2 6∈ Y or ϕi (x1 an1 ) ≤ ϕi (x2 an2 ). If i > k, player I plays j1 ≤ i −k and player II plays j2 ≤ j1 . If j1 = j2 = 0, player I plays n1 , player II plays n2 , and player II wins ⇔ x1 an1 ∈ Y and either x2 an2 6∈ Y or ϕk (x1 an1 ) ≤ ϕk (x2 an2 ). If j1 = j2 = n + 1, player II wins ⇔ x1 amn ∈ Y and either x2 amn 6∈ Y or ϕpn (x1 amn ) ≤ ϕpn (x2 amn ). The cases j1 > j2 = 0 and j1 > j2 = n + 1 are similar, except that “<” replaces “≤”, and player II plays n2 before player I plays n1 when j2 = 0. There is a similar game for verifying that x1 ∈ g(Y ) and either x2 6∈ g(Y ) or øi (x1 ) < øi (x2 ). A standard play of our game for verifying x1 ∈ g(Y ) and either x2 6∈ g(Y ) or øi (x1 ) ≤ øi (x2 ) is a play in which i ≤ k or j1 = j2 . The terminal condition of a standard play is hx1 an1 , x2 an2 , ii if i ≤ k, hx1 an1 , x2 an2 , ki if i > k and j1 = 0, and hx1 amn , x2 amn , pn i if j1 = n + 1. Lemma 3.1. Let ϕ ~ be a putative scale on Y . Let ø ~ = Supk ϕ. ~ Let f satisfy (a), (b), and (c) as in Lemma 2.1. Let xi ∈ g(Y ) for each i ∈ ù (where g : 1 → {0, 1} and g(0) = 1). Assume that øf(i) (xi ) ≥ øf(i) (xi+1 ) for all i. Let ói , i = 0, 1, . . ., be winning strategies for the game for verifying xi+1 ∈ g(Y ) and either xi 6∈ g(Y ) or øf(i) (xi+1 ) ≤ øf(i) (xi ). For each n ∈ ù there are numbers n0 , n1 , . . . such that limi ni = n and there is a function f ∗ : ù → ù such that f ∗ satisfies (a), (b), (c), (d), and (e) of Lemma 2.1 and such that hxi+1 ani+1 , xi ani , f ∗ (i)i is the terminal condition of a standard play according to ói for each i. Proof. Since the norms øi (xj ) are eventually non-decreasing as j increases, they are eventually constant. Let j(i) be given by pj(i) = i and mj(i) = n. For each t > k, let u(t) be such that u(t) > u(t − 1) if t > k + 1, f(u(t)) > k + j(t), and øk+j(t) (xj ) = øk+j(t) (xu(t) ) for all j ≥ u(t). Set ni = n for all i ≥ u(k +1). For u(t +1) > i ≥ u(t), let player I play j1 = j(t) + 1 against ói . This is legal, since f(i) ≥ f(u(t)) > k + j(t) = k + j1 − 1 so f(i) − k ≥ j1 . Since øk+j(t) (xi+1 ) = øk+j(t) (xi ), øk (xi ) = øk (xi+1 ) and ϕpj (xi amj ) = ϕpj (xi+1 amj ) for all j < j(t). This ói , a winning strategy, cannot call for player II to play j2 < j(t) + 1. Hence j2 = j(t) + 1 and the terminal condition of this standard play is hxi+1 amj(t) , xi amj(t) , pj(t) i = hxi+1 an, xi an, ti. Assume that ni+1 is defined, where k < f(i) and i < u(k + 1). Let player I play j1 = 0 against ói . player II must respond with j2 = 0. Let player I play ni+1 as his n1 . Let ni be player II’s response via ói . The terminal condition of

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this standard play is then hxi+1 ani+1 , xi ani , ki. Suppose ni+1 is defined and f(i) ≤ k. Let player I play ni+1 and player II play ni as given by ói . The terminal condition of this standard play is hxi+1 ani+1 , xi ani , f(i)i. Now define f ∗ (i) = f(i) if f(i) ≤ k, f ∗ (i) = k if k < f(i) and i < u(k + 1), and f ∗ (i) = t if u(t) ≤ i < u(t +1). Since f(u(t)) > k +j(t) > mj(t) = t, f ∗ (i) is always ≤ f(i). ⊣ Lemma 3.1 has the same relations to Moschovakis’ [Mos83] Supremum Lemma as Lemma 2.1 has to the Infimum Lemma. Lemma 3.2. Let ϕ ~ be a putative scale on Y and let ø ~ = Supk ϕ ~ . For i0 ∈ ù, if ϕ ~ is i0 -lsc, so is ø. ~ Proof. Let hxj : j ∈ ùi converge to x with each xj ∈ g(Y ) and let the norms øi (xj ) be eventually constant. Let n ∈ ù. For each i, the norms ϕi (xj an) are eventually constant as j increases. Thus x an ∈ Y and ϕi0 (x an) ≤ limj ϕi0 (xj an). Since ϕ ~ is a putative scale, ϕi (x an) ≤ a limj ϕi (xj n) for each i ≤ i0 . Since this holds for all n, it follows that x ∈ g(Y ) and, from the definition of øi0 , that øi0 (x) ≤ limj øi0 (xj ), provided that øi (x) ≤ limj øi (xj ) for all i such that i ≤ k and i ≤ i0 . Fix such an i and let øi (xj0 ) = øi (xj ) for all j ≥ j0 . We must show that sup{ϕi (x an) : n ∈ ù} ≤ sup{ϕi (xj0 an) : n ∈ ù}. We already know, since i ≤ i0 , that ϕi (x an) ≤ limj ϕi (xj an). Let j ∗ ≥ j0 be such that ϕi (xj ∗ an) ≥ ϕi (x an). For some m, ϕi (xj0 am) ≥ ϕi (xj ∗ an), and the proof is complete. ⊣ §4. Game norms. Let Y ⊆ (ã+α) ù and let g : α → {0, 1}. Let ϕ be a norm on Y . We define a norm ø = g(ϕ) on g(Y ). Let x1 , x2 ∈ ã ù. We consider a game of length 2α, played as follows. If White moves at â, i.e. if g(â) = 0, then player II moves at 2â and chooses z1 (â), and player I moves at 2â + 1 and chooses z2 (â). If Black moves at â, then player I chooses z1 (â) at 2â and player II chooses z2 (â) at 2â + 1. Player II wins the game if x1 az1 ∈ Y and either x2 az2 6∈ Y or ϕ(x1 az1 ) ≤ ϕ(x2 az2 ). (Our game and the lemmas that follow are obvious generalizations of [Mos80, 6E].) We define x1 4 x2 to hold if and only if x1 , x2 ∈ g(Y ) and player II has a winning strategy for the associated game. If we can show that 4 is a prewellordering, then our norm ø is essentially defined. Note: We shall let ø have range an initial segment of the ordinals. Lemma 4.1. Assume all the games used in defining 4 are determined. There is no infinite sequence x0 , x1 , x2 , . . . of members of g(Y ) such that xi 64 xi+1 for all i.

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Proof. Suppose such a sequence exists. Let ó be a strategy for White witnessing that x0 ∈ g(Y ). Let ói , for each i, be a winning strategy for player I for witnessing that xi 64 xi+1 . We shall define zi ∈ α ù such that z0 is a play according to ó and hzi , zi+1 i is a play according to ói , for each i. Suppose each zi ↾â is defined for some â < α, with z0 ↾â consistent with ó and each hzi ↾â, zi+1 ↾âi consistent with ói . If White moves at â, let z0 (â) be given by ó and, inductively, let zi+1 (â) be given by ói . If Black moves at â, let zi (â) be given by ói . Since z0 is according to ó, x0 az0 ∈ Y . Since hzi , zi+1 i is according to ói , we have by induction that zi+1 ∈ Y and ϕ(xi+1 azi+1 ) < ϕ(xi azi ). This is a contradiction. ⊣ Lemma 4.2. Assume that all games used in defining 4 are determined. 4 is a prewellordering of g(Y ). Proof. x 4 x since otherwise xi = x contradicts Lemma 4.1. Similarly, a failure of (x 4 x ′ or x ′ 4 x) contradicts Lemma 4.1. If x1 4 x2 and x2 4 x3 , then composing the two strategies in the obvious way gives x1 4 x3 . Lemma 4.1 implies directly that the relation (x1 4 x2 ∧ x2 64 x1 ) is wellfounded. ⊣ For any x1 and x2 we call the associated game the game for verifying x1 ∈ g(Y ) and either x2 6∈ g(Y ) or ø(x1 ) ≤ ø(x2 ). Note that player II has a winning strategy for the game if and only if x1 ∈ g(Y ) and either x2 6∈ g(Y ) or ø(x1 ) ≤ ø(x2 ). We now define the game for verifying x1 ∈ g(Y ) and either x2 6∈ g(Y ) or ø(x1 ) < ø(x1 ). This game is played just as the one for x2 ∈ g(Y ) and either x1 6∈ g(Y ) or ø(x2 ) ≤ ø(x1 ) except that player I wins if and only if x1 az1 ∈ Y and either x2 az2 6∈ Y or ϕ(x1 az1 ) < ϕ(x2 az2 ). Note: Unlike the games of §2 and §3, it is a winning strategy for player I, not for player II, which “verifies” the relation in question. Lemma 4.3. Assume all relevant games are determined. Player I has a winning strategy for the game for verifying x1 ∈ g(Y ) and either x2 6∈ g(Y ) or ø(x1 ) < ø(x2 ) if and only if x1 ∈ g(Y ) and either x2 6∈ g(Y ) or ø(x1 ) < ø(x2 ). Proof. If player I has a strategy for this game clearly x1 ∈ g(Y ). If x2 ∈ g(Y ) and ø(x2 ) ≤ ø(x1 ), we can play player II’s strategy witnessing this against player I’s strategy and get a contradiction. Suppose x1 ∈ g(Y ) and either x2 6∈ g(Y ) or ø(x1 ) < ø(x2 ) and suppose also that player II has a winning strategy for the game. We can compose this strategy with player II’s strategy witnessing x1 4 x1 , getting a strategy witnessing that x2 4 x1 . ⊣

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§5. The main construction. Let Y ⊆ (ã+α) ù and let g : α → {0, 1} with α countable. Let ϕ ~ = hϕi : i ∈ ùi be a putative scale on Y . Let â 7→ kâ be a one-one function from α into ù. For â ≤ α, let gâ (ä) = g(â + ä) for â + ä < α. Let Xâ = gâ (Y ). Xâ ⊆ (ã+â) ù. For â1 < â2 ≤ α, let gâ1 ,â2 = gâ1 ↾ä where ä is the least ordinal such that â1 + ä = â2 . Note that Xâ1 = gâ1 ,â2 (Xâ2 ). We define a relation ⊳ on ù × (α + 1) as follows: If â < α and kâ < n, then hm, â + 1i ⊳ hn, âi for all m ≤ n. If â < α and kâ ≥ n, let ä be minimal such that ä = α or kä < n. hn, äi ⊳ hn, âi. Otherwise ⊳ never obtains. Lemma 5.1. ⊳ is a well-founded relation. Proof. Suppose hn0 , â0 i ⊲ hn1 , â1 i ⊲ · · ·. Since n0 ≥ n1 ≥ n2 ≥ · · ·, ni = n for all i ≥ some i0 . Since â0 < â1 < · · ·, let â < sup{âi : i ∈ ù} be such that kä ≥ n for all ä such that â ≤ ä < sup{âi : i ∈ ù}. Let i be such that i ≥ i0 and âi ≥ â. Since ni = n and kä ≥ n for all ä with âi ≤ ä ≤ âi+1 , hni+1 , âi+1 i ⊳ 6 hni , âi i. ⊣ The construction which follows will depend on certain games’ being determined. Assume for the rest of this section that all relevant games are determined. By induction on ⊳ we define norms øiâ on Xâ for i ∈ ù and â ≤ α. Let øiα = ϕi for all i ∈ ù. If kâ < i and g(â) = 0, let øiâ = (Inf kâ ø ~ â+1 )i . This is well defined since it â+1 â+1 depends only on ø0 , . . . , øi . If kâ < i and g(â) = 1, let øiâ = (Supkâ ø ~ â+1 )i . If kâ > i, let øiâ = gâ,ä (øiä ) where ä > â is minimal such that ä = α or kä < i. Lemma 5.2. Let â < α and x1 , x2 ∈ Xâ . For all i ∈ ù, if g(â) â øi (x1 ) ≤ øiâ (x2 ) ⇔ (Inf kâ ø ~ â+1 )i (x1 ) ≤ (Inf kâ ø ~ â+1 )i (x2 ), and if â â then øi (x1 ) ≤ øi (x2 ) ⇔ (Supkâ ø ~ â+1 )i (x1 ) < (Supkâ ø ~ â+1 )i (x2 ).

= 0 then g(â) = 1

Proof. For i > kâ it is immediate from the definition that øiâ is identical with the corresponding infimum or supremum norm. Assume then that i ≤ kâ . For some ä > â, øiâ = gâ,ä (øiä ). It is easily seen that gâ,ä (øiä ) = gâ,â+1 (gâ+1,ä (øiä )) = gâ,â+1 (øiâ+1 ). Since i ≤ kâ , the veriâ+1 fication game for gâ,â+1 (øi ) is identical with that for (Inf kâ (ø ~ â+1 ))i or â+1 (Supkâ (ø ~ ))i , depending on whether g(â) = 0 or g(â) = 1. ⊣ Note: øiâ may not be identical with the corresponding infimum or supremum norm when i ≤ kâ , simply because the infimum and supremum norms need not have range an initial segment of the ordinals. Had we wished, we could have modified the definitions to make this so.

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Lemma 5.3. If kä ≥ i for all ä such that â1 ≤ ä < â2 then

øiâ1

=

gâ1 ,â2 øiâ2 .

By induction on ⊳ we define games for verifying x1 ∈ Xâ and either x2 6∈ Xâ or øiâ (x1 ) ≤ øiâ (x2 ), and also games for verifying that x1 ∈ Xâ and x2 6∈ Xâ or øiâ (x1 ) < øiâ (x2 ). If kâ ≥ i, let ä > â be minimal such that ä = α or kä < i. First play the game given by the fact that øiâ = gâ,ä øiä (reversing the roles of player I and player II in the < case). If hz1 , z2 i is a play of this game, then player II wins just in case x1 az1 ∈ Xä and either (x2 az2 ) 6∈ Xä or ø ä (x1 az1 ) <(resp. ≤) ø ä (x2 az2 ). Now play the game for verifying this fact. If kâ < i, we first play the finite game given by the fact that øiâ is an infimum or supremum norm defined from ø ~ iâ+1 . The winning conditions for this game involve an inequality on øiâ+1 , for some i ′ ≤ i or else a numerical inequality. ′ In the latter case, the game terminates. In the former case, continue by playing the appropriate game. A standard play of the game for verifying x1 ∈ Xâ and either x2 6∈ Xâ or øiâ (x1 ) ≤ øiâ (x2 ) is a play in which all the plays of constituent games for infimum or supremum norms are standard. The terminal condition of a standard play is defined in the obvious ways. We also speak of a standard partial play of such a game and of the terminal condition of such a partial play. Lemma 5.4. If ϕ ~ is 0-lsc so is ø~0 . Proof. Let hxj : j ∈ ùi converge to x and suppose that each xj ∈ g(Y ) and that the øi0 (xj ) are eventually constant as j increases. Thinning the sequence if necessary, we may suppose that øi (xi ) = øi (xj ) for each j ≥ i. For a contradiction, let ó be a winning strategy for player I for the game for verifying x0 ∈ g(Y ) and either x 6∈ g(Y ) or (g(ϕ0 ))(x0 ) < (g(ϕ0 ))(x). Note that g(ϕ0 ) = ø00 . Let ói be a winning strategy for player II for the game for verifying xi+1 ∈ g(Y ) and either xi 6∈ g(Y ) or øi0 (xi+1 ) ≤ øi0 (xi ). We shall define z, z0 , z1 , . . ., each ∈ α ù, functions f â : ù → ù for â ≤ α, and standard plays w0 , w1 , . . ., with each wi consistent with ói , such that the following conditions hold: i. hzi : i ∈ ùi converges to z. ii. hz, z0 i is a play consistent with ó. iii. For each α ≤ â and each i ∈ ù, the terminal condition of the corresponding part of the play wi is hxi+1 azi+1 ↾â, xi azi ↾â, f â (i)i. iv. v. vi. vii.

f 0 (i) = i for all i. f â (i) ≤ f â (i + 1) for all â ≤ α and all i ∈ ù. Rangef â is unbounded for all â ≤ α. f â+1 (i) ≥ min{f â (i), kâ }.

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viii. For each i, f â (i) is a non-increasing function of â. ix. If ë ≤ α is a limit ordinal, then f ë (i) = inf{f â (i) : â < ë} for all i ∈ ù. â

â

Suppose z â = z↾â, z0 = z0 ↾â, . . ., f â , and the appropriate parts, wi , of the â â wi are defined. Suppose hzi : i ∈ ùi converges to z â , (z â , z0 ) is consistent with ó, the wiâ are consistent with ói , and iii, v, and vi hold at â, and f â (i) ≤ i for all i ∈ ù. Suppose first that g(â) = 0. iii, v, vi and f â (i) ≤ i guarantee that the hypotheses of Lemma 2.1 are satisfied, with Y replaced by Xâ+1 , k replaced by kâ , with (the appropriate fragments of) the ói , and with f(i) = f â (i). Let n, n1 , n2 , . . . be as given by Lemma 2.1. Let n0 be the move given by ó at (z â an, z0â ). Let f ∗ be as given by Lemma 2.1. Let f â+1 (i) = f ∗ (i). Let wiâ+1 be wiâ followed by the play given by Lemma 2.1. Let z(â) = n and zi (â) = ni . Lemma 2.1 guarantees that hziâ+1 : i ∈ ùi converges to z â+1 , the wiâ+1 are consistent with the ói , iii, v, and vi hold at â + 1, f â+1 (i) ≤ f â (i) ≤ i for all i, and f â+1 (i) ≥ min{f â (i), kâ }. Next suppose that g(â) = 1. We proceed just as in the first case, using Lemma 3.1 in place of Lemma 2.1. We let n be the move given by ó at (z â , z0â ). We omit the details. Finally suppose that ë < α is a limit ordinal, that z ë , the zië , the wië , and â the f â for â < ë, are defined. Suppose hzi : i ∈ ùi converges to z â for each â â < ë, (z â , z0 ) is consistent with ó for â < ë, the wiâ are consistent with ói for â < ë, iv holds, iii, v, vi, and vii hold for â < ë, ix holds for limit ordinals < ë, and each f â (i) is non-increasing as a function of â < ë. It follows that hzië : i ∈ ùi converges to z ë , hz ë , z0ë i is consistent with ó, and each wië is consistent with ói . Define fië by ix. It is readily seen that iii and v hold at ë. To see that vi holds at ë, let n ∈ ù. We must show f ë (i) ≥ n for some i. Let â < ë be such that kä ≥ n for every ä such that â ≤ ä < ë. Let i be such that f â (i) ≥ n. For â ≤ ä < ë, f ä+1 (i) ≥ min{f ä (i), kä } by v and vii. Thus induction and ix give that f ä (i) ≥ n for all ä < ë. Hence f ë (i) ≥ n. We have given the construction and verified that it has the required properties. Now let us prove the lemma. Since hz, z0 i is a play consistent with ó, x0 az0 ∈ Y and either x az 6∈ Y or ϕ0 (x0 az0 ) < ϕ0 (x az). Since each wi is consistent with ói and the terminal condition of wi is hxi+1 azi+1 , xi azi , f α (i)i, it follows by induction that xi azi ∈ Y and ϕf α (i) (xi+1 azi+1 ≤ ϕf α (i) (xi azi ) for each i ∈ ù. Since hzi : i ∈ ùi converges to z and hxi : i ∈ ùi converges to x, it follows, since ϕ ~ is 0-lsc, that x az ∈ Y and ϕ0 (x az) ≤ limi ϕ0 (xi azi ) ≤ a ⊣ ϕ0 (x0 z0 ). This is a contradiction. Theorem 5.5. If ϕ ~ is a scale, so is ø~0 .

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Proof. Let hn, âi be ⊳-minimal such that ϕ ~ â is not n-lsc. If kâ < n, it â+1 follows from Lemmas 2.2 and 3.2 that ϕ ~ is not n-lsc. Since hn, â + 1i ⊳ hn, âi this is a contradiction. If kâ ≥ n, let ã > â be the least ordinal such that ã = α or kã < n. Apply Lemma 5.4 with Y replaced by Xã , g replaced by ã gâ,ã , and hϕi : i ∈ ùi replaced by høn+i : i ∈ ùi. Since ø ~ â is not n-lsc, this ã gives that ø ~ is not n-lsc. Since hn, ãi ⊳ hn, âi, we have a contradiction. ⊣ §6. Canonical winning strategies. Let g, Y , â 7→ kâ , ϕ ~ , hø ~ â : â ≤ αi, and hXâ : â ≤ αi be as in §5, with ϕ ~ a scale on Y . Let x ∈ g(y). We define the canonical strategy for White as follows: At position z â with g(â) = 0, if x az â 6∈ X â , White plays z(â) = 0. If a â x z ∈ Xâ , player I plays the smallest n such that x az â an ∈ Xâ+1 and, for all m such that x az â am ∈ Xâ+1 , økâ+1 (x az â an) ≤ økâ+1 (x az â am). â â Theorem 6.1. Assume all games involved in the definition of the ø ~ â are determined. The canonical strategy is a winning strategy. Proof. We prove the following stronger fact. Let z â be any position consistent with the canonical strategy. Let n ∈ ù and ã ≥ â be such that, for all ä with â ≤ ä < ã, kä ≥ n. We define a game G(z â , n, ã) whose length is 2ñ, where ñ is the least ordinal such that â + ñ = ã. Let z1â = z2â = z â . For â ≤ ä < ã, if g(ä) = 1, player I plays z1 (ä) and then player II plays z2 (ä). If â ≤ ä < ã, and g(ä) = 0, player II must play z1 (ä) = the move given by the canonical strategy at z1ä ; then player I plays z2 (ä). Player II wins just in case x az1ã ∈ Xã and either x az2ã 6∈ Xã or ønã (x az1ã ) ≤ ønã (x az2ã ). We shall prove that player II has a winning strategy for G(z â , n, ã). Suppose this is false for some hã, â, ni and choose the lexocographically least hã, â, ni. Choose some z â witnessing this fact. We first show that ã is a limit ordinal > â. If ã = â, then z â 6∈ Xâ . Let ã ′ = â, let n ′ = 0, and let â ′ = 0. Player I can win G(z 0 , n ′ , ã ′ ) by playing z1 (ä) = z â (ä) and playing z2 (ä) according to some strategy witnessing x ∈ X0 . This contradicts the minimality of hã, â, ni unless â = 0, which is clearly impossible. If â < ã = ä + 1, let player II play a winning strategy for G(z â , n, ä). By the definition of the canonical strategy, the play must reach a winning position hz1ä , z2ä i in G(z â , n, ã). Let us then consider the case ã is a limit ordinal > â. We choose a winning strategy ô for player I for G(z â , n, ã) as follows. If some z ã extending z â and consistent with the canonical strategy is such that x az ã 6∈ Xã , let player I play z1 (ä) = z ã (ä). By the argument of the last paragraph, z â is won for White, so let player I play a z2 (ä) according to a winning strategy for White. If every position z ã extending z â which is consistent with the canonical strategy satisfies x az ã ∈ Xã , let player I play an arbitrary winning strategy

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for G(z â , n, ã). Note that, in either case, every play hz1ã , z2ã i consistent with ô satisfies x az2ã ∈ Xã and either x az1ã 6∈ Xã or ønã (x az1ã ) > ønã (x az2ã ). Let z0â = z â and â0 = â. Suppose inductively we have defined â0 < · · · < â â â âi < ã and z0 j , z1 j , . . . , zj j , z âj for each j ≤ i, and strategies ói for j < i such that a. b. c. d.

â

â

zj j ⊆ zj j+1 ⊆ · · · ⊆ zjâi ; z â0 ⊆ z â1 ⊆ · · · ⊆ z âi ; (z âi , z0âi ) is consistent with ô; â zj j ∈ Xâj for all j ≤ i; â

â

â

â

j+1 j+1 j+1 e. For all j < i, øn+j (x azj+1 ) ≤ øn+j (x azj j+1 ), ój is a winning strategy

âi for player II witnessing this fact, and (zj+1 , zjâi ) is consistent with ój . Furthermore, kä ≥ n + j for all ä such that âj ≤ ä < ã. âi âi f. x az âi ∈ Xâi and øn+1 (x aziâi ). (x az âi ) ≤ øn+i

Let âi+1 > âi be such that âi+1 < ã and kä ≥ n + i + 1 for all ä such that âi+1 ≤ ä < ã. By the minimality of ã, let ôi be a winning strategy for âi+1 âi+1 as follows: (zjâi+1 , zj+1 ) player II for G(z âi , n + i, âi+1 ). We get z0âi+1 , . . . , zi+1 â

â

i+1 will be consistent with ój , for all j < i. (zi+1 , zi i+1 ) will be consistent with some strategy for player II witnessing f. (In particular, ziâi+1 will extend z âi .) âi+1 (z âi+1 , zi+1 ) will be consistent with ôi . (z âi+1 , z0âi+1 ) will be consistent with ô. Since ô is a strategy for player I and all other strategies are for player II, the âi+1 reader will easily check that the z0âi+1 , . . . , zi+1 , z âi+1 are determined uniquely, âi+1 âi+1 âi+1 once the strategy witnessing f is chosen. Since øn+i (x azi+1 ) ≤ øn+i (x aziâi+1 ), we may now complete the construction by choosing ói witnessing this fact. Now let z ã ⊇ z âi for each i and let zjã ⊇ zjâi for each i ≥ j. Since the ój are ã ã ã ã winning strategies, we have x azj+1 ∈ Xã and øn+j (x azj+1 ) ≤ øn+j (x azjã ) for each j. By the properties of ô, we have x az0ã ∈ Xã and ønã (x az0ã ) < ønã (x az ã ). Since hx azjã : j ∈ ùi converges to x az ã (note that zi+1 ↾âi = z↾âi ), we must have x az ã ∈ X ã and ønã (x az ã ) ≤ limj ønã (x azjã ) ≤ ønã (x az0ã ). This is a contradiction. ⊣

§7. Definability. For pointclasses Γ and game types g, we wish to define a pointclass g(Γ) and prove theorems such as Scale(Γ) ⇒ Scale(g(Γ)). Since Γ is to be a pointclass in the sense of [Mos80], we shall define g(Γ) only under the following assumption: There is a well-ordering R of a subset A of ù of order type α, where g : α → {0, 1}, such that, if â 7→ kâ is the isomorphism (∗) between (α, <) and (A, R), and g(k ˆ â ) = g(â), then Γ is closed under preimages by functions recursive in (R, g). ˆ

THE REAL GAME QUANTIFIER PROPAGATES SCALES

221

Let â 7→ kâ be a one-one function from an ordinal α into ù. Let R be the induced well-ordering of a subset of ù. For z ∈ ù ù, let zR∗ ∈ α ù be given by zR∗ (â) = z(kâ ). Let g : α → {0, 1}. Let ã be a pointclass satisfying (*). For simplicity we shall assume that pointclasses are collections of sets each of which is a subset of ã ù, where ã < ù 2 but ã is not fixed. X ⊆ ã ù belongs to g(Γ) just in case there is an R witnessing that Γ satisfies (*) and a Y ∈ Γ such that, for the associated â 7→ kâ , X = g({x azR∗ : x az ∈ Y }). Theorem 7.1. Let α be a countable limit ordinal. Let g : α → {0, 1} be such that, for every limit ordinal ë < α and every n ∈ ù, {g(ë + m) : m ≥ n} = {0, 1}. (In other words, White and Black both make infinitely many moves in every ù-block.) Suppose Γ satisfies (*) and that all games involved in defining g(Γ) are determined. Then Scale(Γ) implies Scale(g(Γ)). Proof. Let Y, R witness that X ∈ g(Γ). Let â 7→ kâ be the associated function. Let ϕ ~ be a scale on Y as given by Scale(Γ). Let ϕi∗ (x azR∗ ) = a ∗ ϕi (x z). ϕ~ is a scale on Y ∗ with X = g(Y ∗ ). Let ø~â , â ≤ α be defined as in §5. We must show that the relations a. x1 ∈ X ∧ (x2 6∈ X or øi0 (x1 ) ≤ øi0 (x2 )) b. x1 ∈ X ∧ (x2 6∈ X or øi0 (x1 ) < øi0 (x2 )) are in Γ. (We take these relations to be subsets of (ù+ù+1) ù.) Since the two cases are similar, we consider only ≤. By inserting dummy moves where necessary, we can make a hold just in case player II wins G(x1 , x2 , i), where G(x1 , x2 , i) is a game of type g. The winning conditions of a play of this game are of one of the forms n1 ≤ n2 n1 < n2 x1 az1 ∈ Y ∧ (x2 az2 6∈ Y or ϕj (x1 az1 ) ≤ ϕj (x2 az2 ) x1 az1 ∈ Y ∧ (x2 az2 6∈ Y or ϕj (x1 az1 ) < ϕj (x2 az2 ) where z1 , z2 and n1 , n2 or j are determined from the play in the obvious way. We must find an R′ and a Y ′ which witnesses that the property that player II has a winning strategy for G(x1 , x2 , i) belongs to g(Γ). We do this as follows: If the âth move in the new game corresponds to ∗ ∗ (ä), we let kâ′ = 4kä +2. picking z1R (ä), we let kâ′ = 4kä . If it corresponds to z2R We let the other non-dummy moves corresponding to ä be pk1ä , pk2ä , and pk3ä and pk4ä if needed, where pi is the i + 1st prime. We let the dummy moves corresponding to â be pk5ä , pk6ä , etc. The corresponding R′ and gˆ ′ are recursive in (R, g). ˆ If zR∗ ′ is a play of the game, z1 and z2 are clearly recursive uniformly in (R, g) ˆ and z. We need to be able to find which of the four winning conditions

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hold and the value of j or (n1 , n2 ) as a function of (x1 , x2 , i, z) recursive in (R, g). ˆ Note that, given m = kâ and k ∈ ù, we can find effectively from R the q such that q = kä , where ä is the least ordinal ≥ â such that kä < k (if it exists, and we can determine whether it exists). Repeating this procedure at most i times, beginning with k = i and m = k0 , we can find all the places in the game G(x1 , x2 , i) where the numbers j1 , j2 as in §2 and §3 are played. This allows us to compute the desired information. We omit the details. ⊣ Corollary 7.2. Let g be the type of the real game, i.e., let g : ù 2 → {0, 1} with g(â) = 0 ⇔ â = ù · 2k + n for some k, n ∈ ù. Assume that all integer games of length ù 2 in which White moves at exactly the even ordinals and whose payoffs are in Γ (in the obvious sense) are determined. If Scale(Γ) then Scale(g(Γ)). Proof. Let Y witness X ∈ g(Γ). As in the proof of Theorem 7.1, we get a scale on X which belongs to g ′ (Γ), where g ′ : ù 2 → {0, 1} and g ′ (â) = 0 ⇔ â is even. But g ′ (Γ) = g(Γ). To see this, replace the g ′ game by a g game as follows: Replace each ù-block by two ù-blocks. White plays a strategy for the next ù moves of the original game and then Black chooses a play consistent with the strategy. ⊣ We could prove move complicated definability theorems by letting the ϕi belong to different classes Γi . We could also prove a generalization of Corollary 7.2 for real games of arbitrary countable length. We could also prove definability results for our canonical strategies. Since there are no ideas involved beyond those already presented and those of [Mos80] and [Mos83], we shall do none of this. REFERENCES

Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Yiannis N. Moschovakis [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. [Mos83] Scales on coinductive sets, this volume, originally published in Kechris et al. [Cabal iii], pp. 77–85. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES, CA 90095, USA

E-mail: [email protected]

LONG GAMES

JOHN R. STEEL

The hypothesis that definable games are determined has proven very powerful in its realm, the realm of reals and definable sets of reals. For example, ZFC + ADL(R) seems to yield a “complete” theory of L(R), in the same way that ZFC alone yields a “complete” theory of L. This success makes it natural to investigate stronger forms of definable determinacy, and the universes larger than L(R) which these might civilize. One might hope to ultimately bring determinacy techniques to bear on questions involving quantification over arbitrary sets of reals, for example, the question of the prewellordering property for Π2n . Various papers, in particular Becker [Bec85], Blass [Bla75], Martin [Mar83B], Solovay [Sol78B], and Woodin (unpublished), have contributed to this investigation. These papers have been concerned with games of length strictly less than ù1 , on ù or on R. In this paper we shall go a bit further and consider certain clopen (i.e., decided after countably many moves) games of length ù1 . In §1 we show that the game quantifiers associated to these clopen games propagate scales, and in §2 we show that the games have canonical winning strategies. Of course, both results require the determinacy of the games in question. Our methods here extend those of Moschovakis [Mos80, Chapter 6], who proved these “third periodicity” theorems for game of length ù on ù, and of Martin [Mar83B] who extended Moschovakis’ proof to games of length less than ù1 on ù or R. In §3 we show that the determinacy of clopen games of length ù1 with payoff Π11 in the codes implies the existence of a natural e inner model in which all games of length less than ù1 on R (with arbitrary payoff) are determined. This section makes use of the results of §1 and §2, but not of their proofs. It also makes heavy use of unpublished seminar notes of Woodin, which show how to construct such a model if one exists; our contribution is just to show that some form of definable determinacy implies its existence. We also indicate in §3 how to construct variants of Woodin’s model satisfying stronger forms of determinacy for arbitrary payoff. In §4 we consider the problem of proving the determinacy hypotheses we have been using, and obtain a partial result using the methods of Blass [Bla75] and The author was partially supported by National Science Foundation grant DMS-3802555. The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

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Martin [Mar75]. Along the way we show that the inner model of §3 satisfies the determinacy of certain games on ℘(R). Finally, §5 is devoted to remarks and questions. As an expository device we work in ZF + DC throughout, and state our additional hypotheses as we need them. By R we mean ù ù, the Baire space. What concepts we take for granted, in particular that of a scale, are explained in [Mos80]. §1. Some game quantifiers which propagate scales. Let us call T ⊆ a tree if ∀p ∈T ∀â ∈ dom(p) (p↾â ∈ T ). Associated to such a T is [ α ù : p 6∈ T & ∀â ∈ dom(p) (p↾â ∈ T )}. E = {p ∈

S

α

α<ù1

ù

α<ù1

Suppose we are given some such T with associated E, and suppose that some A ⊆ E is given. We then have a game G(A, T ): player I and player II alternate playing natural numbers, player I moving first at limit ordinals. The game is over when they reach a position p ∈ E, in which case player I wins iff p ∈ A. (It is convenient to agree that if p is a position, then for all α ∈ dom(p), p(α) = hm, ni where m is player I’s αth move and n is player II’s αth move.) Let aT A = {p ∈

ù

ù : p ∈ T & player I has a winning strategy in G(A, T ) starting from p}.

Of course, if p ∈ E =⇒ dom(p) < ù1 , so that T has no ù1 -branches, then G(A, T ) is just the general form for a clopen (that is, decided after countably many moves) game of length ù1 . We want to restrict this notion a bit. Let WO be the set of (codes of) wellorders of ù. Let F : (T ∪ E) ∩ {p : dom(p) ≥ ù} → WO, where dom(p) = |F (p)| = order type of F (p) for all p ∈ dom(F ). For p ∈ T ∪ E such that dom(p) ≥ ù, let p∗ = hF (p), xi where
x(n) = p(|n|F (p) ) = p ordinal rank of n in F (p) . (Here hF (p), xi ∈

ù

ù × ùù ≈

ù

ù.) For any S ⊆ dom(F ), we let

S ∗ = {p∗ : p ∈ S}.

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Now let C (i, j, k, y) iff ∃p ∈ dom(F ) y = p∗ & |i|F (p) = |j|F (p↾α) ,


where α = |k|F (p) .

We call F a scaled coding of T if both T ∗ and C admit HOD(R) scales. Example 1.1. Fix n ≥ 1. Let n [
o α ù : ∀â ≤ dom(p) (Lâ [p↾â], ∈, p↾â) is not Σn admissible . T = p∈ α<ù1

For q ∈ T ∪ E, dom(q) ≥ ù, let F (q) = the first wellorder of ù of order type dom(q) constructed in L[q]. Then T ∗ and C are Π11 , so that F is a scaled coding of T . We shall now prove our general scale propagation theorem. We remark afterward on refinements of the theorem involving weaker determinacy hypotheses and better definability estimates on the scale produced. Theorem 1.2. Assume that all clopen games of length ù1 with HOD(R) payoff are determined. Let T admit scaled coding, and suppose that A ⊆ E is such that A∗ admits a HOD(R) scale. Then aT A admits a HOD(R) scale. Proof. We assume that if p ∈ T and α ∈ dom(p), then p(α)1 ∈ {0, 1}; that is, player II can only play 0’s and 1’s in G(A, T ). This is no loss of generality. Let F be a scaled coding of T , and ñ~ a HOD(R) scale on the associated C . This gives us some useful norms on A∗ relating “global and local codes” of ordinals. For p∗ ∈ A∗ , let
ϑi0 (p∗ ) = n, ñ(i)0 (n, (i)1 , (i)2 , p∗ ) where n is unique such that C (n, (i)1 , (i)2 , p∗ ). Remarks. (a) We code elements of ù <ù by prime powers, so that hn0 . . . nk i = 2n0 +1 · 3n1 +1 · · · , and (i)k = (exponent of pk in i) − 1. Let (0)k = 0. (b) ϑi0 (p∗ ) is an ordinal gotten by using the lexicographic order of ù × ran ñ~. Let also, for p∗ ∈ A∗ ,
ϑi1 (p∗ ) = n, ñ(i)0 ((i)1 , n, (i)2 , p∗ )
where n is unique such that C (i)1 , n, (i)2 , p∗ if any such n exists (i.e., if |(i)1 |F (p) < |(i)2 |F (p) ). Let ϑi1 (p∗ ) = 0 if no such n exists. Let ó ~ be a HOD(R) scale on T ∗ , and for p∗ ∈ A∗ let
ϑi2 (p∗ ) = ó(i)0 (p↾α)∗

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where α = |(i)1 |F (p) , and ϑi3 (p∗ ) = |i|F (p) and ϑi4 (p∗ ) = n, where |n|F (p) = i. Finally, let ϑ~i5 be a very good scale on A∗ , and set, for p∗ ∈ A∗ , øi (p∗ ) = hdom(p), ϑ00 (p∗ ), . . . , ϑ05 (p∗ ), ϑ10 (p∗ ), . . . , ϑ15 (p∗ ), . . . , ϑi0 (p∗ ), . . . , ϑi5 (p∗ )i. We now describe some games which lead to a scale on aT A. As in Moschovakis’ proof that the ordinary game quantifier propagates scales, we compare positions p, q ∈ aT A by playing out G(A, T ) from each position simultaneously on two boards. This assigns an ordinal value to each such position. The new ingredient is that in these comparison games the players must now make additional moves. These moves reflect the ordinal value they assign to one-move variants of intermediate positions they reach during the game. So let p, q ∈ aT A, and let k ∈ ù. We shall define a game Gk (p, q). The players in Gk (p, q) are F and S. They play on two boards, the p board and the q board, and make additional moves lying on neither board. On the p board S plays G(A, T ) from p as player I while F plays as player II. On the q board F plays G(A, T ) from q as player I while S plays as player II. Play is divided into rounds; we now describe the typical round. Round α. (a) F makes player I’s αth move on q board, then S makes player I’s move on the p board. (b) F now proposes some i such that 0 ≤ i ≤ k and (i)0 ∈ {0, 1}. (c) S either accepts i or proposes some i ′ , 0 ≤ i ′ < i and (i ′ )0 ∈ {0, 1}. (d) Let t ≤ k be the least proposal made during (b) and (c): Case 1. t 6= 0. Then F and S must play (t)0 as player II’s αth move on the p and q boards respectively. Case 2. t = 0. Then F plays any m ∈ {0, 1} as player II’s αth move on the p board, after which S plays any m ′ ∈ {0, 1} as player II’s αth move on the q board. This completes round α. Remarks. (a) We call the 0-proposal “freedom”. (b) Our description of round α is valid only if neither board has reached a position in E. As soon as one board reaches a position in E, F and S start simply playing G(A, T ) in their proper roles on the other board, until it

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too reaches a position in E. Gk (p, q) ends when both boards have reached a position in E. (c) A position in Gk (p, q) is a function u : α → ù, where u(â) codes the (up to) 6 moves during round â. Suppose now that u is a run of Gk (p, q) and that r ⊇ p and s ⊇ q are the runs of G(A, T ) produced in u on the two boards. Let e ∈ ù be the least n such that, letting α = |(n)0 |F (r) , either α ≥ dom(s) or (n)1 was the least proposal made during round α. Let α = |(e)0 |F (r) . If either α ≥ dom(s) or F proposed (e)1 during round α, then S wins u iff øe (r ∗ ) ≤ øe (s ∗ ). If S proposed (e)1 during round α, then S wins u iff øe (r ∗ ) < øe (s ∗ ). Remarks. (a) We call e the critical number of u, and write e = crit(u). Notice that crit(u) exists, and in fact crit(u) ≤ h0, ki, for every run u of Gk (p, q). (b) Our convention is that øe takes value ∞ off of A∗ , that ∞ ≤ ∞, and that x < ∞ iff x ∈ Ord. So, for example, if S proposed e1 then S loses unless r ∈ A. For p and q in aT A, let p ≤k q iff S has a winning strategy in Gk (p, q). The next lemma implies that ≤k is a prewellorder. Lemma 1.3. Let p0 ∈ aT A, and suppose that for all n ≥ 0, Σn is either a winning strategy for F in Gk (pn , pn+1 ) or a winning strategy for S in Gk (pn+1 , pn ). Then only finitely many Σn ’s are for F . Proof. Fix a winning strategy ô for player I from p0 in G(A, T ). Assume toward a contradiction that infinitely many Σn ’s are for F . We shall construct runs un according to Σn such that un and un+1 agree on a common play rn+1 ⊆ pn+1 on the pn+1 board, and the play r0 ⊆ p0 on the p0 board is according to ô. (Notice Σn always plays as player I on the pn+1 board and player II on the pn board.) The definition is by induction on rounds. Suppose we have un ↾α for all n, and consequently rn ↾ù + α for all n. (Here rn ↾ù = pn .) Suppose also that rn ↾ù + α ∈ T for all n; that is, no board has reached a position in E. We now define round α of the un ’s. Let a0 = ô(r0 ↾ù + α) and an+1

( Σn (un ↾α aan ) = Σn (un ↾α)

if Σn is for S if Σn is for F

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be the αth moves for player I given by ô and the Σn ’s. We must now define the proposal phase of un (α), for all n. We represent a proposalacceptance/counterproposal by a pair (b, c) of numbers. Claim 1.4. There are a t ≤ k, and n0 , and pairs (bn , cn ) for n ∈ ù, such that (i) un ↾α ahan , an+1 , bn , cn i is according to Σn (if Σn is for S; otherwise un ↾α ahan+1 , an , bn , cn i is according to Σn ), and (ii) hb, ci settles on freedom for n < n0 and on t for n ≥ n0 , and (iii) for infinitely many n, hbn , cn i involves Σn proposing t. Proof (Sketch). Let t0 ≤ k be least such that infinitely many Σ’s which are F -strategies will now propose t0 . Suppose that tn is given. If all but finitely many Σ’s which are S strategies will accept a tn proposal, stop the induction and set t = tn . Otherwise let tn+1 < tn be such an infinitely many Σ’s for S counterpropose tn+1 when their opponent proposes tn . Since tn+1 < tn , we eventually get t. One can check that t works. (If t0 = t, the Σ’s verifying (iii) are for F . Otherwise, the Σ’s verifying (iii) are for S.) ⊣ Now let t, n0 , h(bn , cn ) : n ∈ ùi satisfy the claim. We imagine (bn , cn ) as the proposal phase of un (α). We want finally an αth move dn for player II on the pn board. Case 1. t > 0. Then let dn = (t)0 and dn =

(

for n ≥ n0

Σn (un ↾α ahan , an+1 , bn , cn , dn+1 i) Σn (un ↾α ahan+1 , an , bn , cn i)

if Σn is for S if Σn is for F

Σn (un ↾α ahan , an+1 , bn , cn , dn+1 i) Σn (un ↾α ahan+1 , an , bn , cn i)

if Σn is for S if Σn is for F

for n < n0 . Case 2. t = 0. Let dn =

(

Since infinitely many Σn ’s are for F , this definition makes sense. Finally, we set ( han , an+1 , bn , cn , dn+1 , dn i if Σn is for S un (α) = han+1 , an , bn , cn , dn , dn+1 i if Σn is for F .

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229

Thus in any case un (α) and un+1 (α) agree that rn+1 (α) = han+1 , dn+1 i, while r0 (α) = ha0 , d0 i according to ô. We can continue to define un (α) this way until we reach an α such that ri ↾ù + α ∈ E for some i. This must happen. Fix the first such α, and fix i. Now since ô is a winning strategy for player I and each Σn is a winning strategy for F or S in Gk , we see that rn ↾ù + α ∈ aT A for all n. (Cf. the payoff for Gk and our convention x < ∞ for x ∈ Ord.) But then since the first component in any øe (r ∗ ) is dom(r), rn ↾ù + α ∈ E for all n ≥ i. Now let e ≤ h0, ki be least such that e = crit(un ) for infinitely many n ≥
i. 3 ∗ So e ≤ crit(un ) for cofinitely many n ≥ i, so that ϑ(e) (r ↾ù + α) is n 0 eventually constant as n → ∞. Write rn = rn ↾ù + α, and let 3 â = eventual value of ϑ(e) (r0∗ ) = |(e)0 |F (rn ) . 0

Thus infinitely many, and hence by construction cofinitely many, un ’s settle on the (e)1 proposal round at â. For infinitely many n, Σn is responsible for the proposal. Thus ∗ øe (rn+1 ) ≤ øe (rn∗ )

for cofinitely many n, while ∗ øe (rn+1 ) < øe (rn∗ )

for infinitely many n. This is a contradiction.

⊣

Corollary 1.5. ≤k is a prewellorder of aT A. Proof. Reflexive: if p k p then p, p, p, . . . violates Lemma 1.3. Connected: if p k q and q k p then pqpqpq . . . violates Lemma 1.3. Transitive: if p ≤k q ≤k r, and p k r, then prqprqprq . . . violates Lemma 1.3. Wellfounded: Clear from Lemma 1.3. ⊣ Actually, reflexivity, connectedness, and transitivity could be proved by more direct finite diagrams. Now for p ∈ aT A, let ϕk (p) = ordinal of p in ≤k . Lemma 1.6. ϕ ~ is a semiscale on aT A. T Proof. Let pn → p as
n → ∞, pn ∈ a A for all n, and ∀i ϕi (pn ) eventually constant as n → ∞ . By thinning the sequence of pn ’s we may assume we have a winning strategy Σn for S in Gn (pn+1 , pn ) for all n. Let ô be a winning strategy for player I in G(A, T ) from p0 , and, towards a contradiction, let ó be a winning strategy for player II in G(A, T ) from p. We need a definition. Suppose rn ∈ T ∪ E for n ∈ ù, and for all k

|k|F (rn ) is eventually constant = αk ,

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and rn (αk ) is eventually constant as n → ∞. Let ð : â → {αk : k ∈ ù} be the enumeration of {αk : k ∈ ù} in increasing order, and define r : â → ù by
r(ã) = eventual value of rn ð(ã) as n → ∞. We write then r = lim∗n→∞ rn . This notion of “convergence in the codes” is more useful than pointwise convergence in what follows. (lim∗n→∞ rn = r means just that rn∗ converges to r ∗ in a certain scale.) We now define by induction on rounds runs un of Gn (pn+1 , pn ) according to Σn . We arrange that un and un+1 agree on a common play rn+1 ⊇ pn+1 for the pn+1 board, and that the play r0 ⊇ p0 on the p0 board is by ô. So assume that un ↾α, hence rn ↾ù + α, is given for all n. (Let rn ↾ù = pn .) Let a0 = ô(ro ↾ù + α) and an+1 = Σn (un ↾α aan ) be the moves for player I on the various boards generated by ô and the Σn ’s. If an is eventually constant, say an = a eventually, and if lim∗n→∞ rn ↾ù + α = r for some r which is a play by ó, let d = ó(r aa). Otherwise, let d = 0. Now for any n let in be the largest i such that hd, ii ≤ n and Σn (un ↾a aan ahd, ii) = “accept”, if such an i exists. Let in = 0 otherwise. Case 1. in → ∞ as n → ∞. Pick n0 such that in > 0 for n ≥ n0 . The proposal pair in un (α) is (bn , cn ) where bn = hd, in i for n ≥ n0 and bn = 0 = freedom for n < n0 , and cn = accept for all n. Let ( d if n ≥ n0 dn = Σn (un ↾a ahan , an+1 , bn , cn , dn+1 i) if n < n0 and set un (α) = han , an+1 , bn , cn , dn+1 , dn i. Case 2. Otherwise. Then there is a hd, ii such that infinitely many Σn ’s reject hd, ii. Just as in the claim in Lemma 1.3, we get a t < hd, ii, an n0 , and pairs (bn , cn ) such that

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(i) un ↾α ahan , an+1 , bn , cn i is according to Σn , (ii) hbn , cn i settles on freedom for n < n0 , and on t for n ≥ n0 , and (iii) for infinitely many n, (bn , cn ) involves Σn proposing t. Subcase A. t > 0. Then we set ( (t)0 if n ≥ n0 dn = a Σn (un ↾α han , an+1 , bn , cn , dn+1 i) if n < n0 . Subcase B. t = 0. Define S = {hd0 , . . . , dk i : ∀i ≤ k (di ∈ {0, 1}) and ∀i < k di = Σi (ui ↾α ahai , ai+1 , bi , ci , di+1 i)}. Clearly S is an infinite, finitely branching tree. (This was the reason we restricted player II’s plays in G(A, T ) to {0, 1}.) Let f be an infinite branch of S, and set di = f(i). Finally, in Case 2 we set un (α) = han , an+1 , bn , cn , dn+1 , dn i. We can continue to define un (α) this way until we reach, as we must, an α such that ri ↾ù + α ∈ E for some i. Fix the least such α, and fix i. As before, rn ↾ù + α ∈ a T A for all n, so rn ↾ù + α ∈ A for all n ≥ i. Let un = un ↾α and rn = rn ↾ù + α. We claim crit(un ) → ∞ as n → ∞. If not, let e be least such that e = crit(un ) for infinitely many n. So e ≤ crit(un ) for cofinitely many n, and 3 ϑ(e) (rn∗ ) = |(e)0 |F (rn ) is eventually constant as n → ∞. Let â be this constant 0 value. Then infinitely many un ’s settle on the (e)1 proposal at round â. Thus Case 2 must apply at round â, so by construction cofinitely many un ’s settle on (e)1 at round â, and for infinitely many n, Σn is responsible for (e)1 . But then øe (rn+1 )∗ ≤ øe (rn∗ ) for cofinitely many n, while øe (rn+1 )∗ < øe (rn∗ ) for infinitely many n, a contradiction. Since crit(un ) → ∞ as n → ∞, øe (rn∗ ) is eventually constant as n → ∞, for all e. Thus rn∗ converges in ϑ~3 , ϑ~4 , and ϑ~5 , so that lim∗n→∞ rn = r for some r ⊇ p such that r ∈ A. We are done if we show that r is a play according to ó. If not, let â be least so that r(â) is not according to ó. Let â = |k|F (r)

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and let ã be the eventual value of |k|F (rn ) as n → ∞. We claim that lim∗n→∞ rn ↾ã = r↾â. This follows from the convergence of rn∗ in ϑ~0 and ϑ~1 : if ä < ã, then the code for ä relative to F (rn ↾ã) stabilizes as n → ∞ iff the code for ä relative to F (rn ) stabilizes as n → ∞. Since r ∗ converges in ϑ~2 , we get that r↾â ∈ T . Since r ∗ converges in ϑ~5 , which is very good, the an ’s defined at round ã are eventually constant = a. But then at round ã, d = ó(r↾â aa). Moreover, Case 1 must apply at round ã, since otherwise crit(un ) has a finite lim inf. Thus rn (ã) = ha, d i for all sufficiently large n. Since ϑ~5 is very good, r(â) = ha, d i. But then r↾â + 1 is according to ó, a contradiction which completes the proof of Lemma 1.6. ⊣ (Lemma 1.6) Lemma 1.6 completes the proof of Theorem 1.2, since a well known construction produces a HOD(R) scale on any set which carries a HOD(R) semiscale. ⊣ (Theorem 1.2) Given a scaled coding F of T , and a pointclass Γ of a set of reals, let aT,F Γ = {aT A : A∗ ∈ Γ}. The proof of Theorem 1.2 shows that if T and F are “reasonable” and Γ has the semiscale property, so does aT,F Γ. For example, let [
α ù : ∀â ∈ dom(p) (Lâ [p↾â], ∈, p↾â) is not admissible }, Tn = {p ∈ α<ù1

and let Fn be the scaled coding of Tn described at the beginning of this section. Let us write aΣn Γ = aTn ,Fn Γ, and agree that Γ-ADΣn iff G(A, Tn ) is determined whenever A∗ ∈ Γ. (For “℘(R)-ADΣn ” we write simply “ADΣn ”. Corollary 1.7. Let n ≥ 1, and assume Γ-ADón , where Γ is closed under ∀R . Suppose all Γ sets admit Γ semiscales. Then all aΣn Γ sets admit aΣn Γ scales, so that all aΣn Γ relations admit aΣn Γ uniformizations. Proof. Suppose A∗ ∈ Γ. Let us trace through the proof of Theorem 1.2. We can take ϑi0 . . . ϑi5 to be Γ norms. We must modify øi slightly to get a Γ norm; let øi′ (p∗ ) = hϑ05 (p∗ ), øi (p∗ )i.

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Finally, let Gk′ (p, q) be just like Gk (p, q) except that S cannot win Gk (p, q) unless he wins as player I on the p board. The proof of Theorem 1.2 goes through with these modifications (cf. [Mos80, Chapter 6]) and yields a aΣn Γ semiscale ϕ~′ on aΣn A. One can easily check that, since aΣn Γ is closed under ⊣ real quantification, the scale of the tree of ϕ~′ is in fact a aΣn Γ scale. We would like to point out two curious features of the proof of Theorem 1.2. First, it handles directly only games where player II must play from {0, 1}. Second, the verification that pù ∈ aT A when pn → pù mod ϕ ~ is indirect: we do not construct a strategy for player I in G(A, T ) from pù , but instead defeat a strategy for player II. Is there a more direct proof avoiding these devices? §2. Canonical strategies. We shall construct definable winning strategies for the games of the form G(A, T ), where T admits a scaled coding and A∗ a HOD(R) scale. The strategies are in some sense “best”, as in Moschovakis [Mos80, Chapter 6]. However, the games Gk (p, q) are not adequate to evaluate what’s best for player I; the problem is there is no satisfactory way to decide which k to use in evaluating a given position. So we use the game Gù (p, q), which is like Gk (p, q) except that no bound is put on the size of proposals F may make. Now Gù (p, q) leads to a probably illfounded value order on player I’s possible next moves, but we can avoid that problem by considering directly only games where player I must play from {0, 1}. (Curiouser and curiouser!) Once again, we prove our most general theorem first, then state its sharpened form from games ending at the first Σn admissible relative to the play as a corollary. If Σn is a strategy for G(A, T ), where T admits a scaled coding, then Σ∗ = {p∗ : p is a position according to Σ}. Theorem 2.1. Assume all clopen games of length ù1 with HOD(R) payoff are determined. Suppose T admits a scaled coding, and A ⊆ E is such that A∗ admits a HOD(R) scale. Then if player I wins G(A, T ) he wins via a strategy Σ such that Σ∗ admits a HOD(R) scale.
Proof. We assume without loss of generality that p ∈ T & ä ∈ dom(p) ⇒
p(ä)0 ∈ {0, 1} & p(ä)1 ∈ {0, 1} ; that is both players must play 0 or 1 in G(A, T ). In order to conform to the notation of §1, we assume we have a fixed p0 ∈ T such that dom(p0 ) = ù and player I wins G(A, T ) from p0 : the canonical Σ we are to construct must win from p0 . This is no loss of generality. ~0 . . . ϑ ~ 5 , and ø Let ñ~, ϑ ~ be the families of norms on A∗ defined from a scaled coding of T and a scale on A∗ just as in the proof of Theorem 1.2. For p, q ∈ T such that dom(p) = dom(q) we define a game Gù (p, q). The definition is the same as that of Gk (p, q) for k ∈ ù, except that: (a) we do not require dom(p) = ù, or even that p or q are winning positions for player I in G(A, T )

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(this assumption played no role in the definition of Gk (p, q) anyway), and (b) we allow F to propose any t ∈ ù, not just t ≤ k, when it’s his turn to propose in Gù (p, q) (If S rejects t, he must still counterpropose a t ′ < t.) Now, let for p ⊇ p0 ,   0 if ∀a ∈{0, 1} ∃b ∈{0, 1} such that S has a winning strategy Σ(p) = in Gù (pah0, ai, pah1, bi),   1 otherwise. The determinacy of Gù and its symmetry imply that if Σ(p) = 1, then


∀a ∈ {0, 1} ∃b ∈{0, 1} S has a winning strategy in Gù (pah1, ai, pah0, bi) .

Thus Σ(p) = i ⇒ S has a winning strategy in H (p), where H (p) is the game in which F plays a, then S plays b, then F and S play out Gù (pahi, ai, pah1 − i, bi). It is easy to see that Σ∗ is of the form aS B, for some S admitting a scaled coding and B such that B ∗ admits a HOD(R) scale. By Theorem 1.2, then, it is enough to show that Σ is a winning strategy for player I in G(A, T ) from p0 . So let q ∈ E be an arbitrary play according to Σ; we want to show q ∈ A. Fix a winning strategy ô for player I in G(A, T ) from p0 . For ä ∈ dom(q), let Σä be a winning strategy for S in H (q↾ä). We represent a position or completed run of H (q↾ä) by a function u : α → ù, where α ≥ ä and u↾ä = q↾ä, u(ä) = h1 − q(ä)0 , q(ä)0 , a, bi where a, b are the first two moves of H (q↾ä), and u(ä + 1 + ç) codes the (up to) six moves of round ç in Gù (q↾ä ahq(ä)0 , ai, q↾ä ah1 − q(ä)0 , bi). If u is a position or run of H (q↾ä), then the lower board of u is r, where r↾ä = q↾ä, r(ä) = hq(ä)0 , u(ä)2 i, and r(ä + 1 + ç) = hu(ä + 1 + ç)1 , u(ä + 1 + ç)4 i. Similarly, the upper board of u is s, where s↾ä = q↾ä, s(ä) = h1 − q(ä)0 , u(ä)3 i, and s(ä + 1 + ç) = hu(ä + 1 + ç)0 , u(ä + 1 + ç)5 i. Notice that from position u of H (q↾ä) we can recover ä; ä is the least α ∈ dom(u) such that u(α)0 6= q(α)0 . Let us write ä = ä(u). Let us call a sequence huâ : â < ãi a diagram if ã ≤ dom(u0 ) and (a) for â < ã, uâ is a position in H (q↾ä) according to Σä , where ä = ä(uâ ), and (b) letting râ be the upper board of uâ for â < ã, we have: r0 according to ô, and râ+1 is the lower board of uâ for â + 1 < ã. Our plan is to construct a diagram huâ : â < ãi such that for any limit ë ≤ ã, crit(uâ ) → ù as â → ë. We also arrange that r0 ∈ E, that rë = lim∗â→ë râ if ë < ã is a limit, and that q = lim∗â→ã râ if ã is a limit, while q is the lower board of uã−1 if ã is a successor. The existence of such a diagram easily implies q ∈ A. [r0 ∈ A since r0 is by ô. But then râ ∈ A for â < ã by induction: since râ and râ+1 are the upper and lower boards of uâ , which is by some Σä ,

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râ ∈ A ⇒ râ+1 ∈ A. If ë < ã is a limit, then since crit(uâ ) → ù as â → ë, râ∗ converges in ϑ~3 , ϑ~4 , ϑ~5 to some r ∗ in A∗ ; moreover r = rë since lim∗â→ë râ = rë . So râ ∈ A for â < ã; repeating the argument we get q ∈ A.] We obtain the desired diagram by means of a sequence Dα = huâα : â < ãα i of diagrams defined by induction on α. We maintain by induction that α < ó ⇒ u0α ⊆ u0ó . In fact, if n < min(ù, ãα ) and α < ó, then n < ãó and unα ⊆ unó . (On the other hand, we may well have α < ó such that uùα and uùó are defined and incompatible. We are really building a tree of approximations to the desired uù , “continuously” associating to each branch of this tree a tree of approximations to uù+ù , etc. This is because we do not know definitely even initial segments of the eventual rù , rù+ù , etc., as we build our diagram. For notational simplicity, however, we shall suppress explicit mention of these trees. If huâ : â < ãi is any diagram, then h(uâ , aâ ) : â < ãi is an enlarged diagram if a0 = ô(r0 ) and αâ+1 = Σä (uâ ahaâ i) for â + 1 < ã and ä = ä(uâ ). Given such an enlarged diagram, let   the least t such that (t)0 ∈ {0, 1} and iâ = Σä (uâ ahaâ , aâ+1 , ti) = reject, where ä = ä(uâ ),   ù, if no such t exists. For ë ≤ ã a limit, we say h(uâ , aâ ) : â < ãi accepts readily at ë iff limâ→ë iâ = ù. Given an enlarged diagram h(uâ , aâ ) : â < ãi with ã = â + 1, define aã by: aã = Σä (uâ ahaâ i), where ä = ä(uâ ). Lemma 2.2. Suppose h(uâ , aâ ) : â < ãi is as enlarged diagram which accepts readily at all limit ë ≤ ã. Let d ∈ {0, 1}. Then there are bâ , câ , and dâ , for â < ã, such that, setting dã = d , (a) (uâ ahaâ , aâ+1 , bâ , câ , dâ+1 , dâ i) is by Σä , ä = ä(uâ ), for â < ã, (b) dâ → dë as â → ë, for ë ≤ ã a limit, and (c) bâ → ù as â → ë, for ë ≤ ã a limit câ = accept, for all but finitely many â. Proof. By induction on ã. The successor step is easy, so let ã be a limit. Let ð : ã ֒→ ù be determined by r0 . Let iâ , for â < ã, be as in the definition of ready acceptance. Let ç < ã be such that hd, 0i ≤ iâ for ç ≤ â (so Σä(uâ ) accepts hd, 0i for ç ≤ â.) Let ë be the largest limit ≤ ç. For ç ≤ â < ã, let bâ = the largest t ≤ max(ð(â), iâ ) such that (t)0 = d and Σä accepts t after uâ ahaâ , aâ+1 , ti, where ä = ä(uâ ),

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and câ = accept,

dâ = d.

For ë ≤ â < ç, let bâ = 0 = freedom,

câ = accept

and dâ = Σä (uâ ahaâ , aâ+1 , bâ , câ , dâ+1 i), for ä = ä(uâ ). Finally, the induction hypothesis with d = dë will give us the desired bâ , câ and dâ for â < ë. ⊣ (Lemma 2.2) We shall use tacitly the fact that there is a function which, given a sequence of satisfying the hypothesis of the lemma and a d ∈ {0, 1}, produces a sequence satisfying its conclusion. (We don’t have AC!) We are ready to define our approximations Dα = huâα : â < ãα i to the desired diagram; the definition is by induction on α. α = 0: We may assume q is not by ô; otherwise q ∈ A and we’re done. So let ä be least such that ô(q↾ä) 6= q(ä)0 . Set ã0 = 1, and u00 = q↾ä ah1 − q(ä)0 , q(ä)0 , q(ä)1 , Σä (q↾ä ah1 − q(ä)0 , q(ä)0 , q(ä)1 i)i. S ç α > 0: Set v0 = ç<α u0 . We use sâ for the upper board of vâ . If s0 ∈ E, then our induction on α stops; Dα is undefined. Otherwise, let a0 = ô(s0 ). We shall define vâ and aâ by induction on â. Suppose we have vâ and aâ for â < ã, and that h(vâ , aâ ) : â < ãi is an enlarged diagram. Suppose also the following four conditions are met. (1) ã limit ⇒ h(vâ , aâ ) : â < ãi accepts readily at ã. (2) If ã is a limit, then limâ→ã aâ and lim∗â→ã sâ exists, moreover p0 ⊇ lim∗â→ã sâ , (3) Let s = lim∗â→ã sâ if ã is a limit, and s be the lower board of vã−1
otherwise. Then ∃ä ∈ dom(s) ∩ dom(q) s(ä) 6= q(ä) ; moreover, if ä is the least such ordinal, then s(ä)0 6= q(ä)0 . (4) For ϑ ∈ dom(s0 ), define tϑ by tϑ (0) = ϑ,

tϑ (â + 1) = tϑ (â) for â + 1 ≤ ã,

and tϑ (ë) = |e|F (r) , where r = lim∗â→ã sâ and |e|F (sâ ) = tϑ (â) for all sufficiently large â < ë,

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for ë ≤ ã a limit; tϑ (ë) is undefined if no such e ∈ ù exists. (One should visualize tϑ as the ϑth “column” of the diagram hvâ : â < ãi. Not all columns extend to ã, since not all contribute to ∗-limits all the way down.) Let Cã = {ϑ : tϑ (ã) is defined}. Let s, ä be as in (3). (Our assumptions imply dom(s) = {tä (ã) : ϑ ∈ Cã }.) Fix ϑ (unique) such that ä = tϑ (ã). Then we require that for ϑ ≤ ó ≤ ñ and ó, ñ ∈ Cã ã < min(ãó , ãñ ) (i.e., uãó , uãñ are defined), uãó ⊆ uãñ , and s↾(tó (ã) + 1) is the upper board of uãó . If conditions (1)–(4) are met, then we set aã = lim aâ , â→ã

and vã =

[ {uãó : ó ∈ Cã & tó (ã) ≥ ä}.

Notice that h(vâ , aâ ) : â < ã + 1i remains an enlarged diagram. If one of (1)–(4) fails, then vã is undefined. This must happen at some ã ≤ supó<α ãó since at least (4) will fail. So suppose ã is at least such that vã is undefined. We shall define Dα by taking cases on which of (1)–(4) fail at ã. Case 1. (1) fails at ã. So ã is a limit. Set ãα = ã. Now since (1) fails, we get a t ∈ ù and a cofinal B ⊆ ã such that for â ∈ B, there is a b such that (b)0 ∈ {0, 1} and Σä counterproposes t when F proposes b after vâ ahaâ , aâ+1 i, where ä = ä(vâ ). Let t be least such that a cofinal B ⊇ ã exists, and fix such a B of order type ù. Subcase A. t > 0. There is an ç < ã such that for ç ≤ â < ã and ä = ä(vâ ), Σä accepts t after vâ ahaâ , aâ+1 i; this follows from the minimality of t. Let ð : ã ֒→ ù, be determined by s0 . For ç ≤ â ≤ ã, let bâ = some b witnessing â ∈ B, if â ∈ B, câ = (reject, t), if â ∈ B, bâ = the largest i ≤ max(t, ð(â)) such that (i)0 = (t)0 and Σä accepts i after vâ ahaâ , aâ+1 i, where ä = ä(vâ ), if â 6∈ B, câ = accept, if â 6∈ B, and dâ = (t)0 .

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Let ë be the largest limit ordinal ≤ ç. For ë ≤ â ≤ ç, set bâ = 0 = freedom,

câ = accept,

and dâ = Σä (vâ ahaâ , aâ+1 , bâ , câ , dâ+1 i), for ä = ä(vâ ). Finally, since h(vâ , aâ ) : â < ãi accepts readily at all limit ë′ ≤ ë, we may apply Lemma 2.2 with d = dë to generate bâ , câ and dâ for â < ë. Then let uâα = vâ ahaâ , aâ+1 , bâ , câ , dâ+1 , dâ i for â < ã = ãα . Subcase B. t = 0. Let iâ be as in the definition of ready acceptance, and let  
 S = ó < ã : ∃â ó ≤ â < ó + ù & â ∈ B or iâ ≤ max(h0, 0i, h1, 0i) Then S has order type ù, as otherwise (1) fails at some ë < ã. Let ( some b witnessing â ∈ B, if â ∈ B bâ = 0, if â ∈ S − B and

( (reject, 0), if â ∈ B câ = accept, if â ∈ S − B.

Let {ói : i ∈ ù} be the increasing enumeration of S, and consider 
U = hd0 . . . dk i ∈ 2ù : ∀i < k if ó = ói and ä = ä(vó ) , then di = Σä (vó ahaó , aó+1 , bó , bó , có , dó+1 i) U is an infinite tree on {0, 1} so we have an f ∈ all k. Set dói = f(i),

ù

2 such that f↾k ∈ U for

for i ∈ ù.

Finally, suppose â ∈ ã − S. Let ó ∈ S be least such that â < ó. (ó is a limit.) Set bâ = the largest i ≤ max(iâ , ð(â)) such that (i)0 = dó and Σä accepts i after vâ ahaâ , aâ+1 i, where ä = ä(vâ ), and câ = accept,

dâ = dó .

Now, we let, for â < ã = ãα uâα = vâ ahaâ , aâ+1 , bâ , câ , dâ+1 , dâ i. This completes the definition of Dα in Case 1.

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Case 2. (1) holds and (2) fails at ã. Since h(vâ ), aâ : â < ãi accepts readily at all limit ë ≤ ã, Lemma 2.2 applied with d = 0 yields bâ , câ , and dâ for â < ã. Set ãα = ã, and uâα = vâ ahaâ , aâ+1 , bâ , câ , dâ+1 , dâ i for â < ã. Case 3. (1) and (2) hold but (3) fails at ã. Let s be as in (3). Let a = aã if ã is a successor, a = limâ→ã
aâ otherwise. Subcase A. q ⊆ s, or ∃ä ∈ dom(s) ∩ dom(q) s(ä) 6= q(ä) , but s(ä)0 = q(ä)0 for the least such ä. In this case, proceed exactly as in Case 2. Subcase B. s $ q and q(dom(s))0 = a. In this case apply Lemma 2.2 with d = q(dom(s))1 to get bâ , câ and dâ for â < ã. Set ãó = ã, and uâα = vâ ahaâ , aâ+1 , bâ , câ , dâ+1 , dâ i for â < ã. Subcase C. s $ q and q(dom(s))0 = 1 − a. This is the only case in which we set up a new board. Let ãα = ã + 1. Set uãα = s aha, 1 − a, q(ä)1 , Σä (s aha, 1 − a, q(ä)1 i)i, where ä = dom(s). Now use the lemma with d = Σä (s aha, 1 − a, q(ä)1 i) to define uâα for â < ã. Case 4. (1)–(3) hold, but (4) fails at ã. Let ãα = ã, and apply Lemma 2.2 with d = 0 to get uâα for â < ã. This completes the inductive definition of the Dα ’s. Let vâα , sâα be the vâ and sâ occurring in the definitions of Dα , α > 0. Let M = huâ : â < ãi be a diagram with boards râ , â < ã. We call M good iff lim∗â→ë râ exists for all limit ë ≤ ã, and lim∗â→ë râ = rë for all limit ë ≤ ã. If M = huâ : â < ãi is good, we define tϑM for ϑ ∈ dom(u0 ) to be the ϑth “column” of M , as in condition (4) above: tϑM (0) = ϑ, tϑM (â + 1) = tϑM (â) for â + 1 ≤ ë, and tϑM (ë) = |e|F (r) , where ë ≤ ã is a limit, r = lim∗â→ë râ , and |e|F (râ ) = tϑM (â) for all sufficiently large â < ë. (ë 6∈ dom(tϑM ) if no such e exists.) Let also CãM = {ϑ : tϑM (ã) is defined}. S We now construct the desired diagram. Let u0 = {u0α : u0α is defined}, and note that the upper bound r0 of u0 is in A. Now suppose M = huâ : â < ãi is given, and that (a) crit(uâ ) → ù as â → ë, for any limit ë ≤ ã, (b) M is a good diagram, and (c) for â < ã, if α ∈ CâM and tαM (â) ≤ ä(uâ ), then uâα = uâ ↾(tαM (â) + 1). (So uâα is defined.) Let r = lim∗â→ã râ if ã is a limit, and let r be the lower board of uã−1 otherwise. If r = q then (a) and (b) guarantee that M is the desired diagram.

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Otherwise, since r, q ∈ E, they are incompatible; let ä be least such that r(ä) 6= q(ä). We define uã by [ uã = {uãα : α ∈ CãM & tα (ã) ≥ ä}; we must see that (a), (b) and (c) remain true. The following claims insure this. Claim 2.3. If α ∈ CãM , â < ã, and tαM (â) < ä(uâ ), then tαM (ã) < ä. Proof. By induction on ç ≥ â we see that tαM (ç) = tαM (â) and rç ↾tαM (ç) = q↾tαM (ç). If ç is a successor, this follows from the definition of “diagram” and “column”. If ç is a limit, then since α ∈ CçM , the rϑ -code of tαM (ϑ) is ~ 1 and ϑ ~ 2 , this means eventually constant as ϑ → ç. Since rϑ∗ converges in ϑ ∗ M M M rç ↾tα (ç) = limϑ→ç rϑ ↾tα (ϑ) = q↾tα (â). We can repeat the above argument with ã for ç and r for rç to get r↾tαM (ã) = q↾tαM (ã), so tαM (ã) < ä. ⊣ (Claim 2.3) Claim 2.4. If α ∈ CãM and tαM (ã) = ä, then ãα = ã + 1 and Case 3C held in the definition of Dα . Moreover r↾tαM (ã) + 1 is the upper board of uãα . Proof. By Claim 2.3, tαM (â) ≥ ä(uâ ) for â < ã. So by hypothesis (c) on M, uâα = uâ ↾tαM (â) + 1, for â < ã ≤ ãα . Also vâα = uâ ↾tαM (â), and sâα = râ ↾tαM (â) for â < ã. Now condition (1) holds at ã in the definition of the vâα ’s, as otherwise our construction in Case 1 guarantees that crit(uâ ) ≤ he, ti for infinitely many â < ã, where e is the eventual râ code of tαM (â) and t is as in Case 1. Condition (2) holds: lim∗â→ã râ ↾tαM (â) = r↾tαM (ã) exists, so lim∗â→ã sâα exists. Further, aâα = uâα (tαM (â))0 = uâ (tαM (â))0 = râ (tαM (â))0 for â < ã, since râ∗ converges ~5 , which is very good, limâ→ã a α = r(tαM (ã))0 . in ϑ â

Let s be as in condition (3); then s = r↾(tαM (ã)) = r↾ä. Thus (3) fails at ã in the definition of Dα , and Case 3 holds. Since s ⊇ r ⊇ q, 3A cannot hold. Let a be as in Case 3. Then a = r(ä)0 . If a = q(dom(s))0 = q(s)0 , then the construction in 3B guarantees r(ä)1 = q(ä)1 , so r(ä) = q(ä), contrary to the definition of ä. Thus 3B cannot hold, and 3C does. The rest of Claim 2.4 is obvious by now. ⊣ (Claim 2.4) Claim 2.5. Let α ∈ CãM and tαM (ã) > ä; then (1)–(4) hold at ã in the definition of Dα . Moreover, dom(uã )α = tαM (ã) + 1 is the upper board of uãα . Finally, if ç < α, ç ∈ CãM , tçM (ã) > ä, then uãç ⊆ uãα . Proof. This is a tedious induction on α which we leave to the interested reader. The main point is that if α ∈ CãM , then letting N = hvâα : â < ãi, CãN = CãM ∩ α and tçN = tçM ↾ã + 1 for all ç < α. ⊣ (Claim 2.5)

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It is clear from Claim 2.5 that if we define uã as above, then (a), (b), and (c) remain true for huâ : â < ã + 1i. Lastly, we must see that (a), (b), and (c) are preserved at limit ã. Now (c) is trivial at limits, and for (b) we need only worry that lim∗â→ã râ may not exist; however, this limit must exist granted (a). So it is enough to check (a). If (a) fails, then we have he, ti such that he, ti = lim inf â→ã crit(uâ ). Now râ ∈ A for â < ã by (a)–(c) below ã, and so øk (râ ) converges as â → ã for all k ≤ he, ti. Thus |e|F (rç ) is eventually constant, say for â 6= ç. Let tαM (ç) = |e|F (rç ) , so that tαM (â) = |e|F (râ ) for all â ≥ ç, â < ã. [Though we haven’t yet shown M = huâ : â < ãi is good, we can define tαM and CαM S S by: tαM = â<ã tαM ↾â and CαM = â<ã CαM ↾â .] Now by (c) below ã we have uâα = uâ ↾(tαM (â) + 1) for all â < ã. (In particular vâα is defined for â < ã.) If vãα is defined, then ã < ãα , and then we see from the construction in Cases 1– 4 that the least proposal in uâα (dom(uâα ) − 1) goes to ù as â → ã. (In this case, our current ã is less than the ã referred to in the case hypothesis.) But this proposal is t cofinally often. So vãα is undefined. In Cases 2– 4 apply in the definition of Dα , then again the least proposal in uâα (dom(uâα ) − 1) goes to ù as â → ã. So Case 1 applies. But then our construction guarantees that Σä(uâ ) is responsible for the t proposal cofinally often in uâα (dom(uâα ) − 1). Setting i = he, ti, we get a ϑ < ã such that øi (rç ) ≤ øi (râ ) for all ç ≥ â ≥ ϑ, and øi (rç+1 ) < øi (rç ) for infinitely many ç ≥ ϑ, a contradiction. (The first inequality requires that ø ~ be a scale, not just a semiscale.) This completes the proof of Theorem 2.1. ⊣ (Theorem 2.1) Recall that [
 α ù : ∀â ∈ dom(p) (Lâ [p↾â], ∈, p↾â) is not Σn admissible . Tn = p ∈ α<ù1

Corollary 2.6. Let n ≥ 1, and assume Γ-ADΣn , where Γ is closed under ∀R and has the scale property. Then if player I has a winning strategy in G(A, Tn ), where A∗ ∈ Γ, player I has a winning strategy Σ such that Σ∗ is aΣn Γ. Proof. The proof of this is implicit in the proof of Theorem 2.1.

⊣

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§3. An inner model of ADΣn . One standard justification for deriving consequences of ZF + AD + DC is that, assuming definable determinacy, this theory has interesting inner models. In particular, if all games of length ù on ù with payoff in L(R) are determined, then L(R) |= ZF + AD + DC. In this section we shall provide an analogous justification for deriving consequences of ZF + ADΣn + DC. For α ≤ ù1 , let Γ-ADα be the assertion that all games of length α on ù whose payoff set is in Γ are determined. ADα is the same assertion but 2 with no restriction on payoffs. ADù is the weakest form of full determinacy stronger than that ADù = AD. Blass and Mycielski [Bla75] have shown 2 ADù ⇐⇒ ADR , while Solovay [Sol78B] has shown that ZF + ADR + DC proves Con(ZF + AD + DC). We shall give in detail only the construction of a model of ZF + ADR + DC; even this seems to involved the machinery of long games and scales developed in §1 and §2. We indicate how to modify the construction in order to get models of ZF+ADΣn +DC at the end of the section. We shall rely heavily on work of Woodin (unpublished). Woodin constructs “from below” a class M , and shows that if there is any model of ZF+ADR +DC containing al reals and ordinals, then M is the smallest such model. So our task is to show that some amount of definable determinacy implies M |= ZF + ADR + DC. By Woodin’s work on M , it suffices for this to show that every set of reals in M admits a definable scale. We shall show using a Friedman game that every (∆21 )M set is aΣ2 Π11 . Corollary 2.6 then implies every set of reals e implies that such sets admit definable scales. e Corollary 1.7 in M is aΣ2 Π11 , and e In the end, we require ∆12 -ADΣ2 to show that M |= ZF + ADR + DC. e In §4 we shall present some evidence that in fact ZFC + HOD(R)-ADΣ1 is too weak to construct an inner model of ZF + ADR + DC. We believe that ∆12 -ADΣ2 is very close to being the weakest definable determinacy hypothesis e yielding an inner model of ADR . Some terminology: a filter F on ℘ù1 (X ) = {A ⊆ X : A is countable} is normal iff F is closed under diagonal intersection (Ax ∈ F for all x ∈ X ⇒ {A ∈ ℘ù1 (X )
: ∀x ∈ A (A ∈ Ax )} ∈ F) and fine iff ∀x ∈ X {A ∈ ℘ù1 (X ) : x ∈ A} ∈ F . A set C ⊆ ℘ù1 (X ) is a club iff C is closed under countable increasing unions and ∀A∈ ℘ù1 (X )∃B ∈ C (A ⊆ B). The club filter on ℘ù1 (X ) consists of all A ⊆ ℘ù1 (X ) such that C ⊆ A for some club C; it is normal and fine. (Normality uses AC.) We say ù1 is X -supercompact iff there is a normal, fine ultrafilter on ℘ù1 (X ). One of the basic consequences of ADR , due to Solovay, is that the club filter on ℘ù1 (R) is a normal ultrafilter, so that ù1 is R-supercompact [Sol78B]. We proceed to the main result of this section. It is convenient at this point to add full AC to the metatheory ZF + DC of this paper; this makes possible some simple manipulations of club sets in the proof to follow. (We doubt that AC is actually necessary.)

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Theorem 3.1 (ZFC). Assume ∆12 -ADΣ2 . Then there is an inner model containing all reals and ordinals ande satisfying ZF + ADR + DC + “ ù1 is ℘(R)supercompact” + “every set of reals admits a scale”. Proof. We shall define a slight variant of Woodin’s model (in order to get ℘(R)-supercompactness.) Let M0 = Vù+1 = the set of rank ≤ ù and Më =

[

Mâ , for ë a limit.

â<ë

Now suppose Mα is given. If Mα 6|= ZF− + “℘(℘(R)) exists”, set Mα+1 = {a ⊆ Mα : a is 1st order definable over (Mα , ∈) from parameters}. If Mα |= ZF− + “℘(℘(R)) exists” + ¬AD, then set Mα+1 = Mα (i.e., stop the construction). Finally, suppose Mα |= ZF− +“℘(℘(R)) exists”+ AD, and let ã = ϑMα = sup of lengths of prewellorders of R in Mα . Case 1. cf(ã) = ù. Pick any sequence hAn : n < ùi such that ∀n(An ∈ ℘(R) ∩ Mα ), but hAn : n < ùi 6∈ Mα . (By Wadge, this means ∀B ∈ (℘(R) ∩ Mα )∃n (B ≤W An ). ) Set Mα+1 = {a ⊆ Mα : a is first order definable from parameters over (Mα , ∈, hAn : n < ùi)}. (By Wadge, Mα+1 is independent of the hAn : n < ùi chosen.) Case 2. cf(ã) > ù. Let X = ℘(R) ∩ Mα , and let Fα be the club filter on ℘ù1 (X ). Subcase (a). F is not an ultrafilter over ℘(℘ù1 (X )) ∩ Mα . (That is, there is an A ⊆ ℘ù1 (X ), A ∈ Mα , such that neither A nor ℘ù1 (X ) − A is in F .) Then set Mα+1 = Mα . Subcase (b). Otherwise. Then set Mα+1 = {a ⊆ Mα : a is first order definable from parameters over (Mα , ∈, Fα ∩ Mα )}.

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This completes the definition of the Mα ’s. Clearly they constitute an increasing sequence of transitive sets. Set [ M= Mα . a∈Ord

Let also F(α, A) iff Case 2 applied at α and A ∈ Fα , and Mα = (Mα , ∈, F ∩ Mα ). The point is that Mα has the information it needs to define hMâ : â < αi. Thus there is a fixed formula defining hMâ : â < αi over Mα for all α. There is also a natural sentence expressing “I am an Mα ”. Lemma 3.2. Every (∆21 )M , set of reals is aΣ2 Π11 . e e Proof. Let S be (∆21 )M , where we have dropped the real parameter for convenience. Say S(x) ⇐⇒ M |= ∃A ⊆ R ϕ(A, x) and −S(x) ⇐⇒ M |= ∃A ⊆ R ø(A, x), where ø and ϕ have real quantifiers only. We must recursively associate to any x ∈ R an ADΣ2 type game Gx such that S(x) iff player I wins Gx . For convenience, we shall make the individual moves of Gx reals rather than natural numbers; for games of the ADΣ2 variety this affects nothing of importance. Let x be given. We describe the payoff of Gx by specifying the rules of play. The rules governing round α are defined by induction on α. Round 0. Player I must play a real coding transitive structure A0 such that x ∈ |A0 | and A0 |= “I am an Mα ” & ∃A ⊆ R ϕ(A, x) & ∀A ⊆ R¬ø(A, x) Player II must then play a real coding in a transitive B0 with x ∈ |B0 | and B0 |= “I am an Mα ” & ∃A ⊇ R ø(A, x) & ∀A ⊇ R¬ϕ(A, x). Failure by one of the players is violation of the rules at 0. Round α + 1. If neither player has violated the rules at or before α, then we shall have transitive structures Aα and Bα at the end of round α. Player I must now play the code of a transitive Aα+1 such that RBα ⊆ Aα+1 and an elementary iα : Aα → Aα+1

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with the following properties: Suppose Aα |= “case 2 occurred at stage ä”; that is, Aα |= ∃A F(ä, A). Let C = (℘(R) ∩ Mä )Aα . Then for A ⊆ (℘ù1 (℘(R ∩ Mä ))Aα , player I must arrange A |= F(ä, A) iff iα [C ] ∈ iα (A) (in particular, iα [C ] ∈ |Aα+1 |, and is countable in Aα+1 ). Further, suppose An ∈ ℘(R)Aα for all n < ù, and hAn : n ∈ ùi ∈ |Bα |. Then player I must arrange hiα (An ) : n < ùi ∈ |Aα+1 |. Failure to meet these requirements is a violation of the rules at α + 1 by player I. On his α + 1st move player II must play a Bα+1 and a jα : Bα → Bα+1 meeting requirements completely symmetric to those on Aα+1 and iα+1 . If he doesn’t, player II violates the rules at α. Round ë, ë limit. If no one has violated the rules before ë we shall have direct limit systems hAα , iαâ : α < â < ëi and hBα , jαâ : α < â < ëi. Set Aë = direct limit of hAα , iαâ : α < â < ëi Bë = direct limit of hBα , jαâ : α < â < ëi. Then player I violates the rules at ë if Aë is illfounded, and player II violates the rules at ë if Bë is illfounded. If there is no violation, we assume Aë and Bë are transitive. The reals played during round ë are meaningless for Gx . This completes the rules of Gx . The first player to violate these rules loses Gx . In case of a tie, player I loses. We shall now show that by the time we reach the first Σ2 admissible relative to the play, someone must lose. Claim 3.3. Let p : ë × ù → ù be a partial play of Gx , where ë is a limit ordinal. Suppose in p neither player has violated the rules at or before ë. Then Lë [p] is not Σ2 admissible. Proof. (Here n 7→ p(α, n) codes the reals played by player I and player II at round α in p.) Let p be a counterexample to the claim. Let hAα , iαâ : α < â < ëi and hBα , jαâ : α < â < ëi be the systems produced in p by player I and player II respectively. Let Aë and Bë be their transitive direct limits, and iαë , jαë (for α < ë) the natural maps.

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Now RAë =

[

RAα =

α<ë

[

RB α = RB ë .

α<ë

On the other hand, Aë |= ∃A ⊆ R ϕ(A, x) & ∀A ⊆ R¬ø(A, x) while Bë |= ∃A ⊆ R ø(A, x) & ∀A ⊆ R¬ϕ(A, x), so ℘(R)Aë * Bë and ℘(R)Bë * Aë . Thus we have a ä ∈ OrdAë ∩ OrdBë such that Aë Bë MäAë = MäBë and Mä+1 6= Mä+1 .

Clearly then, MäAë |= ZF− & AD. Let Aë )

ϑ = ϑ(Mä

Bë

= ϑ(Mä ) .

We claim that both Aë and Bë satisfy “cf(ϑ) > ù”. For suppose e.g., Aë satisfies “cf(ϑ) = ù”. Pick hAn : n < ùi in |A| such that A satisfies: ∀B ∈ Mä ∩ ℘(R) ∃n(B ≤W An ). Then for α < ë sufficiently large, say α ≥ α0 , we have a hAαn : n < ùi ∈ |Aα | such that iαë (hAαn : n < ùi) = hAn : n < ùi. Now An ∈ |Bë | for all n. So for fixed n, we have for all sufficiently large, say α ≥ α0 , a Bnα ∈ |Bα | such that jαë (Bnα ) = An . Notice that if α is a limit then RAα = RBα so if α is a limit such that both Aαn and Bnα are defined, then Aαn = Bnα . Let S = {α < ë : Lα [p↾α] ≺Σ1 Lë [p]}. Then S is club in ë and Lë [p, S] is admissible. By our observation above, for each n < ù we can find a ân > α0 and Aânn ∈ |Bân |. By the stability of ân , this means ân ≥ α0 and Aânn = Bnân . Since the map n 7→ ân is ∆1 (Lë , [p, S]), we e have a limit ã such that ã > ân for all n. Now the requirements on player II at round ã + 1 imply that hjã,ã+1 (Bnã ) : n < ùi ∈ |Bã+1 | and since ã > ân for all n, jã+1,ë (hjã,ã+1 (Bnã ) : n < ùi) = hAn : n ∈ ùi. Thus hAn : n < ùi ∈ |Bë |, and Bë satisfies cf(ϑ) = ù. Thus both Aë and Bë Aë Bë think Mä+1 comes from Mä via Case 1 using hAn : n < ùi, so Mä+1 = Mä+1 , a contradiction.

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Thus cf(ϑ) > ù in both Aë and Bë . Pick A ∈ MäAë = MäBë such that Aë |= F(ä, A) and Bë |= ¬F(ä, A). (Such an A must exist.) Pick α0 large enough that iα−1 (A), iα−1 (ä), jα−1 (A) 0 ,ë 0 ,ë 0 ,ë −1 and jα0 ,ë (ä) all exist. Let −1 íã = iãë (ä) −1 ìã = jãë (ä) −1 Bã = iãë (A) −1 Cã = jãë (A)

for α0 ≤ ã < ë. Then h(íã , ìã , Bã , Cã ) : ã < ëi is ∆1 over Lë [p]. B A Now if X ∈ ℘(R)∩Míã ã , then there is a â > ã, â ∈ S, and a Y in ℘(R)∩Mìââ such that iãë (X ) = jâë (Y ). Notice that for â ∈ S, iãë (X ) = jâë (Y ) iff iãâ (X ) = Y . By a closure argument then, we can find â ∈ S, â > α0 , such that A

B

℘(R) ∩ Míâ â = ℘(R) ∩ Mìââ . Call this set C . Then iâ,â+1 [C ] is countable in Aâ+1 , and jâ,â+1 [C ] is countable in Bâ+1 , so iâ+1,ë (iâ,â+1 [C ]) = iâë [C ] and jâ+1,ë (iâ,â+1 [C ]) = jâë [C ]. But â is stable, so jâë (X ) = iâë (X ) for all X ∈ C , so iâë [C ] = jâë [C ]. But by our rules iâ,â+1 [C ] ∈ Bâ+1 jâ,â+1 [C ] 6∈ Câ+1 . Hence iâë [C ] ∈ A and jâë [C ] 6∈ A a contradiction. This proves the claim. ⊣ (Claim 3.3) To finish the proof of Lemma 3.2, we must show that player I has a winning strategy in Gx iff S(x). So suppose S(x). Pick an α such that Mα |= ZF− & ∃A ⊆ R ϕ(A, x) & ∀A ⊆ R¬ϕ(A, x).

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Then we can find a club C ⊆ ℘ù1 (Mα ) such that if P ∈ C, then (P, ∈, F ∩ P) ≺ Mα and (A, ä ∈ P & F(ä, A)) implies (℘(R) ∩ Mä ∩ P) ∈ A. [Proof. For each ä, A ∈ Mα such that F(ä, A), pick a club Cä,A ⊆ A. Define f(ä, A, Y ) = some P ∈ Cä,A such that Y ⊆ P for Y ∈ ℘ù1 (℘(R) ∩ Mä ). Let C be the set of all Q ∈ ℘ù1 (Mα ) such that Q is closed under f and Skolem functions for Mα .] Now for P ∈ C, let ðp : Ap ∼ = (P, ∈, F ∩ P) be the inverse of the collapse. Player I should play in Gx so that for all â Aâ = APâ , for some Pâ ∈ C and Pâ ⊆ Pâ+1 ,

Pë =

[

Pâ for ë limit

â<ë

and iâ,â+1 = ðP−1 ◦ ðP â . â+1 It is clear that the properties of C guarantee that he can play this way forever and violate no rules in doing so. The proof that if ¬S(x) then player II has a winning strategy in Gx is entirely symmetric. ⊣ (Lemma 3.2) By Lemma 3.2, there must be a game of the form G(A, T2 ) where A∗ ∈ Π11 , e which has no winning strategy in M . (This just means M 6|= Π11 -ADΣ2 .) For otherwise the universal aΣ2 Π11 set of reals is (∆21 )M , and hence ethere is a fixed (∆21 )M set universal for (∆21 )M sets, an absurdity. By Theorem 2.1, then, there is a aΣ2 ∆12 set S which is enot in M . Let ìebe the club filter on ℘ù1 (R). Let Q be a aΣ2 ∆12 set which codes a scale e on S. Claim 3.4. L(Q, ì, R) |= AD + “ì is an ultrafilter”. Proof. Let N0 = Vù+1 and Në =

[ â<ë

Nâ , for ë limit.

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Given Nα such that Nα 6|= ZF− , let Nα+1 = {A ⊆ Nα : A is first order definable over (Nα , ∈, Q) from parameters}. Given Nα such that Nα |= ZF− , but Nα 6|= AD or ì is not an ultrafilter over ℘(℘ù1 (R)) ∩ Nα , then set Nα+1 = Nα . Otherwise, let Nα+1 = {A ⊇ Nα : A is first order definable over (Nα , ∈, Q, ì ∩ Nα ) from parameters}. S

Let N = α∈Ord Nα . An argument similar to the proof of Lemma 3.2 but 2 much simpler shows that every (∆21 )N set of reals is aù ,R Π11 (Q), where aα,R is e to games of length αeon R, and Π1 (Q) is the game quantifier corresponding 1 e the least pointclass containing Q and closed under ∀R , ∩, ∪, and continuous substitution. Thus there is a Π11 (Q) game of length ù 2 on R which is not determined in N . Now Martine[Mar83B] shows that every such game has a 2 winning strategy which is aù ,R ∆12 (P), where P is aΣ2 ∆12 and codes a scale on e e Q. (The relevant games of length ù 2 on R are determined by ∆12 -ADΣ2 : any game of length α < ù1 on R with aΣ2 ∆12 payoff is determined.) eThus there is 2 e a aù ,R ∆12 (P), hence a aΣ2 ∆12 , set not in N . Let B be such a set. Let C ⊆ R, e ; and C codes upewinning strategies for all games of length ù on R B ≤W C with payoff ≤W B; we can take C to be aΣ2 ∆12 by [Mar83B]. e Subclaim 3.5. L(C, R) |= AD. Proof. It is enough to see C # exists and is aΣ2 ∆12 . Now every (Σ21 )L(C,R) e D 6∈ L(C, R). Let α be set is aù,R Π11 (C ), so we have a aΣ2 ∆12 set D such that e e least such that Lα (D, R) |= KP; then Lα (D, R) |= AD since aΣ2 ∆12 is closed under “inductive in”. Work in Lα (D, R). Then by Wadge, if Xe ⊆ R and X ∈ L(C, R), X ≤W D. Thus there is a measurable cardinal greater than ΘL(C,R) , so C # exists. Since Lα (D, R) |= C # exists, C # does exist and is in Lα (D, R), hence aΣ2 ∆12 . ⊣ (Subclaim 3.5) e To finish the proof of the claim, let Nα′ = NαL(C,R). By induction on α we see that Nα′ = Nα , ℘(R) ∩ N ⊆ {X : X ≤W B}, Nα |= AD, and ì is an ultrafilter over ℘(℘ù1 (R)) ∩ Nα . This yields the claim at once. ⊣ (Claim 3.4) Now notice that there is a canonical ð : R ։ {B : B ≤W Q}, ð ∈ L(Q, ì, R). Suppose now Mα ∈ L(Q, R, ì);

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then by Wadge ℘(R) ∩ Mα ⊆ {B : B ≤ Q} so that ð includes a canonical ðα : R ։ ℘(R) ∩ Mα = Xα . Suppose Case 2 occurs in the definition of Mα+1 . For any A ⊆ ℘ù1 (Xα ), let A∗ = {A ∈ ℘ù1 (R) : ðα [A] ∈ A}; then A∗ is in L(Q, R, ì) if A is. Also, if A∗ ∈ ì, then A ∈ Fα , while if ℘ù1 (R) − A∗ ∈ ì, then ℘ù1 (Xα ) − A ∈ Fα . [For this reason we seem to need AC in V . Pick ó : Xα → R such that ó ◦ ðα = id. Then if C ⊆ ℘ù1 (R) is club, then {A ∈ ℘ù1 (Xα ) : ó −1 (A) ∈ C} is club in ℘ù1 (Xα ); moreover A ∈ A iff ó −1 (A) ∈ A∗ .] It follows that Fα is an ultrafilter over ℘(℘ù1 (Xα )) ∩ Mα ; moreover Fα is uniformly-in-Mα definable over L(Q, R, ì). It is now easy to see by induction that Mα ∈ L(Q, R, ì) for all α and that Mα 6= Mα+1 for all α. In fact, the function α 7→ Mα is definable over L(Q, R, ì). Thus M satisfies ZF + AD + “í is a normal ultrafilter over ℘(℘ù1 (R))”, where í is the club filter on ℘ù1 (℘(R)∩M ) restricted to M . Since M |= cf(ϑ) > ù by construction, M |= DC (cf. [Sol78B]). Finally, every set of reals in M is ≤W S since M ⊆ L(Q, R, ì). Thus each such set has a scale in L(Q, R, ì). Woodin (unpublished) shows that this implies each set in M has a scale in M . So M |= “Every set of reals has a scale”, and by Martin [MarB] and unpublished work of Woodin, M |= ∀α < ù1 (ADα ). ⊣ (Theorem 3.1) Notice that we could have augmented the construction of M by throwing in canonical winning strategies for games of the form G(A, T1 ), A ∈ M . This would give an M ∗ having the properties of M and satisfying ADΣ1 in addition. [The proof of Lemma 3.2 must be modified as follows: suppose we have an A in Aα ∩ Bα such that A |= player I wins G(A, T1 ) and Bα |= player II wins G(A, T1 ). Then the players of Gx take time out to play G(A, T1 ) and produce thereby a q ∈ ET1 . We then require q ∈ iα,α+1 (A) and q 6∈ jα,α+1 (A). With this modification, the proof of Theorem 3.1 goes through.] A similar argument shows Theorem 3.6. Assume ∆12 -ADΣn+1 , where n ≤ 1. Then there is an inner e model containing all reals and ordinals, and satisfying ZF+ADΣn +DC+“every set admits a scale” + “ù1 is Vù+n+1 -supercompact”. In fact, it seems likely that the proof of Theorem 3.1 will adapt to show: if all clopen games of length ù1 with HOD(R) (or even, “∆12 in the codes”) payoff e ZF + DC + “Every set are determined , then there is an inner models satisfying of reals admits a scale” + “G(A, T ) is determined whenever T admits a scaled coding”. If so, then this is nearly as well as one can do in the direction of models of strong forms of full determinacy without modifying the notion of a winning strategy; cf. §5.

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§4. A determinacy proof. The basic problem in this area is to prove that determinacy of the long games we have been considering from ZFC together with some large cardinal hypothesis. One method for doing this involves reducing the long games to games of length ù on higher type objects, as in the Blass2 Mycielski proof that ADR implies ADù . In this section we shall present such that a reduction for games of the form G(A, T1 ), that is, games ending at the first admissible relative to the play. We shall also prove the determinacy of the length of ù games to which we reduce within the theory ZF+AD+DC+“Every set admits a scale” + “ù1 is ℘(R)-supercompact”. Thus this theory proves ADΣ1 . Now Woodin has recently shown, in ZFC + ∃κ(κ is supercompact), that there is an inner model of ZF + AD + DC + “Every set admits a scale” containing all reals and ordinals. It seems likely that his techniques will/do produce a model satisfying “ù1 is ℘(R)-supercompact” as well. If so, then we have a proof from ZFC + ∃κ(κ is supercompact) of Γ-ADΣ1 , where Γ is the class of sets in Woodin’s model (and thus a proof of e.g., Π11 -ADΣ1 ). e Martin [MarB] and Woodin (unpublished) have independently shown that ZF + AD + DC + “Every set admits a scale” proves ∀α < ù1 (ADα ). Thus Π11 -ADα , for α < ù1 , follows from the existence of supercompact cardinals. e Martin’s idea seems likely to yield, with more work, a proof of ADΣ1 from ZF + AD + DC + “Every set admits a scale”; if so, then Π11 -ADΣ1 follows e from ZFC + ∃κ (κ is supercompact). However, the ideas presented in this section seem more promising than Martin’s approach when it comes to proving Π11 -ADΣn , n ≥ 2, from ZFC + ∃κ (κ is supercompact). e Suppose T ⊆ S α α<ù1 ù, and ð : T → ù is such that ∀p, q ∈T (p ⊆ q ⇒ ð(p) 6= ð(q)); then we shall call ð a continuous coding of T . Our determinacy proof actually applies to game of the form G(A, T ), where T admits a continuous coding. It is an easy exercise to show that T1 admits a continuous coding, as do the trees for the games ending at the first recursively inaccessible, recursive Mahlo, . . . , relative to play. It is also easy to see that T2 , the tree for games ending at the first Σ2 admissible relative to the play, does not admit a continuous coding. We begin with the higher type games to which we shall reduce our long games. Let X be any set, and A ⊆ ù X . By GA we mean the game length ù on X with payoff set A. We call GA determined if one of the players has a winning quasi-strategy. We call A Suslin if for some ordinal κ and tree T on X × κ, A = p[T ] = {f ∈ ù X : ∃g ∈ ù κ ∀n(f↾n, g↾n) ∈ T }. We call A co-Suslin if ù X \ A is Suslin. The author learned the proof of the following lemma from A.S. Kechris; it is due to him. By OD(X ) we mean the class of sets ordinal definable from finitely many elements of X . Lemma 4.1. Assume ZF + AD + DC. Suppose ù1 is X -supercompact, as witnessed by the ultrafilter ì. Suppose that A ⊆ ù X is Suslin and co-Suslin,

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as witnessed by the trees T and U . Then there is a OD({ì, T, U }) winning quasi-strategy for GA . Proof. For C ∈ Pù (X ), let GC A be just like GA , except that the player’s moves are restricted to lie in C . Then GC A is determined by AD. For any C such that player I wins GC A , we can define a canonical “best” winning quasi-strategy ΣC for player I in GC A , as in Moschovakis [Mos80, Chapter 6]. (In order to make the strategy canonical, we must make it “quasi”.) ΣC is OD({T, C }) uniformly in C . Similarly, if player II wins GC A we get a canonical winning quasi-strategy for ôC for player II which is OD({U, C }), uniformly in C . Now suppose player I wins GC a for ì–a.e. C . Define Σ for player I in GA by: p is according to Σ iff for ì–a.e. C , p is according to ΣC . Σ is OD({T, ì}), and it is easy to check that Σ is a winning quasi-strategy ô for player I in GA . Similarly, if player II wins GC A for ì–a.e. C , we get an OD({U, ì}) winning quasi-strategy ô for player II in GA . ⊣ The most natural reduction of continuously coded long games to games of length ù takes us to games on R × ℘(R). Unfortunately, there is a very simple A ⊆ ù (R × ℘(R)) which is not Suslin in models of AD, namely ~ : ën · xn (0) ∈ A0 }. Thus we must take care to make Lemma 4.1 A = {(~ x , A) applicable. We do this by reducing to a game on R × S, where S = {(~ ϕ , ø) ~ : ϕ ~ and ø ~ are scales, and p(~ ϕ ) = R − p(~ ϕ )}. Here p(~ ϕ ) is the projection of ϕ ~ , that is the common domain of the norms ϕi , i ∈ ù. For g ∈ ù S, let p(g) = h, where h(i) = p(g(i)0 ) for all i ∈ ù. For A ⊆ ù (R × S), let
p(A) = {(f, h) : ∃g (f, g) ∈ A & h = p(g) }; thus p(A) ⊆ ù (R × ℘(R)). We say that A ⊆ ù (R × A) is projection-invariant iff whenever p(B) = p(A), then B ⊆ A. Now one can easily modify the example of the last paragraph to obtain an A ⊆ ù (R × S) which is not Suslin in models of AD; however there is no simple projection-invariant A with this property. Conjecture 4.2. There is an inner model of ZF + AD + DC + “Every projection-invariant A ⊆ ù (R × S) is Suslin”. Such a model must of course satisfy “Every set of reals admits a scale”. The model M of §3 may verify the conjecture; the author is not sufficiently in command of the relevant work of Woodin (unpublished) to decide. Fortunately, we shall need the conjecture only for reasonable simple A. Let B ⊆ R and C ⊆ ù (R × ℘(R)); we call C B-projective iff C is the smallest class

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of relations containing the basic relations ~ iff xim ∈ Aj , R(~ x1 . . . x ~ n , A) ~ iff xim ∈ B, S(~ x1 . . . x ~ n , A) and ~ iff xim (j) = ℓ, Q(~ x1 . . . x ~ n , A) for all i, j, ℓ, m in ù, and closed under countable intersection, countable union, complement, and quantification over ù R. The proof of the following lemma is implicit in that of Moschovakis’ 2nd periodicity theorem, so we omit it. Lemma 4.3. Assume ZF+ AD+ DC. Suppose B ⊆ R is Suslin and co-Suslin via the trees T and U . Let A ⊆ ù (R×S) be projection-invariant, and suppose p(A) is B-projective. Then A is Suslin via a tree in OD({T, U }). Theorem 4.4. Assume ZF+AD+DC+“Every set of reals admits a scale”+ “ù1 is ℘(R) supercompact”. Then G(A, T ) is determined whenever T admits a continuous coding. In particular, ADΣ1 holds. Proof. Let ð be a continuous coding of T . There is a scaled coding F of T induced by ð. Let ðˆ = {(p∗ , n) : ð(p) = n}. Let B ⊆ R be such that ð, ˆ T ∗ , ET∗ , A∗ , C , and their complements all have scales Wadge reducible to B. (Here C = C (i, j, k, y) is as the definition of “scaled coding ”.) Let ì be a normal, fine ultrafilter on ℘ù1 (R × S); ì exists since we can map R × S into ℘(R) in a 1-1 way. Since we have AD + “Every set has a scale”, ℘(R) * OD({S}) ∪ R) for any set S. Thus for any set S there is an A ⊆ R such that any set of reals in OD({S} ∪ R) has a scale Wadge reducible to A. Let H0 = OD({B} ∪ {ì} ∪ R) and Hn+1 = OD({ì} ∪ {A} ∪ R), where A Wadge-minimal such that every set of reals in Hn has a scale ≤W A. We now describe a game G∗ on R × S auxilliary to G(A, T ). The game G∗ is quite similar to the auxilliary game in Martin’s proof of Borel determinacy S in [Mar75]. A set Σ ⊆ α<ù1 α ù is a tree if ∀p ∈Σ ∀â ∈ dom(p) (p↾â ∈ Σ). Play in G∗ is divided into rounds; before beginning a round we have a G∗ position and an associated G position. For convenience we assume that if p ∈ E, then dom(p) is a limit. Round 0. We begin with G∗ and G-positions ∅. (a) Player I plays α ∈ ù, then player II plays b ∈ ù. Let e be least in ù − {ð(ha, bi)}; we call e critical for round 0.

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(b) Player I now plays a (~ ϕ , ø) ~ in S ∩ H1 such that p(~ ϕ ) = Σ is a subtree of T (Σ = ∅ is o.k.). Player II may either accept or reject Σ. Player II accepts: Player I is now obliged to keep the G-position in Σ henceforth. Player II is obliged to insure that e 6= ð(p↾â) for any â ≤ dom(p) and any G-position reached henceforth. Player II rejects: Player II must play a q ∈ Σ extending ha, bi and such that e = ð(q). No one incurs any obligations. The new G position is ha, bi if player II accepts and q if player II rejects. Round n, n > 0. We have a G∗ position r with associated G position p. (a) Player I plays a ∈ ù, player II plays b ∈ ù. Let e be least in ù − {ð((paha, bi)↾â) : â ≤ dom(p) + 1}; e is critical for round n. (b) Player I now plays a (~ ϕ , ø) ~ in S ∩ Hn+1 such that p(~ ϕ ) = Σ is a (set of reals coding a) subtree of T . Player II may either accept or reject Σ. Player II accepts: Player I is now obliged to keep the G position in Σ henceforth. Player II is obliged to insure that if q is a G position later reached, then e 6= ð(q↾â) for all â < dom(q). Player II rejects: Player II must play a q ∈ Σ extending paha, bi and such that e = ð(q). No obligations are incurred. The new G position is paha, bi if player II accepts, and q if player II rejects. The first player to fail to meet any one if his obligations loses G∗ . If both players fail first at the same round, then the one violating the obligation incurred earliest loses, with player I’s obligation incurred at round n coming after those incurred by player II at rounds m < n and before those incurred by player II at rounds m ≥ n. Suppose both players meet all obligations incurred in G∗ . After ù moves they have produced a G position p. If p ∈ E, then player I wins G∗ iff p ∈ A. Otherwise, notice that ð(p) was critical at some round, and player II accepted at that round, so that p violates player II’s obligation incurred at that round (and no earlier player II obligations). So if p 6∈ E, and ð(p) was critical at round n, then player I wins G∗ iff p violates no player I obligation incurred at some round m ≤ n. Clearly we may regard G∗ as a game on R × S. Since the relation y ∈ Hn is OD({B, ì}), then there is a tree U1 on R × S such that U1 ∈ OD({B, ì}) and r ∈ [U1 ] iff r is a play of G∗ in which player I loses at no finite stage. Similarly, there is an OD({B, ì}) tree U2 on R × S × ù such that r ∈ p[U2 ] iff r is a play of G∗ in which player II loses at some finite stage. Finally, there is a projection invariant B ⊆ ù (R × S) such that p(B) is hð, ˆ T ∗ , E ∗ , A∗ C i-projective and whenever r is a play of G∗ where no one loses at a finite stage, r ∈ B iff r is a win for player I at stage ù.

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It follows from Lemma 4.3 that B is Suslin via some OD({B, ì}) tree U3 . But then
r is a win for player I in G∗ iff r ∈ [U1 ] & (r ∈ p[U2 ] or r ∈ p[U3 ]) so that G∗ = GA , where A is Suslin via an OD({B, ì}) tree. Similarly, A is coSuslin via an OD({B, ì}) tree, so Lemma 4.1 gives a winning quasi-strategy for G∗ which is OD({B, ì}). Case 1. Player I had a winning quasi-strategy in G∗ . We shall construct a winning S strategy Γ for player I in G(A, T ). Notice that there is a set of reals not in n Hn by DC, and thus there is a map from S R onto n Hn ∩ S. By uniformization we can thus convert player I’s winning ˆ quasi-strategy in G∗ into a full winning strategy; call it Γ. We define Γ from Γˆ by associating inductively to each p we reach following Γ a G∗ position pˆ which is according to Γˆ and whose associated G position is p. We arrange that in pˆ player II has met all his obligations. Suppose we have reached p ∈ T by following Γ, and we have the associated ˆ Let Γ( ˆ p) pˆ according to Γ. ˆ = a; then set Γ(p) = a. Now suppose player II responds b in G(A, T ). Case A. paha, bi meets all player II obligations incurred in p. ˆ a ˆ In this case, let (~ ϕ , ø) ~ = Γ(pˆ ha, bi), and set aha, bi) = p (p\ ˆ aha, b, (~ ϕ , ø), ~ accepti.

Case B. Otherwise. Let the first player II obligation violated by paha, bi be incurred at round n (i.e., in p(n)). ˆ Let p(n) ˆ = hc, d, (~ ϕ , ø), ~ accepti. Set aha, bi) = p↾n (p\ ˆ ahc, d, (~ ϕ , ø), ~ reject, paha, bii.

ˆ (Notice paha, bi ∈ p(~ ϕ ), as otherwise pˆ aha, bi is a loss for Γ.) In either case, the relation between p and pˆ still holds between paha, bi and aha, bi). (p\ Finally, we must define pˆ for p a position of limit length reached by followd ing Γ. Let ë = dom(p). Notice that if n is least such that (p↾â)(n) 6= d \ \ (p↾â + 1)(n), then (p↾â)(n)3 = accept and (p↾â + 1)(n)3 = reject. So d q(n) = limâ→ë (p↾â)(n) exists for all n. If p ∈ E, set pˆ = q. In this ˆ case pˆ is by Γ, so p ∈ A, and we have verified that Γ wins. If p 6∈ E, let e = ð(p). Then e was critical at some round, say round n, of q. Let q(n) = hc, d, (~ ϕ , ø), ~ accepti. Set pˆ = q↾nahc, d, (~ ϕ , ø), ~ reject, pi.

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ˆ which wins for player I in G∗ , p ∈ p(~ Since q was by Γ, ϕ ). Thus pˆ is a position ˆ according to Γ in which player II meets all his obligations, with associated position p, as desired. This defines Γ. We have also verified Γ wins G(A, T ) for player I. Case 2. Player II has a winning quasi-strategy in G∗ . We shall construct a winning strategy Γ for player II in G(A, T ). Let Γˆ be a winning quasi-strategy for player II in G∗ which is OD({B, ì}). Let ó : R ։ {q : q is a G∗ position}. If ó(x) = q we call x an index of q. We shall define Γ by associating inductively to each G position p we reach ˆ together with a real indexing the following Γ in G∗ position pˆ according to Γ, p. ˆ We arrange that in pˆ player I has met all his obligations and that p is the G-position associated to p. ˆ Suppose we have reached p by following Γ, and we have pˆ and x such that ó(x) = p. ˆ Suppose player I now plays a ∈ ù in G(A, T ). Let b ∈ ù be least ˆ set Γ(pahai) = b. Now let q = paha, bi; we want such that pˆ aha, bi is by Γ; to define qˆ and an index for q. ˆ Case A. q violates no player I-obligation in p. ˆ In this case, let [ α Σ = {r ∈ ù : if â ≤ dom(r), then no position of the form α<ù1 a

pˆ ha, b, (~ ϕ , ø), ~ reject, r↾âi, with (~ ϕ , ø) ~ ∈ S ∩ Hn+1 , ˆ n = dom(p), ˆ is in accord with Γ}. ˆ ∈ Hn . So there is Then Σ ∈ Hn , where n = dom(p), ˆ since pˆ ∈ Hn and Γ ˆ a (~ ϕ , ø) ~ ∈ S ∩ Hn+1 such that Σ = p(~ ϕ ). Now Γ must accept Σ after pˆ aha, bi by the definition of Σ, so we set qˆ = pˆ aha, b, (~ ϕ , ø), ~ accepti, where (~ ϕ , ø) ~ ∈ S ∩ Hn+1 , Σ = p(~ ϕ ), and the index of qˆ is obtained from x by the appropriate uniformizing function. (We ought to have fixed at the outset a function h : R → R such that if ó(x) = pˆ has the properties above, and Σ is defined as above, then ó(h(x, a)) = qˆ is related to Σ and pˆ as above.) Case B. Otherwise. Let n be such that the player I-obligation incurred in p(n) ˆ is violated by q. Then pˆ = hc, d, (~ ϕ , ø), ~ accepti, where p(~ ϕ ) = Σ is as in Case A, and q 6∈ Σ ~ ñ~) ∈ S ∩Hn+1 , while q↾â ∈ Σ for â < dom(q). By definition of Σ, there is a (ϑ, ~ ñ~), reject, qi is in accord with Γ. ˆ Using uniformization, such that p↾n ˆ ahc, d, (ϑ, ~ ñ~) together with an index of we can pick such a (ϑ, ~ ñ~), reject, qi. qˆ = p↾n ˆ ahc, d, (ϑ,

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Finally, suppose we reach a position p of limit length following Γ; we must d define pˆ and an index of same. As in Case 1, q(n) = limâ→dom(p) (p↾â)(n) ∗ ˆ If p ∈ E, then since exists for all n, so that q is a completed run of G by Γ. p is the run of G(A, T ) associated to q, p 6∈ A, so that Γ has won. If p 6∈ E, then let ð(p) = e, and let n be such that e is critical at round n in q. Since Γˆ wins for player II, q violates some player I obligation incurred in q(m) for some m ≤ n. Then let ~ ñ~), reject, qi, pˆ = q↾mahc, d, (ϑ, together with an index of p, ˆ be obtained as in Case B above. This completes the definition of Γ and the verification that it wins for player II. ⊣ (Theorem 4.4) There is a natural strategy for iterating the reduction achieved in Theorem 4.4, and thereby proving Π11 -ADΣn (assuming, ultimately, ZFC + ∃κ(κ is supercompact)). It seems one emust strengthen the hypothesis “Every set has a scale” of Theorem 4.4 (as well as requiring ù1 be ℘ù1 (Vù+ù -supercompact)) in order to do this. The first step of the strengthening is given by the conjecture mentioned in this section. The proof of the Kechris-Woodin theorem of §3 adapts at once to show Corollary 4.5. ZF + AD + DC + “Every set has a scale” + “ù1 is ℘(R)supercompact” provides the consistency of ZFC + HOD(R)-ADΣ1 . Thus HOD(R)-ADΣ1 would not have sufficed to produce the model M of §3. §5. Questions. The reader who has arrived at this section by slogging through the intermediate ones will probably have his own list of questions by now. We list here only a few salient ones. Ad §1,Sthe natural question is whether definable scales extend still further. For p ∈ α<ù1 α ù, let Np = {f ∈ andScall a set A ⊆ ù of α<ù1 α ù such that ù1

open-Π11

A=

e

ù1

ù : p ⊆ f},

if there is a Π11 set A∗ of codes for elements e

[ {Np : ∃x ∈ A∗ (x codes p)}.

If CH holds, then aù1 (closed-Π11 ) is just Σ21 , and by Moschovakis’ argument, e some form of definable determie Does Π21 has the prewellordering property. e nacy, e.g., HOD(R)-ADù1 , imply that all aù1 (open-Π11 ) sets of reals admit e definable scales? 1 Ad §2, does every length ù1 game with open-Π1 payoff won by player I have a definable winning strategy (assuming e.g.,e HOD(R)-ADù1 ?). There is a simple such game won by player II for which player II has no definable

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winning strategy. It is a part of the folklore: player I and player II take a nap until, at some move α, player I awakens with a start and asks player II to produce x ∈ WO such that |x| = α. If player I never awakens, he loses, otherwise player II must comply or lose. For all we know, however, every length ù1 game with HOD(R) payoff has a definable pseudo-strategy, where S is a pseudo-strategy winning GA for (say) player I if [ S: ( α ù × WOα ) → ù, α<ù1

where WOα = {x ∈ WO : |x| = α}, and whenever g(α) ∈ WOα for all α < ù1 ,
∀α f(α)0 = S(f↾α, g(α)) ⇒ f ∈ A, for all f ∈ ù1 ù. Does HOD(R)-ADù1 guarantee HOD(R) pseudo-strategies for length ù1 with HOD(R) payoff? (The notion of a pseudo-strategy is due independently to Woodin.) Ad §3, notice that the folklore example above shows ZF + AD + (open-Π11 )ADù1 is inconsistent. A slight modification shows that ZF + AD impliesethe existence of a non-determined “clopen-Π11 ” game of length ù1 . [At any move â before he awakes, player I must pay efor continuing to nap by playing an x ∈ WOâ . Notice that, in contrast to the games admitting scaled codings, the game is not clopen in V, only in models of AD.] Thus §3 goes nearly as far as possible in producing inner models with full winning strategies for games not limited by definability of the payoff. Can we produce stronger inner models using pseudo-strategies? Another question suggested by §3: what is the consistency strength of ADR + “Θ is regular”, or even ADR + cf(Θ) > ù1 , vis-a-vis the long game hierarchy? Does ∆12 -ADΣ2 give us a model of ADR + cf(Θ) > ù1 ? e the obvious question: does ZFC+∃κ(κ is supercompact) Finally, §4 suggests Σ prove Π11 -AD n for all n < ù? Woodin has shown in ZFC + ∃κ(κ is supercome all aù1 (closed-Π1 ) sets are Lebesgue measurable. This suggests pact) that 1 1 might follow from ZFC + ∃κ(κ is supercompact). that in fact (open-Π11 )-ADùe We conclude byementioning the strongest form of determinacy we know (other than the inconsistent forms). There are three parameters entering into the description of a class of games: complexity of payoff, complexity of individual moves, and length. The known limitations in each direction are: (a) payoff : there is a non-determined game on {0, 1} of length ù. (GaleStewart) The natural response is to limit the payoffs considered to OD(R). (b) moves: there is a non-determined game on ℘(R) of length ù whose payoff is OD. [player I plays a non-determined A ⊆ ù ù, then player I and player II play out GA .] (folklore) The natural response is to limit the moves to be OD(R).

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(c) length: there is a non-determined game on {0, 1} with OD payoff of length ù1 + ù. [In first ù1 moves player I must describe an uncountable A ⊆ ù ù with no perfect subset. In the next ù moves, player I and player II play the perfect set game for A.] (Galvin, Laver) The natural (most generous) response is to limit the intermediate positions to be OD(R). (This subsumes our response to (b).) This leads to a “maximum determinacy” principle, or MD. MD. Let G be a game on OD(R) which ends as soon as the players reach a position which is not OD(R), with the winner declared according to some OD(R) payoff condition. Then G is determined. Superficially, anyway, MD allows games of any ordinal length. Is MD consistent? Is it good for anything? REFERENCES

Howard S. Becker [Bec85] A property equivalent to the existence of scales, Transactions of the AMS, vol. 287 (1985), pp. 591–612. Andreas Blass [Bla75] Equivalence of two strong forms of determinacy, Proceedings of the American Mathematical Society, vol. 52 (1975), pp. 373–376. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. Donald A. Martin [MarB] Games of countable length, to appear. [Mar75] Borel determinacy, Annals of Mathematics, vol. 102 (1975), no. 2, pp. 363–371. [Mar83B] The real game quantifier propagates scales, this volume, originally published in Kechris et al. [Cabal iii], pp. 157–171. Yiannis N. Moschovakis [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. Robert M. Solovay [Sol78B] The independence of DC from AD, In Kechris and Moschovakis [Cabal i], pp. 171–184. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720, USA

E-mail: [email protected]

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JOHN R. STEEL

§1. Introduction. We shall call a set T an ù1 -tree if an only if [ α ù, T ⊆ α<ù1

and whenever p ∈ T , then p↾â ∈ T for all â < dom(p). (This diverges a bit from standard usage.) The set of ù1 -branches of T is [T ] = {f ∈

ù1

ù : ∀α < ù1 (f↾α ∈ T )}.

[T ] is just a typical closed set in the topology on ù1 ù whose basic neighborhoods are the sets of the form Np = {f ∈ ù1 ù : p ⊆ f}, where p : α → ù for some countable α. Associated to T is a closed game G(T ) on ù of length ù1 : at round α in G(T ), player I plays mα , then player II plays nα . Letting f(α) = hmα , nα i for all α < ù1 , we say that player II wins the run of G(T ) determined by f iff Q f ∈ [T ]. (We code sequences from ù by prime powers, so that hn0 , . . . , nk i = i≤k pini +1 , and the decoding is given by setting (n)i = k if the exponent of pi in n is k + 1 where pi is the ith prime.) Of course, not all such closed games are determined; indeed, every game of length ù can be regarded as a clopen game of length ù1 . However, the determinacy of G(T ) for definable T does follow from a large cardinal hypothesis, in virtue of the following beautiful theorem of Itay Neeman: Theorem 1.1 (Neeman, [Nee, Nee04]). Suppose that for any real x, there is a countable, ù1 + 1-iterable mouse M such that x ∈ M and M |= ZFC− + “there is a measurable Woodin cardinal”; then G(T ) is determined whenever T is a ù1 -tree such that T is definable over hHù1 , ∈i from parameters. The definability restriction on T is equivalent to requiring that T be coded by a projective set of reals. For that reason, we shall call the conclusion of Neeman’s theorem ù1 -open-projective determinacy. Neeman’s proof works, under a natural strengthening of its large cardinal hypothesis, whenever T is The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

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coded by a Hom∞ set of reals. In this form, Neeman’s result obtains as much determinacy from large-cardinal-like hypotheses as has been proved to date.1 Associated to any game we have a game quantifier: Definition 1.2. Let T be an ù1 -tree; then aI (T ) = {p ∈

ù

ù : p is a winning position for player I in G(T )},

aII (T ) = {p ∈

ù

ù : p is a winning position for player II in G(T )}.

and

Definition 1.3. We define aù1 (open-analytical) to be the class {aI (T ) : T is an ù1 -tree which is definable over hHù1 , ∈i} and aù1 (closed-analytical) to be the class {aII (T ) : T is an ù1 -tree which is definable over hHù1 , ∈i}. Clearly, the determinacy of the games in question implies that the pointclasses aù1 (open-analytical) and aù1 (closed-analytical) are duals. The following observations relate these pointclasses to more familiar ones. Proposition 1.4. For any A ⊆ ù ù , the following are equivalent: (1) A is aù1 (closed-analytical), (2) there is a Σ21 formula ϑ(v) such that for all x ∈ ù ù , x ∈ A iff Col(ù1 , R) ϑ(x), ˇ where Col(ù1 , R) is the partial order for adding a map from ù1 onto R with countable conditions. Proposition 1.5. If CH holds, then aù1 (closed-analytical) = Σ21 . Corollary 1.6. If CH and ù1 -open-analytical determinacy hold, then aù1 (open-analytical) = Π21 . We shall omit the easy proofs of these results. It is natural to ask whether ù1 -open-projective determinacy implies the pointclasses aù1 (open-analytical) and aù1 (closed-analytical) are well-behaved from the point of view of descriptive set theory. In this note we shall extend Moschovakis’ periodicity theorems to this context, and show thereby that ù1 open-projective determinacy implies that aù1 (open-analytical) has the Scale 1 Literally speaking, Neeman’s theorem uses a mouse existence hypothesis, rather than a large cardinal hypothesis. It is not known how to derive this mouse existence hypothesis from any true large cardinal hypothesis, because it is not known how to prove the required iterability. Presumably, if there is a measurable Woodin cardinal, then there are mice of the sort needed in Neeman’s theorem.

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Property, and that there are canonical winning strategies for the ù1 -openprojective games won by player player I. By Corollary 1.6, adding CH to our hypotheses gives the Scale Property for Π21 . Our arguments are quite close to those of [Ste88, §§ 1& 2], which prove the same results for the quantifiers associated to certain clopen games of length ù1 . Therefore, in this paper we shall only describe the simple changes to [Ste88] which are needed for the full theorems on open games, and refer the reader to [Ste88] for the long stretches of the proofs which agree in every detail. §2. The Prewellordering Property. Moschovakis’ argument easily yields the prewellordering property for aù1 (open-analytical). Theorem 2.1. If ù1 -open-projective determinacy holds, then the pointclass aù1 (open-analytical) has the prewellordering property. Proof. Let T be an ù1 -tree which is definable over hHù1 , ∈i, and let B = aI (T ). We define a prewellordering on B by means of a game G(p, q) which compares the values for player I of positions p, q ∈ B. In G(p, q) there are two players, F and S, and two boards, the p-board and the q-board. The play takes place in rounds, the first being round ù. In round α ≥ ù of G(p, q): (a) (b) (c) (d)

F S F S

plays as player I in round α of G(T ) on the q-board, then plays as player I in round α of G(T ) on the p-board, then plays as player II in round α of G(T ) on the p-board, then plays as player II in round α of G(T ) on the q-board.

This play produces f ⊇ p on the p-board and g ⊇ q on the q-board, with f, g ∈ ù1 ù. Player S wins this run of G(p, q) iff for some α < ù1 , f↾α 6∈ T but for all â < α, g↾â ∈ T. In other words, S must win G(T ) as player I on the p-board, and not strictly after F wins as player I (if he does) on the q-board. Put p ≤∗ q ⇔ p, q ∈ A and S has a winning strategy in G(p, q). Let G0 (p, q) be the same as G(p, q), except that a run hf, gi such that neither player has won as player I (i.e., such that f ∈ [T ] and g ∈ [T ]) is a win for S, rather than F . Lemma 2.2. Let p0 ∈ aI (T ), and suppose that for all n ≥ 0, Σn is either a winning strategy for F in G(pn , pn+1 ), or a winning strategy for S in G0 (pn+1 , pn ); then only finitely many Σn are for F . Proof. Let hpn : n < ùi and hΣn : n < ùi be as in the hypothesis. Let ô be winning for player I in G(T ) from p0 .

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Note that in any case, Σn plays for player II on the pn board, and for player I on the pn+1 board. Playing ô and the Σn ’s together in the standard game diagram, we get hun : n < ùi such that for all n, (1) un is a run according to Σn of the appropriate game (G(pn , pn+1 ) if Σn is for F , and G0 (pn+1 , pn ) if Σn is for S), and (2) un and un+1 have a common play rn+1 on their pn+1 -boards, (3) the play r0 on the p0 -board of u0 is according to ô. As usual, un (α) is determined by induction on α, simultaneously for all n. Because ô was winning for player I, we have r0 6∈ [T ]. It follows by induction that rn 6∈ [T ] for all n. Let αn be least such that rn ↾αn 6∈ [T ]. Since Σn won its game, we have that αn+1 ≤ αn if Σn is for S, and αn+1 < αn if Σn is for F . Thus only finitely many Σn are for F . ⊣ (Lemma 2.2) Corollary 2.3. ≤∗ is a prewellorder of B. Proof. (1) Reflexive: if ¬p ≤∗ p, then hp, p, p, . . .i violates Lemma 2.2. (2) Transitive: if p ≤∗ q ≤∗ r, but ¬p ≤∗ r, then hr, q, p, r, q, p, r, . . .i violates Lemma 2.2. (3) Connected : otherwise hp, q, p, q, . . .i violates Lemma 2.2. (4) Wellfounded : clear from Lemma 2.2. ⊣ Corollary 2.4. If q ∈ B, then for any p, p ≤∗ q ⇔ player II has a winning strategy in G0 (p, q). Proof. Otherwise, let pn = q if n is even, and pn = p if n is odd. Let Σn be winning for S in G0 (p, q) if n is even, and let Σn be winning for F in G(p, q) if n is odd. We obtain a contradiction to Lemma 2.2. ⊣ So if q ∈ B, then we have p ∈ B ∧ p ≤∗ q ⇔ S has a winning strategy in G(p, q) ⇔ S has a winning strategy in G0 (p, q). The first equivalence shows {p : p ≤∗ q} is uniformly aù1 (open-analytical) in q, the second shows that it is uniformly aù1 (closed-analytical) in q. Thus ≤∗ determines a aù1 (open-analytical) norm on B. ⊣ (Theorem 2.1) §3. The Scale Property. In order to prove the scale property for aù1 (openanalytical), we need to add to the comparison games of Theorem 2.1 additional moves. As in [Ste88], these additional moves reflect the value which the players assign to one-move variants of positions they reach during a run of the game. One needs that such positions are canonically coded by reals in order to make sense of this. For this reason, in [Ste88] we demanded that there be a function F assigning to possible positions p ∈ T a wellorder F (p) ∈ WO such that F (p) has order type dom(p). What is new here is just the realization that one can relax this a bit, by demanding only that our game tree T incorporates the

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rule that player player I must provide a code of dom(p) as soon as p has been reached, before player II is required to make any further moves. Theorem 3.1. If ù1 -open-projective determinacy holds, then the pointclass aù1 (open-analytical) has the scale property. Proof. Let Tˆ be an ù1 -tree which is definable over hHù1 , ∈i, and let B = a (Tˆ ). We assume without loss of generality that in G(Tˆ ), player II plays only from {0, 1}. We may as well also assume that player I and player II only move in G(Tˆ ) at finite rounds, or at rounds of the form ùç + ù where ç ≥ 1. Finally, we assume that for ç ≥ 1, player player I must produce a code of ùç + ù in rounds ùç + 1 through ùç + ù. Formally, for any r ∈ <ù1 ù, let I

qr (n) = r(n) if ç < ù and qr (ù + ç) = r(ù + ùç + ù) for all ç ≥ 0 such that ù + ùç + ù ∈ dom(r). Set wrç (n) = r(ùç + n + 1)0 , for ç such that ù + ùç + ù ≤ dom(r). For r ∈

<ù1

ù, we put r ∈ A iff

(a) dom(r) = ù + ùî + ù, for some î, (b) ∀ç ∈ dom(qr )(qr ↾ç ∈ Tˆ , but qr 6∈ Tˆ ), and (c) if ùç + ù ≤ dom(r), then wrç ∈ WO∗ and |wrç | = ùç + ù. Here WO∗ is the set of reals x coding wellorders of ù of length |x| > ù, and such that |0|x = ù. (This last very technical stipulation simplifies something later.) For r ∈ <ù1 ù, put r ∈ T ⇔ ∀î ≤ dom(r)(r↾î 6∈ A). Thus in G(T ), player I is just trying to reach a position in A. Claim 3.2. B = aI (T ). Proof. This is clear. It is worth noting that we use the Axiom of Choice at this point, however. ⊣ (Claim 3.2) As we said, our only new idea here is to work with T , rather than Tˆ , in defining the comparison games producing a scale on B. We shall use the notation of the proof of [Ste88, Theorem 1.2] as much as possible. The counterpart of F there is given by: dom(F ) consists of those r ∈ <ù1 ù which satisfy (a) and (c) above, and for such r F (r) = wrç , where dom(r) = ùç + ù. r ∗ = hF (r), xi,

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where x(n) = r(|n|F (r) ). For D ⊆ dom(F ), we let D ∗ = {r ∗ : r ∈ D}. Let C (i, j, k, y) ⇔ ∃r ∈ dom(F )(y = r ∗ ∧ |i|F (r) = |j|F (r↾α)) where |k|F (r) = α = ùî + ù, for some î. Note that C and F are ∆12 . Let ñ~ be a ∆12 scale on C , and let ó ~ be an analytical scale on T ∗ . Let ϑik , for 0 ≤ k ≤ 4 and i < ù, be the norms on A∗ defined from ñ~ and ó ~ exactly as in the proof of [Ste88, Theorem 1.2]. Let D(k, y) ⇔ ∃r ∈ dom(F )(y = r ∗ ∧ ∃î ≥ 1(|k|F (r) = ùî + ù)), let ~í0 , ~í1 be an analytical scales on D and ¬D, and set  0 íi (k, y), when D(k, y), 5 ϑhk,ii (y) = íi1 (k, y), when ¬D(k, y). ~ 6 be a very good scale on A∗ , and set, for p∗ ∈ A∗ , Let ϑ øi (p∗ ) = hdom(p), ϑ00 (p∗ ), . . . , ϑ06 (p∗ ), . . . , ϑi0 (p∗ ), . . . , ϑi6 (p∗ )i, or rather, let øi (p∗ ) be the ordinal of this tuple in the lexicographic order. Just as in [Ste88], the norms øi determine the values we assign to runs of the comparison games which yield a scale on B. Let p, q ∈ aI (T ), and let k ∈ ù. We shall define a game Gk (p, q). The players in Gk (p, q) are F and S. They play on two boards, the p board and the q board, and make additional moves lying on neither board. On the p board, S plays G(T ) from p as I, while F plays as player II. On the q board, F plays G(T ) from q as player I, while S plays as player II. Play is divided into rounds, the first being round ù. We now describe the typical round. We shall require that the moves in Gk (p, q) on the p and q boards which are meaningless for G(T ) be 0. So in round α of Gk (p, q), for α = ù or α a limit of limit ordinals, F and S must each play 0’s on both boards. We make the technical stipulation that F always proposes k at round ù. (This will make it more convenient to describe the payoff condition of Gk (p, q).) No further moves are made in round α, for α = ù or α a limit of limits. Round α, for α a successor ordinal: (a) F plays as player I in round α of G(T ) on the q-board, then (b) S plays as player I in round α of G(T ) on the p-board, then (c) F, S play 0 as player II in round α of G(T ) on the p, q-boards, respectively. Round α, when ∃î ≥ 1(α = ùî + ù): (a) F makes player I’s α-th move on the q board, then S makes player I’s α-th move on the p-board. (b) F now proposes some i such that 0 ≤ i ≤ k and (i)0 ∈ {0, 1}. (c) S either accepts i, or proposes some j such that 0 ≤ j < i and (j)0 ∈ {0, 1}.

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(d) Let t ≤ k be the least proposal made during (b) and (c): Case 1: t 6= 0. Then F and S must play (t)0 as player II’s ù + α-th move on the p and q boards respectively. Case 2: t = 0. Then F plays any m ∈ {0, 1} as player II’s ù + α-th move on the p board, after which S plays any n ∈ {0, 1} as player II’s ù + α-th move on the q board. This completes our description of round α. Play in Gk (p, q) continues until one of the two boards reaches a position in A. If this never happens, then F wins Gk (p, q). If one of the two boards reaches a position in A strictly before the other board does, then the player playing as player I on that board wins Gk (p, q). We are left with the case that we have a run u of Gk (p, q) and r, s with p ⊆ r and q ⊆ s the runs of G(T ) on the two boards, and r, s ∈ A. Letting â = dom(r) = dom(s), we must have â = ùî + ù for some î ≥ 1. Let e ∈ ù be the least n such that for some α ≥ ù, |(n)0 |F (r) | = α, and (n)1 was the least proposal made during round α. (Our technical stipulations guarantee that h0, ki is such an n.) We call e the critical number of u, and write e = crit(u); thus crit(u) ≤ h0, ki. Let α = |(e)0 |F (r) . If F proposed (e)1 during round α, then S wins u iff øe (r ∗ ) ≤ øe (s ∗ ). If S proposed (e)1 during round α, then S wins u iff øe (r ∗ ) < øe (s ∗ ). This completes our description of Gk (p, q). Let G0k (p, q) be just like Gk (p, q), except that if neither board reaches a position in A after ù1 moves, then it is S who wins, rather than F . Lemma 3.3. Let p0 ∈ aI (T ), and suppose that for all n ≥ 0, Σn is either a winning strategy for F in Gk (pn , pn+1 ), or a winning strategy for S in G0 (pn+1 , pn ); then only finitely many Σn are for F . We omit the proof of Lemma 3.3, as it is a direct transcription of the corresponding lemma in [Ste88]. For p, q ∈ aI (T ), we put p ≤k q iff player II has a winning strategy in Gk (p, q). Corollary 3.4. (a) ≤k is a prewellorder of aI (T ). (b) For q ∈ aI (T ), S has a winning strategy in Gk (p, q) iff S has a winning strategy in G0k (p, q). (c) ≤k determines a aù1 (open-analytical) norm on aI (T ). Proof. Just as in the proofs of Corollaries 2.3 and 2.4.

⊣

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Now for p ∈ aI (T ), let ϕk (p) = ordinal of p in ≤k . Lemma 3.5. ϕ ~ is a semiscale on aI (T ). Proof. The proof is very close to that of [Ste88, Lemma 1.6], so we just indicate the small changes needed. Let pn → p as n → ∞, with pn ∈ aI (T ) for all n. Let ô be a winning strategy for player I in G(T ). Suppose pn+1 ≤n pn , as witnessed by the winning strategy Σn for S in Gn (pn+1 , pn ), for all n. We must show that p ∈ aI (T ). Suppose toward contradiction that ó is a winning strategy for player II in G(T ).2 If rn ∈ dom(F ) for all n, then we write r = lim∗n→∞ rn for the same notion of “convergence in the codes” as defined in [Ste88]. We define by induction on rounds runs un of Gn (pn+1 , pn ) according to Σn . We arrange that un and un+1 agree on a common play rn+1 extending pn+1 on the pn+1 board, and that the play r0 extending p0 on the p0 board is by ô. So assume that un ↾α and rn ↾α are given for all n. If α = ù or α is a limit of limit ordinals, we set rn (α) = h0, 0i, and thus un (α) = h0, 0, n, 0, 0i, for all n. None of these moves count for anything, of course. If α is a successor ordinal, then only player I’s move in G(T ) counts. Let a0 = ô(r0 ↾(α)), and an+1 = Σn ((un ↾α)ahan i) fill in the α-th column of player I’s plays in our diagram, and set rn (α) = han , 0i, and thus un (α) = han , an+1 , 0, 0i, for all n. Finally, suppose α = ùî + ù for some î ≥ 1. Again, let a0 = ô(r0 ↾(α)), and an+1 = Σn ((un ↾α)ahan i) for all n. If an is eventually equal to some fixed a, and if lim∗n→∞ rn ↾(α) = r, where r is a play by ó of length ùî + ù for some î ≥ 1, then we set d = ó(r ahai). If not, then we set d = 0. We proceed now exactly as in [Ste88]. For any n, let in be the largest i such that hd, ii ≤ n and Σn ((un ↾α)ahan , hd, iii) = “accept”, if such an i exists. Let in = 0 otherwise. 2 Unfortunately, there are some typos in the proof of [Ste88, Lemma 1.6] which confuse ô with ó at various points.

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Case 1: in → ∞ as n → ∞. Pick n0 such that in > 0 for n ≥ n0 . The proposal pair in un (α) is hbn , cn i, where bn = hd, in i for n ≥ n0 and bn = 0 = freedom for n < n0 , and cn = accept for all n. Let dn = d if n ≥ n0 , and dn = Σn ((un ↾α)ahan , an+1 , bn , cn , dn+1 i) if n < n0 . Case 2: Otherwise. Again, we define the un (α) exactly as in [Ste88]. We shall not repeat the definition here in this case. This completes the definition of the un and rn . Since r0 is a play by ô, r0 ↾α ∈ A for some α. Letting α0 be the least α such that rn ↾α ∈ A for some n, we have rn ↾α0 ∈ A for all sufficiently large n because the Σn ’s won for S. To save notation, let us assume rn ↾α0 ∈ A for all n. Note that α0 = ùî + ù for some î ≥ 1. Let us write un = un ↾α0 and rn = rn ↾α0 for all n. Claim 3.6. crit(un ) → ∞ as n → ∞. Proof. See [Ste88]. The only additional point here is that if e = crit(un ) for infinitely many n, then (e)0 6= 0. ⊣ (Claim 3.6) ∗ It follows from the claim that for all e, øe (rn ) is eventually constant as n → ∞. From this we get that lim∗n→∞ rn = r, for some r ∈ A such that p ⊆ r. We shall show that r is a play by ó, so that ó was not winning for player II in G(T ), a contradiction. Let â be least such that r↾(â + 1) is not by ó. Fix ç a limit ordinal such that â = ç + ù; such an ç must exist because ó is for player II, who only moves at stages of the form ç + ù. Let â = |k|F (r) and α = eventual value of |k|F (rn ) as n → ∞. ~5 , we have that α = ì + ù for some limit Since the rn∗ converge in the scales ϑ ordinal ì. We now look at how column α of our diagram was constructed. ~ 0 and ϑ ~ 1 , we have Since rn∗ converged in ϑ lim∗n→∞ rn ↾α = r↾â. ~ 6 , which is very good, the an ’s defined at round α Since the rn∗ converge in ϑ are eventually constant, with value a = r(â)0 . But then at round α in the construction we set d = ó((r↾â)ahai). Moreover, Case 1 must have applied at α, as otherwise crit(un ) would have a finite lim inf. Thus rn (ã) = ha, d i for ~6 is very good, r(â) = ha, d i, so that r↾(â + 1) all sufficiently large n. Since ϑ is by ó, a contradiction. ⊣ (Lemma 3.5)

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~ be the Let U be the tree of the semiscale ϕ ~ given by Lemma 3.5, and let ϑ ù1 scale of U . One can easily check that, since a (open-analytical) is closed under real quantification, the ϑi are aù1 (open-analytical) norms, uniformly in i. ⊣ (Theorem 3.1) There are some awkward features of our proof of Theorem 3.1. First, it only applies directly to games in which player II is restricted to playing from {0, 1}. Second, our comparison games seem to only yield a semiscale directly, and not a scale. This is connected to the fact that we only show that if pn → p modulo our semiscale, then player II has no winning strategy in G(T ) from p; we do not construct a winning strategy for player I from p. One could probably obtain a more direct proof by bringing in the construction of definable winning strategies for player I. In this connection, one has Theorem 3.7. Assume that ù1 -open-projective determinacy holds, and that player I has a winning strategy in G(T ), where T is an ù1 -tree which is definable over hHù1 , ∈i; then player I has a aù1 (open-analytical) winning strategy in G(T ). One can prove Theorem 3.7 by modifying the proof of [Ste88, Theorem 2.1], in the same way that we modified the proof of [Ste88, Theorem 1.2] in order to prove Theorem 3.1. REFERENCES

Alexander S. Kechris, Donald A. Martin, and John R. Steel [Cabal iv] Cabal seminar 81–85, Lecture Notes in Mathematics, no. 1333, Berlin, Springer, 1988. Itay Neeman [Nee] Games of length ù1 , Journal of Mathematical Logic, to appear. [Nee04] The determinacy of long games, de Gruyter Series in Logic and its Applications, vol. 7, Walter de Gruyter, Berlin, 2004. John R. Steel [Ste88] Long games, this volume, originally published in Kechris et al. [Cabal iv], pp. 56–97. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720, USA

E-mail: [email protected]

PART II: SUSLIN CARDINALS, PARTITION PROPERTIES, HOMOGENEITY

SUSLIN CARDINALS, PARTITION PROPERTIES, HOMOGENEITY INTRODUCTION TO PART II

STEVE JACKSON

§1. Introduction. In this paper we survey the basic concepts and results pertaining to Suslin cardinals, partition properties, and homogeneous trees. This paper in part serves as an introduction to the papers in this section. We explain the basic material, and also survey some of the more recent developments as they pertain to these concepts. Throughout, by “real” we mean an element of the Baire space ù ù. We fix recursive bijections (a0 , . . . , an−1 ) 7→ ha0 , . . . , an−1 i from n ù to ù which are reasonable, say increasing in each argument. We let t 7→ ((t)0 , . . . , (t)n ) denote the recursive inverse map. These maps induce recursive bijections between n ( ù ù) and ù ù, and we use the same notation hα0 , . . . , αn−1 i and (α)i to denote these bijections. There is also a recursive bijection between ù ù ( ù) and ù ù, and we again use the same notation hα0 , α1 , . . . i, (α)i to denote it; which bijection we are referring to will be clear from the context. Our base theory throughout is ZF + DC, although we will frequently be assuming some form of determinacy as well. We let AD abbreviate the axiom of determinacy—the statement that every two player integer game is determined. We remind the reader that the basic properties of ordinals and cardinals remain valid assuming just ZF with the notable exception that successor cardinals no longer need be regular. By a tree on set X we always mean a tree in the descriptive set theoretic sense, that is, T ⊆ <ù X and if x ~ = (x0 , . . . , xn−1 ) ∈ T then x ~ ↾m = (x0 , . . . , xm−1 ) ∈ T for all m ≤ n. We write lh(~ x ) to denote the length of the sequence x ~ . We say y~ extends x ~ if lh(~ y ) ≥ lh(~ x ) and y↾(lh(x)) = x ~ . As is customary, we abuse notation slightly and consider elements of a tree on X × Y as pairs (~ x, y ~ ) where lh(~ x ) = lh(~ y ). By an infinite branch through T we mean a x ~ = (x0 , x1 , . . . ) ∈ ù X such that x ~ ↾n ∈ T for all n. We say T is ill-founded if it has an infinite branch, and otherwise say T is wellfounded. We let [T ] denote the set of infinite branches through T . If T is a tree on X1 × X2 × · · · × Xn , we let pi [T ] denote the projection of [T ] onto the ith coordinate, that is, x ~ ∈ pi [T ] iff ∃~ y1 , . . . , y ~n−1 (~ y1 , . . . , y~i−1 , x ~ , y~i+1 , . . . , y~n−1 ) ∈ [T ]. We write The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

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p[T ] for p1 [T ]. If T is a tree on X × Y and s = (x0 , . . . , xn−1 ) ∈ <ù X , we define Ts = {t ∈ ≤n Y : (s↾ lh(t), t) ∈ T }. We also use this notation for trees on general products X1 × · · · × Xn and allow s ∈ <ù (Xi ), with the obvious meaning. If T is an illfounded tree on X , and ≺ is a wellordering of X , it makes sense to speak of the left-most branch ℓ of T . This is simply the infinite branch of T which is ≺-lexicographically less than any other infinite branch. If ≺ is a wellordering of X and T is a tree on X , then ≺ induces a natural linear order on T called the Kleene-Brouwer order. It is defined by x ~
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y ∈ A we have (x ∈ A ∧ ϕ(x) ≤ ϕ(y)) ⇔ x ≤Γ y ⇔ x ≤Γ˘ y. e e It is easy to see that Definition 1.1 implies this alternate definition, and the alternate definition implies that of Definition 1.1 if Γ is closed under ∧, ∨ (see [Mos80]). The prewellordering property of Γ is a estrong way of saying that e every Γ set is a union of ∆ sets. e e A regular norm on a set A can be identified with a prewellordering  of A, that is a reflexive, transitive, connected (i.e., ∀x, y ∈ A (x  y ∨ y  x)) binary relation on A whose strict part ≺ is wellfounded, where x ≺ y ⇔ (x  y ∧ ¬y  x). We call A the domain of the prewellordering. We let Θ denote the supremum of the lengths of the prewellorderings of R. It is a standard fact (see [Mos80]) that PWO(Γ) ⇒ Red(Γ) if Γ is closed e e e under disjunction. Also, Red(Γ) ⇒ Sep(Γ˘ ), and if Γ has a universal set, then e e e Red(Γ) ⇒ ¬ Sep(Γ) (these are all ZF results). e to [Van78B] for the basic facts about Wadge degrees Weerefer the reader assuming AD. We mention a few of the results of significance for us. Assume AD for the remainder of this discussion. Wadge’s Lemma asserts that for any A, B ⊆ ù ù that either A ≤W B or B ≤W ù ù − A. One consequence is that a pointclass has a universal set iff it is non-selfdual. In view of Wadge’s Lemma, It is natural to consider the equivalence class [A]W of sets Wadge equivalent to A if A is self-dual, that is A ≡W ù ù − A, and pairs of equivalence classes ([A]W , [ ù ù − A]W ) when A is non-selfdual. Defined this way, Martin showed that the Wadge degrees are wellfounded. Moreover, in [Van78B] it is shown that the dual and non-selfdual degrees alternate, and at limit ordinals of cofinality ù there is a self-dual degree, and a non-selfdual degree at ordinals of uncountable cofinality. For A ⊆ ù ù, let |A|W denote the Wadge rank of A. By a L´evy pointclass we mean a non-selfdual pointclass Γ closed under e either ∃R or ∀R (or both). Note that if Γ is non-selfdual and closed under ∃R , e then Γ is closed under countable unions, and likewise for ∀R and countable e ˘ intersections. For suppose each AL n ∈ Γ. Let B ∈ Γ − Γ. By Wadge’s Lemma, e e e each An ≤W B. It follows that n An ≤W B, where the join is defined by L S L a x ∈ n An iff ∃y (y(0)ax ∈ n An ) n An = {n x : x ∈ An }. But then L which suffices as {(x, y) : y(0)ax ∈ n An } ∈ Γ. e Definition 1.2. For Γ a (possibly selfdual) pointclass, let e o(Γ) = sup{|A|W : A ∈ Γ}. e e Let ä(Γ) be the supremum of the lengths of the ∆ prewellorderings of ù ù, e e where ∆ = Γ ∩ Γ˘ . e e e If ∆ is closed under ∨ then ä(∆) is a limit ordinal, and if ∆ is closed under e e ù. If Γ is any pointclass and e ϕ is a regular countable unions then cf(ä(∆)) > e e

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Γ-norm on A ∈ Γ, then |ϕ| ≤ ä(Γ), as any initial segment of the norm e e prewellordering.e The following frequently used lemma corresponds to a ∆ e (see 4C.14 of [Mos80]) is a partial converse to this.

Lemma 1.3 (Moschovakis). Assume Γ has a universal set, and Γ is closed e e ). under ∀R , ∨. If ϕ is a Γ-norm on a Γ-universal set A, then |ϕ| = ä(Γ e e e So, assuming AD, this holds for any non-selfdual pointclass Γ and any e Γ-norm on a set A ∈ Γ − Γ˘ . e In [KSS81] the following e e is shown. It shows that for sufficiently closed pointclasses ∆, o(∆) and ä(∆) are the same. e e e Theorem 1.4 (AD). Let ∆ be selfdual and closed under ∃R , ∧. Then o(∆) = e lengths of the ∆ wellfounded relations on ùeù. ä(∆) = the supremum of the e e We note that the hypothesis that ∆ is closed under ∧ is almost redundant in e Theorem 1.4; it only rules out the case where ∆ = Γ ∪ Γ˘ for some non-selfdual e e e Γ closed under quantifiers. e Steel [Ste81B] showed that for a non-selfdual Γ one of Sep(Γ), Sep(Γ˘ ) e e e holds. Van Wesep [Van78A] showed that Sep(Γ) and Sep(Γ˘ ) cannot both e e hold. Thus we have:

Theorem 1.5 (AD). For any non-selfdual pointclass Γ exactly one of Sep(Γ), e e Sep(Γ˘ ) holds. e For Γ a L´evy class, Kechris, Solovay, Steel [KSS81] and Steel [Ste81A] show e the following.

Theorem 1.6 (AD). Let Γ be non-selfdual, closed under ∃R or ∀R , and e assume ∆ = Γ ∩ Γ˘ is not closed under wellordered unions. Then exactly one e e e ˘ of PWO(Γ), PWO(Γ) holds. e e The technical hypothesis that ∆ is not closed under wellordered unions holds, for example, if every Γ sete is ∞-Borel (see Definition 2.11 below). e It follows therefore from the slight strengthening AD+ of AD introduced by Woodin (see [Woo99]). In fact, [KSS81] and [Ste81A] completely analyze the prewellordering property for L´evy classes. If Γ is a L´evy class, following [KSS81] and [Ste81A] e consider

ä0 = sup{o(∆) : ∆ is closed under ∃R , ∧, ¬, and ∆ ⊆ Γ}. e e e e Let ∆0 be the selfdual class of sets of Wadge rank less than ä0 . It is easily e that ä is a limit ordinal. If cf(ä ) > ù, then there is a non-selfdual checked 0 0 Γ closed under ∀R with ∆0 = Γ0 ∩ Γ˘ 0 (see [Ste81A] for details). Γ is then in e projective hierarchy over e Γ e. If Γe is not closed under ∃R , then we e generate the 0 0 e e this hierarchy by applying quantifiers to Γ0 and/or Γ˘ 0 . If Γ0 is closed under e e e

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quantifiers, we must apply quantifiers to Γ0 ∧ Γ˘ 0 . We call the class Γ0 at the e where cf(ä ) > ù,ewe have base of this hierarchy a Steel pointclass. Inethe case 0 PWO(Γ0 ), and the prewellordering property propagates up the hierarchy by e periodicity. [If Γ0 is not closed under quantifiers then PWO(∃R Γ0 ), PWO(∀R ∃R Γ0 ), etc. e e e If Γ0 is closed under quantifiers, then PWO(∃R (Γ0 ∧ Γ˘ 0 ), PWO(∀R ∃R (Γ0 ∧ Γ˘ 0 ), e e e e e etc. See [KSS81] and [Ste81A] forSdetails.] If cf(ä0 ) = ù, then we let Γ0 = ù ∆0 , the collection of countable unions of e e prewellordering sets in ∆0 . Then PWO(Γ0 ), and propagates up by periodicity. e e R R R Namely, we have PWO(∀ Γ0 ), PWO(∃ ∀ Γ0 ), etc. e of a projective hierarchy. We call the pointclass Γ0eas above the base e Concerning the reduction property we have the following. Steel shows in [Ste81B] that for Γ non-selfdual with ∆ closed under ∧, that Red(Γ) or Red(Γ˘ ) e all Γ at the baseeof a projective hierarchy. Inspecting e e holds. This includes the 0 e possible projective hierarchies shows that this also includes all L´evy classes except the case where Γ = Γ0 ∧ Γ˘ 0 (or the dual of this class) and Γ0 is closed e eIn this e case we can argue directly that e Red(Γ) under real quantification. e holds assuming again the technical hypothesis that ∆0 is not closed under e wellordered unions. For in this case we may assume without loss of generality that PWO(Γ0 ). Let κ = o(∆0 ). e e 0 = S Coding Lemma (see Theorem 1.8 below) it follows that Γ S From the e∈ Γ Γ . Suppose A = A ∩A , B = B ∩B are in Γ , with A , B ∆ = 0 0 1 0 1 0 0 0 0 κ κ S S e e e e and A1 , B1 ∈ Γ˘ 0 . Write A0 = α<κ Cα , B0 = α<κ Dα , with Cα , Dα ∈ ∆0 . e e Let [ [ A′ = A1 ∩ (Cα ∩ ( ù ù − (Dâ ∩ B1 ))), α<κ ′

B = B1 ∩

[ α<κ

â<α

(Dα ∩ ( ù − ù

[

(Câ ∩ A1 ))).

â≤α

˘ 0 is closed under < κ unions, and it follows From the Coding Lemma, Γ e ′ ′ that A , B are in Γ, and they are easily seen to reduce A, B. e Corollary 1.7 (AD). Let Γ be non-selfdual, closed under ∃R or ∀R , and e assume ∆ = Γ ∩ Γ˘ is not closed under wellordered unions. Then exactly one e e ˘e of Red(Γ), Red(Γ) holds. e e The Moschovakis Coding Lemma is a basic tool in determinacy theory. It requires full AD, even for prewellorderings of short length. It can be stated in several different forms, one of which is the following. Theorem 1.8 (Coding Lemma). Assume AD. Let Γ be a non-self-dual e pointclass closed under ∃R , ∧. Suppose ≺ is a Γ wellfounded relation on e

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ù. Then for any R ⊆ dom(≺) × ù ù such that ∀x ∈ dom(≺) ∃y R(x, y), there is an A ⊆ dom(≺) × ù ù, A ∈ Γ, which is a choice set for R. That is, e 1. ∀α < |≺| ∃x ∈ dom(≺) ∃y [|x|≺ = α ∧ A(x, y)]. 2. ∀x, y [A(x, y) ⇒ R(x, y)]. ù

Proof (sketch). Let U ⊆ ( ù ù)3 be in Γ and universal for the Γ subsets e of minimal length soe that the of ù ù × ù ù. We may assume that ≺ is chosen theorem fails. It follows that ä is a limit. For ä < |≺|, we say u ∈ ù ù is a ä choice code if (1) above holds for all α < ä using Uu in place of A, and in place of (2) we use ∀x, y [Uu (x, y) ⇒ |x|≺ < ä ∧ R(x, y)]. By minimality of |≺| it follows that for all ä < |≺| that there is a ä choice code. Consider the integer game where player I plays out u ∈ ù ù and player II plays out v, and player II wins iff whenever u codes a ä choice set for some ä, then v codes a ä ′ choice set for some ä ′ > ä. If player I had a winning strategy then we would get a Σ11 set S ⊆ ù ù such that each u ∈ S is a ä choice code e ∀ä < |≺| ∃ä ′ > ä ∃u ∈ S (u is a ä ′ choice set). We for some ä < |≺|, and can then form the “union” of S by: A(x, y) ⇔ ∃u ∈ S Uu (x, y). A then easily satisfies (1) and (2) above. Suppose player II had a winning strategy ô. We attempt to define a Γ relation Uε (x, y) (by defining its code ε) such e if U (x, y) then y is a ä choice code for some that dom(Uε ) = dom(≺) and ε ä ≥ |x|≺ . Using the Recursion Theorem (which holds for any Γ with a e universal set, hence from AD for any non-self-dual Γ), let ε ∈ ù ù be such that e Uε (x, u) ⇔ u = ô(f(ε, x)), where f is a continuous function from the s-m-n Theorem such that Uf(ε,x) (y, z) ⇔ ∃x ′ ≺ x ∃u [Uε (x ′ , u) ∧ Uu (y, z)]. We clearly have ∀x ∈ dom(≺) ∃y Uε (x, y). Induction on |x|≺ also shows that Uε (x, y) implies that y is a ä choice code for some ä ≥ |x|≺ . Finally, define A by A(x, y) ⇔ ∃x ′ ∃u [Uε (x ′ , u) ∧ Uu (x, y)]. Then A satisfies (1), (2). ⊣ One frequently arising case is when the relation ≺ is the strict part of a prewellordering  with both ≺ and  in Γ. This will be the case if  is the prewellordering from a Γ-norm on a Γe set. Also, it frequently occurs e is, if R(x, y), x  x ′ , and that the R of Theorem 1.8 eis invariant, that ′ ′ x  x, then R(x , y). In this case we may view R as assigning a nonempty set to each ordinal α < |≺|. Note that the A of Theorem 1.8 may also be taken to be invariant, as given A from Theorem 1.8 we may define A′ (x, y) ⇔ ∃x ′ [(x ′  x) ∧ (x  x ′ ) ∧ A(x ′ , y)]. Definition 1.9. If  is a prewellordering (or equivalently a norm), and A ⊆ ||, we say A is ∆ in the codes (for some pointclass Γ) if there are Γ, Γ˘ e D such that ∀x ∈ dom() (|x| e ∈ A ⇔ C (x)e ⇔ e (respectively) sets C and  D(x)). If Γ is as in Theorem 1.8,  is a prewellordering with  and ≺ in Γ, and e A ⊆ eë = ||, then A is ∆ in the codes. This follows by applying Theorem e

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1.8 to the characteristic function of A. If Γ is non-self-dual, closed under ∀R , e say that any A ⊆ ä(Γ) is ∆ in the and PWO(Γ), then we can improve this to e e is ∆ e1 R codes (as opposed to ∆(∃ Γ)). For example, any subset of ä 12n+1 2n+1 in e e e e the codes (see Definition 2.6 below). To see this, note that if Γ is closed under ∃R then we are done by Theorem 1.8. Otherwise, let ϕ beea Γ norm on a e ∨. Then by Γ-complete set P. Assume for the moment Γ is also closed under e eù ù Lemma 1.3, ϕ is onto ä(Γ). Let U ⊆ ù × ù × ù be universal for Γ˘ . Given e e A ⊆ ä(Γ), say y ∈ ù ù codes A↾â (for â < ä(Γ)) if e e Uy (z, a) ⇔ (z ∈ P ∧ ϕ(z) < â ∧ z ∈ A ∧ a = 1) ∨ (z ∈ P ∧ ϕ(z) < â ∧ z ∈ / A ∧ a = 0). From the Coding Lemma, for all â < ä(Γ) there is a y coding A↾â. Play e II plays y, and player II wins the integer game where player I plays x, player iff [x ∈ P ⇒ ∃â > ϕ(x) (y codes A↾â)]. player II wins by boundedness as a winning strategy for player I would give a Σ11 set S ⊆ P coding cofinally e in ä(Γ) many ordinals, from which we would compute P ∈ Γ˘ by x ∈ P ⇔ e e ϕ(x) ∈ A iff ∃y ∈ S x ≤Γ˘ y. If ô is winning for player II and x ∈ P, then Uô(x) (x, 1) iffe¬Uô(x) (x, 0). If Γ is closed under ∀R but not ∃R , PWO(Γ), and e Γ is not closed under ∨, then Γe is a Steel pointclass at the base of a projective e e hierarchy. The analysis of [Ste81A] shows that there is a Γ prewellordering on a Γ complete set of length ä(Γ) which is Σ11 bounded, andethe above argument e applies, using a Γ universal e e the coding. We summarize this in then set to do e the following corollary. Corollary 1.10 (AD). If Γ is non-self-dual, closed under ∀R , and PWO(Γ), e codes with respect to a Γ-norm on a Γ-complete e then every A ⊆ ä(Γ) is ∆ in the e e e e set. §2. Suslin Cardinals. We first introduce the notion of a Suslin cardinal. Definition 2.1. For α ∈ Ord, X a set, and A ⊆ ù X , we say A is α-Suslin if there is a tree T on X × α such that A = p[T ]. We say κ ∈ Ord is a Suslin cardinal if there is an A ⊆ ù ù which is α-Suslin but not â-Suslin for any â < α. Being α-Suslin only depends on |α|, the cardinality of α, so it is immediate that a Suslin cardinal is a cardinal. We say A ⊆ ù X is co-α-Suslin if the complement ù X − A is α-Suslin. We say A is Suslin if it is κ-Suslin for some κ and likewise for co-Suslin. We let Ξ denote the supremum of the Suslin cardinals. We might have Ξ = Θ, in which case every set has a scale, or might have Ξ < Θ. In the latter case it is not known assuming AD whether Ξ must be a Suslin cardinal, although the Suslin cardinals are closed below Ξ.

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From AD+ it follows that Ξ is a Suslin cardinal if Ξ < Θ (see Theorem 2.13 below; see also Lemma 2.20 for more about Ξ). Definition 2.2. For κ ∈ Ord we let S(κ) be the pointclass of κ-Suslin sets. It is easy to see that S(κ) is a (boldface) pointclass (i.e., is closed under Wadge reduction) and is closed under ∃R , countable unions and intersections. For the case of interest when κ is a Suslin cardinal, Kechris [Kec81B] shows, assuming AD, that S(κ) is non-selfdual. In fact, Kechris shows that for any κ, S(κ) is non-selfdual provided the Suslin cardinals are closed below κ. From Theorem 2.13 below, this always holds except perhaps in the case when κ ≥ Ξ and Ξ is not a Suslin cardinal (in which case S(κ) is the collection of all Suslin sets). Suslin representations are closely related to scales. The notion of a scale, and the scale property, was introduced by Moschovakis. It was introduced originally as a distillation of the key ingredients in the Novikov-Kondoˆ proof of Π11 uniformization, but it quickly evolved into one of the main structural e notions in descriptive set theory. We recall the definition. The reader can also consult [KM78B] for further discussion. Definition 2.3. A semi-scale hϕn : n ∈ ùi on a set A ⊆ ù X (X a set) is a collection of norms ϕn on A such that if hxm : m ∈ ùi ⊆ A is a sequence of points in A converging coordinate-wise (i.e., in the product of the discrete topology on X ) to x ∈ ù X , and for all n, ϕn (xm ) is eventually constant, then x ∈ A. We say hϕn i is a scale if it in addition satisfies the lower semi-continuity property: ∀n ϕn (x) ≤ limm→∞ ϕn (xm ). A (semi)-scale ϕ ~ on A is a good (semi)-scale if whenever xm ∈ A and for all n ϕn (xm ) is eventually constant, then x = limm→∞ xm exists (and thus x ∈ A). A (semi)-scale is called very good if it is good and whenever x, y ∈ A and ϕn (x) ≤ ϕn (y), then ϕi (x) ≤ ϕi (y) for all i < n. We say ϕ ~ is an α semi-scale, etc., if all norms map into α. The existence of an α-semiscale on A ⊆ ù X is easily equivalent to X being α-Suslin. For if A = p[T ], T a tree on X × α, then we can define a semiscale ϕ ~ on A by ϕn (~ x ) = the nth coordinate of the leftmost branch ℓx~ of Tx~ . Conversely, given an α-semiscale ϕ ~ on A, define the tree of the semiscale Tϕ by: ((x0 , . . . , xn−1 ), (â0 , . . . , ân−1 )) ∈ Tϕ iff ∃~ x ∈ ù X extending (x0 , . . . , xn−1 ) with ϕ0 (~ x ) = â0 , . . . , ϕn−1 (~ x ) = ân−1 . Using the definition of semi-scale we easily get that A = p[T ]. In the main case of interest, when X = ù, we can start from a α-Suslin representation for A ⊆ ù ù and produce a very-good α ′ -scale on A, where α ′ has the same cardinality as α. This is done by letting, for x ∈ A, ϕn (x) = hℓx (0), x(0), . . . , ℓx (n − 1), x(n − 1)i where ℓx again denotes the leftmost branch of Tx and hç0 , . . . , çk−1 i denotes the rank of the tuple in the

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lexicographic ordering on α k . It is easy to check that the ϕn form a very-good scale. With a little extra effort we can in fact take α ′ = α as the next lemma shows (see also [KM78B]). Lemma 2.4. For any α ≥ ù and A ⊆

ù

ù, the following are equivalent.

1. A is α-Suslin. 2. A admits an α-very-good scale. Proof. Suppose A is α-Suslin. Then A is κ-Suslin where κ = |α|, so we may assume that α = κ. Let T be a tree on ù ×κ with A = p[T ]. First assume cf(κ) > ù. Define T ′ on ù × κ by: ((x(0), . . . , x(n − 1)), (â0 , . . . , ân−1 )) ∈ T ′ iff â0 ≥ max{â1 , . . . , ân−1 } and ((x(0), . . . , x(n − 2)), (â1 , . . . , ân−1 )) ∈ T . Clearly A = p[T ′ ] as well. Let ϕ ~ be the very-good scale from T ′ as above where now hâ0 , x(0), . . . , ân−1 , x(n − 1)i denotes rank in the lexicographic order on tuples from κ 2n whose first entry is the maximum of the entries. Clearly ϕn (x) < κ, and this defines a very-good scale. Suppose cf(κ) = ù. Let κ = supn κn , each κn ≥ ù. Define T ′ to consist of all pairs ((x(0), . . . , x(n − 1)), (â0 , . . . , ân−1 )) where âi < κi , and (â0 , . . . , ân−1 )) is an initial segment of a sequence of the form (i0 , 0, 0, . . . , 0, ã0 , i1 , 0, 0, . . . , 0, ã1 , . . . ) where i0 , i1 , · · · ∈ ù, there are exactly ik zeros after ik , and (~ x , ~ã ) ∈ T . Clearly A = p[T ′ ]. Using lexicographic order on those (â0 , . . . , ân ) satisfying ∀i âi < κi now produces a very-good κ-scale on A. ⊣ We recall now the definition of a Γ-scale, a fundamental notion in descriptive e set theory introduced by Moschovakis. Definition 2.5. A scale ϕ ~ on A is said to be a Γ-scale (or Γ-very good scale, e the scalee property if every etc.) if all the norms ϕn are Γ-norms. We say Γ has e e A ∈ Γ admits a Γ-scale. e e Note that if ϕ ~ is a Γ-scale on A, then its regularization ϕ~′ is also a Γ-scale e on A (it is easily still ea scale and the relations <∗n , ≤∗n are the same). Also, if Γ has the scale property and is closed under ∧, ∨, then every A ∈ Γ admits a e -very good scale since if ϕ Γ ~ is a Γ-scale on A, then we can define aevery good e scale by øn (x) = hϕ0 (x), x(0), .e. . , ϕn (x), x(n)i (it is important to order by the ordinal first). The original significance of the scale property was that it provides definable uniformizations. For example, if Γ has the scale property and Γ is closed e ù ù × ù ù in Γ has a Γ uniformization e under ∧, ∨, and ∀R , then every A ⊆ e ′ (x, y))). e We refer the (i.e., a Γ set A′ ⊆ A such that ∀x (∃yA(x, y) ⇔ ∃!yA e reader to [Mos80] for the proof. Also, if A admits a very good scale all of whose norms are in a pointclass Γ, then A has a uniformizing function f e

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which is aΓ-measurable, that is, the relation R(x, n, m) ⇔ f(x)(n) = m is in e aΓ. eAnother significance of Γ-scales is that they give extra information about the Suslin representation ofethe set A. If A admits a Γ-scale ϕ ~ and all the ϕn are regular (without loss of generality), then the treeeTϕ of the scale is a tree on ù × ä(Γ) (and possibly on a smaller ordinal as well). This is because each e ≤ ä by definition of ä and the fact that all initial segments of ϕn has length the ϕn prewellordering lie in ∆. An important special case of this concerns the e projective ordinals. Definition 2.6. Let ä 1n = ä(Σ1n ) = the supremum of the lengths of the ∆1n e e prewellorderings of ù ù.e

In particular, every regular Π12n+1 scale on a Π12n+1 set maps into ä 12n+1 . On e scale must map onto e each norm of the e ä1 . the other hand, from Lemma 1.3 2n+1 e As an immediate corollary we have:

Corollary 2.7. If ϕ ~ is a regular Π12n+1 scale on a Π12n+1 set, then all of the e e 1 norms map onto ä 2n+1 . e We review now briefly the basic facts about transferring scales and the scale property to higher pointclasses. As we mentioned above, S(κ) is closed under ∃R , so if every set in Γ admits a scale, the same is true for ∃R Γ. The e that Second Periodicity Theorem eof Moschovkis [Mos80] shows assuming AD ù ù R if A ⊆ ù × ù admits a scale then so does ∀ A. In fact, the proof also works for A ⊆ ù X × ù ù, X any set. Concerning the scale property we have the following. If Γ is closed under ∀R and every Γ set admits a Γ-very good e the scale property. In particular, e e scale, then ∃R Γ has if Γ is closed under e the scale property, then ∃R Γ has the scalee property (and is ∀R , ∧, ∨ and has closed under ∧, ∨). The scale on A ∈ ∃R B, Be ∈ Γ, is obtained by taking the “infimum” of the very good scale on B. We refere the reader to [Mos80] for details. One version of the Second Periodicity Theorem says that if Γ is closed e under ∃R , ∧ ∨ and has the scale property, then assuming ∆ determinacy ∀R Γ e e has the scale property. Both of these scale transfer results are contained in the following somewhat more general form of the Second Periodicity Theorem (see [Mos80]). Theorem 2.8 (Moschovakis). Assume every set in Γ admits a very good ∗,ϕ e scale S ϕ ~ , with the norm relations ≤∗,ϕ n ,
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pointclasses Π12n+1 , Σ12n+2 have the scale property. We will use this in a moment e Susline cardinals below the supremum of the ä 1 . Steel [Ste83B] to analyze the n ~ with ϕn e∈ ù · (n + 1)-Π11 . shows that every Σ11 set admits a very-good scale ϕ e e It follows from Theorem 2.8 that every Π12n set admits a scale all of whose e a2n−1 is the 2n − 1 iterate of the norms lie in a2n−1 ù · k-Π11 for some k (here e integer game quantifier). We mention two other scale transfer results. Martin [Mar83B] shows that under a suitable determinacy hypothesis (the determinacy of certain games on ù of countable length), the integer game quantifier aα of countable length α propagates scales and the scale property. That is, if Γ is closed under ∧, ∨ and has the scale property and if A ⊆ ( ù ù × ù α ) is in eΓ, then aα A also has the e precise definitions and scale property. We refer the reader to [Mar83B] for the statements. In particular, assuming ADR if Γ (as above) has the scale property then so does aR Γ. Martin’s proof is purelye game theoretic, generalizing the e methods of the periodicity theorems. If one doesn’t care about the definability of the scales, Martin and Steel [MS83] by completely different methods showed that just assuming AD, if A admits a scale then aR A also admits a scale. Their method is related to the homogeneous tree construction. Two important consequences of a κ-Suslin representation are the KunenMartin Theorem and a κ + -Borel representation for the set. First we recall the Kunen-Martin Theorem. Theorem 2.9 (Kunen, Martin). Let ≺ be a κ-Suslin wellfounded relation on ù ù. Then |≺| < κ + , where |≺| denotes the rank of ≺. Proof. Let T be a tree on ù × ù × κ with ≺ = p[T ]. Let U be the wellfounded tree consisting of finite ≺-decreasing sequences (x0 , . . . , xn ), that is, xn ≺ · · · ≺ x1 ≺ x0 . Easily ≺ and U have the same rank. To each x ~ = (x0 , . . . , xn ) ∈ U assign ð(~ x ) = (x0 ↾(n + 1), . . . , xn ↾(n + 1), ℓ(x1 , x0 )↾(n+1), . . . , ℓ(xn , xn−1 )↾(n + 1)), where ℓ(y, z) ∈ ù κ is the leftmost branch of Ty,z . If y ~ extends x ~ , we view ð(~ y ) as extending ð(~ x ) in a natural way. ð gives an order-preserving map of U into a wellfounded relation on (essentially) κ. Thus |≺| = |U | < κ + . ⊣ We recall the notion of a κ-Borel set (see also [KM78B]). We must be careful working in ZF to distinguish between the “effective” and “non-effective” versions of the definition. The next definition is the non-effective version. Definition 2.10. Bκ is the smallest collection containing the open sets and closed under complements and S wellordered unions (equivalently intersections) of length < κ. We let B∞ = κ Bκ . By an effective κ-Borel code S = hT, ϕi we mean a wellfounded tree T on a ë < κ and a map ϕ which assigns to each terminal node s of T a basic open set ϕ(s) in ù ù. Such an S builds up an element of Bκ in a natural manner.

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S Namely, extend ϕ to all of S by ϕ(s) = ù ù − t≺s ϕ(t), the union ranging over t extending s with lh(t) = lh(s) + 1. Then set ϕ(S) = ϕ(∅) (∅ is the maximal node of T ). Sometimes we use minor variations of this definition, like allowing each node of the tree to be either a “union” or an “intersection” node, in the obvious sense. It is easy to see that the relation x ∈ ϕ(S) is absolute between transitive models containing x, S. Definition 2.11. We say A ⊆ ù ù is effectively κ-Borel if there is an effective κ-Borel code S for A (i.e., A = ϕ(S)). We say A is ∞-Borel if it is effectively κ-Borel for some κ. Woodin has given another characterization of ∞-Borel sets. Namely, A is ∞-Borel iff there is an S ⊆ Ord and a formula ϕ such that ∀x (x ∈ A ⇔ L[S, x] |= ϕ(S, x)). Of course, any effectively κ-Borel set is in Bκ , but the converse is not clear in ZF (or ZF + AD). Suslin representations give effective Borel codes according to the following lemma (see also [KM78B]). Lemma 2.12. If A is κ-Suslin or co-κ-Suslin then A is effectively κ ++ -Borel. In fact, A is (effectively) a κ + union of sets which are κ + -Borel. If cf(κ) > ù then every κ-Suslin set is an effective κ union of κ-Borel sets. Proof. Let A = p[T ], T a tree on ù × κ. For s ∈ <ù κ and α < κ + , let B(s, α) = {x : |s|Tx ≤ α}, where |s|Tx denotes the rank of s in the tree Tx if s is in the wellfounded part of Tx , is undefined if s is in the illfounded part / Tx . So, terminal nodes of of Tx , and for convenience we set |s|Tx = 0 if s ∈ the tree Tx have rank 0 as do all further extensions. Clearly every B(s, 0) is clopen. For α > 0 we have: \ [ B(s, α) = B(s aç, â). ç<κ â<α

It follows κ + -Borel. Since ù ù − S by induction that all B(s,ùα) are (effectively) + A = α<κ+ B(∅, α), we have that ù − A is a κ union of κ + -Borel sets. Note that if x ∈ A and α is greater than the supremum of the |s|Tx for s in the wellfounded part of Tx , then (1) the root node ∅ of Tx is not wellfounded of rank ≤ α and (2) for all s ∈ Tx if s does not have rank ≤ α in Tx then there is an immediate extension s aç which does not have rank ≤ α in Tx . Conversely, (1) and (2) imply Tx is illfounded. Thus, [ [ \ ˘ aç, α))] ˘ (B(s, α) ∪ B(s A= [B(∅, α) ∩ α<κ +

s∈ <ù κ

ç<κ

This shows A is also a κ + S union of κ + -Borel sets. If cf(κ) > ù then p[T ] = α<κ p[T ↾α], and the last statement of the lemma follows. ⊣

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The converse to Lemma 2.12 is not in general true. Assuming AD+V=L(R), there is a largest Suslin class, Σ21 , but every set of reals is ∞-Borel (we discuss this in more detail below). e The following result is a fundamental results about the Suslin cardinals. It was first proved by Steel from the assumption AD + V=L(R) (cf. [Ste83A]), and then proved by Woodin from AD. Woodin’s proof introduced several new notions and methods including the concept of a strong ∞-Borel code. Theorem 2.13 (Steel, Woodin). Assume AD. Then the Suslin cardinals are closed below their supremum. Woodin has isolated a natural strengthening of AD called AD+ which implies that the Suslin cardinals are closed below Θ (see [Woo99]). Thus, AD+ implies S(κ) is always non-self-dual (this answers one of the questions of [Kec81B]). In fact Woodin shows that assuming AD, AD+ is equivalent to the Suslin cardinals being closed below Θ. Though we do not prove the Steel-Woodin Theorem here, we define the notion of a strong ∞-Borel code, which is crucial for their analysis. If S = hT, ϕi is an ∞-Borel code and C ⊆ Ord, by S↾C we mean the ∞-Borel code obtained by restricting to the subtree of T generated by C (i.e., the set of nodes s ∈ <ù C ). If s is not terminal in T but is terminal in T ↾C , we set ϕ(s) = ∅ in computing ϕ for T ↾C (in practice we may assume C has the property that if s ∈ <ù C is non-terminal in T then s is non-terminal in T ↾C so this minor annoyance does not arise). Definition 2.14 (Woodin). A strong ∞-Borel code is an ∞-Borel code S with the property that player II wins the following ordinal game GS : player I and player II alternate playing ordinals αi . Let C = hαi : i ∈ ùi. Player II wins the run iff ϕ(S↾C ) ⊆ ϕ(S). Note that for C countable, S↾C is just an ordinary Borel code (actually, isomorphic to one in an obvious sense). Lemma 2.15 (Woodin). A ⊆ ù ù is Suslin iff A has a strong ∞-Borel code. In fact, if A has a strong ∞-Borel code of size κ, then A is κ-Suslin. Proof. Suppose first that A = p[T ] where T is a tree on ù × κ. Let S ⊆ κ + (S ⊆ κ if cf(κ) > ù) be the ∞-Borel code for A produced by Lemma 2.12. We show that S is actually a strong ∞-Borel code. In fact, we show that for all C ⊆ Ord that ϕ(S↾C ) ⊆ ϕ(S). Suppose x ∈ ϕ(S↾C ). We produce an f ∈ ù Ord with (x, f) ∈ [T ]. The top node ∅ is a union node of S which writes S as a κ + union of codes Sα . Thus, for some α ∈ C we have x ∈ ϕ(Sα ↾C ). The code Sα is the conjunction of a code D for ˘ B(∅, α) and E which is a conjunction over s ∈S <ù κ of codes Es for S a code a ˘ aç, α). Fs is a ˘ ç, α). Let Fs denote the code for ç B(s B(s, α) ∪ ç B(s ˘ aç, α). Now x ∈ ϕ(D↾C ) and taking κ disjunction of codes Gsaç for B(s

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s = ∅ we have x ∈ ( ù ù − ϕ(D↾C )) ∪ ϕ(F∅ ↾C ). Thus, x ∈ ϕ(F∅ ↾C ). Thus there is an f(0) ∈ C such that x ∈ ϕ(Gs0 ↾C ), where s0 = hf(0)i. Since x ∈ ϕ(E↾C ) and f(0) ∈ C , we have x ∈ ϕ(Es0 ↾C ). Since x ∈ ϕ(Gs0 ↾C ), we also have x ∈ ϕ(Fs0 ↾C ). Thus there is an f(1) ∈ C such that letting s1 = hf(0), f(1)i we have x ∈ ϕ(Gs1 ↾C ). Since s1 ∈ <ù C and x ∈ ϕ(E↾C ), we have x ∈ ϕ(Es1 ↾C ). Continuing, we produce f as claimed. Suppose next that S ⊆ κ is a strong ∞-Borel code for A. Fix a winning strategy ô for player II in the game GS . It is straightforward to construct a tree T on ù × κ such that if (x, f) ∈ [T ] then C := {f(0), f(1), . . . } is closed under ô and x ∈ ϕ(S↾C ), and furthermore, for any C closed under ô and x ∈ ϕ(S↾C ) there is an f enumerating C with (x, f) ∈ [T ]. [The definition of T dovetails the requirements that ran(f) be closed under ô and x ∈ ϕ(S↾ ran(f)). The latter construction is similar to the proof that every Borel set is ù-Suslin.] If x ∈ [T ] then for some f ∈ ù κ, (x, f) ∈ [T ] and so x ∈ ϕ(S↾C ) ⊆ ϕ(S) = A since C = {f(0), f(1), . . . } is closed under ô and hence can be enumerated by a run of the game Gs following ô. If x ∈ ϕ(S), then there is c.u.b. set of C ∈ ℘ù1 (κ) for which x ∈ ϕ(S↾C ). If we choose C to also be closed under ô, then we can find an f enumerating C with (x, f) ∈ [T ], so x ∈ p[T ]. ⊣ The theory of Suslin cardinals and scales is closely related to the theory of wellordered unions of pointclasses. We recall a few of the results of this theory. In [JM83] it is shown that if Γ is closed under ∃R , ∀R and PWO(Γ), e then Γ is closed under wellordered unions. In [KSS81] it is show that if Γe is e e R R closed under ∃ but not ∀ , closed under countable unions and intersections, and PWO(Γ), then Γ is closed under wellordered unions. In [Jac07A] it is e last result e generalizes to the case Γ not closed under countable noted that this e unions and intersections. Thus we have:

Theorem 2.16 (AD). If Γ is a non-selfdual pointclass closed under ∃R and e PWO(Γ), then Γ is closed under wellordered unions. e e When the prewellordering property falls on the side closed under ∀R , we have the following result of Martin. Theorem 2.17 (Martin). Assume AD. Let Γ be non-selfdual, closed under e ∀R , ∧, ∨, and PWO(Γ). Then ∆ is closed under < ä(Γ) length unions and e e e intersections. Proof. Suppose α < ä(Γ) be least such that ∆ is not closed under α e e length unions. By the Coding Lemma, every α union of ∆ sets is in Γ˘ . e e By Wadge’s Lemma it follows that some Γ˘ -complete set, and hence every S e ˘ , write A = Γ˘ set is an α-union of ∆ sets. If A ∈ Γ â<α Aâ , where each S e e e Aâ ∈ ∆. For x ∈ A, let ϕ(x) = ìâ (x ∈ Aâ ). Since Γ˘ = α ∆, we get e e e S that the norm relations <∗ϕ , ≤∗ϕ are in Γ˘ [for example, <∗ϕ = â<α Bâ where e

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Bâ = {(x, y) : x ∈ Aâ ∧ y ∈ / Aâ }]. This shows PWO(Γ˘ ), a contradiction e since not both Γ, Γ˘ can have the prewellordering property. ⊣ e e In some cases we can show that a stronger principle than Theorem 2.16 applies, namely, there are no long increasing sequences of Γ sets. In [JM83] it is shown that any increasing or decreasing sequence of Σe12n sets must have e < ä 1 ). More length < ä 12n (in particular, any Π12n prewellordering has length 2n e e e generally, the following was shown: Theorem 2.18 (Jackson, Martin). Assume AD. Let κ be a Suslin cardinal with cf(κ) > ù. Then there is no increasing or decreasing sequence of S(κ) sets of length κ + .

Interestingly, Hjorth has obtained improvements to some of these results using completely different methods, namely inner model theory. In [Hjo96], Hjorth shows that there is no ù2 sequence of distinct Σ12 sets. In [Hjo01], Hjorth generalizes this to show that there is no ùn+2 esequence of distinct a ù · n-Π11 sets (and this is shown from Π12 -determinacy). Wheneκ is a Suslin cardinal with cf(κ)e > ù then in fact S(κ) has the scale property (cf. [JM83]). However, there are cases where we can establish the non-existence of long sequences without scales. This is the content of the next result, due to Chuang, which uses also the methods of [JM83] (cf. [Chu82], [Jac07A]). Theorem 2.19 (Chuang). Assume AD. Let Γ be non-selfdual, closed under e ∀R , countable unions and intersections and PWO(Γ ). Then there is no ä(Γ)+ e e increasing or decreasing sequence of Γ sets. e In particular, if Γ is non-selfdual and closed under quantifiers then there e are no o(∆)+ increasing or decreasing sequences of Γ sets. e Recall Ξ denotes the supremum of the Suslin cardinals. The next lemma shows that the largest Suslin cardinal, if it exists, must be a limit of Suslin cardinals. Lemma 2.20 (AD). If Ξ is a Suslin cardinal, then the Suslin cardinals are unbounded below Ξ, and Ξ is a regular cardinal. Furthermore, S(Ξ) (the pointclass of Ξ-Suslin sets) is closed under ∃R , ∀R , countable unions and intersections and has the scale property with norms onto Ξ. Proof. Let Γ = S(Ξ). By Kechris [Kec81B] Γ is non-selfdual, and closed e Γ is also closed under ∀R under ∃R , and ecountable unions and intersections. R as otherwise by second periodicity ∀ Γ would be aestrictly larger pointclass admitting scales. Let ä = ä(Γ) = o(∆e) by Theorem 1.4. From the Coding e we could e Lemma, ä ≤ Ξ (if Ξ < ä, then code any tree T on ù × Ξ via a pointclass in ∆, which would then compute p[T ] ∈ ∆, a contradiction). One e e of Γ, Γ˘ has the prewellordering property by Theorem 1.6, and so Theorem 2.19 e e

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applies to either Γ or Γ˘ . In either case, there is no ä + increasing sequence e e ˘ and let A = p[T ] with T a of Γ sets. Suppose ä < Ξ. Let A ∈ Γ − Γ e e regular e tree on ù × Ξ. Let ϕ ~ be the corresponding scale, with κn = sup ϕn . Since A is not α-Suslin for any α < Ξ, we have that supn κn = Ξ. Using ϕ0 , . . . , ϕn we easily get a κn increasing sequence of Γ sets, and so κn < ä + . e Ξ being a cardinal. Since cf(ä + ) > ù, Ξ = supn κn < ä + . This contradicts So, Ξ = ä. Easily, cf(ä) > ù. Again letting S A, T be as above, we have that A is a ä union of ∆ sets (since p[T ] = α<ä p[T ↾α] as cf(ä) > ù). This e shows PWO(Γ) since if PWO(Γ˘ ), then Γ˘ would be closed under wellordered e e e unions by 2.16, and then A ∈ Γ˘ . So Γ is closed under wellordered unions. It e e follows that ä is regular, as otherwise by the coding lemma we could compute any Ξ union of ∆ sets to be in ∆, using the closure properties of Γ; however e ∆ is not closed under ä unionsefrom PWO(Γ). To see Scale(Γ),elet A ∈ Γ e e be the least α < e Ξ such that e with A = p[T ] as above. For x ∈ A, let ø0 (x) ø0 (x) ø0 (x) x ∈ p[T ↾α]. Let øi+1 (x) = hø0 (x), ℓ0 (x), . . . , ℓi (x)i, where ℓjα (x) is the jth coordinate of the leftmost branch of Tx ↾α. It is straightforward to check that høi : i ∈ ùi is a scale on A. Moreover, all of the norm relations <∗øn , ≤∗øn are easily expressible as ä unions of ∆ sets, hence they are e Γ relations. e Let ë be the supremum of the Suslin cardinals which are less than Ξ. Suppose towards a contradiction that ë < Ξ. First suppose that ë is not a Suslin cardinal (this case cannot actually arise by Theorem 2.13, but we argue directly here). Again let A = p[T ], where A ∈ Γ − Γ˘ and T a tree on ù × Ξ. S e for some ë′ < ë (letThis writes A = α<Ξ Aα , where each Aα is ëe′ -Suslin ting Aα = p[T ↾α], which is α-Suslin, and hence ë′ -Suslin for some ë′ < ë by definition of ë). Since Ξ is regular, for some fixed ë1 < ë we have that A is a Ξ union of ë1 -Suslin sets. If Γ1 = S(ë1 ), then either PWO(Γ1 ) or e e PWO(Γ˘ 1 ), and so either Γ1 or ∃R Γ˘ 1 is closed under wellordered unions. In e e e either case this shows A ∈ ∆, a contradiction. If ë is a Suslin cardinal, the e We have A is a Ξ union of ë-Suslin sets now, argument is almost identical. R ˘ and either S(ë) or ∃ S(ë) is closed under wellordered unions which gives the same contradiction. ⊣ We are now ready to classify the Suslin cardinals within the supremum of the ä 1n . Various parts of the following theorem are due to Kechris, Martin, and eMoschovakis; we refer the reader to [Mos80] or [Kec78A] for a more detailed accounting. Theorem 2.21 (Kechris, Martin, Moschovakis). Assume AD. All of the ä 1n e are regular. ä 12n+2 = (ä 12n+1 )+ , and ä 12n+1 = (ë2n+1 )+ , where ë2n+1 is a cardinal e e e e e of cofinality ù. The Suslin cardinals below supn ä 1n are exactly the ë2n+1 , ä 12n+1 . e e e Also S(ë2n+1 ) = Σ12n+1 , S(ä 12n+1 ) = Σ12n+2 . e e e e

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Proof (sketch). Let ϕ be a Π12n+1 norm on a Π12n+1 -complete set P. Thus ϕ is onto ä 12n+1 . If f : α → ä 12n+1e were cofinal andeα < ä 12n+1 , then the Coding e α would show that e Lemma applied to f and a ∆e12n+1 prewellordering of length e there is a Σ12n+1 set S ⊆ P which coded cofinally many ordinals. This would compute Pe ∈ Σ12n+1 , a contradiction (x ∈ P ⇔ ∃y ∈ S (x ≤Σ12n+1 y)). Thus, e e ä 12n+1 is regular. e Every Π1 1 1 2n+1 set, and hence every Σ2n+2 set is ä 2n+1 -Suslin from the scale e e e 1 property for Π2n+1 . From the Kunen-Martin Theorem it follows that every e relation has length < (ä 12n+1 )+ . On the other hand, CorolΣ12n+2 wellfounded e e is Π1 lary 1.10 shows that every wellorder of ä 12n+1 2n+1 in the codes, and thus e e 1 there are Π2n+1 prewellorderings of any length < (ä 12n+1 )+ . Hence ä 12n+2 = the supremume of the length of the Σ12n+2 wellfoundederelations = theesupremum e = (ä 12n+1 )+ . of the lengths of the Π12n+1 prewellorderings e e , then the Coding Lemma 1 If f : α → ä 2n+2 were cofinal with α < ä 12n+2 e 1 e 1 would give a Σ2n+2 set S of codes for Σ2n+2 wellfounded relations (say via e these together into a single Σ1 e set U ). We could put a Σ12n+2 universal 2n+2 e wellfounded relation: (x, y) ≺ (x ′ , y ′ ) iff x = x ′ ∈ S ∧ U (y, y ′ ). eThis x

contradicts the Kunen-Martin Theorem. Thus, ä 12n+2 is regular. e as an easy computation A Π12n+1 -complete set P cannot be < ä 12n+1 -Suslin e e from the Coding Lemma would then show that P ∈ Σ12n+1 . Thus, ä 12n+1 is e ä 1 -Suslin e set is a Suslin cardinal. The Coding Lemma shows that every 2n+1 e Σ12n+2 , and thus S(ä 12n+1 ) = Σ12n+2 . e e Define ë e 1 2n+1 to be the least cardinal κ such that some Π2n -complete set is e e 1 κ-Suslin. So, Σ2n+1 ⊆ S(ë2n+1 ). From the scale property for Π12n+1 it follows e e e scale all of whose norms are in ∆12n+1 . Each that every Π12n admits a (regular) e of these norms must therefore have length < ä 12n+1 , and thus soe does their e supremum. This shows ë2n+1 < ä 12n+1 . By definition, ä 12n+1 is the supremum e e the Kunen-Martin 1e of the lengths of the ∆2n+1 prewellorderings, and from e + 1 Theorem each of these has length < ë+ 2n+1 . Thus, ä 2n+1 ≤ ë 2n+1 , and hence e e e + 1 ä 2n+1 = ë2n+1 . e Finally, e we show that cf(ë 1 2n+1 ) = ù. If cf(ë 2n+1 ) > ù, then every Σ2n+1 e e e set would be a ë2n+1 union of sets each of which was < ë2n+1 -Suslin. By e , this would show that every Σ1 e definition of ë2n+1 2n+1 set was a ë 2n+1 union of e ⊣ ∆12n+1 , a contradiction. Σ12n sets, and ehence by Theorem 2.17 would be e e e The previous theorem can be generalized to obtain a classification theorem for the Suslin cardinals from AD. We refer the reader to [Jac07A] for a precise statement and proof. Assuming AD, the values of the projective ordinals ä 1n have been determined. e of ä 1 through ä 1 were The answer is stated in the next theorem (the values 1 4 e e

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known earlier; see [Kec78A] for details and history). The proof uses a theorem of Martin (see Theorem 4.23 below) and a certain algebra of “descriptions.” We refer the reader to [Jac07A] for an introduction to this theory and [Jac88], [Jac99] for more detailed proofs. Theorem 2.22 (Jackson). Assume AD. Then ä 11 = ù1 , ä 12 = ù2 , and for e and w(m+1) = e w(1) = ù n ≥ 1, ä 12n+1 = ùw(2n−1)+1 , ä 12n+2 = (ä 12n+1 )+ where e e w(m) e ù (ordinal exponentiation). §3. Partition Properties. Partition properties play an important role in determinacy theory. Establishing the strong partition relation on the odd projective ordinals ä 12n+1 is a critical step in the inductive analysis of the projective e sets (and similarly for a ways beyond). The fact that from AD there are arbitrarily large below Θ cardinals with the strong partition property has useful consequences, for example, it gives a proof that every Suslin, co-Suslin ordinal game is determined. Kechris and Woodin [KW83] have shown, working over the base theory ZF + DC + V=L(R), the equivalence of AD with the existence of arbitrarily large below Θ cardinals with the strong partition property. We first recall the basic definitions. We let S α (for S ⊆ Ord) denote the ˝ set of increasing functions from α to S. Recall the Erdos-Rado partition notation: we write κ → (ë)ñì to mean that for every partition F : κ ñ → ì of the increasing functions from ñ to κ into ì pieces, there is a set A ⊆ κ of size ë which is homogeneous for the partition, that is, there is an α < ì such that for all f ∈ Añ , F (f) = α. We call ñ the exponent on the partition relation. If we omit the subscript, we mean ì = 2. Assuming choice, no cardinal can satisfy a partition relation with infinite exponent (see [Kan94, Proposition 7.1]), however assuming AD very strong partition relations are possible. Since even κ → (κ)2 implies (by an easy argument) that κ is regular, we henceforth assume that κ denotes a regular cardinal. Definition 3.1. We say κ has the weak partition property if for all ë < κ we have κ → (κ)ë . We say κ has the strong partition property if κ → (κ)κ . For proving partition relations from AD and also in deriving consequences from them, it is more more convenient to adopt an alternative form of the partition relations. The alternate form allows us to get homogeneous sets which are c.u.b. in κ, but we must fix the “type” of the function f : ë → κ. For (everywhere) discontinuous functions (i.e., for all limit α, f(α) > supâ<α f(â)) this amounts to specifying the uniform cofinality of f, which we make precise in the following definition. Definition 3.2. Let g : ë → κ. We say f : ë → κ has uniform cofinality g if there is a function f ′ with domain {(α, â) : â < g(α)} which is strictly

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increasing in the second argument such that for all α < κ we have f(α) = supâ
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Definition 3.4. A (ë, κ) coding map is a function ϕ : ù ù → ℘(ë × κ) such that for every function f : ë → κ, there is an x ∈ ù ù with ϕ(x) = f. The intention is to code functions f : ë → κ by reals, but in general ϕ(x) is not required to be a function or to have domain ë. If ϕ is a (ë, κ) coding map, we define for each α < ë and â < κ the sets Rα,â and Rα . We define x ∈ Rα,â ⇔ ϕ(x)(α, â) ∧ ∀â ′ < κ (ϕ(x)(α, â ′ ) ⇒ â ′ = â). Also, x ∈ Rα ⇔ ∃â < κ (x ∈ Rα,â ). Definition 3.5. Let κ be a regular cardinal, ë ≤ κ. We say κ is ë-reasonable if there is a non-selfdual pointclass Γ closed under ∃R , and a (ë, κ) coding e map ϕ satisfying (where ∆ = Γ ∩ Γ˘ ): e e e 1. ∀α < ë ∀â < κ Rα,â ∈ ∆. R 2. Suppose α < ë, A ∈ ∃e ∆, and A ⊆ Rα . Then ∃â0 < κ ∀x ∈ A ∃â < e â0 Rα,â (x). We say κ is reasonable if it is κ-reasonable. If there is a unique â < κ such that ϕ(x)(α, â), then we write ϕ(x)(α) for this â.

Theorem 3.6 (Martin). Assume AD. If κ is ù·ë-reasonable, then κ → (κ)ë . Proof. We will show below that ∆ is closed under < κ unions and intersece applications this will also follow from tions; we assume this for now. In most the argument showing reasonableness. We refer below to the sets Rα , Rα,â of Definition 3.5. Fix a partition P : κ ë → {0, 1}. Play the integer game where player I plays out x ∈ ù ù, player II plays out y ∈ ù ù. If there is a least ordinal α < ù · ë such that x ∈ / Rα or y ∈ / Rα , then player II wins provided x ∈ / Rα . Otherwise, let fx , fy : ù·ë → κ be the functions they determine (e.g., fx (α) = ϕ(x)(α)). Define in this case fx,y : ë → κ by fx,y (α) =

sup

max(fx (α ′ ), fy (α ′ )).

α ′ <ù·(α+1)

Player II then wins iff P(fx,y ) = 1. Assume without loss of generality that player II has a winning strategy ô. For α < ù · ë and â < κ, define x ∈ Sα,â ⇔ ∀α ′ ≤ α ∃â ′ ≤ â (x ∈ Rα ′ ,â ′ ). Thus, Sα,â ∈ ∆ by (1) of Definition 3.5 and the closure of ∆ under < κ unions e e ] ∈ ∃R ∆. Now, and intersections. Hence, for all α < ù · ë and â < κ, ô[S α,â e ô[Sα,â ] ⊆ Rα as ô is winning for player II. Thus, è(α, â) := sup{ϕ(x)(α) : x ∈ ô[Sα,â ]} < κ, from (2) of Definition 3.5. Let C ⊆ κ be the set of points closed under è, and C ′ ⊆ C the set of limit points of C . Suppose F : ë → C ′ is of the correct type; we show that P(F ) = 1. Let x be such that ϕ(x) determines a function fx : ù · ë → C such that

INTRODUCTION TO PART II

F (â) =

sup

293

fx (â ′ ). We may assume fx (â) ≥ â for all â. Let y = ô(x).

â ′ <ù·(â+1)

Since for all â < ù · ë we have x ∈ Sâ,fx (â) , it follows from the the definition of C that ϕ(y) determines a function fy : ù · ë → κ such that fy (â) ≤ NC (fx (â)) ≤ fx (â + 1) for all â, where NC (ã) denotes the least element of C greater than ã. Thus, F = fx,y , so P(F ) = 1. We sketch the argument that ∆ is closed underS< κ unions. Suppose not, e and let ä < κ be least such that some union A = Aα is not in ∆. Note that e α<ä S R0 = R0,ã is a κ union of ∆ sets, and R0 ∈ / ∃R ∆. Suppose first PWO(Γ). e e e ã<κ Then Γ is closed under well-ordered unions by Theorem 2.16. Thus A ∈ Γ, S e and inefact by Wadge’s Lemma Γ = ä ∆ (the pointclass of ä-unions of sets S e e in ∆). Also, R0 ∈ Γ and so R0 = Sα for some Sα ∈ ∆. Since κ is regular, e e e α<ä one of the Sα ⊆ R0 must be “unbounded” in κ, a contradiction to ù · ëreasonableness. So assume PWO(Γ˘ ), and thus from periodicity PWO(Γ1 ), e e where Γ1 = ∃R Γ˘ . From Theorem 2.16, S Γ1 is closed under well-ordered e e e unions, and so R0 ∈ Γ1 . We cannot have ä ∆ = Γ, as otherwise Martin’s S e e e argument (Theorem 2.17) shows PWO(Γ). It follows that ä ∆ ⊇ Γ˘ , and so S R S R e SeS , with e α ä ∃ ∆). Thus, R0 = ä ∃ ∆ ⊇ Γ1 (and hence actually Γ1 = e e e e α<ä each Sα ∈ ∃R ∆. As before, this contradicts reasonableness. ⊣ e Perhaps the most fundamental partition result from AD is the strong partition relation on ù1 (recall ä 11 = ù1 ), a result due to Martin which we record e next. Theorem 3.7 (Martin). Assume AD. Then ù1 → (ù1 )ù1 .

First note that the weak partition relation on ù1 follows fairly immediately from Theorem 3.6. For suppose ë < ù1 . Taking a bijection ð : ë → ù, we define the (ë, ù1 ) coding map ϕ by ϕ(x)(α, â) iff (x)ð(α) ∈ WO and |(x)ð(α) | = â. It is straightforward to check that this coding is reasonable, and the weak partition relation on ù1 follows. An immediate consequence (considering partitions of functions f ∈ (ù1 )ë , with ë = 1) is that the c.u.b. filter on ù1 is a countably additive ultrafilter (i.e., a measure) on ù1 . Another easy partition argument shows that this measure is normal (note one cannot directly appeal to Fodor’s Lemma as this requires choice). The n-fold product of this normal measure is the measure induced from the weak partition property of ù1 and functions f : n → ù1 . We let W11 denote the normal measure on ù1 , and W1n the n-fold product. To get the strong partition relation on ù1 requires a coding of the functions from ù1 to ù1 . A first attempt might be to use the coding of functions (viewed as subsets of (ù1 )2 ) given by the Coding Lemma, Theorem 1.8. This, however, is not sufficient to satisfy the conditions of Definition 3.5. Martin’s original

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proof (see [Kec78A]) used the fact that from AD every subset of ù1 lies in L[x] for some real x, and a c.u.b. set of (Silver) indiscernibles exists for each L[x]. Since Martin’s proof, several other proofs of the strong partition relation on ù1 have been found. Kechris [Kec77] gives a proof using generic codes for countable ordinals. Using their theory of generic codes for uncountable 1 ordinals, Kechris and Woodin [KW07] have shown that ä 12n+1 → (ä 12n+1 )eä 2n−1 e (whichebuilds on for all n ≥ 1. In [Jac90] a more combinatorial proof is given some of the ideas in Kunen’s proof of the weak partition relation on ä 13 ). This proof, which involves analyzing the measures on ù1 , is currently the eonly one known to generalize to the higher odd projective ordinals. Finally, in [JM04] a variation of the latter proof is given which avoids the complete analysis of measures. Whether this argument can be generalized to higher cardinals is not known. The inductive analysis of the projective hierarchy does allow the analysis of measures on ù1 to be generalized to the higher odd projective ordinals, and thus to get a coding a coding map for the functions f : ä 12n+1 → ä 12n+1 which e e [Jac99]): is reasonable (using Γ = Σ12n+1 ). Thus we have (cf. [Jac88], e e 1 Theorem 3.8 (Jackson). Assume AD. For all n, ä 12n+1 → (ä 12n+1 )eä 2n+1 . e e This analysis extends a ways beyond the projective hierarchy. For example if Σ1α , α < ù1 enumerates the first ù1 non-self-dual pointclasses closed under ∃Re, and if ä 1α denotes the supremum of the lengths of the ∆1α prewellordere all odd α (where limit ordinals are considerede even as usual), ings, then for ä 1α has the strong partition property. The analysis can be propagated fure ther (and the analogs of the odd ä 1α continue to have the strong partition e first inaccessible cardinal the induction property), but somewhere around the breaks down. In [Jac91] results are shown which imply that an inductive “from below” analysis of the measures cannot succeed through κ R , the ordinal of the inductive sets (the Wadge rank o(Γ) of the smallest non-selfdual e Thus, the following remains pointclass Γ closed under real quantification). e open.

Conjecture 3.9 (AD + DC). Let Γ be non-self-dual, closed under ∀R , counte able unions and intersections, and PWO(Γ ). Let ä be the supremum of the e lengths of the ∆ prewellorderings. Then ä → (ä)ä . e In fact, this conjecture is open in general even assuming Γ has the scale e property, in which case ä is a regular Suslin cardinal. By a result of Martin and Paris (see [Kec78A]), all of the even projective ordinals, ä 12n have the weak partition property, but not the strong. There are e cardinals below the projective ordinals. other regular From the inductive analysis, there are 2n+1 − 1 regular cardinals strictly between ä 12n+1 and ä 12n+3 . The smallest of these is ä 12n+2 = (ä 12n+1 )+ . The e e e e

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regular cardinals strictly between ä 12n+1 and ä 12n+3 are precisely the ultrapowers of ä 12n+1 by the normal measureseon ä 12n+1 ,ewhich in turn correspond to the e regular cardinals below ä 12n+1 . For eexample, between ä 13 and ä 15 they are e e ä 14 = ℵù+2 , ℵù·2+1 , and ℵe ù ù +1 . All of these regular cardinals satisfy the same 1 e infinite exponent partition relations, that is, they satisfy κ → (κ)eä 2n+1 but 1 κ 9 (κ)eä 2n+2 (see [Jac]). Although the detailed inductive analysis mentioned above breaks down fairly early in the L(R) hierarchy, it is still possible to obtain partition results higher up. The Kechris-Woodin generic coding arguments mentioned earlier [KW07] show that if Γ is non-self-dual, closed under ∀R , countable unions and intersections, andePWO(Γ), and ä is the supremum of the lengths of the ∆ prewellorderings, and if ë e< ä is a Suslin cardinal with ∀R Së ⊆ ∆, then e e ä → (ä)ë . Also, if ä corresponds to a “sufficiently closed” pointclass, then ä will have the strong partition property. Results of this form are proved in [KKMW81]. We mention a result along these lines. Theorem 3.10 (AD). Let Γ be non-self-dual, closed under ∀R , PWO(Γ), e ∃ ∆ ⊆ ∆, and assume also eΓ is closed under ∨. Then ä(∆) has the strong e e e e partition property. R

The proof of Theorem 3.10 uses the “Uniform Coding Lemma,” a version of the Coding Lemma which says roughly that given a prewellordering  and a relation R ⊆ dom() × ù ù, there is a choice set A ⊆ R (in the sense of Theorem 1.8) such that the initial segments Aα of A are uniformly Σ11 over the initial segments α of . We refer the reader to [KKMW81] for aeprecise statement and proof. We make some observations which put Theorem 3.10 in context. Following Steel [Ste81A], let C = {o(∆) : ∆ is self-dual ∧ ∃R ∆ ⊆ ∆}. e e e e So, C is c.u.b. in Θ and consists of places in the Wadge hierarchy where we are at the base of a projective-like hierarchy. By an earlier remark, it doesn’t matter in the definition of C if we add that ∆ is closed under finite unions and intersections. Also, an argument using theeCoding Lemma shows that every κ ∈ C is a cardinal. Steel shows in [Ste81A] that if κ ∈ C and cf(κ) > ù, then there is a non-selfdual pointclass Γ closed under ∀R with PWO(Γ) such that o(∆) = κ e e e (where ∆ = Γ ∩ Γ˘ ). This Steel pointclass satisfies all of the hypotheses e e e of Theorem 3.10 except perhaps the assumption that Γ is closed under ∨. e disjunction with [Ste81A] also shows that a Steel pointclass Γ is closed under e ∆ sets iff Γ is closed under disjunction iff Γ is closed under countable unions. e may thus e rephrase Theorem 3.10 as follows. e We

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Corollary 3.11. If κ ∈ C is regular and the corresponding Steel pointclass Γ (i.e., κ = o(∆) where ∆ = Γ ∩ Γ˘ ) is closed under ∨, then κ has the strong e e e e e partition property.

From [Ste81A, Theorem 2.1] it follows that if κ is regular and a limit of Suslin cardinals, then the corresponding Steel pointclass Γ is closed under ∨. Also, κ ∈ C and κ = ä(∆). Thus we have the following. e e Corollary 3.12. If κ is a regular limit of Suslin cardinals, then κ has the strong partition property. Steel conjectures in [Ste81A] that if κ ∈ C is regular then the corresponding Steel pointclass Γ is closed under disjunctions. We state this explicitly as: e Conjecture 3.13. For κ ∈ C , κ is regular iff κ has the strong partition property.

It is conceivable that this conjecture could hold even if some of the Steel pointclasses for regular κ are not closed under disjunction. Although Steel’s conjecture is still open, it does hold for a fairly large initial segment of κ’s. Kechris [Kec81B] showed that for A ⊆ ù ù contained within the inductive sets, A is ë-Suslin, where ë is the length of some projective over A prewellordering. Thus for κ ≤ κ R (the Wadge rank of the inductive sets) in C , with Steel pointclass Γ, every ∆ set is ë Suslin for some ë < κ. From [Ste81A] again it e under ∨. Thus, the above conjecture is true for κ ≤ κ R . follows thateΓ is closed e In fact, from Steel’s analysis of scales in L(R) [Ste83A] Kechris’ result can be extended to all κ up to the least non-trivial Σ1 gap, in the terminology of [Ste83A] (the least such gap is strictly larger than κ R ). Thus we have: Corollary 3.14. If κ ∈ C and κ is less than or equal to the least α beginning a non-trivial Σ1 gap, then κ is regular iff κ has the strong partition property. Finally, as discussed in [KKMW81], the pointclasses Γ satisfying the hye potheses of Theorem 3.10 are in abundance below Θ. We mention one more application of the strong partition property. If κ has the strong partition property we can, following Steel, introduce a generalized notion of Mahloness on κ. We consider stationary sets S ⊂ κ, and we assume for convenience that S contains only ordinals of uncountable cofinality. Recall that S is said to be thin if for every α ∈ S, S ∩ α is not stationary in α. For every stationary S, its set of thin points S ′ is also stationary. We define an equivalence relation ∼ on the thin stationary sets by S ∼ T iff there is a c.u.b. C ⊆ κ such that C ∩ S = C ∩ T . We define S ≺ T iff there is a c.u.b. C such that for all α ∈ C ∩ T , S is stationary in α. Fact 3.15 (Steel). If κ → (κ)κ , then ≺ is a well-order of the ∼-equivalence classes of the thin stationary subsets of κ.

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Proof (sketch). Given thin stationary sets S, T , consider the partition P of functions f : κ → κ of the correct type according to whether αf (S) < αf (T ), αf (S) = αf (T ), or αf (S) > αf (T ), where αf (S) is the least limit point of f in S. It is easy to check that a homogeneous C would give respectively S ≺ T , S ∼ T , or T ≺ S. ⊣ It is shown in [Jac91] that for Γ non-selfdual and closed under real quane Mahlo order greatly exceeding κ. tification, κ = o(Γ) has a generalized e §4. Homogeneous Trees. The notion of a homogeneous tree arose independently in the work of Kunen and Martin. Kechris and Martin independently (cf. [Kec81A]) then formulated the general notion of a homogeneous tree. This concept, and the related notion of a weakly homogeneous tree, plays an important role in some of the proofs of determinacy from large cardinals as well as in various theorems in the AD world. Thus, the notion is important in both the ZFC and ZF + AD contexts. If f : X → Y is a map from the set X to the set Y , and ì is a measure (or filter, ultrafilter, etc.) on X , then we define f(ì) to be the measure on Y n given by f(ì)(A) = ì(f −1 [A]). For X a set and m ≤ n, let ðm : Xn → Xm n denote the projection map ðm (s) = s↾m. Definition 4.1. We say a tree T on ù ×X is homogeneous if there is a family of measures hìs : s ∈ <ù ùi satisfying: 1. Each ìs is a measure on Ts , that is, ìs (Ts ) = 1. n 2. If s ≤ t (that is, t extends s), then ðm (ìt ) = ìs where n = lh(t), m = lh(s). 3. For every x ∈ ù ù, if Tx is illfounded then for any sequence A1 , A2 , . . . with ìx↾n (An ) = 1, there is an f ∈ ù X such that for all n, (x↾n, f↾n) ∈ An . We say T is ä-homogeneous if the measures ìs may be taken to be ä-complete. We say T is <ä-homogeneous if T is ë homogeneous for all ë < ä. We say A ⊆ ù ù is homogeneously Suslin (or ä-homogeneously Suslin, etc.) if A = p[T ] for some homogeneous T . The last property (3) says that T appears “homogeneous,” at least as far as restricting to measure one sets goes. This property is also equivalent to saying that the direct limit Mx of the ultrapowers by the measures ìx↾n is wellfounded (using the natural embeddings from the ìs ultrapower to the ìt ultrapower when s ≤ t). To see this, first suppose that (3) holds for the ìx↾n . If Mx were illfounded, then there would be functions gn : Tx↾n → Ord and measure one sets An with respect to ìn such that ∀α ~ ∈ An+1 gn+1 (α) ~ < gn (α↾n). ~ Let f ∈ ù X be as in (3) for these An . Then g1 (f↾1) > g2 (f↾2) > . . . , a contradiction. Suppose next that the direct limit Mx is wellfounded. If (3) failed, then then let A1 , A2 , . . . be measure one sets with respect to the

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ìx↾n with no f ∈ ù X as in (3). Let V be the tree of all α ~ ∈ ù X such that ∀n ≤ lh(α) ~ α ~ ↾n ∈ An . So, V is wellfounded. Note that the set of α ~ ∈ X n such that α ~ ∈ V has ìx↾n measure one by (2). For such α ~ define gn (α) ~ = |α|T , the rank of α ~ in T . We then have [g1 ]ìx↾1 > [g2 ]ìx↾2 > . . . , a contradiction to the wellfoundedness of Mx . It is easy to check that if A is homogeneously Suslin and B ≤W A, then B is also homogeneously Suslin [if the function reducing B to A is Lipschitz, this is clear, if it is just continuous, one must pad using dummy principal measures on the integers; details are left to the reader]. One use of homogeneous trees is that they provide a means of giving determinacy. The basic method used is “integrating out” one of the players ordinal moves. This method goes back to Martin’s proof of Π11 determinacy from a e measurable cardinal. Lemma 4.2 (Martin). Suppose A = p[T ] where T is a homogeneous tree on ù × ë for some ë ∈ Ord. Then A is determined. Proof. Consider the auxiliary game G ∗ played as follows: I II

(x(0), α0 )

(x(2), α1 , α2 ) x(1)

(x(4), α3 , α4 ) . . . x(3)

x(5) . . .

Here x(i) ∈ ù and αi ∈ Ord. Player I must play so that for all i, ((x(0), . . . , x(2i)), (α0 , . . . , α2i )) ∈ T otherwise player II wins. If player I meets this requirement, then player I wins. G ∗ is a closed game for I, so is determined. If player I wins G ∗ , then clearly player I wins the game G for the set A by simply ignoring the extra ordinal moves that the strategy gives. So suppose ô ∗ is a winning strategy for player II in G ∗ . We define a strategy ô for player II in G as follows. Given a position p = (x(0), . . . , x(2i)) with player II to move, let ô(p) be the integer a such that for ìx↾(2i+1) almost all (α0 , . . . , α2i ), ô ∗ will respond with a when player I plays (x(0), x(2), . . . , x(2i)) and (α0 , . . . , α2i ). This is well-defined by the countable completeness of the measure ìx↾(2i+1) . Suppose x ∈ ù ù is a run according to ô. We must show that x ∈ / A. Suppose x ∈ A = p[T ]. For each i, let A2i+1 be a ìx↾(2i+1) measure one set such that if player I plays (x(0), . . . , x(2i)) and (α0 , . . . , α2i ) in G ∗ , then ô ∗ responds with x(2i + 1). By homogeneity, let f ∈ ù Ord be such that f↾(2i + 1) ∈ A2i+1 for all i. Then x together with f gives an infinite run following ô ∗ , a contradiction to ô ∗ being a winning strategy for player II. ⊣ Variations of the above argument are also frequently used. For example, we have the following. Lemma 4.3. Let G be the ordinal game where player I plays integers x(2i) and ordinals α(i), and player II plays integers x(2i +1), and ordinals â(i) < ä producing x ∈ ù ù, α ~ ∈ ù Ord, and â~ ∈ ù ä:

INTRODUCTION TO PART II

I (x(0), α0 ) II

(x(2), α1 ) (x(1), â0 )

(x(3), â2 )

299 (x(4), α2 ) . . . (x(5), â3 ) . . .

Assume the payoff set for player I is of the form ~ ∧ A(x) F (x, α, ~ â) where F ⊆ ù ù × ù Ord × ù ä is closed and A = p[T ] where T is homogeneous with ä + -complete measures. Then G is determined. Proof. Let U be a tree on ù × Ord ×ä such that F = [U ]. Let G ∗ be the auxiliary closed game for player I where player I also makes moves ã0 , ~ (ã1 , ã2 ), (ã3 , ã4 ), . . . and player I must play so that (x↾i, α ~ ↾i, â↾i) ∈ U and ∗ (x↾i, ~ã↾i) ∈ T for all i. Again, if player I wins G then player I wins G. If ô ∗ is a winning strategy for player II in G ∗ , we again integrate ô ∗ to define a strategy ô for player II in G. This is well-defined using the ä + -completeness of the measures ìx↾(2i+1) . The proof that ô is winning for player II in G is exactly as before. ⊣ For the purposes of propagating Suslin representations, and for other purposes, a weaker notion, that of a weakly homogeneous tree is useful. We will have that the weakly homogeneously Suslin sets are existential quantifications of the homogeneously Suslin sets. There are several different definitions of a tree being weakly homogeneous. These turn out to be almost equivalent assuming AC (see Lemma 4.7 below), but not necessarily just in ZF. However, assuming AD essentially every tree turns out to be weakly homogeneous by the stronger definition anyway (see Theorem 4.11 below). The first version we give is the stronger form of the definition. The following definition, which we take as our official one, essentially says that a tree T on ù×X is weakly homogeneous if there is a homogeneous tree T ′ on (ù×ù)×X which is isomorphic to T via a tree isomorphism which is the identity on the first coordinate. Thus, in weak homogeneity we have the option of splitting the tree at each node into countably many pieces, and using a different measure for each. In the actual definition we allow the option of a node in T ′ to split only finitely often on the second coordinate. Definition 4.4. A tree T on ù × X is weakly homogeneous if there are measures ìs,t on lh(s) X and non-empty sets As,t ⊆ lh(s) X defined for all s ∈ <ù ù and some t ∈ <ù ù with lh(s) = lh(t) and satisfying: 1. The set of (s, t) for which ìs,t , As,t are defined is a tree. Furthermore, ìs,t is a measure on Ts , and As,t ⊆ Ts . 2. If (s ′ , t ′ ) extends (s, t) then ìs ′ ,t ′ projects to ìs,t and As ′ ,t ′ ⊆ As,t × lh(s ′ )−lh(s) X. 3. For every (s, t) for which ìs,t , As,t are defined, and every s ′ extending s with lh(s ′ ) = lh(s) + 1, the sets As ′ ,t ′ partition Ts ′ ∩ (As,t × X ).

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4. For all x, y ∈ ù ù such that for all n ìx↾n,y↾n is defined, if there is an f ∈ ù X such that ∀n f↾n ∈ Ax↾n,y↾n , then the direct limit Mx,y of the ultrapowers by the ìx↾n,y↾n is wellfounded. Note that by (3) it follows that if Tx is illfounded, say (x, f) ∈ [T ], then there is a y ∈ ù ù such that for all n f↾n ∈ Ax↾n,y↾n . Then by (4) the direct limit of the ìx↾n,y↾n ultrapowers is wellfounded. This shows that the above definition implies the following, which is also frequently taken to be the definition. In this version we don’t split the tree, just the measures. Definition 4.5. A tree T on ù × X is weakly homogeneous if there is a family of measures ìs,t , defined for all s ∈ <ù ù and some t ∈ <ù ù with s, t ∈ <ù ù with lh(s) = lh(t) satisfying: 1. The set of (s, t) for which ìs,t is defined is a tree. Each ìs,t is a measure on lh(s) X and ìs,t (Ts ) = 1. 2. If (s ′ , t ′ ) extends (s, t) and ìs ′ ,t ′ is defined, then ìs ′ ,t ′ projects to ìs,t . 3. For every x ∈ ù ù, if Tx is illfounded then there is a y ∈ ù ù such that the measures ìx↾n,y↾n are all defined and the direct limit Mx,y of the ultrapowers by the measures ìx↾n,y↾n is wellfounded. Again, the conclusion of (3) is equivalent to saying that there is a y ∈ ù ù such that whenever Bx↾n,y↾n ⊆ Tx↾n have ìx↾n,y↾n measure one, then there is an f ∈ ù X such that ∀n f↾n ∈ Bx↾n,y↾n . We say A ⊆ ù ù is weakly homogeneously Suslin if A = p[T ] for some weakly homogeneous tree T . The following is essentially immediate from the strong form of the definition of weakly homogeneous tree. Lemma 4.6. A ⊆ ù ù is weakly homogeneously Suslin iff we can write A(x) ⇔ ∃y B(x, y) where B is homogeneously Suslin. In fact, for every ordinal α, A is the projection of a weakly homogeneous tree on ù × α iff we can write A(x) ⇔ ∃y B(x, y) where B is the projection of a homogeneous tree on (ù × ù) × α. The conclusion of Lemma 4.6 is still true using the weaker definition of weakly homogeneous tree, Definition 4.5, provided we assume ZFC according to the following lemma, due to Woodin. Lemma 4.7 (ZFC). Let T be a tree on ù × X which is weakly homogeneous according to the weaker Definition 4.5. Then there is a subtree T ′ ⊆ T which is weakly homogeneous according to the stronger Definition 4.4 and such that p[T ′ ] = p[T ]. Furthermore T and T ′ use the same set of measures in their homogeneity systems. Proof. Let T be a tree on ù × X satisfying Definition 4.5. Recall that the completeness of any measure is a measurable cardinal, and in ZFC these must be inaccessibles. Thus the measures ìs,t (as in Definition 4.5) are all (2ù )+ complete. We need to define the sets As,t so that the ìs,t together with the As,t

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satisfy Definition 4.4 for a subtree T ′ . Fix s, t ∈ <ù ù with lh(s) = lh(t). For every x, y ∈ ù ù extending s, t, if the direct limit Mx,y of the ultrapowers by the measures ìx↾n,y↾n is illfounded, let Ax,y ⊆ Tx↾n be ìx↾n,y↾n measure one n sets such that for no f ∈ ù X do we have ∀n f↾n ∈ Ax,y n (we use AC here as well). Let \ x,y As,t = Alh(s) , x,y

the intersection taken over all x, y extending s, t with Mx,y illfounded. So, ìs,t (As,t ) = 1. We may further shrink the As,t (keeping them measure one) so that: (1) if t1 , t2 are of the same length and incompatible then As,t1 ∩ As,t2 = ∅ and (2) if (s ′ , t ′ ) immediately extends (s, t), then As ′ ,t ′ ⊆ As,t × X . [By re-indexing the measures ìs,t we may assume that whenever t1 ⊥ t2 then ìs,t1 6= ìs,t2 . For each such pair there is a B = Bs,t1 ,t2 with ìs,t1 (B) = 1, ìs,t2 (B) = 0. We intersect As,t1 with all such Bs,t1 ,t2 . This ensures (1). Then by induction on lh(s) = lh(t), intersect As,t with As,¯ t¯ × κ, where (s, t) is an ¯ This ensures (2).] Let T ′ ⊆ T consist of those immediate extension of (s, ¯ t). (s, α) ~ such that for some t ∈ <ù ù with lh(s) = lh(t) we have α ~ ∈ As,t . To see p[T ′ ] = p[T ], let x ∈ p[T ]. From Definition 4.5, let y ∈ ù ù be such that the direct limit Mx,y by the ìx↾n,y↾n measures is wellfounded. Since the Ax↾n,y↾n have ìx↾n,y↾n measure one, it follows that there is an f ∈ ù X such that ∀n f↾n ∈ Ax↾n,y↾n . Thus x ∈ p[T ′ ]. To see that T ′ is weakly homogeneous according to Definition 4.4, fix x, y ∈ ù ù such that for all n ìx↾n,y↾n is defined, and suppose f ∈ ù X with ∀n f↾n ∈ Ax↾n,y↾n . We must show that the direct limit Mx,y of the ìx↾n,y↾n ultrapowers is wellfounded. If not, then the measure one sets Ax,y n are defined x,y ù and Ax↾n,y↾n ⊆ Ax,y X n for all n. By definition of the An there is no g ∈ x,y such that ∀n g↾n ∈ An . But this contradicts the assumption on f. ⊣ In fact, in ZFC we can simplify the definition of weakly homogeneous tree a bit further in the following, due also to Woodin. Lemma 4.8 (ZFC). A tree T on ù × X is weakly homogeneous iff there is a countable set M of measures hìi : i ∈ ùi, with each measure on j X for some j, such that whenever Tx is illfounded then there is a sequence ìn1 , ìn2 , . . . from M with each ìni (Tx↾i ) = 1, each ìni+1 projects to ìni , and with the direct limit of the ultrapowers by the ìni wellfounded. Proof. It is clear that Definition 4.5 implies the statement from the lemma. To see the other direction, let M be the countable family of measures for the tree T as in the statement of the lemma. Define the measures ìs,t ∈ M inductively as follows. If ìs,t is defined, ìs,t (Ts ) = 1, and s ′ is an immediate extension of s, then we let ìs ′ ,t ′ enumerate all the measures ì ∈ M which are ′ measures on lh(s ) X which project to ìs,t and such that ì(Ts ′ ) = 1. Suppose now that Tx is illfounded. Let ìn1 , ìn2 , . . . be as in the statement of the lemma.

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Then there is a y ∈ ù ù such that for all i, ìni = ìx↾n,y↾n . By assumption the direct limit of the ultrapowers by the measures ìx↾n,y↾n is defined and wellfounded. ⊣ We next show how weak homogeneity propagates Suslin representations. This is the construction of the Martin-Solovay tree, a useful construction in many contexts. Given a weakly homogeneous tree T on ù × κ, the MartinSolovay construction produces a tree T ′ on ù × ë, for some ë ∈ Ord, with p[T ′ ] = ù ù − p[T ]. It suffices to assume T is weakly homogeneous according to the weaker Definition 4.5. Definition 4.9 (Martin-Solovay Tree). Let T be a tree on ù × κ which is weakly homogeneous according to Definition 4.5 with measures ìs,t , each of which we may assume is a measure on lh(s) κ. We define the Martin-Solovay tree T ′ as follows. Let (si , ti ) be an enumeration of all (s, t) ∈ ( <ù ù)2 with lh(s) = lh(t) such that any proper extension of any (si , ti ) is enumerated at a later stage. Define (s, α ~ ) ∈ T ′ iff there is an f : Ts → κ + which is orderpreserving with respect to the Kleene-Brouwer ordering <sBK on Ts such that for all i < lh(s), αi = [f i ]ìsi ,ti . Here f i is defined by f i = f ↾ {~ã : (si , ~ã) ∈ T } if si is an initial segment of s and ìsi ,ti is defined; otherwise αi = 0. Variations on this definition are frequently used. For example, we can consider functions f : Ts → ä for ä other than κ + , and we can also impose restrictions on the “type” of the functions f used. We might, for example, require the functions to be of the correct type (Definition 3.2). We let ms(T, ä) denote the Martin-Solovay tree constructed from T using function into ä of the correct type. We write ms(T, ä, ì ~ ) if we want to display the homogeneity measures being used. Theorem 4.10 (ZF + DC). Let T be a weakly homogeneous tree (according to Definition 4.5), and let T ′ be the corresponding Martin-Solovay tree. Then p[T ′ ] = ù ù − p[T ]. Proof. First suppose x ∈ ù ù is such that x ∈ / p[T ], that is, Tx is wellfounded. Thus the Kleene-Brouwer <xBK on Tx is a wellordering of length < κ + . Let f : <xBK → κ + be order-preserving. For each i such that si is an initial segment of x and ìsi ,ti is defined, let again f i = f ↾ {~ã : (si , ~ã) ∈ T }. For such i let αi = [f i ]ìsi ,ti , and for other i let αi = 0. Then by definition of T ′ , (x↾n, α↾n) ~ ∈ T ′ for all n, so x ∈ p[T ′ ]. Suppose next that x ∈ p[T ]. We must show that Tx′ is wellfounded. Suppose ′ Tx were illfounded, and let (x, α) ~ ∈ [T ′ ]. For each j ∈ ù, let fj : Tx↾j → κ + be order-preserving with respect to <x↾j BK such that for all i < j such that si is an initial segment of x and ìsi ,ti is defined we have αi = [fji ]ìsi ,ti . For

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each such i let Bi have ìsi ,ti measure one such that for all j1 , j2 > i and all ~ã ∈ Bi , fji 1 (~ã) = fji 2 (~ã). Since Tx is illfounded, let y ∈ ù ù be such that all ìx↾n,y↾n are defined and the direct limit Mx,y of these ultrapowers is wellfounded. Consider Bi1 , Bi2 , . . . , where (sik , tik ) = (x↾k, y↾k). Since Mx,y is wellfounded, there is a g ∈ ù κ such that ∀n g↾n ∈ Bin . From the definition of the Bi we now have that f i0 (hg(0)i) > f i1 (hg(0), g(1)i) > f i2 (hg(0), g(1), g(2)i) > . . . , a contradiction. Here f ik (~ã) denotes the common value of fjik (~ã) for j > ik , which is well-defined for ~ ã ∈ Bik . For example, f i0 (hg(0)i) = fii00+1 (hg(0)i) = i0 i1 fi1 +1 (hg(0)i) > fi1 +1 (hg(0), g(1)i). The first two equalities follow from g(0) ∈ Bi0 , and the last from fi1 +1 being order-preserving. ⊣ Martin and Woodin [MW07] have shown the remarkable result that from AD every tree T on ù × κ for κ less than the supremum of the Suslin cardinals is weakly homogeneous, even according to the stronger Definition 4.4. We record this in the following theorem. Theorem 4.11 (Martin, Woodin). Assume AD. Let κ be less than the supremum of the Suslin cardinals. Then every tree on ù × κ is weakly homogeneous in the strong sense of Definition 4.4. Martin originally proved this theorem from ADR [MarD], and then Woodin modified and extended the argument to work from AD. The assumption that κ is below the largest Suslin cardinal (if there is one) is necessary if one is assuming just AD [If κ is the largest Suslin cardinal, and then S(κ) is nonselfdual by [Kec81B] and thus a tree on ù × κ projecting to a S(κ)-complete set A cannot be weakly homogeneous as otherwise by Lemma 4.10 ù ù − A would be Suslin]. ADR is equivalent over AD to “every set of reals is Suslin” by a theorem of Woodin, and so this case does not arise from ADR . Woodin has proved a “ZFC-analog” of Theorem 4.11. This theorem, or variations on it, play an important role in some of the proofs of determinacy from large cardinals. Definition 4.12. ä is a Woodin cardinal if for every f : ä → ä there is a κ < ä closed under f and an elementary embedding j : V → M with Vj(f)(κ) ⊆ M . An equivalent definition (cf. [MS89]) is that for every A ⊆ Vä there is a κ < ä which is “A-strong,” that is, for every ë < ä there is an embedding j : V → M with Vë ⊆ M and such that j(A) ∩ Vë = A ∩ Vë . Theorem 4.13 (Woodin). Let ä be a Woodin cardinal. Let T be a tree on ù × α, for α ∈ Ord. Then there is a κ < ä such that in the generic extension V[G], where G is generic for coll(ù, κ), T is < ä weakly homogeneously Suslin.

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We refer the reader to [Lar04] or [SteA] for a proof, which is vaguely similar to that of Theorem 4.11. Homogeneous trees are important for propagating Suslin representations according to Theorem 4.10. An important question is whether the resulting Martin-Solovay tree is itself homogeneous. Again, this question is important in both the AD and ZFC contexts. In the AD context, partition properties are used to establish the homogeneity of the Martin-Solovay tree. This is the content of the next theorem. Theorem 4.14. Let T be a weakly homogeneous tree on ù × κ in the (weaker) sense of Definition 4.5. Assume ä > κ and ä → (ä)κ . Then the Martin-Solovay tree ms(T, ä) is homogeneous via ä-complete measures. The following variation is also useful. Theorem 4.15. Let T be a weakly homogeneous tree on ù × κ in the sense of Definition 4.5, and assume T has the property that whenever ((a0 , . . . , an ), (α0 , . . . , αn )) ∈ T then α0 ≥ max{α1 , . . . , αn }. Assume κ → (κ)κ . Then the Martin-Solovay tree ms(T, κ) is homogeneous. Furthermore the measures may be taken to be κ + complete provided the following holds: All of the homogeneity measures ì for T give bounded subsets of κ measure 0 and there is a non-selfdual pointclass Γ closed under ∃R such that (1) The supremum of the e length of the ∃R ∆ wellfounded relations is κ and (2) There is a prewellordering e R of length κ which is ∃ ∆-bounded. e Remark. The extra pointclass hypotheses to guarantee κ + -completeness of the measures follows from κ being reasonable (according to Definition 3.5). Since the reasonableness of κ is the only known hypothesis to give the strong partition relation on κ, in practice this hypothesis is always satisfied. For assuming κ is reasonable, (1) follows by considering hR0,â : â < äi for ä < κ. Since ∆ (using Γ from reasonableness) is closed under < κ unions (this was e of Theorem 3.6), this gives a ä-length prewellordering in shownein the proof ∆. (2) follows by considering hR0,â : â < κi which gives an ∃R ∆-bounded e e does not prewellordering from the definition of reasonableness. The author + know if the κ -completeness of the measures follows from just the strong partition property at κ. The two proofs are almost identical, so we just prove the latter. Proof. Note that whenever Tx is wellfounded the rank of any node in the Kleene-Brouwer ordering on Tx is less than κ. Thus the proof of Theorem 4.10 goes through and shows that p[T ] = ù ù − p[T ′ ] where T ′ = ms(T, κ) is the Martin-Solovay tree. Recall (s, α) ~ ∈ T ′ iff there is a f : <sBK → κ of the correct type which represents α ~ in the sense that for all i < lh(s) αi = [f i ]ìsi ,ti (if si is an initial segment of s and ìsi ,ti is defined). The strong partition property of κ gives a natural

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measure ís then on Ts′ : we define A ⊆ Ts′ to have ís measure one if there is a c.u.b. C ⊆ κ such that for any f : <sBK → C of the correct type, α ~ ∈ A where αi = [f i ]ìsi ,ti . It is clear that if s ′ extends s then ís ′ projects to ís (since a subfunction of a function of correct type is also of correct type). By definition of T ′ we have ís (Ts′ ) = 1. To verify homogeneity, fix x ∈ ù ù with Tx′ illfounded, so Tx is wellfounded, and let An be measure one sets withT respect to vx↾n . Let Cn ⊆ κ be c.u.b. witnessing An has measure one. Let C = n Cn , so C is c.u.b. in κ. Since Tx is wellfounded and every node in Tx has rank < κ in the KleeneBrouwer order on Tx , there is an order-preserving map f : Tx → C which is of the correct type. If we let αi = [f i ]ìsi ,ti (if si is an initial segment of s and ìsi ,ti is defined; αi = 0 otherwise), then (x, α) ~ ∈ [T ′ ] and ∀n (x↾n, α↾n) ~ ∈ An . The + κ -completeness of the measures ís follows from the next general lemma. ⊣ Lemma 4.16. Assume κ → (κ)κ , and let ì be a measure on κ which gives every bounded subset of κ measure 0. Let í be the measure on jì (κ) defined by: í(A) = 1 iff there is a c.u.b. C ⊆ κ such that for all f : κ → C of the correct type, [f]ì ∈ A. Then assuming there is a pointclass Γ as in e Theorem 4.15, í is a κ + -complete measure. Proof. The strong partition relation on κ easily gives that í is a measure. To show κ + -completeness, suppose ä ≤ κ is least suchSthat there is a sequence hAα : α < äi with í(Aα ) = 0 for all α < ä but í( α<ä Aα ) =S1. We first show that ä < κ. Suppose ä = κ. Fix a c.u.b. C0 ⊆ κ witnessing α<κ Aα has í measure one. For f : κ → C0 of the correct type let α(f) be the least α < κ such that [f]ì ∈ Aα . Consider the partition P1 : we partition pairs (α, f) where α < κ and f : κ → κ of the correct type with f(0) > α according to whether α ≥ α(f) (more formally we are considering functions from the order type 1 ⊕ κ into κ; of course 1 ⊕ κ ∼ = κ). We claim that on the homogeneous side the stated property holds. Suppose not, and let C be homogeneous for the contrary side. We may assume C ⊆ C0 . Let f : κ → C be of the correct type, and let α ≥ α(f) with α ∈ C . It is easy to see that there is an f ′ : κ → C such that (1) f ′ (0) > α, (2) f ′ is of the correct type, and (3) f ′ (â) = f(â) for all â > â0 for some fixed â0 < κ. [Let â0 > α be a closure point of ran(f), that is, â0 is the â0 th element of ran(f). Let f ′ (â) = f(â) for â ≥ â0 , and for â < â0 define f ′ (â) = f(α + 1 + â).] Since ì concentrates on unbounded sets, [f ′ ]ì = [f]ì . Thus, α ≥ [f ′ ]ì , contradicting the homogeneity of C for the contrary side. Fix now C1 ⊆ C0 homogeneous for the stated side of P1 . Let α0 = min(C1 ). The argument of the previous paragraph S now shows that for any f : κ → C1 of the correct type, α(f) < α0 . So, í( α<α0 Aα ) = 1, a contradiction. Thus, ä < κ. It suffices therefore to show that í is κ-complete. Again suppose hAα : α < äi were a counterexample. Fix a prewellordering  of

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length κ which is ∃R ∆-bounded. Let D = fld(), and for x ∈ D let |x| denote the rank of x in . eSay ó ∈ ù ù is a code if ó codes a continuous function from ù ù to ù ù (which we also denote by ó) such that ∀x ∈ D (ó(x) ∈ D). For ó a code define fó : κ → κ by fó (α) = inf{|ó(x)| : x ∈ D ∧ |x| = α}. For ó a code let Có ⊆ κ be the c.u.b. set of points closed under fó . For any c.u.b. C ⊆ κ there is a code ó such that Có ⊆ C . To see this, play the Solovay game where player I plays x, player II plays y, and player II wins iff x ∈ D ⇒ (y ∈ D ∧ |y| > NC (|x|)), where NC (α) = the least element of C greater than α. Since Σ11 subsets of D are bounded in the prewellordering, player II has a winning estrategy ó, and we have Có ⊆ C . By assumption, there is an ∃R ∆ wellfounded relation of length ä. From the e is an ∃R ∆ set S ⊆ ù ù consisting of codes Coding Lemma it follows that there and such that for all α < ä there is a ó ∈ S esuch that Có defines a í measure one set contained in Aα . Define f : κ → κ as follows. For α < κ, let f(α) = sup{fó (α) : ó ∈ S}. To see that f(α) < κ, fix x ∈ D with |x| = α. For any ó ∈ S, fó (α) ≤ |ó(x)|. Thus, f(α) ≤ supó∈S |ó(x)|. However {ó(x) : ó ∈ S} is a continuous image of S, and thus is in ∃R ∆ (note that ∃R ∆ is closed under ∧, ∨. This follows by e under quantifiers, that is Γ is at the considering cases as to ewhether ∆ is closed e base of a projective hierarchy, or not). So by boundedness, f(α) < eκ. Let C be the c.u.b. set of points closed under f, and A the í measure one set determined by C . For every α < ä there is a ó ∈ S with Có determining a measure T one set Aó ⊆ Aα . Since f > fó everywhere, C ⊆ Có . Thus A ⊆ Aα . ⊣ So, í( α<ä Aα ) = 1, a contradiction. We note that the ultrapowers jì (κ) of Lemma 4.16 are all cardinals according to the following result of Martin. We refer the reader to [Jac07A] for a proof. Lemma 4.17. Assume κ → (κ)κ . Then for any measure ì on κ, the ultrapower jì (κ) is a cardinal. A related result is the following (see also [Jac07A] for a proof). A measure is said to be semi-normal if it gives every c.u.b. set measure one (every normal measure is semi-normal). Lemma 4.18. Assume κ → (κ)κ . Then for any semi-normal measure ì on κ, the ultrapower jì (κ) is a regular cardinal. From Theorems 4.14, 4.15 we get the following result on the projective ordinals. We assume as hypotheses two facts which end up being true, and

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in the complete analysis are proved by induction along with the following. Recall ä 12n+1 = (ë2n+1 )+ where cf(ë2n+1 ) = ù. e e e 1 Theorem 4.19. Assume ∀n ä 12n+1 → (ä 12n+1 )eä 2n+1 , and ä 12n+1 is closed under e ì on α, j (â) < e and any measure ultrapowers, that is, for any α,eâ < ä 12n+1 ì e of a homogeneous tree on ù ×ë2n+1 ä 12n+1 . Then every Π12n set is the projection e e measures. Every Π12n+1 set is the projectioneof a with (ä 12n−1 )+ -complete e e homogeneous tree on ù × ä 12n+1 with ä 12n+1 -complete measures. Furthermore, e over the measures appearing in the e ë2n+3 = supì jì (ä 12n+1 ) where ì ranges e e 1 homogeneous tree for a Π2n+1 -complete set. e Proof. If we assume inductively that every Π12n set is the projection of a e set is the projection of a homogeneous tree on ù × ë2n+1 , then every Σ12n+1 e e ù×ë weakly homogeneous tree on 2n+1 . e From Theorem 4.14 and the closure of ä 12n+1 under ultrapowers, every e tree on ù × ä 1 1 1 Π2n+1 set is the projection of a homogeneous 2n+1 with ä 2n+1 e e e complete measures. Moreover, we may easily arrange it so this tree satisfies the hypothesis of Theorem 4.15. Likewise, starting from such homogeneous trees on Π12n+1 sets, we get that every Σ12n+2 set is the projection of a weakly e Theorem 4.15 it follows that every e homogeneous tree on ù × ä 12n+1 and from e Π12n+2 set is the projection of a homogeneous tree on ë := supì jì (ä 12n+1 ), e e where the supremum ranges over the measures ì in the weakly homogeneous 1 tree for a Σ2n+2 set (equivalently the measures in a homogeneous tree for a Π12n+1 set).eFurthermore, the measures in this homogeneous tree for the Π12n+2 e are e will be (ä 1 )+ -complete (the pointclass hypotheses of Theorem 4.15 set 2n+1 e 1 1 satisfied by Γ = Σ2n+1 ). In particular, every Σ2n+3 set is ë-Suslin. This shows e e other hand, ë is a countable ë ≥ ë2n+3 . eOn the supremum of cardinals (by e 4.17) each of which is less than ä 1 (by closure under ultrapowers). Theorem 2n+3 e Thus, ë ≤ ë2n+3 . ⊣ e In fact, the families of measures necessary to compute ë2n+3 can be simplified considerably. Assuming the strong partition relationeon the ä 12n+1 and the closure of the ä 12n+1 under ultrapowers, we inductively define the efollowing e Let W i denote the i-fold product of the normal measure families of measures. 1 on ù1 . We identify dom(W1i ) with ù1 by the ordering

(α1 , α2 , . . . , αi )
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2 −1,i induced by the weak partition relation on ä 12n+1 , functions f : dom(S2n−1 )→ e 2n −1,i 1,i 1 ä 2n+1 , and the measure S2n−1 . Let S2n+1 be the measure defined just as e j,i , for 2 ≤ j ≤ 2n+1 − 1 be the family S11,i , replacing ä 11 by ä 12n+1 . Let S2n+1 e e of measures on ë2n+3 defined using the strong partition relation on ä 12n+1 , e → ä1 e functions f : ä 12n+1 2n+1 of the correct type and the following measure ì e e on ä 12n+1 : let í denote the (j − 1)th measure in the list W1i , S11,i , W3i , S31,i , S32,i , 3,ie S3 , W5i , . . . . Then ì is the measure induced by the weak partition relation on ä 12n+1 , functions f : dom(í) → ä 12n+1 , and the measure í. If j = 2, then e we regard W1i as a measure on ù1 bye the ordering n

(α1 , α2 , . . . , αi )
Lemma 4.20. For any measure ì on ä 12n+1 occurring in a homogeneous tree e j (ä 1 ) ≤ j i (ä 1 ). Thus, on a Π12n+1 set, there is an i such that ì 2n+1 W2n+1 2n+1 e e e i ë2n+3 = supi jW2n+1 (ä 12n+1 ). e e We also have the following “local version”. Lemma 4.21. For any measure ì occurring in a homogeneous trees for a i , or Π1m set, for m < 2n + 1, there is a measure í in one of the families W2k+1 ej,i 1 S2k+1 for k < n such that for a c.u.b. set of α < ä 2n+1 we have jì (α) ≤ jí (α). e In fact both of the above lemmas hold for all measures ì on ä 12n+1 or e to the ë2n+1 respectively. The above lemmas reduce the computation of ë2n+3 e e i , computation of certain iterated ultrapowers by the canonical measures W2m+1 j,i S2m+1 . These are analyzed through the use of “descriptions,” finitary objects which describe how to build equivalence classes of functions with respect to these measures. The reader can consult [Jac07A] for an introduction to this theory, [Jac99] for a complete account of the first inductive step of this analysis (including the computation of ä 15 and the proof of the strong partition relation e the general case. on ä 13 ), and [Jac88] for details on e We mention one more important application of homogeneous trees in determinacy theory, particularly in the theory of the projective ordinals. This is constructing the “Martin tree,” a generalization of the Kunen tree. We recall first the Kunen tree in the following (see [Kec78A] or [Jac07A] for a proof). Theorem 4.22 (Kunen). Assume AD. There is a tree T on ù × ù1 such that for all f : ù1 → ù1 there is a x ∈ ù ù with Tx wellfounded and such that for all infinite α < ù1 , f(α) < |Tx ↾α|. The Kunen tree can be used as the basis for developing the first level of the projective hierarchy analysis, that is, computing ä 13 , proving the strong e on ä 1 . The same partition relation on ä 11 , and the weak partition relation 3 e e

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proof gives the Kunen tree on the higher ä 12n+1 . That is, there is a tree T on ù × ä 12n+1 such that for all f : ä 12n+1 → eä 12n+1 there is a x ∈ ù ù with Tx e for all α ∈ C with cf(α) = ù e and a c.u.b. C ⊆ ä 1 e such that wellfounded 2n+1 e we have f(α) < |Tx ↾α|. The Kunen tree, however, only provides almost everywhere domination with respect to points of cofinality ù. The higher levels of the analysis require the generalization of the Kunen tree which works at general cofinalities; this is the Martin tree. Theorem 4.23 (Martin). Assume AD. There is a tree T on ù × ä 12n+1 such e and a that for any f : ä 12n+1 → ä 12n+1 there is a x ∈ ù ù with Tx wellfounded e e c.u.b. C ⊆ ä 12n+1 such that for all α ∈ C , f(α) < |Tx ↾ (supì jì (α))|. Here e ranges over the measures occurring in homogeneous trees on the supremum Π1m sets for m < 2n + 1. e We refer the reader to [Jac07A] or [Jac99] for a proof. The ZFC analog of Theorem 4.14 for propagating homogeneity is the Martin-Steel Theorem [MS89]. We state their theorem as follows. Theorem 4.24 (Martin, Steel). Let T be a weakly homogeneous tree with ä complete measures, where ä is a Woodin cardinal. Then the Martin-Solovay tree T ′ is < ä homogeneous. Weakly homogeneous trees are closely related to absoluteness. To see this, first recall the following standard, easily checked fact. Fact 4.25. Let ì be a κ-complete measure on X , and let V[G] be a generic extension of V by a poset P of size < κ. Then in V[G], ì′ := {A ⊆ X : ∃B ∈ V (B ⊆ A ∧ ì(B) = 1)} is a κ-complete measure. Furthermore, if Y ∈ V then for every f : X → Y in V[G] there is a g : X → Y in V which agrees with f on a ì measure one set. The following lemma is an immediate consequence of the previous fact, the definition of the Martin-Solovay tree and Theorem 4.10. Lemma 4.26. Let T be a weakly homogeneous tree with κ-complete measures and T ′ = ms(T ). If P is a poset with |P| < κ then in V[G] we also have that T ′ = ms(T ). In particular, V[G] |= (p[T ′ ] = ù ù − p[T ]). Woodin has shown a converse of Lemma 4.26. Specifically he has shown the following (see [Lar04] or [SteA] for a proof). Theorem 4.27 (Woodin). Assume ZFC. Suppose T , U are trees, ä is a Woodin cardinal, and for all posets P of size ≤ ä we have V[G] |= (p[T ] = ù ù − p[U ]). Then T is <ä-weakly homogeneous. The absolutely complementing property of the previous theorem is frequently made into the following definition (cf. [FMW92]).

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Definition 4.28. A ⊆ ù ù is κ-universally Baire if there are trees T , U with A = p[T ] such that for any poset P with |P| < κ we that V[G] |= (p[T ] = ù ù − p[U ]). One significance of universal Baireness is that the usual forcing arguments for showing regularity properties of analytic set can be made to work from the assumption of universal Baireness. More precisely, in [FMW92] it is shown that if A is (2ℵ0 )+ -universally Baire then A has the standard regularity properties such as the Baire property, Lebesgue measurability, and being completely Ramsey. In particular, assuming ZFC, every weakly homogeneously Suslin set has these properties. Kechris, however, has shown [Kec88B] that weak homogeneity has consequences beyond these usual regularity properties. Specifically he shows: Theorem 4.29 (Kechris). Suppose A ⊆ ù1 has a code set A∗ ⊆ WO which is weakly homogeneously Suslin. Then A ∈ L[x] for some x ∈ ù ù. Theorem 4.29 was improved in [FMW92] to work from the assumption that A is universally Baire (assuming the existence of a measurable cardinal). A weakly homogeneously Suslin representation for A ⊆ ù ù gives an unambiguous way of interpreting A in (small) generic extensions according to the following lemma. Lemma 4.30. Let A = p[T ] = p[U ] where T , U are weakly homogeneously Suslin with κ-complete measures. Then for any poset P with |P| < κ we have that V[G] |= (p[T ] = p[U ]). Proof. Let T ′ , U ′ be the Martin-Solovay trees for T , U respectively. In V, p[T ] ∩ p[T ′ ] = ∅ and p[U ] ∩ p[U ′ ] = ∅, and by absoluteness these intersections remain empty in V[G]. Suppose x ∈ V[G] and x ∈ p[T ] − p[U ]. Then, in V[G], x ∈ p[T ] ∩ p[U ′ ] as U , U ′ project to complements in V[G] by Lemma 4.26. By absoluteness there is an x ∈ V with x ∈ p[T ] ∩ p[U ′ ] which is impossible since in V, p[U ′ ] = ù ù − p[U ] = ù ù − A = ù ù − p[T ]. ⊣ Finally along these lines we mention that weak homogeneity allows an extension of Lemma 4.26 from complements to “projective in” (and a ways beyond). Lemma 4.31. Let A ⊆ ù ù and B be projective in A, say B(x) ⇔ ϕ(x, A) where ϕ is a Σ1n formula over A. Suppose A = p[T ], B = p[U ] where e homogeneous trees with κ-complete measures. Assume T , U are weakly also that every set projective in A is κ-weakly homogeneously Suslin. Let P be a poset with |P| < κ. Let A¯ = (p[T ])V[G] , B¯ = (p[U ])V[G] . Then in ¯ ¯ V[G] |= (∀x (B(x) ⇔ ϕ(x, A)). Proof. Let T1 be weakly homogeneous (with κ-complete measures) and p[T1 ] = ù ù − p[T ] = ù ù − A = p[T ′ ], where T ′ = ms(T ). We claim that

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in V[G] we also have p[T1 ] = p[T ′ ]. Let x ∈ V[G] and suppose first that x ∈ p[T1 ] − p[T ′ ]. Then by Lemma 4.26 x ∈ p[T1 ] ∩ p[T ]. By absoluteness there is a y ∈ V with y ∈ p[T1 ] ∩ p[T ]. But in V, p[T1 ] = ù ù − p[T ], a contradiction. Likewise if x ∈ p[T ′ ] − p[T1 ] then x ∈ p[T ′ ] ∩ p[T1′ ] where T1′ = ms(T1 ). By absoluteness again, there is a y ∈ V with y ∈ p[T ′ ] ∩ p[T1′ ]. However, in V p[T1′ ] = ù ù − p[T1 ] = p[T ] = ù ù − p[T ′ ], a contradiction. Let A2 be defined by A2 (x) ⇔ ∃y ¬A(hx, yi). In ZF there is a simple operation which takes a tree W projecting to a set B and produces a tree U˜ projecting to the existential quantification B2 (x) ⇔ ∃y B(hx, yi). Let T˜ ′ , T˜1 be the resulting trees constructed from T ′ and T1 . So in both V and V[G] we have p[T˜ ′ ] = ∃R p[T ′ ] = ∃R p[T1 ] = p[T˜1 ]. Also, U2 := T˜1 will be weakly homogeneous since T1 is. We have thus produced a weakly homogeneous U2 such that p[U2 ] = ∃R ¬ p[T ] in both V and V[G]. From Lemma 4.30 it doesn’t matter which weakly homogeneous U2 projecting to ∃R ¬A in V we take. This then proves the lemma in the case where ϕ(x, A) = ∃y¬A(hx, yi). The general case now follows by repeating the argument. ⊣

REFERENCES

Howard S. Becker [Bec81] Determinacy implies that ℵ2 is supercompact, Israel Journal of Mathematics, vol. 40 (1981), no. 3– 4, pp. 229–234. Chen-Lian Chuang [Chu82] The propagation of scales by game quantifiers, Ph.D. thesis, UCLA, 1982. Qi Feng, Menachem Magidor, and W. Hugh Woodin [FMW92] Universally Baire sets of reals, In Judah et al. [JJW92], pp. 203–242. Gregory Hjorth [Hjo96] Two applications of inner model theory to the study of Σ12 sets, The Bulletin of Symbolic e Logic, vol. 2 (1996), no. 1, pp. 94–107. [Hjo97] Some applications of coarse inner model theory, The Journal of Symbolic Logic, vol. 62 (1997), no. 2, pp. 337–365. [Hjo01] A boundedness lemma for iterations, The Journal of Symbolic Logic, vol. 66 (2001), no. 3, pp. 1058–1072. Stephen Jackson [Jac] Non-partition results in the projective hierarchy, to appear. [Jac88] AD and the projective ordinals, In Kechris et al. [Cabal iv], pp. 117–220. [Jac90] A new proof of the strong partition relation on ù1 , Transactions of the American Mathematical Society, vol. 320 (1990), no. 2, pp. 737–745. [Jac91] Admissible Suslin cardinals in L(R), The Journal of Symbolic Logic, vol. 56 (1991), no. 1, pp. 260–275. [Jac99] A computation of ä 15 , vol. 140, Memoirs of the AMS, no. 670, American Mathematical e Society, July 1999. [Jac07A] Structural consequences of AD, In Kanamori and Foreman [KF07].

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Stephen Jackson and Donald A. Martin [JM83] Pointclasses and wellordered unions, In Kechris et al. [Cabal iii], pp. 55–66. Stephen Jackson and Russell May [JM04] The strong partition relation on ù1 revisited, Mathematical Logic Quarterly, vol. 50 (2004), no. 1, pp. 33– 40. H. Judah, W. Just, and W. Hugh Woodin [JJW92] Set theory of the continuum, MSRI publications, vol. 26, Springer-Verlag, 1992. Akihiro Kanamori [Kan94] The higher infinite, Springer-Verlag, Berlin, 1994. Akihiro Kanamori and Matthew Foreman [KF07] Handbook of set theory into the 21st century, Springer, 2007. Alexander S. Kechris [Kec77] AD and infinite exponent partition relations, Circulated manuscript, 1977. [Kec78A] AD and projective ordinals, In Kechris and Moschovakis [Cabal i], pp. 91–132. [Kec81A] Homogeneous trees and projective scales, In Kechris et al. [Cabal ii], pp. 33–74. [Kec81B] Suslin cardinals, κ-Suslin sets, and the scale property in the hyperprojective hierarchy, this volume, originally published in Kechris et al. [Cabal ii], pp. 127–146. [Kec88B] A coding theorem for measures, this volume, originally published in Kechris et al. [Cabal iv], pp. 103–109. Alexander S. Kechris, Eugene M. Kleinberg, Yiannis N. Moschovakis, and W. Hugh Woodin [KKMW81] The axiom of determinacy, strong partition properties and nonsingular measures, this volume, originally published in Kechris et al. [Cabal ii], pp. 75–100. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer, 1981. [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Alexander S. Kechris, Donald A. Martin, and John R. Steel [Cabal iv] Cabal seminar 81–85, Lecture Notes in Mathematics, no. 1333, Berlin, Springer, 1988. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. [KM78B] Notes on the theory of scales, this volume, originally published in Cabal Seminar 76–77 [Cabal i], pp. 1–53. Alexander S. Kechris, Robert M. Solovay, and John R. Steel [KSS81] The axiom of determinacy and the prewellordering property, In Kechris et al. [Cabal ii], pp. 101–125. Alexander S. Kechris and W. Hugh Woodin [KW83] Equivalence of determinacy and partition properties, Proceedings of the National Academy of Sciences of the United States of America, vol. 80 (1983), no. 6 i., pp. 1783–1786. [KW07] Generic codes for uncountable ordinals, this volume, originally circulated manuscript, 2007. Paul B. Larson [Lar04] The stationary tower: Notes on a course by W. Hugh Woodin, University Lecture Series (AMS), vol. 32, American Mathematical Society, Providence, RI, 2004.

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Donald A. Martin [MarD] Weakly homogeneous trees, Circulated manuscript. [Mar83B] The real game quantifier propagates scales, this volume, originally published in Kechris et al. [Cabal iii], pp. 157–171. Donald A. Martin and John R. Steel [MS83] The extent of scales in L(R), this volume, originally published in Kechris et al. [Cabal iii], pp. 86–96. [MS89] A proof of projective determinacy, Journal of the American Mathematical Society, vol. 2 (1989), pp. 71–125. Donald A. Martin and W. Hugh Woodin [MW07] Weakly homogeneous trees, this volume, 2007. Yiannis N. Moschovakis [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. John R. Steel [SteA] The derived model theorem, Available at http://www.math.berkeley.edu/∼ steel. [Ste81A] Closure properties of pointclasses, In Kechris et al. [Cabal ii], pp. 147–163. [Ste81B] Determinateness and the separation property, The Journal of Symbolic Logic, vol. 46 (1981), no. 1, pp. 41– 44. [Ste83A] Scales in L(R), this volume, originally published in Kechris et al. [Cabal iii], pp. 107– 156. [Ste83B] Scales on Σ11 -sets, this volume, originally published in Kechris et al. [Cabal iii], pp. 72– 76. Robert Van Wesep [Van78A] Separation principles and the axiom of determinateness, The Journal of Symbolic Logic, vol. 43 (1978), pp. 77–81. [Van78B] Wadge degrees and descriptive set theory, In Kechris and Moschovakis [Cabal i], pp. 151–170. W. Hugh Woodin [Woo99] The axiom of determinacy, forcing axioms, and the nonstationary ideal, De Gruyter Series in Logic and its Applications, Walter de Gruyter, Berlin, 1999. DEPARTMENT OF MATHEMATICS P.O. BOX 311430 UNIVERSITY OF NORTH TEXAS DENTON, TX 76203-1430, USA

E-mail: [email protected]

SUSLIN CARDINALS, κ-SUSLIN SETS AND THE SCALE PROPERTY IN THE HYPERPROJECTIVE HIERARCHY

ALEXANDER S. KECHRIS

Let Θ = sup{î : î is the length of a prewellordering of the set of reals R (= ù ù)}. Let κ < Θ be an infinite cardinal. The class S(κ) of κSuslin sets has some well-known closure properties, i.e., it is closed under continuous substitutions, countable intersections and unions, and existential quantification over R. We investigate in §§1 and 2 the question whether S(κ) is R-parametrized (i.e., whether S(κ) admits universal sets), assuming S AD. Let us call a cardinal κ Suslin iff there is a new κ-Suslin set i.e., S(κ) \ ë<κ S(ë) 6= ∅. Let κ0 , κ1 , κ2 , . . . , κî , . . . be the increasing enumeration of the Suslin cardinals (the first few of them are κ0 = ù, κ1 = ù1 , κ2 = ùù , κ3 = ùù+1 . . . ). We show in §1 that S(κ) is R-parametrized iff there is a largest Suslin cardinal ≤ κ. Thus S(κ) is R-parametrized for all κ < Θ iff the sequence hκî i is normal (and has a largest element if bounded below Θ). We conjecture that this is indeed the case, and in §2 we verify this conjecture at least below κ R = the first non-hyperprojective ordinal, so that S(κ) is always R-parametrized for κ ≤ κ R . In §3 we study the hyperprojective hierarchy (Σ1î , Π1î , ∆1î }î<κR and, using e e e throughout this AD again, we establish that the scale property propagates hierarchy following the pattern established for the prewellordering property in [KSS81]. In particular, it follows that if Γ is a projective-like pointclass, e contained in the inductive sets, then either Γ or Γ˘ has the scale property. This e e smooth propagation of scales breaks down immediately past the inductive sets and this “gap phenomenon” is discussed briefly in §4. Finally §5 contains a number of open problems, conjectures and remarks related to the preceding work. This paper draws heavily on the terminology, notation and results established in [KSS81]. We refer also to [Mos80] for the basic results from descriptive set theory that we use. Our underlying theory is ZF+DC, and any further assumptions (mainly AD) are explicitly indicated. Of course those of our results which are also absolute Research partially supported by NSF Grant MCS79-20465 and an A. P. Sloan Foundation fellowship. The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

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for the inner model L(R), although stated as proved from AD, need only ADL(R) and in fact (in most cases) much weaker forms of definable determinacy, like Hyperprojective Determinacy, etc. An interested and patient reader should have no trouble figuring out how much determinacy is needed in each case. §1. Suslin cardinals and κ-Suslin sets. Let κ be an infinite ordinal. A set A ⊆ R is κ-Suslin if there is a tree T on ù × κ such that A = p[T ] ≡ {α ∈ R : ∃f ∈ ù κ(hα, fi ∈ [T ])}. Similar definitions apply to pointsets A ⊆ X contained in arbitrary product spaces. Denote by S(κ) the pointclass of κSuslin pointsets. It is immediate that S(κ) = S(κ ′ ), where κ ′ is the cardinality of κ, so that it is enough to consider only S(κ), when κ is a cardinal. The class S(κ) has some well-known closure properties which we summarize below. Proposition 1.1. The pointclass S(κ) is closed under continuous substitutions, countable unions and intersections and ∃R . The obvious question now is whether S(κ) is R-parametrized, i.e., has universal sets. We provide below a sufficient condition for this to be true. For this we need to introduce first the following notion. S Definition 1.2. A cardinal is called Suslin iff S(κ) \ ë<κ S(ë) 6= ∅ i.e., there is a κ-Suslin set which is not ë-Sousin for any ë < κ. Assuming AD, let κ0 , κ1 , κ2 , . . . , κî , î < Θ, enumerate the Suslin cardinals. Thus κ0 = ù, κ1 = ù1 = ä 11 , κ2 = ùù , κ3 = ùù+1 = ä 13 , . . . . e e The only Suslin cardinals below supn<ù ä 1n are ë2n+1 , ä 12n+1 , where (ë2n+1 )+ = e e ä 12n+1 . We have now the following main eresult. e e Theorem 1.3 (AD). Let κ be a Suslin cardinal. Then S(κ) is R-parametrized. Proof. Assume not, towards a contradiction. Then, by Wadge, S(κ) ≡ Λ is closed under complements, so also under ∀R , i.e., it is strongly closed in the sense of [KSS81, §2.3]. By the results of that paper (especially 2.4.1) if M $ Λ then there is a pointclass Γ, with M $ Γ $ Λ such that Γ is closed under wellordered unions. We shall use this fact repeatedly below. Note S first that it is sufficient to show that cf(κ) > ù. Because then letting M = î<κ S(î), if A ∈ S(κ), then A is a wellordered union of sets in M (if S T is a tree on ù × κ, then p[T ] = î<κ p[T ↾î]). Let M $ Γ $ Λ be closed under wellordered unions. Then S(κ) = Λ ⊆ Γ, a contradiction. Note now that if ë = sup{î : î is the length of a Λ prewellordering of R}, then ë has cofinality > ù, since Λ is closed under countable unions. Thus it is enough to prove the lemma below to obtain the desired contradiction.

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Lemma 1.4. ë = κ. Proof of Lemma 1.4. If κ < ë, then there is a Λ-norm ϕ : R ։ κ. By the Moschovakis Coding Lemma we can code trees on ù × κ within Σ11 (≤ϕ , <ϕ ), e where x ≤ϕ y ⇔ ϕ(x) ≤ ϕ(y) and similarly for <ϕ . Thus S(κ) ⊆ Σ11 (≤ϕ , <ϕ ) $ Λ, e

a contradiction. S So κ ≥ ë. Pick now A ∈ S(κ) \ î<κ S(î). Since A is κ-Suslin, A carries a κ ′ -scale hϕn : n ∈ ùi, where κ ′ has the same cardinality as κ. We can also assume that hϕn i is regular i.e., each ϕn maps A onto an initial segment of the ordinals. For n ∈ ù, let ìn = length of ϕn . Clearly κ ≤ supn∈ù ìn , so it will be enough to show that for each n, card(ìn ) ≤ ë. So fix n ∈ ù. For any î < ìn , let Aî = {α ∈ A : ϕn (α) = î}. Then if T is the tree associated with the scale hϕn i, i.e., T = {hα(0), ϕ0 (α), . . . , α(m − 1), ϕm−1 (α)i : m ∈ ù, α ∈ A} and for î ≤ ìn we let T≤î = {ha0 , î0 , . . . , am−1 , îm−1 i ∈ T : în ≤ î} and similarly for T<î , we have by the semicontinuity property of scales that Aî = p[T≤î ] \ p[T<î ]. Thus Aî ∈ Λ, since p[T≤î ], p[T<î ] are κ ′ -Suslin and Λ is closed under complements. Moreover î 6= ç < ìn implies Aî ∩ Aç = ∅. Put for î < ìn , g(î) = kAî kW ≡ the Wadge ordinal of Aî . Since by [KSS81, Lemma 2.3.1], we have ë = sup{kBkW : B ∈ Λ}, it follows that g : ìn → ë. We claim now that for each ñ < ë, the order type of {î < ë : g(î) < ñ} = g −1 [ñ] is less than ë. Granting let for ñ < ë, fñ : ë ։ ϕ −1 [ñ]. Then if f(ñ, î) = S this, −1 fñ (î), f : ë × ë ։ ñ<ë g [ñ] = ìn , thus card(ìn ) ≤ ë and we are done.

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We prove the claim now: Let for each ñ < ë, N = {A : kAkW < ñ}. Then N $ Λ, so find Γ closed under wellordered unions and such that N $ Γ $ Λ. Define x ≺ y ⇔ ∃î∃î ′ [î, î ′ ∈ g −1 [ñ] ∧ î < î ′ ∧ x ∈ Aî ∧ y ∈ Aî ′ ]. Clearly ≺ is wellfounded, has rank the order type of g −1 [ñ] and belongs to Γ, thus the order type of g −1 [ñ] is less than ë, and we are done. ⊣Lemma 1.4 and Theorem 1.3 By the preceding result the only case when S(κ) is not R-parametrized is when S(κ) = S(ó), where ó S is a limit of Suslin cardinals but is itself not Suslin. In this case S(ó) = κ<ó S(κ) is closed under complements. We conjecture that this never happens, i.e., Conjecture 1.5 (AD + anything reasonable). If κ < Θ is a limit of Suslin cardinals, then κ is a Suslin cardinal. Thus if hκî : î < Ξi is the enumeration of the Suslin cardinals, our conjecture means that for í < Ξ limit κí = supî<í κî , and moreover that either supî<Ξ κî = Θ, i.e., if S∞ =

S

ë

S(ë), then

S∞ = ℘(R) or else Ξ = î + 1 is successor and S∞ = S(κî ). Thus in particular we have the following Conjecture 1.6 (AD + anything reasonable). The pointclass S∞ is R-parametrized, unless S∞ = ℘(R). It has been proved by Martin and Steel that AD + V = L(R) ⇒ S∞ = Σ21 ∧ (Ξ = ä 21 + 1, κä 21 = ä 21 ). e e e e On the other hand [Sol78B] raises the question whether ADR + Θ is regular ⇒ S∞ = ℘(R)? We shall see now in the next section that our first conjecture is verified for κ ≤ κ R (and in fact for a while beyond that, by §4).

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§2. Suslin cardinals below κ R . Our main purpose here is to prove the following. Theorem 2.1 (AD). The class of Suslin cardinals is closed unbounded below κR . As an immediate corollary we have Theorem 2.2 (AD). For each κ ≤ κ R , the pointclass S(κ) is R-parametrized. It will be convenient at this stage to introduce the hyperprojective hierarchy to facilitate the presentation of the proof of Theorem 2.1. In the next section we shall take up a detailed study of the structural properties of the hyperprojective hierarchy itself. First let IND denote the class of inductive over the structure of analysis R g ˘ the pointclass of hyperprojective sets. We pointsets and HYP ≡ IND ∩ IND g g ] define below a hierarchy on HYP. ] Definition 2.3 (AD). For each 1 ≤ î < κ R let the pointclasses Σ1î , Π1î , ∆1î e e e be defined as follows: (i) For î < ù, these are the usual projective pointclasses. ˘1 . (ii) Σ1î+1 = ∃R Π1î , Π1î+1 = Σ î+1 e e e1 e (iii) For limit ë, let Σë be the smallest projective-like pointclass closed under e S ˘ 1. ∃R which contains î<ë Σ1î , and let Π1ë = Σ ë e e e (iv) ∆1î = Σ1î ∩ Π1î . e e e Put also for limit ë, [ Λ1ë = ∆1î . e e î<ë We have of course

Finally define

[ [ Σ1î (= HYP = ∆1î ). e ] î<κR e R î<κ

ä 1î = sup{ç : ç is the rank of a Σ1î well founded relation} e e and for limit í ë1í = supî<í ä 1î . e e (It follows from the results in §3 that also ä 1î = sup{ç : ç is the rank of a ∆1î e e prewellordering}, so that this notation is justified.) We shall need of course crucially below some facts about positive elementary inductive definability on the structure of analysis R; see [Mos74A] here.

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For some monotone operator Φ(x, A), where x varies over a product space X and A over ℘(X ), let Φî be its îth iterate, defined by [ Φî (x) ⇔ Φ(x, Φ<î ), Φ<î = Φç . ç<î

Definition 2.4. For each limit ordinal ë ≤ κ R define the following pointclass INDë = {A : A is Wadge reducible to some Φ<ë , where g Φ is a positive elementary operator on R}. (Recall that A is Wadge reducible to B (A ≤W B) iff there is continuous f with A = f −1 [B]). Thus INDκR = IND. g g We will need the following two lemmas of which the first one is implicit in [Mos74A]. We will sketch their proofs after deriving from them a proof of Theorem 2.1. Lemma 2.5. For each limit ë ≤ κ R , IND is closed under continuous subg stitutions, ∩, ∪, ∪ù , ∃R and is R-parametrized. In fact there is a “universal” positive elementary over R operator Φ(α, A) such that Φ<ë is INDë -complete g to it). If for all limit ë ≤ κ R (i.e., each member of INDë is Wadge reducible g ù cf(ë) > ù, then INDë is also closed under ∩ as well. Finally INDë has the g g prewellordering property. (We have actually put a lot more information in Lemma 2.5 than we need for the proof of Theorem 2.1. It will be used in later sections.) Lemma 2.6 (AD). For each limit ë < κ R , Λ1ë $ INDë $ Λ1ë+ù e e g (The class INDë can be actually computed precisely in the hyperprojective gresults in the next section.) hierarchy by the Granting these two lemmas let us proceed to give the Proof of Theorem 2.1. It is clear that the sequence hκî : î < κ R i is unbounded below κ R . We have to show that it is closed. So let ϑ < κ R be limit. Put S κ = supç<ϑ κç . We have to show that κ is a Suslin cardinal i.e., S(κ) 6⊆ ç<ϑ S(κç ) ≡ Λ. Since Λ is strongly closed and Λ $ HYP, there is a ] limit ordinal ó < κ R with Λ = Λ1ó . e Note that ó ≤ ë1ó ≡ supî<ó ä 1î ≤ κ, where the last bound comes from the e e Kunen-Martin Theorem.

Let now Φ be the universal operator asserted to exist in Lemma 2.5. For each ordinal î ≤ κ R , let hϕnî : n ∈ ùi be the canonical scale on Φî defined by

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the procedure of [Mos78]. Let ñî = supn (lh(ϕnî )). By the definition of hϕnî i it is easy to check that for each limit ordinal í and each m ∈ ù ñí+m < ë1í+ù . e Moreover again by the procedure of [Mos78], for each limit ordinal ϑ, Φ<ϑ admits a scale on max{ϑ, supî<ϑ ñî }, thus Φ<ó admits a scale on max{ó, supî<ó ñî } ≤ ë1ó ≤ κ e ˘ , we have by Lemma 2.6 that i.e., a κ-scale. Since Φ<ó ∈ INDó \ IND g gó ⊣ Φ<ó 6∈ Λ1ó = Λ. Thus Φ<ó ∈ S(κ) \ Λ and we are done. e We conclude this section by giving the proofs of Lemmas 2.5 and 2.6. Proof of Lemma 2.5. The proof of the closure properties follows that of the closure properties of IND as in [Mos74A, Chapter 1]. As an example let us prove closure under ∃Rg : Let A ∈ INDë and let Ψ be a positive elementary g on R operator such that for some continuous f (x, α) ∈ A ⇔ f(x, α) ∈ Ψ<ë .

Let x ∈ B ⇔ ∃α(x, α) ∈ A. Then x ∈ B ⇔ ∃α∃î < ë[f(x, α) ∈ Ψî ] ⇔ ∃î < ë∃α[f(x, α) ∈ Ψî ]. Consider now the following simultaneous induction Ψî1 (â) ⇔ Ψ(â, Ψ<î 1 ) Ψî2 (x) ⇔ ∃α[f(x, α) ∈ Ψ<î 1 ]. Then Ψî1 = Ψî and Ψî2 (x) ⇔ ∃α[f(x, α) ∈ Ψ<î ], so for limit ë, î Ψ<ë 2 (x) ⇔ ∃î < ë∃α[f(x, α) ∈ Ψ ]

⇔ x ∈ B. Now, by the Simultaneous Induction Lemma [Mos74A, 1C.1], there is a positive elementary on R operator Ω and constants x0 such that Ψî2 (x) = Ωî (x0 , x),

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thus x ∈ B ⇔ Ω<ë (x0 , x), so B ∈ INDë and we are done. gprewellordering property we just use again the proof that IND has For the the prewellordering property; see [Mos74A, Chapter 2]. According g to it, to each positive elementary on R operator Ψ we can assign two other positive elementary on R operators Ψ1 , Ψ2 such that if ø is the canonical norm on Ψ∞ associated with the induction, and ≤∗ø , <∗ø its corresponding relations, then for each î we have ø(x) ≤ î ∧ x ≤∗ø y ⇔ Ψî1 (x, y) ø(x) ≤ î ∧ x <∗ø y ⇔ Ψî2 (x, y), so that for limit ë, x ∈ Ψ<ë ∧ x ≤∗ø y ⇔ Ψ<ë 1 (x, y) x ∈ Ψ<ë ∧ x <∗ø y ⇔ Ψ<ë 2 (x, y). thus ø↾Ψ<ë is a INDë -norm and this establishes PWO(INDë ). gthe statement about parametrization:g Finally we prove First recall that by [Mos70A] every positive Π11 relation Ψ(x, A), where e x ∈ X , A ⊆ R can be brought in the form ¯ Ψ(x, A) ⇔ ∀â[R(x, â) ∨ ∃n((â)n ∈ A)] ∧ R(x),

with R, R¯ ∈ Π11 . Similarly every positive Σ11 relation Ψ′ (x, A) is equivalent to e e one in the form ¯ Ψ′ (x, A) ⇔ ∃â[S(x, â) ∧ ∀n((â)n ∈ A)] ∨ S(x),

with S, S¯ ∈ Σ11 . Similar normal forms are valid for positive Π11 or Σ11 relations e eache fixed X , Y e ⊆ Y an arbitrary product space. As a result, for Ψ(x, A) for A there is a positive Π11 (resp. Σ11 ) relation of the form Ψ(ε, x, A) (ε ∈ R, x ∈ e for the positive Π1 (resp. Σ1 ) relations of the X , A ⊆ Y) which eis universal 1 1 e e form Ψ(x, A). According to [Mos74A, Chapter 1, Ex. 1.15] one can find for each positive elementary on R operator Ψ(x, A) an operator Ψ′ (y, x, A) which is a disjunction of a Π11 positive operator and a Σ11 positive operator, and constants y0 e e all limit ë, such that for Ψ<ë (x) ⇔ Ψ′

<ë

(y0 , x).

So it is clear that it is enough to find an operator Φ(ε, α, A) (ε ∈ R, α ∈ R, A ⊆ R × R) which is the disjunction of a positive Π11 and a positive Σ11 operator e e

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such that for all such Ψ(α, X ) (α ∈ R, X ⊆ R) there is ε0 ∈ R (depending on Ψ) so that: Ψ<ë (α) = Φ<ë (ε0 , α), for all limit ë. Note that to achieve this it is sufficient to find Φ as above such that for all Ψ as above there is ε0 (depending on Ψ) such that Φ(ε0 , α, A) ⇔ Ψ(α, {α ′ : A(ε0 , α ′ )}) (Then by a simple induction on î, Φî (ε0 , α) ⇔ Ψî (α).) For that, let Ξ(ä, α, X ), where ä, α ∈ R, X ⊆ R, be universal in the class of disjunctions of positive Π11 and Σ11 formulas. Put then e e Φ(ε, α, A) ⇔ Ξ(ε, α, {α ′ : A(ε, α ′ )}). To verify that this works, fix an appropriate Ψ and let ε0 be such that Φ(α, X ) ⇔ Ξ(ε0 , α, X ). Then Φ(ε0 , α, A) ⇔ Ξ(ε0 , α, {α ′ : A(ε0 , α ′ )} ⇔ Ψ(α, {α ′ : A(ε0 , α ′ )}).

⊣

Proof of Lemma 2.6. By induction on ë. Obvious for ë = ù. If ë = ë′ + ù is a successor limit ordinal, then by induction hypothesis, Λ1ë′ $ INDë′ $ Λ1ë′ +ù = Λ1ë . e e e g Since Λ1ë is strongly closed, it is easy to check that for each positive elementary ′ e on R operator Ψ one has Ψë +n ∈ Λ1ë for each n ∈ ù, thus Ψ<ë ∈ Σ1ë+1 $ Λ1ë+ù . e n ∈ eù, if e 1 $ IND , notice that for any So INDë $ Λ1ë+ù . To see that Λ ë ë e e g g 1 A ∈ Σë′ +n , then A is gotten from some B ∈ INDë′ by repeatedly applying a e g finite number of real quantifications. This is because Λ1ë′ $ INDë′ . Then by gthat for some the Simultaneous Induction Lemma [Mos74A, 1C.1] ite follows ë′ positive elementary on R operator Ψ1 , A ≤W Ψ1 . But by the proof of the ′ prewellordering property for INDë , clearly Ψë1 ∈ INDë , so A ∈ INDë1 . Thus S g Λ1 is closed underg complements g Λ1ë $ INDë . Λ1ë = n Σ1ë′ +n ⊆ INDë and since ë e For theecase when gby gë is the limiteof limit ordinals, it immediatelye follows 1 induction hypothesis that Λë $ INDë . For the other inclusion, notice that by e g our induction hypothesis each member of INDë is the wellordered union of g[KSS81] either Σ1 or Σ1 has 1 1 sets in Λë and thus belongs to Σë+1 (since by ë+1 ë eunions). e e thus is closed under wellordered e the prewellordering property and ⊣ So INDë $ Λ1ë+ù . e g

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§3. The scale property in the hyperprojective hierarchy. As a special case of the results in [KSS81] one can determine the prewellordering pattern in the hyperprojective hierarchy. We do the same thing for the scale property. Moreover we identify the pointclasses S(κ), for κ a Suslin cardinal < κ R , within the hyperprojective hierarchy. It will be convenient to introduce first the following terminology, motivated by the classification of projective-like hierarchies in [KSS81, §4]: For ϑ a limit ordinal < κ R , we shall say that, (i) ϑ is type I iff cf(ϑ) = ù, or equivalently iff the projective-like hierarchy {Π1ϑ , Σ1ϑ+1 , Π1ϑ+2 , Σ1ϑ+3, . . . } is of type I. e e e e (ii) ϑ is of type II iff {Π1ϑ , Σ1ϑ+1 , . . . } is of type II, e 1 ,eΣ1 , . . . } is of type III. (iii) ϑ is of type III iff {Π ϑ+1 ϑ e e John Steel has proved a beautiful result (see [Ste81A]) which allows us to characterize types II and III above in terms of ordinal invariants. The characterization is as follows. Theorem 3.1 (AD, [Ste81A]). A limit cardinal ϑ < κ R is of type III iff κϑ ) is regular (iff ϑ = ë1ϑ (= κϑ ) and ϑ is regular). (Thus ϑ < κ R is of e κ ) is singular.) e type II iff cf(ϑ) > ù and ë1ϑ (= ϑ e We now have the following ë1ϑ (=

Theorem 3.2 (AD). Let ϑ be a limit ordinal < κ R . Then we have, (i) If ϑ is of type I, the following classes have the scale property Π1ϑ , Σ1ϑ+1 , Π1ϑ+2 , Σ1ϑ+3 , . . . , e e e e Moreover, S(κϑ+m ) = Σ1ϑ+m , for all m ≥ 0. Finally, for all m ≥ 0, e (κϑ+m )+ = ä 1ϑ+m and is measurable, e ä 1ϑ+2m+1 = (ä 1ϑ+2m )+ , so κϑ+2m+1 = ä 1ϑ+2m , and e e κϑ+2m has cofinality ù. e (ii) If ϑ is of type II, the following classes have the scale property Moreover,

Σ1ϑ , Π1ϑ+1 , Σ1ϑ+2 , Π1ϑ+3 , . . . . e e e e

S(κϑ+m ) = Σ1ϑ+m , for all m ≥ 0. e Finally, for all m ≥ 0, (κϑ+m )+ = ä 1ϑ+m and is measurable, e 1 ä ϑ+2m+2 = (ä 1ϑ+2m+1 )+ , so κϑ+2m+2 = ä 1ϑ+2m+1 , and e eκ e ϑ+2m+1 has cofinality ù. (iii) If ϑ is of type III, the following classes have the scale property Π1ϑ , Σ1ϑ+1 , Σ1ϑ+2 , Σ1ϑ+3 , . . . . e e e e

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Moreover S(κϑ+m ) = Σ1ϑ+m+1 , for all m ≥ 0. e Finally, for all m ≥ 0, (κϑ+m )+ = ä 1ϑ+m+1 is measurable e ä 1ϑ+2m+1 = (ä 1ϑ+2m )+ , so κϑ+2m = ä 1ϑ+2m , e e has cofinality ù, and e κϑ+2m+1 1 κϑ = ä ϑ and is measurable. e Proof. By induction on ϑ. (i) By induction hypothesisS(or by standard facts about the projective hierarchy when ϑ = ù) Λ1ϑ = ç<ϑ S(κç ). Moreover κϑ = limç<ϑ κç = ë1ϑ S e e has cofinality ù. Let Σ = { n An : For all n, An ∈ Λ1ϑ }. Then, by our e induction hypothesis, Σ has the scale property and is closed under ∃R , so 1 R since Πϑ = ∀ Σ we have, by the Second Periodicity Theorem of [Mos80], that e , Π1 , . . . all have the scale property. Π1ϑ , Σ1ϑ+1 ϑ+2 e κ e= ë1 , there is a ∆1 prewellordering of R of length κ , thus S(κ ) ⊆ e Since ϑ ϑ ϑ ϑ ϑ e from the fact that Λ1 $ S(κ ) and the closure e S(κ ) follows Σ1ϑ . That Σ1ϑ ⊆ ϑ ϑ ϑ e eof S(κ ). Thus Σ1 = S(κ ). The equalities e S(κϑ+m ) = Σ1ϑ+m properties ϑ ϑ ϑ e follow now by the usual arguments as for the projective hierarchy. The esame holds for the proofs of the fact about ä 1ϑ+m and κϑ+m . (A quicker proof that κϑ+2m+2 has cofinality ù makes use ofethe fact that S(κϑ+2m+1 ) = Σ1ϑ+2m+1 is e closed under wellordered unions.) (ii) First notice that if the cofinality of ϑ is bigger than ù, then (∗) S(κϑ ) = INDϑ . g To prove (∗) S first use the fact that each elementSof S(κϑ ) is a wellordered union of sets in ç<ϑ S(κç ), since cf(κϑ ) > ù. But ç<ϑ S(κç ) ⊆ Λ1ϑ , by induction e hypothesis, and Λ1ϑ $ INDϑ by Lemma 2.6. Since INDϑ is projective-like e g g closed under ∃R and has the prewellordering property by Lemma 2.5, it is closed under wellordered unions, thus S(κϑ ) ⊆ INDϑ . For the inclusion INDϑ ⊆ S(κϑ ), just look at the proof of Theoremg 2.1 and note that in the g 1 notation there, Λ = Λϑ by our induction hypothesis, thus ϑ = ó. Since in case ϑ is ofetype II, Σ1ϑ consists (by the analysis of type II hierarchies in [KSS81]) of all wellorderedeunions in Λ1ϑ , we also have in this case that e S(κϑ ) = INDϑ = Σ1ϑ . e g The rest of the conclusions of (ii) will follow routinely once we can show that each set in INDϑ admits an INDϑ -scale, each norm of which has length ≤ κϑ . g For that g let Φ be the universal positive elementary on R operator of î Lemma 2.5. Let hϕn i be the canonical scale on Φî as defined in [Mos78].

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Let also hϕn i be the scale on Φ<ϑ defined there, i.e., ϕ0 (x) = least î < ϑ such that x ∈ Φî ϕn+1 (x) = hϕ0 (x), ϕnϕ0 (x) i. Clearly ϕ0 is a INDϑ -norm. Now we have g x ≤∗ϕn+1 y iff x <∗ϕ0 y ∨ ∃î < ϑ[ϕ0 (x) = ϕ0 (y) = î ∧ ϕnî (x) ≤ ϕnî (y)]. Now it is easy to verify that for each fixed î < ϑ the relation ϕ0 (x) = ϕ0 (y) = î ∧ ϕnî (x) ≤ ϕnî (y) is in Λ1ϑ , thus ≤∗ϕn+1 is the wellordered union for Λ1ϑ relations, thus belongs to e that all these norms have INDϑe. Similarly for <∗ϕn+1 and we are done. The fact g length ≤ κϑ is included in the proof of Theorem 2.1. (iii) By (∗) of case (ii) we know that S(κϑ ) = INDϑ = Σ1ϑ+1 , e g so all the assertions of (iii) follow except that Π1ϑ has the scale property. We e prove this now.

First recall from [KSS81, 2.4] that Π1ϑ is equal to the class of all ∆1ϑ (= Λ1ϑ )e e e bounded wellordered unions of ∆1ϑ sets. e 1 Fix now A ∈ Πϑ and let Φ be the universal positive elementary on R e before. For x ∈ Φ∞ , let |x| ≡ ϕ (x) = least î such that operator considered 0 x ∈ Φî . Put then (ε, x) ∈ C ⇔ x ∈ Φ<ϑ ∧ ε codes a continuous function (≡ fε ), and for (ε, x) ∈ C , let Yε,x = fε−1 [Φ<|x| ]. Then put B = {(ε, x) ∈ C : Yε,x ⊆ A}, so that B ∈ Σ1ϑ+1 = INDϑ . Then there is continuous g such that e g B = g −1 [Φ<ϑ ]. Put finally for î < ϑ, [ Aî = {Yε,x : g(ε, x) ∈ Φ<î ∧ x ∈ Φ<î }. S Clearly A = î<ϑ Aî , Aî ∈ ∆1ϑ and {Aî }î<ϑ is a ∆1ϑ -bounded union of sets in e e ∆1ϑ . Now e α ∈ Aî ⇔ ∃ε∃x[g(ε, x) ∈ Φ<î ∧ x ∈ Φ<î ∧ fε (α) ∈ Φ<|x| ] ⇔ ∃ε∃x[g(ε, x) ∈ Φ<î ∧ x ∈ Φ<î ∧ fε (α) <∗ϕ0 x].

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Let Φ1 be a positive elementary on R operator such that z ∈ Φç ∧ z <∗ϕ0 y ⇔ Φç1 (z, y). Then α ∈ Aî ⇔ ∃ε∃x[g(ε, x) ∈ Φ<î ∧ x ∈ Φ<î ∧ Φ<î 1 (fε (α), x)]. From this it is clear that we can define a map î 7→ hønî i, where for each î < ϑ, the sequence hønî i is a ∆1ϑ -scale on Aî , making use of the canonical scales e <î <î being put, as in S [Mos78], on Φ and Φ1 . This in turn defines the following scale on A = î<ϑ Aî : ø0 (α) = least î such that α ∈ Aî , øn+1 (α) = hø0 (α), ønø0 (α) (α)i. We show that this is a Π1ϑ -scale. Clearly ≤∗øϑ , <∗ø0 are in Π1ϑ (see [KSS81, 2.5]). e e argument for <∗ being similar). Consider now ≤∗øn+1 (the We have øn+1 α ≤∗øn+1 â iff α <∗ø0 â ∨ ∃î < ϑ[∃î ′ ≤ î(ø0 (α) = ø0 (â) = î ′ ′

′

∧ønî (α) ≤ ønî (â))], thus ≤∗øn+1 =<∗ø0 ∪

[

Dî ,

î<ϑ

where Dî =

[

{(α, â) : ø0 (α) = ø0 (â) = î ′ ∧ ønî (α) ≤ ønî (â)} ∈ ∆1ϑ . e ′ î ≤î ′

′

So it is enough to check that hDî : î < ϑi is ∆1ϑ -bounded. For that let S e X ∈ ∆1ϑ , X ⊆ î<ϑ Dî be given. Then e {α : ∃â(α, â) ∈ X } = Y S is in ∆1ϑ and Y ⊆ î<ϑ Aî , so for some î < ϑ, Y ⊆ Aî . But then X ⊆ Dî . e Indeed, if (α, â) ∈ X , then α ∈ Y , thus ø0 (α) = î ′ ≤ î. So also ø0 (â) = î ′ S ′ ′ and since (α, â) ∈ î<ϑ Dî we have that ønî (α) ≤ ønî (â), thus (α, â) ∈ D. ⊣ We have now the following immediate corollaries: Corollary 3.3 (AD). Let Γ be any projective-like pointclass contained in e IND. Then one of Γ or Γ˘ has the scale property. e e g Corollary 3.4 (AD). Let κ be a Suslin cardinal ≤ κ R . Then (i) S(κ) has the scale property iff cf(κ) > ù. ˘ (ii) S(κ) has the scale property iff cf(κ) = ù. (iii) κ + is always measurable.

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(iv) If κ = κç and κ ∗ = κç+1 is the next Suslin cardinal, then (a) If κ has cofinality ù, then κ ∗ = κ + is a measurable cardinal, while (b) If κ has cofinality > ù, then κ ∗ has cofinality ù. Corollary 3.5 (AD). Let κ < κ R be a Suslin cardinal. Then there is a ˘ prewellordering of R in S(κ) ∩ S(κ) of length κ. Finally, define for each A ∈ S∞ , the cardinal κ(A), the Suslin cardinal of A, to be the smallest κ such that A ∈ S(κ). Recall that kAkW denotes the Wadge ordinal of A. We have now the following estimate. Corollary 3.6 (AD). For each A ∈ IND, κ(A) ≤ kAkW . g

Proof. Let κ(A) = κç , ç < κ R (the case κ(A) = κ R is obvious). If ç = ϑ is limit and kAkW < κϑ = ë1ϑ , then A ∈ Λ1ϑ , so κ(A) < ë1ϑ , a contradiction. If e according toethe type of ϑ. ç = ϑ + m, ϑ limit, m > 0,econsider cases ⊣ §4. Gaps in the propagation of scales. Assume AD for the discussion in this section. The uninterrupted propagation of the scale property throughout the hyperprojective hierarchy is disturbed as one goes past the class of inductive sets. If {Γi } is a projective-like hierarchy of type IV (see [KSS81]), then Kechris has e shown that neither Γ1 nor Γ˘1 can have the scale property and Martin substane e tially extended this to show that in fact no Γi or Γ˘i can have the scale property. e projective-like e ∗ ∗ ∗ hierarchy of type In particular, {Π1 , Σ2 , Π3 , Σ∗4 , . . . } is the least e e e e ˘ )), no Σ∗ or Π∗ can have the scale property. IV (so that Π∗1 = ∀R (IND ∨ IND n n e . Compare this with the core IND e 3.3 cannot g beg Thus Corollary extended past responding result about the prewellorderingg property (see [KSS81]), which extends to all Γ ⊆ L(R) and beyond. (In particular Σ∗1 , Π∗2 , Σ∗3 , . . . all have e e e the prewellordering property.) ∗ closed under ∃R Now if Σù denotes S ∗ the smallest projective-like pointclass e ∗ and containing n Σn , then [Mos83] has shown that Πù has the scale property and thus so do Σ∗ù+1e, Π∗ù+2 , . . . , and the familiar pattern for the propagation of scales resumes e again for a while, after this gap of length ù. But later on wider and wider gaps occur, reflecting eventually the “unbounded” gap occurring in L(R) beyond S∞ = Σ21 . We will not however pursue this matter any further e here. Beyond this first occurrence of gaps, other phenomena happen for the first time at the level of κ R . For example, κ R is the least Suslin cardinal κ for which the conclusion of Corollary 3.5 fails and also the least ordinal κ for which there is an A with kAkW = κ but κ(A) > kAkW . (Take A to be a complete co-inductive set.)

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§5. Miscellaneous remarks, questions, and conjectures. We assume again AD throughout this section. 5.1. Closure properties of S(κ). Let κ be a Suslin cardinal. Of course S(κ) is closed under ∃R , but when is it also closed under ∀R ? The following result gives a necessary and sufficient condition when cf(κ) > ù. We need first to define a notion of indescribability of ordinals. For any limit ordinal ë let Bë = {x : ∃î < ë, x ⊆ Lî }. Thus Lë ⊆ Bë and Bë is transitive. Recall that a formula in the language of ZF augmented by extra predicates is Π2 if it has the form ∀x∃yϕ, where ϕ is bounded. We call now an ordinal ë b Π12 -indescribable if for each X ⊆ Lë and each Π2 formula ϕ of the appropriate language, with parameters from Bë , we have (Bë , ∈, X ) |= ϕ ⇒ ∃ϑ < ë, (Bϑ , ∈, X ∩ Lϑ ) |= ϕ. It is easy to check that such ë’s are regular cardinals. We have now Theorem 5.1 (AD). Let κ be a Suslin cardinal of cofinality greater than ù. Then the following are equivalent: (i) S(κ) is closed under ∀R , (ii) κ is b Π12 -indescribable. Proof. (i) ⇒ (ii). Let Γ = S(κ). Then Γ is a Spector class on the structure e length of a ∆ prewellordering of analysis R, so if ä = sup{î : î is the e of R}, by the Companion Theorem of [Mos74A], ä is ethe ordinal of its e companion admissible set M above R. As every admissible set is Π2 reflecting and every set A ⊆ Lä is ∆1 in M , by the Moschovakis Coding Lemma (see [Mos70A]), it is easyeto verify that ä is b Π12 -indescribable (this argument is e due to Moschovakis). So it is enough to show that ä = κ. By Kunen-Martin, ä ≤ κ + and, since e now A ∈ S(κ) \ S e ä is a limit cardinal, ä ≤ κ. Let ç<κ S(ç). Let hϕn i be a e regular κ-scale on A. eWe can assume moreover that hϕn i is good, i.e., ϕi (x) ≤ ϕi (y) ⇒ ∀j ≤ i(ϕj (x) ≤ ϕj (y)).

Fix now n. For î < length(ϕn ), let Kn (î) = sup{ϕj (x) : x ∈ A ∧ ϕn (x) = î}. Then Kn (î) < κ, since if ϕn (x0 ) > î, then for any x with ϕn (x) = î < ϕn (x0 ) we must have ϕj (x) ≤ ϕj (x0 ), ∀j ≤ n, by goodness, but also ϕj (x) < ϕj (x0 ), ∀j ≥ n, by goodness again, so Kn (î) ≤ sup ϕj (x0 ) < κ. Let now T be the tree on ù × κ associated with hϕi i. Let Tn (î) = {ha0 , u0 , . . . , am−1 , um−1 i ∈ T : un ≤ î ∧ ∀i < m, ui ≤ Kn (î)} and similarly for Tn′ (î), replacing un ≤ î by un < î. Then Aîn = {x : ϕn (x) ≤ î} = p[Tn (î)],

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Bnî = {x : ϕn (x) < î} = p[Tn′ (î)], so since Tn (î), Tn′ (î) are trees on Kn (î) < κ, clearly Aîn , Bçî ∈ ∆. Thus e Cnî = {x : ϕn (x) = î} ∈ ∆. e The rest of the argument is as in the proof of the Lemma 1.4. (ii) ⇒ (i) Let T be a tree on ù × κ. Let (x, α) ∈ B ⇔ T (x, α) is not wellfounded. Let x ∈ A ⇔ ∀α(x, α) ∈ B ⇔ ∀α, T (x, α) is not wellfounded ⇔ (Bκ , ∈, x, T ) |= ∀α∃î(T ↾î(x, α) is not wellfounded) (since cf(κ) > ù), ⇔ (Bκ , ∈, x, T } |= ∀α∃î∃f ∈ î ù ∀n(x↾n, α↾n, f↾n) ∈ T ⇒ ∃κ ′ < κ, (Bκ′ , ∈, x, T ↾κ ′ ) |= (∗) (where (∗) is the Π2 formula ∀α∃î∃f ∈ î ù (x↾n, α↾n, f↾n) ∈ T )) ⇒ ∃κ ′ < κ, ∀α[T ↾κ ′ (x, α) is not wellfounded]). Thus x ∈ A ⇔ ∀α[S(x, α) is not wellfounded], where S is a tree on ù 2 × κ ′ . So A ∈ ∀R S(κ ′ ). Since B is an arbitrary element of S(κ), we have that S(κ) ⊆ ∀R S(κ ′ ), so ˘ ′ ), if ∀R S(κ ′ ) ⊆ S(κ) we are done. Otherwise, by Wadge, S(κ) ⊆ ∃R S(κ ′ R˘ ′ so since S(κ) ⊇ S(κ ), we have S(κ) = ∃ S(κ ). Let κ¯ be the largest Suslin cardinal ≤ κ ′ (which must exist since S(κ) is R-parametrized). Then ˘ κ) S(κ) = ∃R S( ¯ and S(κ) ¯ is a projective-like pointclass closed under ∃R , thus R˘ R ¯ = S(κ) 6⊆ ∀ S(κ), ¯ a contradiction. ⊣ ∃ S(κ) Note that from the preceding argument and the fact that for every Spector class Γ on R its associated ordinal ä is Mahlo (see [KKMW81]) we have that e of the equivalent conditions e (i), (ii) above hold then actually κ is also if either Mahlo. As a corollary of Theorem 5.1, we see that κ R is the least b Π12 -indescribable Suslin cardinal. Conjecture 5.2. The assumption cf(κ) > ù is not needed in Theorem 5.1. Question 5.3. Is κ R the least Mahlo b Π12 -indescribable cardinal?

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5.2. The Prewellordering and scale properties for S(κ). By [KSS81], for ˘ each Suslin cardinal κ, either S(κ) or S(κ) has the prewellordering property. From Corollary 3.4 we see that, when κ ≤ κ R , what distingushes the first case from the second is whether cf(κ) > ù or not. Question 5.4. Does this hold also for arbitrary κ? Similarly about the scale property. One fact that we have noticed is that if S(κ) is closed under ∀R , then S(κ) has the prewellordering property. 5.3. Properties of the Suslin cardinals. Question 5.5. Do the properties of Corollary 3.4 (iii), (iv) hold for arbitrary Suslin cardinals? 5.4. About S∞ . According to our conjecture in §2 we expect S∞ either to be all of ℘(R) or else to be R-parametrized. In the latter case S∞ = S(κ), where κ is the largest Suslin cardinal, so, by our remarks in Section 5.2, S∞ (being closed under ∀R ) also has the prewellordering property, i.e., is a Spector class on the structure of analysis R. We mentioned in §1 that assuming AD + V = L(R), S∞ = Σ21 . It is conceivable that in some reasonable theory e that if S is R-parametrized, then S ⊆ Σ2 . It extending AD one can prove ∞ ∞ 1 e S is already known (in AD only) that there are some forbidden values for ∞ when we go past Σ21 . For example, S∞ cannot be Σ2n for n > 1 or Π2n for n ≥ 1, e e e and similarly for Σkn , Πkn for all k ≥ 3, n ≥ 1. Also it cannot be ∆kn for any e e e k + n > 3. These observations are based on the fact that {A : A ∈ S∞ } is Σ21 , e (as a collection of sets of reals). 5.5. Reliable cardinals. According to [Bec79] an ordinal ë is called reliable if there is a regular scale hϕi i on a set A ⊆ R such that {ϕi (α) : i ∈ ù, α ∈ A} = ë. Clearly every Suslin cardinal is reliable. Conjecture 5.6 (AD). At least for cardinals ≤ κ R , the notions of being Suslin and reliable coincide. If one goes back to our analysis of the hyperprojective hierarchy in §3 it is easy to check that the following conjecture (which is motivated by some conjectures in [Kec78C]) implies the preceding one. Conjecture 5.7 (AD). If κ is a Suslin cardinal and S(κ) has the scale property, then any strictly increasing wellordered sequence of sets in S(κ) has length < κ + . This is not even known for κ = ℵ1 , i.e., for S(κ) = Σ12 . It is known by a e different argument that there are no reliable cardinals between ℵ1 and ℵù . But

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it is not known if there are any reliable cardinals between ä 13 = ℵù+1 and the e predecessor of ä 15 . e The proof that there are no reliable cardinals between ℵ1 and ℵù is based on the following more general fact. Proposition 5.8 (AD). Assume ë is reliable and there is a tree V on ù × ë, with sup{rank(V (α)) : V (α) is wellfounded} = ë+ . Then ë+ is regular.

REFERENCES

Howard S. Becker [Bec79] Some applications of ordinal games, Ph.D. thesis, UCLA, 1979. Alexander S. Kechris [Kec78C] On transfinite sequences of projective sets with an application to Σ12 equivalence relations, e North-Holland, 1978, Logic colloquium ’77 (A. Macintyre, L. Pacholski, and J. Paris, editors), pp. 155–160. Alexander S. Kechris, Eugene M. Kleinberg, Yiannis N. Moschovakis, and W. Hugh Woodin [KKMW81] The axiom of determinacy, strong partition properties and nonsingular measures, this volume, originally published in Kechris et al. [Cabal ii], pp. 75–100. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer, 1981. [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. Alexander S. Kechris, Robert M. Solovay, and John R. Steel [KSS81] The axiom of determinacy and the prewellordering property, In Kechris et al. [Cabal ii], pp. 101–125. Yiannis N. Moschovakis [Mos70A] Determinacy and prewellorderings of the continuum, Mathematical logic and foundations of set theory. Proceedings of an international colloquium held under the auspices of the Israel Academy of Sciences and Humanities, Jerusalem, 11–14 November 1968 (Y. Bar-Hillel, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam-London, 1970, pp. 24–62. [Mos74A] Elementary induction on abstract structures, North-Holland, 1974. [Mos78] Inductive scales on inductive sets, this volume, originally published in Kechris and Moschovakis [Cabal i], pp. 185–192. [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. [Mos83] Scales on coinductive sets, this volume, originally published in Kechris et al. [Cabal iii], pp. 77–85. Robert M. Solovay [Sol78B] The independence of DC from AD, In Kechris and Moschovakis [Cabal i], pp. 171–184.

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John R. Steel [Ste81A] Closure properties of pointclasses, In Kechris et al. [Cabal ii], pp. 147–163. DEPARTMENT OF MATHEMATICS CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CA 91125, USA

E-mail: [email protected]

THE AXIOM OF DETERMINACY, STRONG PARTITION PROPERTIES AND NONSINGULAR MEASURES

ALEXANDER S. KECHRIS, EUGENE M. KLEINBERG, YIANNIS N. MOSCHOVAKIS, AND W. HUGH WOODIN

In this paper we study the relationship between AD and strong partition properties of cardinals as well as some consequences of these properties themselves. Let us say that an uncountable cardinal κ has the strong partition property if (∀ì < κ)[κ → (κ)κì ], i.e., if for every partition of all increasing κ-term sequences into fewer than κ parts, there is a set C ⊆ κ of cardinality κ such that all the increasing sequences into C lie in the same part of the partition. We will show in §1 that the axiom of determinacy (AD) implies the existence of unboundedly many cardinals with the strong partition property below Θ, where Θ is the supremum of the ranks of the prewellorderings of the continuum. In §2, we will show that conversely, if there is a cardinal κ with the strong partition property above an ordinal ë, then every ë-Suslin set is determined. Combining these two results we obtain an elegant purely set-theoretic characterization of AD within R+ , the smallest admissible set containing the continuum; namely, in R+ , AD holds if and only if the power set of every ordinal exists, and there are arbitrarily large cardinals with the strong partition property. In §3 we will strengthen the main result of §1 to obtain from AD unboundedly many cardinals κ below ϑ, such that not only κ has the strong partition property but also {ë < κ : ë has the strong partition property} is stationary in κ (thus κ is also Mahlo). Finally, in §4 we will show that every Mahlo cardinal with the strong partition property carries a normal measure concentrating on regular cardinals. Up until now all the normal measures on cardinals κ, produced by AD, were of the “singular” type, i.e., for some regular ë < κ they concentrated on the ordinals of cofinality ë. A.S.K. was partially supported by NSF Grant MCS79-20465 and an A.P. Sloan Foundation Fellowship. E.M.K. was partially supported by NSF Grant MCS78-03744. Y.N.M. was partially supported by NSF Grant MCS78-02989. The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

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In matters of descriptive set theory, we will follow in general the terminology and notation of [Mos80]. We refer to [Kle77] for information concerning partition properties. Our work in this paper takes place in ZF + DC with all other hypothesis (mainly AD) stated explicitly. §1. A partition theorem. In the main result of this section, we will establish that AD implies the existence of many cardinals with the strong partition property. As usual, a space will be any product X = X1 × · · · × Xn of copies of ù and R = ù, a pointset is any subset of a space and a pointclass is any collection Γ of pointsets. A Γ-norm on a pointset P ⊆ X is any function ù

ϕ : P → Ord such that both

≤∗ϕ x x

and

≤∗ϕ <∗ϕ

<∗ϕ

are in Γ, where

y ⇔ x ∈ P ∧ [y 6∈ P ∨ ϕ(x) ≤ ϕ(y)], y ⇔ x ∈ P ∧ [y 6∈ P ∨ ϕ(x) < ϕ(y)].

We say the Γ has the prewellordering property if every pointset in Γ admits a Γ-norm. Recall from [Mos80] that a Spector pointclass closed under ∀R is a pointclass Γ which is closed under recursive substitutions, ∧, ∨, ∃ù , ∀ù and ∀R , which is ù-parametrized and which has the prewellordering property. We will need here a slightly stronger notion. A partial function f: X → ù is Γ-recursive (or in Γ), if Graph(f) = {(x, i) : f(x) = i} is in Γ. We say that Γ is closed under Kleene’s 3 E (deterministic quantification on R), if whenever f : X × R → ù is a Γ-recursive partial function, then the relation P(x) ⇔ (∀α)[f(x, α)↓ ] ∧ (∃α)[f(x, α) = 0] is in Γ. If Γ is closed under Kleene’s 3 E, then (trivially) Γ is closed under ∀R and ∆, ∆ are both closed under ∀R and ∃R ; and if Γ is closed under both ∀R and ∃R ,ethen (again trivially) Γ is closed under Kleene’s 3 E. By [Mos67], if A is any pointset, then the envelope Env(A, 3 E) is defined to be the class of all pointsets which are Kleene-semirecursive in 3 E and (the characteristic function of) A. This is a Spector pointclass closed under 3 E, which furthermore

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contains both A and its complement and is not closed under ∃R . These Kleene envelopes are very important for the applications of our present results, although one need not know any recursion in type-3 in order to follow the proofs in §1. We can now state our main result in this section. Theorem 1.1. Assume AD, let Γ be Spector pointclass closed under 3 E and let κ = o(∆) e = supremum of the ranks of prewellorderings of R in ∆ e be the ordinal associated with Γ; then κ has the strong partition property. This result implies that (granting AD), κ( 3 E) = the Kleene ordinal of the continuum = the ordinal associated with Env( 3 E) has the strong partition property, as does the closure ordinal of the continuum κ R = sup{î : î in the rank of a hyperprojective prewellordering of R}. It also implies that (under AD again), there are arbitrarily large cardinals with the strong partition property below Θ. One very important problem is whether the projective ordinals ä 12n+1 , n ≥ 1 e Clearly (in particular ä 13 ), have the strong partition property, granting AD. e our Theorem 1.1 does not tell us anything in this case, since the Spector pointclasses Γ = Π12n+1 are not closed under 3 E. Using a method of Martin (see [Kec78A, Lemma 11.1]), the proof of Theorem 1.1 is reduced to the problem of finding an appropriate coding (by elements of R) of functions f : κ → κ. Let us first reformulate Martin’s Lemma in a form convenient for our purposes here. Suppose Γ is a Spector pointclass closed under ∀R and with associated ordinal κ, let ϕ: S → κ be a Γ-norm which maps some set S ⊆ R in Γ onto κ and suppose that for each ε ∈ R we have a partial function fε : κ → κ; we will say that ϕ and {fε : ε ∈ R} define a good coding in Γ of the functions on κ to κ if conditions (i)–(iii) below hold, where for ordinals î, ϑ < κ, we let Cî (ε) ⇔ fε (î)↓ , Cî,ϑ (ε) ⇔ (∀î ′ ≤ î)[fε (î ′ )↓ ∧ fε (î ′ ) ≤ ϑ],

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(i) For each fixed î, ϑ < κ, the relation Cî,ϑ is in ∆. e (ii) There is some relation V (ε, α, â) in Γ˘ which computes the values of each e fε relative to the norm ϕ in the following sense: α ∈ S ∧ fε (ϕ(α))↓ ⇒ (∃â)V (ε, α, â) ∧ (∀â){V (ε, α, â) ⇒ [â ∈ S ∧ fε (ϕ(α)) = ϕ(â)]}. (iii) For every total function f : κ → κ, there is some ε ∈ R such that f = fε . Lemma 1.2. Assume AD and let Γ be a Spector pointclass closed under ∀R with ordinal κ, which admits a good coding of the functions on κ to κ; then κ has the strong partition property. Proof. Notice first that a good coding in Γ of the functions on κ to κ has the following additional boundedness property. (ii′ ) If A ∈ Γ˘ and for some î < κ, A ⊆ Cî , then e sup{fε (î) : ε ∈ A} < κ. This is because, by the definitions, if α ∈ S and ϕ(α) = î, then sup{fε (î) : ε ∈ A} ≤ sup{ϕ(â) : (∃ε)[ε ∈ A ∧ V (ε, α, â)]} and the set {â : (∃ε)[ε ∈ A ∧ V (ε, α, â)]} is a subset of S which lies in Γ˘ , e so that the standard boundedness argument for Spector pointclasses closed R under ∀ applies (see [Mos80, 4C.11]). Using this observation, we can prove the lemma by a small modification of the argument given in [Kec78A, Lemma 11.1], whose notation we will use. Let hAç : ç < ìi, where ì < κ, be a partition of κ [κ] into ì many pieces. For each ç < ì consider the game presented there with A replaced by ¬Aç . Call it Gç . If for some ç < ì player I has a winning strategy in Gç , then clearly there is a homogeneous set landing in Aç , so we are done. So assume player II has a winning strategy in each Gç , ç < ì, towards a contradiction. By the Coding Lemma [Mos80, 7D.5] let hSç : ç < ìi be a sequence of sets such that (a) Sç 6= 0, (b) ó ∈ Sç ⇒ ó is a winning strategy in Gç , S (c) ç<ì Sç ∈ Γ˘ , e and put G(î, ϑ) = sup{fó(ε) (î) + 1 : ó ∈

[ ç<ì

Sç ∧ ε ∈ Cî,ϑ }.

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By boundedness again G(î, ϑ) < κ, and from this it is easy to find a closed unbounded set D ⊆ κ such that κ

T

D↑ = D↑ ⊆ ¬Aç , ∀ç < ì;

thus D↑ ⊆ ç<ì ¬Aç = ∅, a contradiction. Here κ D↑ denotes the set of all increasing maps h : κ → D such that h(x) > sup{h(y) : y < x} and there is hînx : n ∈ ùix∈κ such that î0x < î1x < · · · < înx < · · · → h(x). ⊣ Thus to prove Theorem 1.1 it will be sufficient to show that if Γ is a Spector pointclass closed under Kleene’s 3 E, then Γ admits a good coding of the functions on its ordinal κ. The key to this proof is a strong (uniform) version of the Coding Lemma (I), [Mos80, 7D.5], which is implicit in the proof of that result, particularly as that was described in the original paper [Mos70A]. Consider then the customary language of analysis (or second-order number theory), where we have variables n, k, i, . . . over ù and α, â, ã, . . . over R, symbols for 0, 1, =, + and · on ù and application “α(t)” of variables over R on terms. We obtain an extension L(÷) of this language by adding new prime formulas of the form ÷(α, â) ≃ m; in the intended interpretation, ÷ will denote a partial function from R × R into ù in the obvious way. We will often denote a formula of L(÷) by a symbol such as ϕ(÷, x1 , . . . , xn ), which shows explicitly the occurrence of ÷; we will then use (ambiguously) the same symbol ϕ(÷, x1 , . . . , xn ) to denote the relation on ÷, x1 , . . . , xn defined by the formula. Let Σ11 (÷) be the smallest collection of formulas of L(÷) which contains all ordinary Σ11 formulas (with no ÷) and the prime formula ÷(α, â) ≃ m and which is closed under the positive operations ∧, ∨, ∃ù , ∀ù , and ∃R . It is important that we do not allow the negative formula ¬(÷(α, â) ≃ m) in Σ11 (÷). Lemma 1.3. For each Σ11 (÷) formula ϕ(÷, x1 , . . . , xn ), there are (ordinary) formulas ñ(x1 , . . . , xn ) and ó(x1 , . . . , xn , α, â, ã) such that for all ÷, x1 , . . . , xn , Σ11

ϕ(÷, x1 , . . . , xn ) ⇔ ñ(x1 , . . . , xn ) ∨ ∃α∃â∃ã ∀t[÷((α)t , (â)t ) ≃ ã(t)]


∧ ó(x1 , . . . , xn , α, â, ã) .

Proof. The proof of this is very easy, as in [Mos80, 7D.7] or [Mos70A, Lemma 1]—where the argument is not completely correct, as it does not work for the empty partial function ÷. ⊣

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From Lemma 1.3 and the known parametrization theorems for Σ11 , we can obtain easily a uniform (in ÷) parametrization theorem for Σ11 (÷). Lemma 1.4. For each product X = X1 × · · · × Xn of copies of ù and R, there is a fixed Σ11 (÷) formula G X (÷, ε, x) ⇔ G(÷, ε, x), where ε varies over ç, x varies over X and the following are true: (i) If ϕ(÷, x) is any Σ11 (÷) formula with x varying over X, then for some recursive ε and all ÷ and x, ϕ(÷, x) ⇔ G(÷, ε, x). (ii) For each pair of spaces Y, X, there is a recursive function S X,Y = S : R × Y → R, so that for all ÷, x, y (omitting superscripts) G(÷, ε, y, x) ⇔ G(÷, S(ε, y), x). After these preliminary lemmas, fix a prewellordering ≤ on a subset S of R with rank function ñ : S → ë, and let f : ën → ℘(Y) be any function which assigns to n-tuples from ë subsets (possibly empty) of a space Y. A choice set for f is any subset C ⊆ Rn × Y, such that (i) (α1 , . . . , αn , y) ∈ C ⇒ α1 , . . . , αn ∈ S ∧ y ∈ f(ñ(α1 ), . . . , ñ(αn )), (ii) f(î1 , . . . , în ) 6= ∅ ⇒ for each α1 , . . . , αn ∈ S with ñ(α1 ) = î1 , . . . , ñ(αn ) = în , there is some y such that (α1 , . . . , αn , y) ∈ C . In effect, C assigns to each î1 , . . . , în < ë such that f(î1 , . . . , în ) 6= ∅ a non-empty subset of f(î1 , . . . , în ). If α 6∈ S, put by convention ñ(α) = ∞ > ë and consider the partial function ÷ = ÷(≤) encoding ≤: ( 1, if α ∈ S ∧ ñ(α) ≤ ñ(â) ÷(α, â) ≃ 0, if â ∈ S ∧ ñ(â) < ñ(α).

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For each ì < ë consider also the approximations ( 1, if α ∈ S ∧ ñ(α) ≤ ì ∧ ñ(α) ≤ ñ(â), ÷ì (α, â) ≃ 0, if â ∈ S ∧ ñ(â) ≤ ì ∧ ñ(â) < ñ(α). Thus ÷ì ⊆ ÷ì′ if ì ≤ ì′ < ë and [

÷ì = ÷.

ì<ë

If f : ën → ℘(Y) as above and ì < ë, we let fì be the restriction of f to ì, ( f(î1 , . . . , în ), if î1 , . . . , în ≤ ì, fì (î1 , . . . , în ) = ∅, otherwise. Lemma 1.5 (The Uniform Coding Lemma). Assume AD, let ≤ be a prewellordering on a subset S of R with rank function ñ: S ։ ë and associated partial functions ÷ì (ì < ë), let f : ën → ℘(Y) be a function and let G(÷, ε, α1 , . . . , αn , y) be the universal Σ11 (÷) formula of Lemma 1.4. There is a fixed ε ∗ ∈ R such that for all ì < ë the relation Cì (α1 , . . . , αn , y) ⇔ G(÷ì , ε ∗ , α1 , . . . , αn , y) is a choice set for fì . Proof. Proof of this is a modification of the proof of [Mos80, 7D.5]. Take n = 1 for convenience, and in the notation of Lemma 1.5, call ε ∗ a uniform code of a choice set for f (relative to ≤). Assume, towards a contradiction, that this lemma fails and pick the least ë such that for some ≤ and some f as above there is no uniform code ε ∗ for f. It is easy to check that ë is limit. Fix such a counterexample ≤ and f for ë now. Consider then the game where player I plays α and player II plays â and  Player II wins iff α is not a uniform code for any fî , î < ë  (relative to ≤↾î = {(α, â) : α ≤ â ∧ ñ(â) ≤ î}) ∨ [α is a uniform code for some fî and there is ç > î such that â is a uniform code for fç ]. The proof can be now completed as in [Mos80, 7D.5]. ⊣ We can finally produce the desired coding of functions f : κ → κ by elements of R.

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Lemma 1.6. Assume AD, let Γ be a Spector pointclass closed under 3 E and let κ be the ordinal of Γ. Then Γ admits a good coding of the functions on κ to κ. Proof. Choose a set S ⊆ R in Γ and a Γ-norm ϕ: S ։ κ and let ≤ = ≤ϕ be the associated prewellordering, so that in the notation we have established, for ì < κ, ( 1, if α ∈ S ∧ ϕ(α) ≤ ì ∧ ϕ(α) < ϕ(â), ÷ì (α, â) ≃ 0, if â ∈ S ∧ ϕ(â) ≤ ì ∧ ϕ(â) < ϕ(α). (Again ϕ(α) = ∞, if α 6∈ S.) Let G(÷, ε, α, â) be the universal Σ11 (÷) formula of Lemma 1.4, and put for î < κ: fε (î)↓ ⇔ ∀α{[α ∈ S ∧ ϕ(α) = î] ⇒ ∃âG(÷î , ε, α, â)} ∧∀α∀α ′ ∀â∀â ′ {[α, α ′ ∈ S ∧ ϕ(α) = ϕ(α ′ ) = î ∧ G(÷î , ε, α, â) ∧ G(÷î , ε, α ′ , â ′ )]

(∗)

⇒ [â ∈ S ∧ â ′ ∈ S ∧ ϕ(â) = ϕ(â ′ )]}; If fε (î)↓ put fε (î) = the unique æ such that for some α, â ∈ S with ϕ(α) = î, ϕ(â) = æ, we have G(÷î , ε, α, â). We now verify for this ϕ and {fε : ε ∈ R} the conditions (i)–(iii) in the definition of a good coding. Condition (iii) is a direct consequence of the Uniform Coding Lemma, applied to f ∗ (î) = {â : â ∈ S ∧ f(î) = ϕ(â)}. To prove (i) and (ii) notice first that for each fixed î < κ, the relation ÷î (α, â) ≃ m is in ∆, and in fact the following stronger assertion is true: there is a Γ-recursive e function F (α, â, m, ã) such that partial ã ∈ S ⇒ F (α, â, m, ã)↓ ∧

[F (α, â, m, ã) ≃ 1 ⇔ ÷ϕ(ã) (α, â) ≃ m]; this follows directly from the fact that ϕ is a Γ-norm. Now using the hypothesis that Γ is closed under 3 E, it follows immediately that there are relations ˘ α, â, ã) in Γ and Γ˘ respectively, such that P(ε, α, â, ã) and P(ε, ˘ α, â, ã) ã ∈ S ⇒ [P(ε, α, â, ã) ⇔ P(ε, ⇔ G(÷ϕ(ã) , ε, α, â)],

DETERMINACY, PARTITION PROPERTIES, NONSINGULAR MEASURES

341

i.e., the key relation G(÷î , ε, α, â) is in ∆, for each î, uniformly in î. We can establish (ii) immediately by esetting ˘ α, â, α). V (ε, α, â) ⇔ P(ε, To check (i), notice that the relation fε (î)↓ ∧ fε (î) ≤ ϑ is defined by replacing in (∗) above the last clause â ∈ S ∧ â ′ ∈ S ∧ ϕ(â) = ϕ(â ′ ) by â ∈ S ∧ â ′ ∈ S ∧ ϕ(â) = ϕ(â ′ ) ≤ ϑ; for this too we can find some Q, Q˘ in Γ, Γ˘ respectively such that ˘ â ′ , ã) ã ∈ S ⇒[Q(â, â ′ , ã) ⇔ Q(â, ⇔ â ∈ S ∧ â ′ ∈ S ∧ ϕ(â) = ϕ(â ′ ) ≤ ϕ(ã)]. Putting these equivalences together then and using the closure properties of Γ, we find some R, R˘ in Γ, Γ˘ respectively so that ˘ ã, ã ′ ) ã, ã ′ ∈ S ⇒[R(ε, ã, ã ′ ) ⇔ R(ε, ⇔ fε (ϕ(ã))↓ ∧ fε (ϕ(ã)) ≤ ϕ(ã ′ )], from which (i) follows directly.

⊣

§2. Partition properties imply determinacy. Our main goal now will be to show that partition properties of cardinals imply the determinacy of Suslin sets. Combining this and the result of §1 we obtain an elegant set theoretical equivalent of AD within the smallest admissible set above the continuum. What actually comes up in the proof is a relatively weak consequence of the strong partition property which we establish first. Lemma 2.1. If κ → (κ)ì for each ì < κ, then we can associate with each wellordering W of rank ≤ κ a countably additive measure ìW on the set W [κ] of increasing W -term sequences in κ, so that the following coherence property holds: if W ′ ⊆ W is a subordering of W and A ⊆ W [κ], then ìW (A) = 1 ⇒ ìW ′ {f↾W ′ : f ∈ A} = 1. Proof. As usual, let W A↑ (A ⊆ κ) be the set of all increasing maps h from W into A with the property that h(x) > sup{h(y) : y < x} and there is hînx : n ∈ ùix∈W such that for each x ∈ W , î0x < î1x < · · · < înx < · · · → h(x). Then the partition property κ → (κ)ì , ∀ì < κ, easily implies that the following is a (countably additive) measure on W [κ]: ìW (A) = 1 iff there is a closed unbounded C ⊆ κ with

W

C ↑ ⊆ A.

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The coherence property for these measures follows immediately. ⊣ Recall that a set A ⊆ R is called ë-Suslin if there is a tree T on ù × ë such that A = p[T ] = {α ∈ R : ∃f ∈

ù

ë(α, f) ∈ [T ]}.

Theorem 2.2. Let ë be an ordinal and assume that there is a cardinal κ > ë with the property that κ → (κ)ì , ∀ì < κ. Then every ë-Suslin subset of R is determined. Proof. Let A = p[T ], where T is a tree on ù ×ë. For each ∅ 6= ó ∈ <ù ù = the set of all finite sequences from ù, let T ′ (ó) = {u : lh(u)(= m) ≤ length(ó) ∧ (ó↾m, u) ∈ T }. Let
α(0)

α(2)

...

A II

α(1)

α(3)

Player II wins ⇔ α 6∈ A ⇔ T (α) is wellfounded. ∗

In the auxiliary game A , player II makes additional moves as follows: I

α(0)

α(2)

...

A∗ II

α(1), f1

α(3), f3

To win, player II must insure that for each n and for each i ≤ n, f2i+1 is an order preserving map from Wα↾2i+1 into κ, and f1 ⊆ f3 ⊆ · · · ⊆ f2n+1 . Clearly this is a closed game for player II, so it is determined. Actually, since the auxiliary moves by player II come from a set which is not necessarily wellorderable, without the axiom of choice, one can only assert that either player I has a winning strategy or else player II has a “multiple-valued” winning

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strategy or quasistrategy. The easiest way to visualize a winning quasistrategy for player II is as a set Q of sequences hα(0); α(1), f1 ; . . . ; α(2n + 1), f2n+1 i closed under subsequences such that the winning conditions for player II are satisfied and such that (a) ∀α(0)∃α(1), f1 hα(0); α(1), f1 i ∈ Q, and (b) for every hα(0); α(1), f1 ; . . . ; α(2n + 1), f2n+1 i ∈ Q and every α(2n + 2) there is some α(2n + 3), f2n+3 such that hα(0); α(1), f1 ; . . . ; α(2n + 1), f2n+1 ; α(2n + 2); α(2n + 3), f2n+3 i ∈ Q. Since however player I plays only natural numbers, an easy application of DC shows that if player II has a winning quasistrategy he actually has a winning strategy. If now indeed player II has a winning strategy in the auxiliary game A∗ , he clearly has a winning strategy in the original game. So assume player I has a winning strategy ô ∗ in the auxiliary game. We will define a strategy ô for player I in A by using the measures ìó to “integrate out” player II’s auxiliary moves in the usual way. For any ó = hα(0), α(1), . . . , α(2n + 1)i and any order preserving f : T ′ (ó) → κ, let f1 , f3 , . . . , f2n+1 be the auxiliary moves induced by f, i.e., f2i+1 = f↾T ′ (α(0), . . . , α(2i + 1)) (i ≤ n). At position ó of the game A then, have player I play by ô(ó) = m ⇔ for ìó -almost all f, ô ∗ (α(0), α(1), f1 , . . . , α(2n + 1), f2n+1 ) = m. Assume towards a contradiction that player I follows ô but loses, in a run of A which produces the play α = α(0), α(1), α(2), . . . , so that T (α) is wellfounded, i.e., Wα is a wellordering. By the construction, for each n we have a set Bn ⊆

Wα↾(2n+1)

[κ]

of measure 1 in the canonical measure on this space, so that player I’s play by ô in A together with any member of Bn is a partial play by player I in A∗

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which follows ô ∗ . Now by the coherence property of the measures, for each n, the set {f ∈

Wα

[κ] : f↾T ′ (α(0), . . . , α(2n + 1)) ∈ Bn }

has measure 1, so the intersection of all these sets has measure 1 and is not empty; if f is in this set, then α(0), α(1), f↾T ′ (α(0), α(1)), . . . , α(2n + 1), f↾T ′ (α(0), . . . , α(2n + 1)), . . . is a play in A∗ , where player I plays by ô ∗ and loses, contrary to our assumptions. ⊣ Let R+ be the smallest admissible set containing R. By combining Theorems 1.1 and 2.2 we now have Theorem 2.3. The following are equivalent (i) R+ |= AD. (ii) R+ |= ∀κ(℘(κ) exists) and ∀ë∃κ > ë(κ has the strong partition property). Proof. Assume R+ |= AD. That R+ |= ∀κ(℘(κ) exists), follows immediately from the Coding Lemma [Mos80, 7D.5]. That R+ |= ∀ë∃κ > ë(κ has the strong partition property), follows from Theorem 1.1. Conversely assume (ii). By the proof of Theorem 2.2 we have that R+ |= ∀A ⊆ R(A is ë-Suslin for some ë ⇒ A is determined)

(∗)

(To adapt the proof of Theorem 2.2 in this context one needs to recall that for any admissible set A and any open game on a set x ∈ A, if the player that tries to win the open side has a winning quasistrategy, he has one which is in A.) Recall now ([Mos80, 7C] and [Mos74A, Chapter 9]) that the pointsets A ⊆ R in R+ are exactly the hyperprojective ones. So it is enough to show that every hyperprojective pointset A carries a hyperprojective scale. As a warmup let us see how to do this for the projective sets. Since every Σ12 set carries a Σ12 -scale we immediately have that it is determined e therefore by the Second Periodicity Theorem e by (∗). So we have Det(Σ12 ), and [Mos80, 6C.3 and 6C.1]eevery Σ14 set carries a Σ14 -scale. Again by (∗) we have e Theorem, every Σ16 set Det(Σ14 ), thus by another use ofethe Second Periodicity e e carries a Σ16 -scale, etc. We willeprove the general result about hyperprojective sets now, by defining an appropriate hierarchy on these sets and extending the above argument through the transfinite. In the rest of this proof we will follow the terminology and notation of [Mos80, 7C]. Recall that IND is the pointclass of all inductive pointsets. For g each ordinal î and each positive analytical operator φ(x, A), let φ î be its îth

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iterate, i.e., φ î (x) ⇔ φ(x, φ <î ), where φ <î (x) ⇔ ∃ç < îφ ç (x). For each limit ordinal ë < κ R , let INDë = {A : A is the continuous preimage of φ <ë , g for some positive analytical operator φ}. We summarize some basic structural properties of INDë . g Lemma 2.4. For each limit ë < κ R , the pointclass INDë is closed under g continuous substitutions, ∧, ∨, ∪ù , ∃R , has the prewellordering property and is R-parametrized. In fact there is a single “universal” positive analytical operator φ0 (ε, α, A), A varying over subsets of R × R, such that for each limit ë < κ R , φ0<ë is universal for the INDë subsets of R, and moreover for each positive analytical φ(α, B) (with Bg varying over subsets of R) there is a recursive ε such that for all limit ë < κ R , φ <ë (α) ⇔ φ0<ë (ε, α). For a proof see for example [Kec81B]. We define also the pointclass INDë+2n for each n ≥ 0 by the induction g INDë+2n+2 = ∃R ∀R INDë+2n . g g Again these are closed under ∧, ∨, ∪ù , ∩ù , ∃R and are R-parametrized. Let us in fact fix canonical universal sets Uë+2n for each INDë+2n as follows: g Uë = φ0<ë , Uë+2n+2 = {(ε, α) : ∃â∀ãUë+2n (ε, hα, â, ãi)}. These are universal sets in INDë+2n for the INDë+2n subsets of R, but as g a good universal system g build upon them in [Mos80, 3H.1] one can easily X X {Uë+2n }, where Uë+2n is universal for the subsets of a product space X in X , we call ε an INDë+2n -code INDë+2n . If A ⊆ X and x ∈ A ⇔ (ε, x) ∈ Uë+2n gthe closure g of A. One of the basic properties of a good universal system is that properties of INDë+2n are uniform in the INDë+2n -codes; see [Mos80, 3H.2]. g set carries a hyperprojective In order tog show that every hyperprojective scale, we will prove that if φ(α, A) is a positive Σ12 operator, then for each ë < κ R limit, φ <ë carries an INDë -scale and φ ë+n carries a INDë+2n+2 -scale for g g each n ≥ 0. Since every hyperprojective pointset is the continuous preimage î R of φ for some such φ and some î < κ , this will complete our proof. The proof will be by effective transfinite induction, on a suitable coding system for ordinals < κ R , using the Recursion Theorem. Let first ø(w, A), w ∈ R, A ⊆ R, be positive analytical such that if |w| = least î such that ø î (w),

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and W = ø ∞ , then | · | : W ։ κR and there are recursive pointsets Lim ⊆ R, Succ ⊆ R and a total recursive function pd such that w ∈ W ⇒ [Lim(w) ⇔ |w| is limit] ∧ [Succ(w) ⇔ |w| is successor] ∧ [Succ(w) ⇒ | pd(w)| = |w| − 1]. The following lemma will now complete the proof.

⊣

Lemma 2.5. Given φ(α, A) a Σ12 positive operator, there are scales hói<ë i, hóië+n i on φ <ë , φ ë+n respectively, for each limit ë < κ R , and partial recursive functions f0 , g0 , f1 , g1 such that, letting for each scale hói i on a pointset P, S(i, x, y) ⇔ x ≤∗ói y, T (i, x, y) ⇔ x <∗ói y, be its associated relations, we have w ∈ W ⇒ if |w| = ë + n, ë limit, n ≥ 0, then for n = 0 we have that f0 (w), g0 (w) are INDë -codes of the relations g associated with hói<ë i, while for all n, f1 (w), g1 (w) are INDë+2n+2 -codes of the relations associated with hóië+n i. g Proof. The scales are defined as in [Mos78] inductively: ó0<ë (α) = |α|φ = least î such that α ∈ φ î , |α|φ

<ë ói+1 (α) = h|α|φ , ói

(α)i;

while hóië+n i is defined by the Second Periodicity Theorem inductively on n starting from hói<ë i. For this to be a legitimate scale, it is sufficient to know that INDë has the scale property, by an argument similar to that given for the g sets in the early stages of the present proof. In any case, however, projective the associated relations of hói<ë i, hóië+n i are defined independently of this, so if we can prove the second assertion of the lemma about these associated relations, then by induction on ë we have immediately that INDë has the scale g property (recall that every A ∈ INDë is the continuous preimage of φ <ë , where φ can be taken to be Σ12 positiveg analytical), and thus the proof is complete. In order to construct f0 , f1 , g0 , g1 , we shall use effective transfinite induction. It is rather routine to define f1 (w), g1 (w) once all f0 (v), g0 (v), f1 (v), g1 (v) are known for |v| < |w|, thus we can concentrate on f0 (w), g0 (w). So assume |w| = ë and that all f0 (v), g0 (v), f1 (v), g1 (v) are known for |v| < ë.

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We write the first relation associated with hói<ë i (the calculation for the other one is similar): S(i, α, â) ⇔[i = 0 ∧ |α|φ ≤ |â|φ < ë] ∨ {i > 0 ∧ [(|α|φ < |â|φ < ë) |α|φ

∨ (|α|φ = |â|φ < ë ∧ ói

|α|φ

(α) ≤ ói

(â))]}.

By the Stage Comparison Theorem (see [Mos74A]) there are positive analytical φ1 , φ2 , φ3 , φ4 such that |α|φ ≤ |â|φ < ë ⇔ φ1<ë (α, â) ∧ φ <ë (â) |α|φ < |â|φ < ë ⇔ φ2<ë (α, â) ∧ φ <ë (â) |α|φ = |â|φ = |v|<ë ⇔ φ3<ë (α, v) ∧ φ3<ë (â, v) ∧ φ4<ë (v, α) ∧ φ4<ë (v, â) ∧ ø <ë (v). Thus S(i, α, â) ⇔[i = 0 ∧ φ1<ë (α, â) ∧ φ <ë (â)] ∨ {i > 0 ∧ [φ2<ë (α, â) ∧ φ <ë (â)] ∨ (∃v)[φ3<ë (α, v) ∧ φ3<ë (â, v) ∧ φ4<ë (v, α) ∧ |v|

|v|

φ4<ë (v, â) ∧ ø <ë (v) ∧ ói (α) ≤ ói (â)]}. By induction hypothesis, if |v| < ë, say |v| = ë′ + n ′ , we have that f1 (v) is a ′ ′ |v| INDë′ +2n′ +2 code of the first relation associated with hóië +n i = hói i. It is g therefore enough to establish the following fact: There is a positive analytical ϑ(ε, α, w, A), A ⊆ R3 , and ε0 recursive, such that for each î, |w| ≤ î ∧ (ε, α) ∈ U|w|′ ⇔ (ε0 , ε, α, w) ∈ ϑî , where |w|′ = ë + 2n + 2, if |w| = ë + n. By the Simultaneous Induction Lemma ([Mos74A, 1C.1] or [Mos80, 7C.11]), it is actually enough to construct a system ϑ1 , ϑ2 , ϑ3 such that ϑ3î (ε, α, w) ⇔ |w| ≤ î ∧ (ε, α) ∈ U|w ′ | . Let first, by the Stage Comparison Theorem, ÷(ε, α, w, A) be a positive analytical operator such that letting |ε, α|φ0 = least î such that (ε, α) ∈ φ0î , we have |ε, α|φ0 ≤ î ∧ (w 6∈ W ∨ |ε, α|φ0 < |w|) ⇔ ÷ î (ε, α, w).

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The system is now as follows: ϑ1î (w) ⇔ ø(w, ϑ1<î ) ϑ2î (ε, α, w) ⇔ ÷(ε, α, w, ϑ2<î ) ϑ3î (ε, α, w) ⇔ ø(w, ϑ1<î ) ∧ {[Lim(w) ∧ ∃â∀ãϑ2<î (ε, hα, â, ãi, w)] ∨ [Succ(w) ∧ ∃â∀ãϑ3<î (ε, hα, â, ãi, pd(w))]}.

⊣

In view of the preceding result, we are naturally led to pose the following question. Open Problem 2.6. Is it true that in L(R), AD is equivalent to the existence of arbitrarily large below Θ cardinals with the strong partition property? And we conclude this section by proving an extension of Theorem 2.2 for games on ordinals. If A ⊆ ù ë is a pointset on ë, we say that A is í-Suslin if there is a tree T on ë × í such that
f ∈ A ⇔ ∃g ∈ ù í (f, g) ∈ [T ] . We now have Theorem 2.7. Let ë, í be ordinals and assume that there is a cardinal κ > ë, í such that ì

κ → (κ)î , ∀ì, î < κ. If A ⊆

ù

ë is such that both A and ¬A are í-Suslin, then A is determined.

Proof. We try to imitate the argument in the proof of Theorem 2.2. Consider the game G(A) associated with A, and, picking a tree T with p[T ] = A, consider the auxiliary game G∗ (A, T ) defined there. If player I has a winning strategy in G∗ (A, T ) we are done. So assume player II has a winning quasistrategy in G∗ (A, T ). This we cannot necessarily convert into a winning strategy however, since ë will be in general uncountable and we do not have ACë . So we cannot immediately conclude that player I will have a winning strategy for G(A). Here we have to use the fact that ¬A is also í-Suslin. We fix a tree S so that p[S] = ¬A and we consider an auxiliary game G′ (A, S), which is as before except that it is now player I’s responsibility to make the extra moves. If player II has a winning strategy in G′ (A, S), we easily again conclude that player II has a winning strategy in G(A). Else player I has a winning quasistrategy in G′ (A, S). Using DC now it is easy to see that there are runs in G∗ (A, T ) and G′ (A, S) with the same real part α(0), α(1), . . . in which player II (resp. player I) has followed his winning quasistrategy in G∗ (A, T ) (resp. G′ (A, S)). This is clearly a contradiction. ⊣

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From this result and Theorem 1.1 we immediately obtain the following, where we call A ⊆ ù ë Suslin if it is ì-Suslin for some ì. Theorem 2.8. Assume AD. Let ë < Θ and let A ⊆ and ¬A are Suslin. Then A is determined.

ù

ë be such that both A

This strengthens the last conclusion of [Mos81, Theorem 2.2], by removing the restriction that ë ≤ κ R . It also provides an alternative proof of that result, which however does not give the key definability estimates of the original argument that are needed in the rest of that paper. (Note that a set A ⊆ ù ë is Suslin iff it admits uniform semiscales in the terminology of [Mos81].) §3. Mahlo cardinals from determinacy. In this section we prove a strengthening of Theorem 1.1 for Spector pointclasses Γ which are additionally closed under both ∃R and ∀R . The result is as follows. Theorem 3.1. Assume AD, let Γ be a Spector pointclass closed under both ∃R and ∀R and let κ = o(∆) be the ordinal associated with Γ. Then κ has e and moreover {ë : ë < κ ∧ ë has the strong the strong partition property, partition property} is stationary in κ. In particular, κ is Mahlo. Among other things, this implies that there are arbitrarily large cardinals κ with the above properties below Θ. This is because for each pointset A the pointclass IND(A) = all pointsets which are inductive in A, is a Spector pointclass containing both A and its complement and is closed under ∃R and ∀R . Also it follows that κ R is Mahlo and in fact {ë : ë < κ R ∧ ë has the strong partition property} is stationary in κ R (since κ R = o(∆), where e Γ = IND = IND(∅)). Note here that by [Ste81A], κ( 3 E) is not Mahlo. In fact Moschovakis has conjectured that κ( 3 E) is the first (weakly) inaccessible cardinal (granting AD of course). It can be seen that κ R is not the first Mahlo cardinal. It is conjectured that the first Mahlo cardinal is the ordinal of 4 S (= the type 4 superjump), granting AD again (see [Har73] for results about the superjump). We now give the proof of Theorem 3.1. Proof. It is enough by Theorem 1.1 to show that {o(∆∗ ) : Γ∗ is a Spector pointclass closed under 3 E and contained in ∆} e e is stationary in κ = o(∆). e Let f : κ → κ. We shall find Γ∗ as above, such that o(∆∗ ) is closed under e f, i.e., î < o(∆∗ ) ⇒ f(î) < o(∆∗ ). e e

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Fix S ⊆ R and a Γ-norm φ : S ։ κ. Then in the notation of the proof of Lemma 1.6 find ε such that f = fε . From the definition of fε it is obvious that there are two relations R, R˘ in Γ, Γ˘ e e respectively such that ˘ α ∈ S ⇒[R(α, â) ⇔ R(α, â)

(∗)

⇔ â ∈ S ∧ f(φ(a)) = φ(â)]. ˘ (Otherwise To simplify the notation, assume that actually R, R˘ are in Γ, Γ. replace everywhere below Γ by Γ(α0 ) for some appropriate parameter α0 .) A type 3 object is a function 3

F : ù Z × Y → ù,

where Z, Y are product spaces. For example Kleene’s 3 E is the type 3 object 3

E : ùR → ù

given by 3

E(h) =

(

0, if ∃α[h(α) = 0] 1, if ∀α[h(α) 6= 0].

We say that a Spector pointclass Γ is closed under 3 F if for each h : X×Z → ù, a Γ-recursive partial function, the relation P(i, x, y) ⇔ ∀z[h(x, z)↓ ] ∧

3

F (ëzh(x, z), y) = i

3

is in Γ. To each such F we associate the envelop Env( 3 E, 3 F ) = all pointsets which are Kleene-semirecursive in 3 E, 3 F. This is a Spector pointclass closed under 3 E, 3 F and in fact by [Mos74B] it is the smallest one with these properties, thus if Γ is a Spector pointclass closed under 3 E, 3 F , then Env( 3 E, 3 F ) ⊆ Γ. Moreover by [Mos67] if A ∈ Env( 3 E, 3 F ), there is B ∈ Env( 3 E, 3 F ) with x 6∈ A ⇔ ∃α(x, α) ∈ B. This immediately implies that if Γ is a Spector pointclass closed under both ∃R , ∀R and if 3 F is a type 3 object, so that Γ is closed under 3 F , then Env( 3 E, 3 F ) ⊆ ∆. This is the key fact that we will need below. Consider now the following type 3 object 3 F associated with f : κ → κ: 3

F : ù R×R × R2 → ù,

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351

and for each h ∈ ù R×R , α ∈ R  0, if h is the characteristic function of a prewellordering     ≤ on R such that |≤| < κ, α, â ∈ S 3 F (h, α, â) =  ∧ φ(α) < φ(â) < f(|≤|);    1, otherwise. Lemma 3.2. Γ is closed under 3 F . Granting the lemma consider Γ∗ = Env( 3 E, 3 F ). By our preceding remarks, it is enough to show that κ ∗ = o(∆∗ ) is closed e If î < κ ∗ , under f. For that let S ∗ ∈ Γ∗ and φ ∗ : S ∗ ։ κ ∗ be a Γ∗ -norm. let h : R × R → ù be the characteristic function of ≤ = {(α, â) : φ ∗ (α) ≤ φ ∗ (â) < î}. Clearly h is in ∆∗ . Since Γ∗ is closed under 3 F , this implies that e ≺ = {(α, â) : 3 F (h, α, â) = 0} is also in ∆∗ . But e

≺ = {(α, â) : φ(α) < φ(â) < f(î)}.

Thus f(î) is the length of a ∆∗ prewellordering and so f(î) < κ ∗ = o(∆∗ ), e e which is what we wanted to prove. To verify the lemma note that if h : X × R × R → ù has graph in Γ and for some x ∈ X, ∀z[h(x, z)↓ ], then one can check in a ∆(x) way, uniformly in x, whether or not ëzh(x, z) is a characteristic function of a prewellordering of R. If this is the case and we denote by ≤x = ≤ this prewellordering, then by [Mos80, 4C.14] we can find, effectively in x, a ãx = ã ∈ S with φ(ã) > |≤|, and thus using the Coding Lemma [Mos80, 7D.5] we can see that {ä : φ(ä) = |≤|} is in ∆(x), uniformly in x again. From this and (∗) it is immediate that we can check in a ∆(x) way, uniformly in x, whether φ(α) < φ(â) < f(|≤|) is true or not, thus completing the proof that Γ is closed under 3 F . ⊣ §4. Nonsingular measures from AD. It has been known for quite some time that AD implies the existence of many measurable cardinals below Θ (see for example [Kle77] or [Kec78A]). All the normal measures on cardinals κ that were produced, however, were of the “singular” type, i.e., they concentrated on the ordinals of cofinality ë, for some regular ë < κ. It has thus been open whether one could obtain from AD measurable cardinals below Θ that carry measures which concentrate on regular cardinals. Our result below provides (when combined with Theorem 3.1) many such examples below Θ. Theorem 4.1. Assume that κ is a Mahlo cardinal satisfying the strong partition property. Then there exists a nontrivial κ-additive normal measure on κ, giving the regular cardinals less than κ measure 1.

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Proof. Let Q be any stationary set of regular cardinals less than κ. We will construct a normal measure giving Q measure 1. First let us define Qˆ to be the set of those î in Q such that some closed unbounded subset of î is disjoint from Q. Note that Qˆ consists of the difference between Q and the result of applying the Mahlo operation to Q. We can now define a function ìQ : κ 2 → 2 by ˆ ìQ (X ) = 1 iff for some closed unbounded subset C of κ, X ⊇ C ∩ Q. Lemma 4.2. ìQ is a non-trivial κ-additive measure on κ. Proof of Lemma 4.2. Let UQ = {A ⊆ K : ìQ (A) = 1}. We will show that UQ is a nonprincipal κ-additive ultrafilter on κ. It is first important to note that Qˆ is stationary. For given any closed unbounded set C , the least limit point of C in Q is a member of Qˆ ∩ C . Since, now, Qˆ is stationary, UQ must be nonprincipal. Also, clearly, UQ is a filter. Suppose X ⊆ κ is given. We must show that X ∈ UQ or ¬X ∈ UQ , and so let us define a partition F : κ [κ] → 2 by F (Y ) = 1 iff the least limit point of Y which is a member of Q is also a member of X . (Identify here κ [κ] with the set of subsets of κ of cardinality κ.) Let D be a set of cardinality κ homogeneous for F and let us suppose that F [ κ [D]] = {0}. ˆ where Dl.p. denotes the set of limit points of D. Claim. X ⊇ Dl.p. ∩ Q, ˆ Let Cî be a closed Proof of Claim. Suppose î is a limit point of D in Q. unbounded subset of î disjoint from Q. By an interlocking argument, we can thin out D ∩ î and Cî simultaneously to (D ∩ î)′ and Cî′ respectively such that (D ∩ î)′ and Cî′ have the same limit points and such that each is unbounded in î. Since Cî ∩ Q = ∅, there is no limit point of (D ∩ î)′ less than î in Q, and so î is the least limit point of (D ∩ î)′ in Q. Since F ((D ∩ î)′ ∪ (D − î)) = 0, î ∈ X . ⊣ (Claim) By a similar claim, if we had that F [ κ [D]] = {1}, then we would have that ˆ Thus either X or ¬X is in UQ , and so UQ is an ultrafilter. ¬X ⊇ Dl.p. ∩ Q. T Suppose now that Xî ∈ UQ for each î <κ ϑ < κ. We must show that î<ϑ Xî ∈ UQ . Let us define a partition G : [κ] → ϑ + 1 as follows: given Y ∈ κT [κ], let ç be the least limit point of Y which is a member of Q. Then if ç ∈ î<ϑ Xî we define G(Y ) to be ϑ. Otherwise, G(Y ) is the least î < ϑ such that ç 6∈ Xî . Let, now, E be a set of cardinality κ homogeneous for G. Then ifT G[ κ [E]] = {ϑ}, an argument similar to one used above would show that î<ϑ Xî ⊇ T ˆ and hence that El.p. ∩ Q, î<ϑ Xî ∈ UQ . Otherwise, our argument above would show that ¬Xî ⊇ El.p. ∩ Qˆ for some î < ϑ. Since for some closed

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ˆ we would have El.p. ∩ C ∩ Qˆ ⊆ Xî ∩ ¬Xî = ∅. unbounded C, Xî ⊇ C ∩ Q, T ˆ Since El.p. ∩ C ∩ Q is stationary, this is impossible. Thus î<ϑ Xî ∈ UQ . ⊣ Lemma 4.3. ìQ is normal. Proof of Lemma 4.3. Suppose f : κ → κ and ìQ ({î : f(î) < î}) = 1. ˆ and let Let C be a closed unbounded set such that {î : f(î) < î} ⊇ C ∩ Q, ˆ → 2 by us define a partition F : κ [C ∩ Q] F (Y ) = 0 iff the value of f on the least limit point of Y which is a member of Q is less than the least member of Y. Let D be homogeneous for F and of cardinality κ. Claim. F [ κ [D]] = {0}. ˆ the least limit point of D which is Proof of Claim. Since D ⊇ C ∩ Q, a member of Q, ç, is sent by f to some ordinal less than ç. Clearly ç is still the least limit point of D \ (f(ç) + 1) which is a member of Q, and so F (D \ (f(ç) + 1)) = 0. Thus F [ κ [D]] = {0}. ⊣ (Claim) By an argument T similar to one used earlier, the above claim yields that ˆ ⊆ D. Since ìQ is κ-additive, ìQ (f −1 {î0 }) = 1 for some f[Dl.p. ∩ Q] T î0 < D. ⊣ By Lemmas 4.2 and 4.3, ìQ is a κ-additive nontrivial normal measure on ˆ ìQ (Q) = 1, and our proof is complete. κ. Since Q ⊇ κ ∩ Q, ⊣ With a bit of extra effort, we could carry out the above proof starting with any stationary set Q. Thus for uncountable κ satisfying the strong partition property, each stationary set gets measure 1 under some normal measure. For these and further results we refer to [Kle82]. REFERENCES

Leo A. Harrington [Har73] Contributions to recursion theory in higher types, Ph.D. thesis, MIT. Alexander S. Kechris [Kec78A] AD and projective ordinals, In Kechris and Moschovakis [Cabal i], pp. 91–132. [Kec81B] Suslin cardinals, κ-Suslin sets, and the scale property in the hyperprojective hierarchy, this volume, originally published in Kechris et al. [Cabal ii], pp. 127–146. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer. Eugene M. Kleinberg [Kle77] Infinitary combinatorics and the axiom of determinacy, Lecture Notes in Mathematics, vol. 612, Springer-Verlag.

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[Kle82] A measure representation theorem for strong partition cardinals, The Journal of Symbolic Logic, vol. 47, no. 1, pp. 161–168. Yiannis N. Moschovakis [Mos67] Hyperanalytic predicates, Transactions of the American Mathematical Society, vol. 129, pp. 249–282. [Mos70A] Determinacy and prewellorderings of the continuum, Mathematical logic and foundations of set theory. Proceedings of an international colloquium held under the auspices of the Israel Academy of Sciences and Humanities, Jerusalem, 11–14 November 1968 (Y. Bar-Hillel, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam-London, pp. 24–62. [Mos74A] Elementary induction on abstract structures, North-Holland. [Mos74B] Structural characterizations of classes of relations, Generalized recursion theory. Proceedings of the 1972 Oslo symposium (Jens Erik Fenstad and Peter G. Hinman, editors), Studies in Logic and the Foundations of Mathematics, North-Holland, pp. 53–79. [Mos78] Inductive scales on inductive sets, this volume, originally published in Kechris and Moschovakis [Cabal i], pp. 185–192. [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam. [Mos81] Ordinal games and playful models, In Kechris et al. [Cabal ii], pp. 169–201. John R. Steel [Ste81A] Closure properties of pointclasses, In Kechris et al. [Cabal ii], pp. 147–163. DEPARTMENT OF MATHEMATICS CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CA 91125, USA

E-mail: [email protected] DEPARTMENT OF MATHEMATICS STATE UNIVERSITY OF NEW YORK BUFFALO, NY 14260, USA

E-mail: [email protected] DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES, CA 90095, USA

E-mail: [email protected] DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720, USA

E-mail: [email protected]

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ALEXANDER S. KECHRIS AND W. HUGH WOODIN

Historical Remark. This paper was circulated in handwritten form in March 1982 and contained Sections 1-4 below. There is an additional Section 5 containing information about the solution of a problem mentioned in the last paragraph of Section 1. §1. Statements of results. 1.1. Let L(R) be the smallest inner model of ZF containing the set of reals R. Let Θ be the sup of the ordinals which are subjective images of R. Theorem 1.1. Assume ZF+DC. Then the following are equivalent: (1) L(R)  AD; (2) L(R)  ∀ë < Θ∃κ(ë < κ ∧ κ → (κ)κ ); (3) L(R)  ∀ë < Θ∃κ(ë < κ ∧ κ → (κ)ë ). It has been already known, see [KKMW81], that ZF+DC+AD ⇒ ∀ë < Θ∃κ(ë < κ ∧ κ → (κ)κ ). Recall that A ⊆ R is called ë-Suslin if there is a tree T on ù × ë such that A = p[T ] = {α ∈ R : ∃f ∈ ù ë∀n(hα↾n, f↾ni ∈ T )}, and A is called Suslin if it is ë-Suslin for some ë. Again in [KKMW81] it is proved that (in ZF+DC): ∀ë < Θ∃κ(ë < κ ∧ κ → (κ)ë ) ⇒ “Every Suslin set of reals is determined”. Thus Theorem 1.1 is an immediate consequence of Theorem 1.2. Assume ZF+DC. Then the following are equivalent: (1) L(R)  Every Suslin set of reals is determined; (2) L(R)  AD. Preparation of this paper is partially supported by NSF Grants DMS-0455285 (A.S.K.) and DMS-0355334 (W.H.W.). The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

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The proof of Theorem 1.2 relies heavily on work of [Ste83A], which analyzes the propagation of the scale property in L(R), using the fine structure of this inner model. Also in the proof of Theorem 1.2, as well as Theorems 1.8 and 1.10 below, essential use is made of a technique of [Mar83A] for handling finite strings of alternating quantifiers over R via iterated products of the Martin measure on the Turing degrees. There are several strengthenings and corollaries of the preceding theorems. For instance in Theorem 1.2 we can weaken (1) as follows: Call a pointclass Γ reasonable if it is closed under ∨, ∧, bounded number quantification, cone tinuous substitutions and ∃R or ∀R and is R-parametrized. We say that Γ e is determined if every A ∈ Γ, A ⊆ R is determined. Then we have that, in e ZF+DC, the following are equivalent: (1) L(R)  Every reasonable Γ with the scale property is determined; e (2) L(R)  AD. The following is also an immediate corollary of Theorem 1.2. Corollary 1.3. Assume ZF+DC. If there is a cardinal κ with κ ≥ ΘL(R) and κ → (κ)κ (or even κ → (κ)ë , ∀ë < ΘL(R) ), then L(R)  AD. In particular Con(ZF+DC+∃κ(κ ≥ ΘL(R) ∧ κ → (κ)κ )) implies Con(ZF+DC+AD). Note that in Corollary 1.3, κ is assumed to have the partition property in the universe, not necessarily in L(R). Conceivably this result could be used to demonstrate the consistency of ZF+DC+AD from appropriate large cardinal assumptions. The following is a related open problem. Question 1.4. Is ZF+DC+∃κ(κ ≥ Θ ∧ κ → (κ)κ ) consistent? If so, what is its strength, in particular does it imply Con(ZF+DC+ADR )? We describe next some corollaries about the relativization of partition properties to L(R). Corollary 1.5. Let ó be any of the statements ∀ë < Θ∃κ(ë < κ ∧ κ → (κ)κ ) or ∀ë < Θ∃κ(ë < κ ∧ κ → (κ)ë ). Then, in ZF+DC, ó relativizes to L(R) (i.e., ó implies ó L(R) ). Also, in ZF+DC, ó implies L(R)  ù1 → (ù1 )ù1 . The following is a rather curious reflection property of L(R). Corollary 1.6. Assume ZF+DC+V = L(R). Let κ = ä 21 (i.e., the least î e such that Lî (R) ≺1 L(R)). Then, if κ → (κ)κ , we have that ë → (ë)ë for ë many small ë < κ, e.g., ë = ù1 , and also ë → (ë) for cofinally in Θ many ë’s. Finally we examine the relation between partition properties and partition measures in L(R). Let us say that a cardinal κ has a partition measure if

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for each wellordering W of order type ≤ κ there is a countably additive measure ìW on the set W [κ] of increasing W -term sequences from κ with the following coherence property: If W ′ ⊆ W and A ⊆ W [κ] then ìW (A) = 1 implies ìW ′ ({f↾W ′ : f ∈ A}) = 1. Clearly κ → (κ)κ implies that κ has a partition measure, but the existence of a partition measure on κ does not imply that κ → (κ)κ . Steel [Ste80] shows that ZF+DC+AD ⇒ ∀κ < Θ(κ has a partition measure). Also in [KKMW81] it is shown that, in ZF+DC, if ∀ë < Θ∃κ(κ > ë ∧ κ has a partition measure), then every Suslin set is determined. So we have the following Corollary 1.7. The following are equivalent, in ZF+DC, (1) L(R)  ∀ë < Θ∃κ(ë < κ ∧ κ has a partition measure); (2) L(R)  ∀ë < Θ∃κ(ë < κ ∧ κ → (κ)κ ); (3) L(R)  AD. Thus globally in L(R) partition measures are equivalent to partition properties. 1.2. The key new tool in the proof of Theorem 1.2 is a “transfer” theorem of the form (in ZF+DC): Det(Γ1 ) ⇒ Det(Γ2 ), e e where Γ1 , Γ2 are pointclasses with appropriate properties and interrelatione eΓ is “much bigger” than Γ . A basic instance of this type of ships, and 2 1 e theorem cane be stated as follows: For each pointclass Γ closed under continuous substitutions, ∧, ∨, let Σ∗n (Γ) e e be defined as follows, e n Σ∗1 (Γ) = A ⊆ X : For some B ∈ Γ, C ∈ Γ˘ e e e e
o x ∈ A ⇔ ∃y[B(x, y) ∧ C (x, y)] , Π∗n (Γ) = {¬A : A ∈ Σ∗n (Γ)}, e e e e Σ∗n+1 (Γ) = {∃yA(x, y) : A ∈ Π∗n (Γ)}. e e e e A typical, for our purposes, example of such a Γ is Γ = IND, the pointclass e e g of inductive sets of reals, in which case we just write We have now:

Σ∗n ≡ Σ∗n (IND), Π∗n ≡ Π∗n (IND). e e g e e g

Theorem 1.8. Assume ZF+DC. Let Γ be a pointclass closed under continuous substitutions, ∧, ∨, ∃R , ∀R . If Γ hase the prewellordering and uniformizae that for all n < ù, we have Det(Σ∗ (Γ)), tion properties, then Det(∆) implies n e e e where as usual ∆ = Γ ∩ Γ˘ . e e e

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Corollary 1.9. Assume ZF+DC. Then Det(HYP) implies that for all n < ] ˘ ). ù, we have Det(Σ∗n ) (where HYP = IND ∩ IND e g g ] [Har78] and [MarA] have shown that, in ZF+DC, Det(Π11 ) is equivalent to e Det(< ù 2 -Π11 ). Abbreviate for convenience e Mn ≡ ù · n-Π11 . e Let a be the game quantifier and for any pointclass Γ let e am Γ = aaa . . . a Γ , e | {z } e m

where aΓ = {aαA(x, α) : A ∈ Γ}. Another instance of the “transfer” theorem eprovides a generalization ofethe Harrington-Martin result to all even levels of the projective hierarchy. Theorem 1.10. Assume ZF+DC. Then for k ≥ 1, Det(∆12k ) ⇔ ∀n < ù(Det(a2k−1 Mn )). e For instance, if k = 1 this says that, in ZF+DC, Det(∆12 ) ⇔ ∀n < ù(Det(aMn )). e In particular, combining this with Martin’s result in [Mar78], one sees that, in ZFC, if there is an iterable j : Vë → Vë , then for all n < ù, Det(aMn ) holds. The classes aMn are substantially bigger than Σ12 , but still well within ∆13 . e e the consistency of Det(Π 1 Because it turns out that (Det(ù 2 +1-Π11 ) proves 1) e e (see [MarA]), the Martin-Harrington result is basically best possible. Similarly Theorem 1.10 is basically best possible, since from [KS85] it follows that Det(a2k−1 (ù 2 + 1-Π11 )) proves the consistency of ∀n < ù(Det(a2k−1 (ù · e n-Π11 ))). e It should be also the case that Theorem 1.10 holds at odd levels as well, i.e., that for k ≥ 1, we have Det(Π12k+1 ) ⇔ ∀n < ù(Det(a2k Mn )) e but at this writing we do not know a proof of that. It is also interesting to see whether the techniques used here can be employed to give a different proof of the Harrington-Martin Theorem (the only known proof of it goes through the theory of sharps, while our methods are direct and purely analytical).

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§2. Proof of Theorem 1.8. We start from the proof of Theorem 1.8, since it will be used in the proof of Theorem 1.2. Assume Det(∆). Let A ⊆ R × eR be a Σ∗n (Γ) game, say n = 2 for notational simplicity. Thus e e A(α, â) ⇔ ∃ã∀ä[R(α, â, ã, ä) ∧ S(α, â, ã, ä)], where R ∈ Γ, S ∈ Γ˘ . Let e e P(i, x) ⇔ [i = 0 ∧ R(x)] ∨ [i 6= 0 ∧ ¬S(x)]. So P ∈ Γ. Let ϕ : P ։ κ be a Γ-norm on P. Note that κ is limit and e e cf(κ) > ù, since otherwise P and therefore R, S and A are in ∆ and there is nothing to e prove. For î < κ, let Rî (x) ⇔ ϕ(0, x) < î, Sî (x) ⇔ ¬[ϕ(1, x) < î]. Thus R=

[

Rî , S =

î<κ

\

Sî ,

î<κ

î < ç ⇒ Rî ⊆ Rç ∧ Sî ⊇ Sç [ \ ë < κ limit ⇒ Rë = Sî , Së = Sî . î<ë

î<ë

Thus A(α, â) ⇔ ∃ã∀ä[∃îRî (α, â, ã, ä) ∧ ∀îSî (α, â, ã, ä)], where from now on, in order to simplify the notation, we agree that ordinal variables î, ç, ë vary over ordinals < κ. Also we agree that: ó denotes a strategy for player I, ô denotes a strategy for player II, ó ∗ â denotes player I’s answer if player II plays â, ô ∗ α denotes player II’s answer if player I plays α. Assume now A is not determined, towards a contradiction. Then ∀ó, ô∃α, â[A(α, ô ∗ α) ∧ ¬A(ó ∗ â, â)], or explicitly ∀ó, ô∃α, â{∃ã∀ä[∃îRî (α, ô ∗ α, ã, ä) ∨ ∀îSî (α, ô ∗ α, ã, ä)] ∧ ∀ã∃ä[∀î¬Rî (ó ∗ â, â, ã, ä) ∧ ∃î¬Sî (ó ∗ â, â, ã, ä)]}.

(1)

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We use below letters c, d, e, . . . to denote Turing degrees. Abbreviate: ∀∗ cP(c) ⇔ ∃c0 ∀c ≥ c0 P(c), where c ≤ d is the usual partial order on the Turing degrees. If ã is a real, let also ã ≤ d mean that ã ≤T ä for any ä of Turing degree d. Note now the following simple implications: ∃ã∀äP(ã, ä) ⇒ ∀∗ c∀∗ d∃ã≤c∀ä≤dP(ã, ä). ∃ã∀äP(ã, ä) ⇒ ∀∗ c∃ã≤c∀∗ d∀ä≤dP(ã, ä) ⇒ ∀∗ c∀∗ d∃ã≤c∀ä≤dP(ã, ä).)

(Proof.

∀ã∃äP(ã, ä) ⇒ ∀∗ c∀∗ d∀ã≤c∃ä≤dP(ã, ä). (Proof.

∀ã∃äP(ã, ä) ⇒ ∀∗ c∀ã≤c∀∗ d∃ä≤dP(ã, ä) ⇒ ∀∗ c∀∗ d∀ã≤c∃ä≤dP(ã, ä), ∗ since ∀n[∀ ePn (e)] ⇒ ∀∗ e[∀nPn (e)].)

Thus we have from (1), using these manipulations: ∀ó, ô∃α, â{∀∗ c∀∗ d[∃ã ≤ c∀ä ≤ d(∃îRî (α, ô ∗ α, ã, ä) ∨ ∀îSî (α, ô ∗ α, ã, ä))] ∗

∗

∧ ∀ c∀ d[∀ã ≤ c∃ä ≤ d(∀î¬Rî (ó ∗ â, â, ã, ä) ∧ ∃î¬Sî (ó ∗ â, â, ã, ä))]}. Note now that ∀∗ c∀∗ dP1 (c, d) ∧ ∀∗ c∀∗ dP2 (c, d) ⇒ ∀∗ c∀∗ d[P1 (c, d) ∧ P2 (c, d)], thus ∀ó, ô∃α, â{∀∗ c∀∗ d[∃ã≤c∀ä≤d(∃îRî (α, ô ∗ d, ã, ä) ∨ ∀îSî (α, ô ∗ α, ã, ä)) ∧∀ã≤c∃ä≤d(∀î¬Rî (ó ∗ â, â, ã, ä)

(2)

∧ ∃î¬Sî (ó ∗ â, â, ã, ä))]}. ¨ Now fix a real x ∈ R. Then by a simple Skolem-Lowenheim argument, we can find a countable set of reals M ⊆ R containing x, closed under pairing and also downward closed under ≤T (i.e., y ∈ M ∧ z ≤T y ⇒ z ∈ M ), such that ∀ó, ô ∈ M ∃α, â ∈ M {∀∗ c∀∗ d[∃ã≤c∀ä≤d(∃îRî (α, ô ∗ d, ã, ä) ∨ ∀îSî (α, ô ∗ α, ã, ä)) ∧∀ã≤c∃ä≤d(∀î¬Rî (ó ∗ â, â, ã, ä) ∧ ∃î¬Sî (ó ∗ â, â, ã, ä))]}.

(3)

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But since M is countable we have again ∀ó, ô ∈ M ∃α, â ∈ M ∀∗ c∀∗ dR(α, â, ó, ô, c, d) ⇒ ∀∗ c∀∗ d∀ó, ô ∈ M ∃α, â ∈ MR(α, â, ó, ô, c, d). Thus finally we conclude that ∀∗ c∀∗ d∀ó, ô ∈ M ∃α, â ∈ M [∃ã ≤ c∀ä ≤ d(∃îRî (α, ô ∗ α, ã, ä) ∨ ∀îSî (α, ô ∗ α, ã, ä)) ∧∀ã ≤ c∃ä ≤ d(∀î¬Rî (ó ∗ â, â, ã, ä)

(4)

∧ ∃î¬Sî (ó ∗ â, â, ã, ä))]. Note now that since the Rî ’s, ¬Sî ’s are increasing and î varies over the ordinal κ of cofinality > ù, we must have ∃ã ≤ c∀ä ≤ d(∃îRî (α, ô ∗ α, ã, ä) ∨ ∀îSî (α, ô ∗ α, ã, ä)) ⇒ ∃ç∃ã ≤ c∀ä ≤ d(Rç (α, ô ∗ α, ã, ä) ∨ ∀îSî (α, ô ∗ α, ã, ä)),

(5)

and similarly ∀ã ≤ c∃ä ≤ d(∀î¬Rî (ó ∗ â, â, ã, ä) ∧ ∃î¬Sî (ó ∗ â, â, ã, ä)) ⇒ ∃ç∀ã ≤ c∃ä ≤ d(∀î¬Rî (ó ∗ â, â, ã, ä) ∨ ¬Sç (ó ∗ â, â, ã, ä)). Applying the same procedure once again, we see that ∀ó, ô ∈ M ∃α, â ∈ M [∃ã ≤ c∀ä ≤ d(∃îRî (α, ô ∗ α, ã, ä) ∨ ∀îSî (α, ô ∗ α, ã, ä)) ∧∀ã ≤ c∃ä ≤ d(∀î¬Rî (ó ∗ â, â, ã, ä) ∧ ∃î¬Sî (ó ∗ â, â, ã, ä))]. ⇒ ∃ç∀ó, ô ∈ M ∃α, â ∈ M [∃ã≤c∀ä≤d(Rç (α, ô ∗ α, ã, ä) ∨ ∀îSî (α, ô ∗ α, ã, ä)) ∧∀ã≤c∃ä≤d(∀î¬Rî (ó ∗ â, â, ã, ä) ∧ ¬Sç (ó ∗ â, â, ã, ä))]. Thus, in particular, ∀ó, ô ∈ M ∃α, â ∈ M [∃ã ≤ c∀ä ≤ d(∃îRî (α, ô ∗ α, ã, ä) ∨ ∀îSî (α, ô ∗ α, ã, ä)) ∧∀ã ≤ c∃ä ≤ d(∀î¬Rî (ó ∗ â, â, ã, ä) ∧ ∃î¬Sî (ó ∗ â, â, ã, ä))]. ⇒

(6)

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∃ç∀ó, ô ∈ M ∃α, â ∈ M [∃ã ≤ c∀ä ≤ d(Rç (α, ô ∗ α, ã, ä) ∨ Sç (α, ô ∗ α, ã, ä)) ∧∀ã ≤ c∃ä ≤ d(¬Rç (ó ∗ â, â, ã, ä)

(7)

∧ ¬Sç (ó ∗ â, â, ã, ä))]. Abbreviate Qç (ó, ô, α, â, c, d) ⇔ ∃ã ≤ c∀ä ≤ d(Rç (α, ô ∗ α, ã, ä) ∨ Sç (α, ô ∗ α, ã, ä)) ∧∀ã ≤ c∃ä ≤ d(¬Rç (ó ∗ â, â, ã, ä) ∧ ¬Sç (ó ∗ â, â, ã, ä)). So from (4) and (7) we have finally ∀∗ c∀∗ d∃ç∀ó, ô ∈ M ∃α, â ∈ M (Qç (ó, ô, α, â, c, d)). Call a countable M ⊆ R x-good if it contains x ∈ R, is closed under pairing, and downward closed under ≤T . We think of M as a real via some appropriate coding. So we have shown that ∀x∃M (M is x-good ∧ ∀∗ c∀∗ d∃ç∀ó, ô ∈ M ∃α, â ∈ M (Qç (ó, ô, α, â, c, d))). Now an easy calculation shows that the relation P(x, M ) ⇔ M is x-good ∧ ∀∗ c∀∗ d∃ç∀ó, ô ∈ M ∃α, â ∈ M (Qç (ó, ô, α, â, c, d)) is in Γ. Since Γ has the uniformization property, there is a function F : R → R in ∆ esuch thatefor each x ∈ R, if e F (x) = Mx , we have P(x, Mx ). Put M(x) = {My : y ≤T x ∧ My is x-good}. Then clearly ∀x∀∗ c∀∗ d∃ç∃M ∈ M(x)∀ó, ô ∈ M ∃α, â ∈ M (Qç (ó, ô, α, â, c, d)). Let D be the set of Turing degrees, and define for each real x ∈ R a partial function fx : D × D → κ by letting fx (c, d) be the least ç such that ∃M ∈ M(x)∀ó, ô ∈ M ∃α, â ∈ M (Qç (ó, ô, α, â, c, d)) if there is one. Thus ∀x∀∗ c∀∗ d(fx (c, d) is defined). Consider now the following game: I α, x0

II x1 , â

(8)

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Player I plays α, x0 ∈ R; player II plays x1 , â ∈ R. Let x = hx0 , x1 i. Then player I wins iff ∀∗ c∀∗ d∃ã ≤ c∀ä ≤ d[Rfx (c,d) (α, â, ã, ä) ∨ Sfx (c,d) (α, â, ã, ä)]. An easy calculation shows that this game is in ∆, so it is determined. We e I has a winning strategy will derive a contradiction from this fact. Say player ó. ¯ (The argument is entirely similar if player II has a winning strategy.) We need first the following Lemma 2.1. ∀v∃w ≥T v∀w ′ ≥T w∀∗ c∀∗ d(fw ′ (c, d) ≥ fw (c, d)). Proof. Assume not, and pick v such that ∀w ≥T v∃w ′ ≥T w¬∀∗ c∀∗ d(fw ′ (c, d) ≥ fw (c, d)). Note that if c0 ∈ D is such that ∀c ≥ c0 ∀∗ d(fw (c, d) is defined ∧ fw ′ (c, d) is defined), then the relation c ≥ c0 ∧ ∀∗ d(fw (c, d) ≥ fw (c, d)) is in ∆, so by using ∆-Turing Determinacy, we have e e ¬∀∗ c∀∗ d(fw ′ (c, d) ≥ fw (c, d)) ⇒ ∀∗ c¬∀∗ d(fw ′ (c, d) ≥ fw (c, d)). Similarly ∀∗ c¬∀∗ d(fw ′ (c, d) ≥ fw (c, d)) ⇒ ∀∗ c∀∗ d¬(fw ′ (c, d) ≥ fw (c, d)) ⇔ ∀∗ c∀∗ d(fw ′ (c, d) < fw (c, d)). So we have ∀w ≥T v∃w ′ ≥T w∀∗ c∀∗ d(fw ′ (c, d) < fw (c, d )). Thus we can choose v ≤T w0 ≤T w1 ≤T w2 ≤T . . . such that ∀n∀∗ c∀∗ d(fwn+1 (c, d) < fwn (c, d)), so that ∀∗ c∀∗ d∀n(fwn+1 (c, d) < fwn (c, d)), a contradiction. By Lemma 2.1, find a real w ≥T ó¯ such that ∀w ′ ≥T w∀∗ c∀∗ d(fw ′ (c, d) ≥ fw (c, d)). Consider now the strategy ó0 for player I in a game of the form I α

II â

⊣

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given by ó0 ∗ â = (ó¯ ∗ hw, âi)0 . Recall that ó¯ ∗ hw, âi = hα, x0 i for some α, x0 and thus α = (ó¯ ∗ hw, âi)0 . Thus, if in the game (8) player II plays w, â, player I answers by ó0 ∗ â, x0 (for some x0 ). Clearly ó0 ≤T w. By the definition of fw (c, d), we have ∀∗ c∀∗ d∃M ∈ M(w)∀ó, ô ∈ M ∃α, â ∈ M (Qfw (c,d) (ó, ô, α, â, c, d)). Since M(w) is a countable set it follows easily from ∆-Turing Determinacy e that ∃M ∈ M(w)∀∗ c∀∗ d∀ó, ô ∈ M ∃α, â ∈ M (Qfw (c,d) (ó, ô, α, â, c, d)).

(We are using here that ∀∗ c∀∗ d∃mR(m, c, d) ⇒ ∃m∀∗ c∀∗ dR(m, c, d), if R is in ∆, a fact which follows from ∆-Turing Determinacy.) So fix M0 ∈ e that e M(w) such ∀∗ c∀∗ d∀ó, ô ∈ M0 ∃α, â ∈ M0 (Qfw (c,d) (ó, ô, α, â, c, d)).

(9)

Since M0 ∈ M(w), M0 is w-good, so in particular, as ó0 ≤T w, ó0 ∈ M0 . Recalling the definition of Qç , we have ∀∗ c∀∗ d∀ó ∈ M0 ∃â ∈ M0 ∀ã ≤ c∃ä ≤ d [¬Rfw (c,d) (ó ∗ â, â, ã, ä) ∧ ¬Sfw (c,d) (ó ∗ â, â, ã, ä)], so also, by taking ó = ó0 , ∀∗ c∀∗ d∃â ∈ M0 ∀ã ≤ c∃ä ≤ d [¬Rfw (c,d) (ó0 ∗ â, â, ã, ä) ∧ ¬Sfw (c,d) (ó0 ∗ â, â, ã, ä)]. But M0 is countable, so, exactly as before, we conclude that ∃â ∈ M0 ∀∗ c∀∗ d∀ã ≤ c∃ä ≤ d [¬Rfw (c,d) (ó0 ∗ â, â, ã, ä) ∧ ¬Sfw (c,d) (ó0 ∗ â, â, ã, ä)]. Thus fix â0 ∈ M0 such that ∀∗ c∀∗ d∀ã ≤ c∃ä ≤ d[¬Rfw (c,d) (ó0 ∗ â0 , â0 , ã, ä) ∧ ¬Sfw (c,d) (ó0 ∗ â0 , â0 , ã, ä)]. Put α0 = ó0 ∗ â0 . Then, for some x0 , I α0 , x0

II w, â0

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365

is a run of the game (8) in which player I followed ó. ¯ Thus, if x = hx0 , wi, we have ∀∗ c∀∗ d∃ã ≤ c∀ä ≤ d[Rfx (c,d) (α0 , â0 , ã, ä) ∨ Sfx (c,d) (α0 , â0 , ã, ä)]. This contradicts immediately the above, if we can only show that ∀∗ c∀∗ d(fx (c, d) = fw (c, d)). This can be proved as follows: First, since w ≤T x, we have from the lemma that ∀∗ c∀∗ d(fw (c, d) ≤ fx (c, d)), so it is enough to check that ∀∗ c∀∗ d(fw (c, d) ≥ fx (c, d)). Since M0 is w-good and â0 ∈ M0 , ó0 ∈ M0 , we also have α0 , x0 ∈ M0 thus x ∈ M0 . So M0 is x-good. Since M0 ∈ M(w), M0 = My for some y ≤T w ≤T x, thus also M0 ∈ M(x). But from (9) ∀∗ c∀∗ d∀ó, ô ∈ M0 ∃α, â ∈ M (Qfw (c,d) (ó, ô, α, â, c, d)), therefore, in particular, ∀∗ c∀∗ d∃M ∈ M(x)∀ó, ô ∈ M0 ∃α, â ∈ M (Qfw (c,d) (ó, ô, α, â, c, d)). But by definition, for almost all c and d, fx (c, d) is the least ç such that (∃M ∈ M(x)∀ó, ô ∈ M ∃α, â ∈ M (Qç (ó, ô, α, â, c, d))), so for almost all c and d, we have fx (c, d ) ≤ fw (c, d), and the proof of Theorem 1.8 is complete. Now consider the case Γ = IND. The preceding proof shows that if IND has the uniformization property, then Det(HYP) implies ∀n < ù(Det(Σ∗n )), i.e., we have a lightface version. However one needs Det(HYP) to prove that ] this lightface IND has the uniformization property. So we can ask whether implication is true outright. At this moment we can only establish this for n = 1, and we do not know if it is true for higher n. The argument is as follows: Assume Det(HYP). Let A(α, â) ⇔ ∃ã[∃îRî (α, â, ã) ∧ ∀îSî (α, â, ã)], be a typical Σ∗1 set, where Rî , Sî are as in the preceding proof. Assume again A is not determined towards a contradiction. Thus ∀ó, ô∃α, â[A(α, ô ∗ α) ∧ ¬A(ó ∗ â, â)].

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Fix a real x ∈ R. Then there is an x-good M such that ∀ó, ô ∈ M ∃α, â ∈ M [A(α, ô ∗ α) ∧ ¬A(ó ∗ â, â)]. Thus, ∀ô ∈ M ∃α ∈ M ∃ã[∃îRî (α, ô ∗ α, ã) ∧ ∀îSî (α, ô ∗ α, â)] ∧∀ó ∈ M ∃â ∈ M ∀ã[∀î¬Rî (ó ∗ â, â, ã) ∨ ∃î¬Sî (ó ∗ â, â, ã)]. From the first conjunct we have again some î0 such that ∀ô ∈ M ∃α ∈ M ∃ã[Rî0 (α, ô ∗ α, ã) ∧ ∀îSî (α, ô ∗ α, ã)].

(∗)

Then from the second conjunct we have ∀ó ∈ M ∃â ∈ M ∀ã[¬Rî0 (ó ∗ â, â, ã) ∨ ∃î¬Sî (ó ∗ â, â, ã)], so that by standard reflection properties of IND (recall that IND is the class of Σ1 over the next admissible of R sets of reals), we have some î1 with ∀ó ∈ M ∃â ∈ M ∀ã[¬Rî0 (ó ∗ â, â, ã) ∨ ¬Sî1 (ó ∗ â, â, ã)]. thus going back to (∗) again we have also ∀ô ∈ M∃α ∈ M∃ã[Rî0 (α, ô ∗ α, ã) ∧ Sî1 (ó ∗ â, â, ã)]. So we have proved that ∃î0 , î1 ∀ó, ô ∈ M ∃α, â ∈ M [∀ã(¬Rî0 (ó ∗ â, â, ã) ∨ ¬Sî1 (ó ∗ â, â, ã)) ∧ ∃ã(Rî0 (α, ô ∗ α, ã) ∧ Sî1 (α, ô ∗ α, ã))]. So ∀x∃î0 , î1 ∃M [M is x-good ∧ ∀ó, ô ∈ M ∃α, â ∈ M (Qî0 ,î1 (ó, ô, α, â))], where Qî0 ,î1 (ó, ô, α, â) ⇔∀ã[¬Rî0 (ó ∗ â, â, ã) ∨ ¬Sî1 (ó ∗ â, â, ã)] ∧ ∃ã[Rî0 (α, ô ∗ α, ã) ∧ Sî1 (α, ô ∗ α, ã)]. Define now f : R → κ by f(x) = least hî0 , î1 i∃M [M is x-good ∧ ∀ó, ô ∈ M ∃α, â ∈ M (Qî0 ,î1 (ó, ô, α, â))], and write f(x) = hf(x)0 , f)x1 i. Then f ∈ HYP, and if we consider the game I α, x0

II x1 , â

where x = hx0 , x1 i; player I wins iff ∃ã[Rf(x)0 (α, â, ã) ∧ Sf(x)1 (α, â, ã)], we obtain a contradiction, exactly as before, noticing that this is now a HYP game.

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§3. Proof of Theorem 1.10. This is a variant of the proof in §2. We take k = 2, n = 2 for notational simplicity. We thus assume below Det(∆14 ). First note that, using ∀α(α # exists), we have that every aM2 sete A can be represented in the form x ∈ A ⇔ L[x] |= ϕ(x, u1 , u2 ), where ϕ is a formula of set theory and u1 , u2 , . . . the uniform indiscernibles; see [Mar83A]. Thus a typical a3 M2 game has the form A(α, â) ⇔ aãaä(L[α, â, ã, ä] |= ϕ(α, â, ã, ä, u1 , u2 )). Assume now, towards a contradiction, that this game is not determined. Thus we have ∀ó, ô∃α, â [aãaä(L[α, â, ã, ä] |= ϕ(α, ô ∗ α, ã, ä, u1 , u2 )) ∧ ¬aãaä(L[ó ∗ â, â, ã, ä] |= ϕ(ó ∗ â, â, ã, ä, u1 , u2 ))]. Now aãP(ã) ⇔ ∃ñ∀εP(ñ · ε), where if ñ is a strategy for player I and ε is what player II plays, ñ · ε is the run of the game with player I following ñ and player II playing ε. Thus aãP(ã) ⇒ ∀∗ c∃ñ ≤ c∀ε ≤ cP(ñ · ε). Abbreviate: aã ≤ c(R(ã)) ⇔ ∃ñ ≤ c∀ε ≤ cR(ñ · ε). Thus we have aãaä(L[α, ô ∗ α, ãä] |= ϕ(α, ô ∗ α, ã, ä, u1 , u2 )) ⇒ ∀∗ c∀∗ daã ≤ caä ≤ d(L[α, ô ∗ α, ã, ä] |= ϕ(α, ô ∗ α, ã, ä, u1 , u2 )). Now note that S(ã) ⇔ aä(L[ó ∗ â, â, ã, ä] |= ϕ(ó ∗ â, â, ã, ä, u1 , u2 )) is a a M2 game, thus in particular ∆14 , so it is determined. So e ¬aãaä(L[ó ∗ â, â, ã, ä] |= ϕ(ó ∗ â, â, ã, ä, u1 , u2 )) 2

⇒ a′ ã¬aä(L[ó ∗ â, â, ã, ä] |= ϕ(ó ∗ â, â, ã, ä, u1 , u2 )), where a′ ã¬P(ã) ⇔ player II has a winning strategy in {ã : P(ã)}. Applying this once more we have ¬aãaä(L[ó ∗ â, â, ã, ä] |= ϕ(ó ∗ â, â, ã, ä, u1 , u2 )) ⇒ a′ ãa′ ä(L[ó ∗ â, â, ã, ä] |= ¬ϕ(ó ∗ â, â, ã, ä, u1 , u2 )).

(10)

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Thus, exactly as before, ¬aãaä(L[ó ∗ â, â, ã, ä] |= ϕ(ó ∗ â, â, ã, ä, u1 , u2 )) ⇒ ∀∗ c∀∗ da′ ã ≤ ca′ ä ≤ d (L[ó ∗ â, â, ã, ä] |= ¬ϕ(ó ∗ â, â, ã, ä, u1 , u2 )), where a′ ã ≤ c has the obvious meaning as in (10), by interchanging the roles of players player I and player II. So finally we have ∀ó, ô∃α, â∀∗ c∀∗ d[aã ≤ caä ≤ d(L[α, ô ∗ α, ã, ä] |= ϕ(α, ô ∗ α, ã, ä, u1 , u2 )) ∧ a′ ã ≤ ca′ ä(L[ó ∗ â, â, ã, ä] |= ¬ϕ(ó ∗ â, â, ã, ä, u1 , u2 ))]. Now fix x ∈ R and find M which is x-good such that ∀∗ c∀∗ d∀ó, ô ∈ M ∃α, â ∈ M [aã ≤ caä ≤ d(L[α, ô ∗ α, ã, ä] |= ϕ(α, ô ∗ α, ã, ä, u1 , u2 )) ∧ a′ ã ≤ ca′ ä ≤ d(L[ó ∗ â, â, ã, ä] |= ¬ϕ(ó ∗ â, â, ã, ä, u1 , u2 ))]. By a simple indiscernibility argument, this implies that ∀∗ c∀∗ d∃î0 , î1 , î2 ∀ó, ô ∈ M ∃α, â ∈ M [aã ≤ caä ≤ d(Lî0 [α, ô ∗ α, ã, ä] |= ϕ(α, ô ∗ α, ã, ä, î1 , î2 )) ∧ a′ ã ≤ ca′ ä ≤ d(Lî0 [ó ∗ â, â, ã, ä] |= ¬ϕ(ó ∗ â, â, ã, ä, î1 , î2 ))]. Here and below ordinal variables î0 , î1 , î2 are supposed to vary over ù1 and it is implicit in the above notation that î1 < î2 < î0 . Abbreviate Qî0 ,î1 ,î2 (ó, ô, α, â, c, d) ⇔ aã ≤ caä ≤ d(Lî0 [α, ô ∗ α, ã, ä) |= ϕ(α, ô ∗ α, ã, ä, î1 , î2 ) ∧ a′ ã ≤ ca′ ä ≤ d(Lî0 [ó ∗ â, â, ã, ä] |= ¬ϕ(ó ∗ â, â, ã, ä, î1 , î2 )). Thus we have shown that ∀x∃M (M is x-good ∧ ∀∗ c∀∗ d∃î0 , î1 , î2 ∀ó, ô ∈ M ∃α, â ∈ M (Qî0 ,î1 ,î2 (ó, ô, α, â, c, d))) Since the relation P(x, M) following ∀x∃M is Σ14 , we have, by the uniformization theorem for Σ14 , a function F : R → R in ∆14 such that if F (x) = Mx , then P(x, Mx ). (That P(x, M ) is Σ14 follows from the fact that the expression R(c, d) following ∀∗ c∀∗ d is Σ12 , thus e ¬∀∗ dR(c, d) ⇔ ∀∗ d¬R(c, d), so ∀∗ dR(c, d) is Π13 , therefore e

∀∗ c∀∗ dR(c, d)

is Σ14 , uniformly in all the parameters involved, i.e. R is Σ14 ). e

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369

Let again M(x) = {My : y ≤T x ∧ My is x-good}. Thus ∀x∀∗ c∀∗ d∃î0 , î1 , î2 ∃M ∈ M(x)∀ó, ô ∈ M ∃α, â ∈ M (Qî0 ,î1 ,î2 (ó, ô, α, â, c, d)). Define as before a partial function by letting fx (c, d) be the least hî0 , î1 , î2 i such that ∃M ∈ M(x)∀ó, ô ∈ M ∃α, â ∈ M (Qî0 ,î1 ,î2 (ó, ô, α, â, c, d)), and let fx (c, d) = hfx (c, d)0 , fx (c, d)1 , fx (c, d)2 i. Then consider the game I α, x0

II x1 , â

where x = hx0 , x1 i; player I wins iff ∀∗ c∀∗ daã ≤ caä ≤ d(Lfx (c,d)0 [α, â, ã, ä] |= ϕ(α, â, ã, ä, fx (c, d)1 , fx (c, d)2 )). This is a ∆14 game, so it is determined, and we proceed as in §2 to derive a contradiction. Note. In the proof of the analog of Lemma 2.1, we make use of Det(Π13 ) e and similarly for the arguments following it. Let us remark that from the proof we just gave, we have the following lightface version of Theorem 1.10: For all k ≥ 1, ∀m(Det(a2k−2 Mm )) ⇒ [Det(∆12k ) ⇔ Det(a2k−1 Mn )], ∀n < ù. In particular, for k = 1, we have Det(∆12k ) ⇔ Det(aMn ), ∀n < ù granting ∀α(α # exists). If our conjecture at the end of § 1 holds, then the above hypothesis should be just equivalent to Det(Π12k−1 ). e

§4. Proof of Theorem 1.2. Assume AD+DC+V=L(R) and every Suslin set of reals is determined. We shall prove by induction on î ∈ Ord, that every set in Lî (R) is determined. Abbreviate this by Det(Lî (R)).

We can assume î ≥ ù + ù. (In [KKMW81], it is actually proved that, under our hypotheses, we have Det(LκR (R)), where κ R is the first R-admissible ordinal.) So assume Det(Lî (R)). We want to prove Det(Lî+1 (R)) We can assume that there is a new set of reals in Lî+1 (R), otherwise there is nothing to prove.

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Following the analysis of [Ste83A], this will be done by considering two main cases on î. From now on our blanket assumptions on î are Det(Lî (R)) and ∃A ⊆ R(A ∈ Lî+1 (R) \ Lî (R)). Case I. There is no ordinal î ′ < î such that Lî ′ (R) ≺R 1 Lî (R), where this notation means that for every Σ1 formula ϕ and every real parameter r, Lî ′ (R) |= ϕ(r) ⇔ Lî (R) |= ϕ(r). Lemma 4.1. There is a partial map p : R ։ Lî (R), whose graph is Σ1 over Lî (R) with only real parameters. e Proof. First note that there is a total function f : î ×î ×R ։ Lî (R) which is Σ1 over Lî (R) with only parameter the largest limit ordinal ë ≤ î −1, if such e i.e., if î is successor. In this case, since ë < î, we have L (R) 6≺R L (R), exists, ë î 1 so there is a Σ0 formula ϕ(r, x) and a real parameter r0 such that Lî (R) |= ∃xϕ(r0 , x) but Lë (R) |= ∀x¬ϕ(r0 , x). Say x0 ∈ Lë+n+1 (R), ë + n + 1 ≤ î, be such that Lî (R) |= ϕ(r0 , x0 ). Let ϑ be a formula such that x0 = {y ∈ Lë+n (R) : Lë+n (R) |= ϑ(y, z0 )}, where z0 ∈ Lë+n (R). Then we have in Lî (R): ë′ = ë ⇔ ë′ is limit ∧ ∃M ∃z∃x(M = Lë′ +n (R) ∧ z ∈ M ∧ x = {y ∈ M : M |= ϑ(y, z)} ∧ ϕ(r0 , z)). So {ë} is Σ1 in Lî (R) with only real parameters. Thus we have that there is total f : î e× î × R ։ Lî (R) which is Σ1 over Lî (R) with only real parameters. e function for L (R) as follows: Let Using f we can define a Σ1 -Skolem î ø(i, x, y) be a universal Σ1 formula. Say ø(i, x, y) ⇔ ∃zϑ0 (i, x, y, z), ϑ0 ∈ Σ0 . Define in Lî (R) g1 (i, x) = ç1 ⇔ ∃ç2 ∃æ1 ∃æ2 [∃r∃sϑ0 (i, f(ç1 , ç2 , r), f(æ1 , æ2 , s)) ∧ ∀(ç1′ , ç2′ , æ1′ , æ2′ )
THE EQUIVALENCE OF PARTITION PROPERTIES AND DETERMINACY

Put S(i, z) =

(

f(g1 (i, (z)0 ), g2 (i, (z)0 ), (z)1 ) 0

371

if (z)1 ∈ R, otherwise.

Here z = h(z)0 , (z)1 , (z)2 i is some Σ1 coding function in Lî (R). Thus S is a partial function with graph Σ1 over Lî (R) involving only real parameters, e such that ∃yø(i, x, y) implies ∃rø(i, x, S(i, hx, ri)).

Thus we have R ⊆ S(ù × R) = M ≺1 Lî (R), where M ≺1 N means that M is a Σ1 substructure of N , i.e., for every Σ1 formula ϕ and every x0 ∈ M, M |= ϕ(x0 ) ⇔ N |= ϕ(x0 ). Collapse now M to ′ some Lç (R) for ç ≤ î. Clearly Lç (R) ≺R 1 Lî (R), so ç = î. Since S = S↾M ′ ∗ is Σ1 over M with only real parameters, the image of S , say S , under the e collapse is Σ1 over Lî (R) with only real parameters and S(ù × R) = Lî (R). e Let p(r) = S ∗ (r(0), ëtr(t + 1)).

Then p : R ։ Lî (R) is a partial map with Σ1 over Lî (R) graph involving only e real parameters. ⊣ We shall also need the following basic result of [Ste83A]: Theorem 4.2. [Ste83A] For any infinite cardinal î, if Det(Lî (R)) holds, the pointclass of sets of reals which are Σ1 over Lî (R) with only real parameters e has the scale property. Let now A ⊆ R belong to Lî+1 (R). Say, for notational convenience, A is Π3 over Lî (R). Then for some ϕ ∈ Σ0 and some z0 ∈ Lî (R): e r ∈ A ⇔ Lî (R) |= ∀x0 ∃x1 ∀x2 ϕ(r, x0 , x1 , x2 , z0 ). Here and below letters r, s, t will be reserved for reals only. Using the projection map p : R ։ Lî (R), this can be written as r ∈ A ⇔ Lî (R) |= ∀r0 ∃r1 ∀r2

k _

[øi (r, r0 , r1 , r2 , s0 ) ∧ ÷i (r, r0 , r1 , r2 , s0 )],

i=1

where øi ∈ Σ1 , ÷1 ∈ Π1 and s0 ∈ R. Let us call Γ the class of all sets of reals which are Σ1 over Lî (R) with e so that Γ has the scale property by Steel’s e Theorem. Let us real parameters, e consider now the following subcases on î. Subcase 1. î is limit and for every Σ0 formula ϕ and every s ∈ R Lî (R) |= ∀r∃xϕ(r, x, s) ⇒ ∃ç < î(Lç (R) |= ∀r∃xϕ(r, x, s)).

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Then clearly Γ is closed under ∧, ∨, ∀R , ∃R , continuous substitutions and e has the scale property, so in particular the prewellordering and uniformization properties. Thus by Theorem 1.2 Det(Γ) ⇒ Det(Σ∗n (Γ)), ∀n < ù. e e e But since every set in Γ is Suslin, we have Det(Γ). So we have Det(Σ∗n (Γ)). But from the precedingeit follows that the typicaleset A ∈ Lî+1 (R) is ine Σ∗ne(Γ) e e for some n, thus we have Det(Lî+1 (R)), and we are done.

Subcase 2. î is limit but Subcase 1 fails. Then there is some Σ0 formula ϕ and some s0 ∈ R such that Lî (R) |= ∀r∃xϕ(r, x, s0 ), but for all ç < î, we have Lç (R) 6|= ∀r∃xϕ(r, x, s0 ). For each real r ∈ R, let h(r) be the least ç < î such that Lç (R) |= ∃xϕ(r, s, x0 ). Then h : R → î is total with Σ1 over Lî (R) graph involving only real parame eters and h is cofinal in î. Thus if B ⊆ R is Γ˘ we have some ø ∈ Σ0 and some e r0 ∈ R such that r ∈ B ⇔ Lî (R) |= ∀xø(r, x, r0 )

⇔ ∀sLh(s) (R) |= ∀xø(r, x, r0 ) ⇔ Lî (R) |= ∀sø ′ (r, s, r0 ), where ø ′ ∈ Σ1 . This shows that Γ˘ ⊆ ∀R Γ = {∀sP(r, s) : P ∈ Γ}. e e e So we see that the set of reals A is in the class ∀R ∃R ∀R Γ. Now Γ is closed e property, e under continuous substitutions, ∧, ∨, ∃R and has the scale so by R the Second Periodicity Theorem (see [Mos80]) ∀ Γ has the scale property, e every set in Γ is Suslin. granting Det(Γ), which is given to us by the fact that e e By the Second Periodicity Theorem again, ∃R ∀R Γ has the scale property, so every set in this class is determined and by one last application of Second Periodicity ∀R ∃R ∀R Γ has the scale property, so A is Suslin, thus determined. So every set in Lî+1 (R) is determined and we are done. Subcase 3. î is successor. ˘ ⊆ ∀R Γ. In Say î = ç + 1. As in Subcase 2 it will be enough to show that Γ e ˘ is a countable intersection of setsein Γ. fact we shall prove here that every set in Γ e e Indeed let B ∈ Γ˘ . Then for some Σ0 formula ϕ and some real parameter e s0 , we have r ∈ B ⇔ Lç+1 (R) |= ∀xϕ(r, x, s0 )

⇔ ∀m∀z[z ⊆ Lç (R) ∧ z is Σm over Lç (R) ⇒ Lç+1 (R) |= ϕ(r, z, s0 )]. e

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Let Bm = {r ∈ R : ∀z[z ⊆ Lç (R) ∧ z is Σm over Lç (R) ⇒ Lç+1 (R) |= ϕ(r, z, s0 )]}. e T Then B = m Bm and each Bm is Σ1 over Lç+1 (R) with real parameters only, since if ø(i, x, y) is universal Σn wee have r ∈ Bm ⇔ Lç+1 (R) |= ∃M ∃æ(M = Læ (R) ∧

∧ ∀x ∈ M ∀iϕ(r, {y ∈ M : ø M (i, x, y)}, s0 )). So our proof in Case I is complete.

⊣ CASE I

′

Case II. There is an ordinal î < î such that Lî ′ (R) ∗

≺R 1

Lî (R).

≺R 1

Let î < î be least such that Lî ∗ (R) Lî (R). Let Γ be the pointclass of all sets of reals which are Σ1 over Lî (R) with only realeparameters. Clearly e of reals which are Σ over L ∗ (R) with only real these are the same as the sets 1 î parameters. We note first that Γ is closed undere∀R : Indeed let ø be Σ1 . Then e Lî ∗ (R) |= ∀rø(r, s) ⇒ Lî (R) |= ∃ç(Lç (R) |= ∀rø(r, s))

⇒ Lî ∗ (R) |= ∃ç(Lç (R) |= ∀rø(r, s)) ⇒ ∃ç < î ∗ (Lç (R) |= ∀rø(r, s)). Thus Lî ∗ |= ∀rø(r, s) ⇔ Lî ∗ (R) |= ∃ç(Lç (R) |= ∀rø(r, s))). So Γ is closed under continuous substitutions, ∧, ∨, ∃R , ∀R and has the scale e by Steel’s Theorem 4.2. property Let now n ≥ 1 be the least integer such that there is a new Σn over Lî (R) set e of [Ste83A]. of reals in Lî+1 (R). We shall need below the second basic result Theorem 4.3. ([Ste83A]) Let î be an infinite ordinal and assume ∃A ⊆ R(A ∈ Lî+1 (R) \ Lî (R)). Let n ≥ 1 be least such that there is A ⊆ R which is Σn over Lî (R) but A 6∈ Lî (R). Assume finally that there is a countable e sequence of Σn formulas ó0 , ó1 , . . . , ordinals ç0 , ϑ0 < î and a real r0 ∈ R such that for no ordinals î ′ < î, ç ′ , ϑ′ < î ′ we have ∀i[Lî (R) |= ói (r0 , ç0 , ϑ0 ) ⇔ Lî ′ (R) |= ói (r0 , ç ′ , ϑ′ )]. Then, if Det(Lî (R)) holds, the pointclass Σ of sets of reals which are Σn over e Lî (R) (with arbitrary parameters) has the e scale property and the complement of every set in Σ is a countable intersection of sets of reals in Lî (R), so in particular in ∀Re Σ. e Since there is a new Σn over Lî (R) subset of R, there is a partial map e L (R) graph. p : R ։ Lî (R) with Σn over î e

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Assume now, towards a contradiction, that there exists an undetermined game in Lî+1 (R). Say, for notational simplicity, it is Σn+2 , so it has the form e A(α, â) ⇔ Lî (R) |= ∃x∀yø(α, â, x, y, z1 ), where ø ∈ Σn , z1 ∈ Lî (R). Using the projection map p, we can write this as follows k0 _ A(α, â) ⇔ Lî (R) |= ∃ã∀ä [øi (α, â, ã, r1 , ç0 , ϑ0 ) ∧ ÷i (α, â, ã, ä, r1 , ç0 , ϑ0 )],

i=1

for some real r1 ∈ R, ç0 , ϑ0 < î, where øi ∈ Σn , ÷i ∈ Πn (here ç0 , ϑ0 are such that z1 is definable in Lϑ0 (R) from ç0 and reals). Since A is not determined, we have as usual that ∀ó, ô∃α, â{∃ã∀äLî (R) |=

k0 _

[øi (α, ô ∗ α, ã, ä, r1 , ç0 , ϑ0 ) ∧ ÷i (α, ô ∗ α, ã, ä, r1 , ç0 , ϑ0 )]

i=1

∧ ∀ã∃äLî (R) |=

k0 ^

[¬øi (ó ∗ â, â, ã, ä, r1 , ç0 , ϑ0 )

i=1

∨ ¬÷i (ó ∗ â, â, ã, ä, r1 , ç0 , ϑ0 )]}.

Then, by the manipulations we used in § 2, we also have ∀x∃M {M is x-good ∧ ∀∗ c∀∗ d∀ó, ô ∈ M∃α, â ∈ M : ∃ã ≤ c∀ä ≤ d(Lî (R) |=

k0 _

[øi (α, ô ∗ α, ã, ä, r1 , ç0 , ϑ0 )

i=1

∧ ∀ã ≤ c∃ä ≤ d(Lî (R) |=

k0 ^ i=1

∧ ÷i (α, ô ∗ α, ã, ä, r1 , ç0 , ϑ0 )]) [¬øi (ó ∗ â, â, ã, ä, r1 , ç0 , ϑ0 ) ∨ ¬÷i (ó ∗ â, â, ã, ä, r1 , ç0 , ϑ0 )]).

We consider now two subcases: Subcase 1. ∀x∃M {M is x-good ∧ ∀∗ c∀∗ d∃î ′ < î∃ç ′ , ϑ′ < î ′ ∀ã, ô ∈ M ∃α, â ∈ M : W0 ∃ã ≤ c∀ä ≤ d(Lî ′ (R) |= ki=1 [øi (α, ô ∗ α, ã, ä, r1 , ç ′ , ϑ′ )∧ ) ÷i (α, ô ∗ α, ã, ä, r1 , ç ′ , ϑ′ )]) Vk0 [¬øi (ó ∗ â, â, ã, ä, r1 , ç ′ , ϑ′ )∨ ∧∀ã ≤ c∃ä ≤ d(Lî ′ (R) |= i=1 ¬÷i (ó ∗ â, â, ã, ä, r1 , ç ′ , ϑ′ )]).

(∗∗)

Let then P(x, M ) be relation following ∀x∃M above. It is clearly in the class Γ, since it is Σ1 over Lî (R) with real parameters only. Thus since e e

THE EQUIVALENCE OF PARTITION PROPERTIES AND DETERMINACY

375

∀x∃MP(x, M ) and Γ has the uniformization property, we can find F : R → R e = M , then P(x, M ) holds. Put, as usual, in ∆ such that if F (x) x x e M(x) = {My : y ≤T x ∧ My is x-good}, and abbreviate by Qî ′ ,ç′ ,ϑ′ (ó, ô, α, â, c, d) the expression (∗∗) above, assuming that ç ′ , ϑ′ < î ′ . Thus we have ∀x∀∗ c∀∗ d∃î ′ ∃ç ′ , ϑ′ < î ′ ∃M ∈ M(x)∀ó, ô ∈ M ∃α, â ∈ M : Qî ′ ,ç′ ,ϑ′ (ó, ô, α, â, c, d). But, since Lî ∗ (R) ≺R 1 Lî (R), we also have ∀x∀∗ c∀∗ d∃î ′ < î ∗ ∃ç ′ , ϑ′ < î ′ ∃M ∈ M(x)∀ó, ô ∈ M ∃α, â ∈ M : Qî ′ ,ç′ ,ϑ′ (ó, ô, α, â, c, d). Let now fx0 (c, d) be the least î ′ < î ∗ such that ∃ç ′ , ϑ′ < î ′ ∃M ∈ M(x)∀ó, ô ∈ M ∃α, â ∈ M (Qî ′ ,ç′ ,ϑ′ (ó, ô, α, â, c, d)), let fx1 (c, d) be the least ç ′ < fx0 (c, d) such that ∃ϑ′ < fx0 (c, d)∃M ∈ M(x)∀ó, ô ∈ M ∃α, â ∈ M (Qfx0 (c,d),ç′ ,ϑ′ (ó, ô, α, â, c, d)), and let fx2 (c, d) be the least ϑ′ < fx0 (c, d) such that ∃M ∈ M(x)∀ó, ô ∈ M ∃α, â ∈ M (Qfx0 (c,d),fx1 (c,d),ϑ′ (ó, ô, α, â, c, d)). Again ∀∗ c∀∗ d(fxi (c, d) is defined), i = 0, 1, 2. Consider now the game I α, x0

II x1 , â

where x = hx0 , x1 i; player I wins iff ∀∗ c∀∗ d∃ã ≤ c∀ä ≤ d(Lfx0 (c,d) (R) |=

k0 _

[øi (α, â, ã, ä, r0 , fx1 (c, d), fx2 (c, d))

i=1

∧ ÷i (α, â, ã, ä, r0 , fx1 (c, d), fx2 (c, d))]). This is a game in ∆, so it is determined (since every set in Γ is Suslin). Then e e we obtain a contradiction exactly as in § 2. Subcase 2. Subcase 1 fails. Then find x0 ∈ R such that for all M which are x0 -good we have ¬∀∗ c∀∗ d∃î ′ < î∃ç ′ , ϑ′ < î ′ ∀ó, ô ∈ M ∃α, â ∈ M (Qî ′ ,ç′ ,ϑ′ (ó, ô, α, â, c, d)).

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ALEXANDER S. KECHRIS AND W. HUGH WOODIN

Then, by the formula just before Subcase 1, find M0 such that M0 is x0 -good and ∀∗ c∀∗ d∀ó, ô ∈ M0 ∃α, â ∈ M0 (Qî,ç,ϑ0 (ó, ô, α, â, c, d)). Then ¬∀∗ c∀∗ d∃î ′ < î∃ç ′ , ϑ′ < î ′ ∀ó, ô ∈ M0 (Qî ′ ,ç′ ,ϑ′ (ó, ô, α, â, c, d)). Since the expression following ¬∀∗ c∀∗ d is Σ1 over Lî (R) with only real pae rameters, it is in Γ, and we have Γ-Turing Determinacy, since every set in Γ is e e e Suslin, so we conclude that ∀∗ c∀∗ d¬∃î ′ < î∃ç ′ , ϑ′ < î ′ ∀ó, ô ∈ M0 ∃α, â ∈ M0 : Qî ′ ,ç′ ,ϑ′ (ó, ô, α, â, c, d), thus ∀∗ c∀∗ d∀î ′ < î∀ç ′ , ϑ′ < î ′ ∃ó, ô ∈ M0 ∀α, â ∈ M0 : ¬Qî ′ ,ç′ ,ϑ′ (ó, ô, α, â, c, d). Fix then c0 , d0 ∈ D such that ∀ó, ô ∈ M0 ∃α, â ∈ M0 (Qî,ç0 ,ϑ0 (ó, ô, α, â, c0 , d0 )) but ∀î ′ < î∀ç ′ , ϑ′ < î ′ ∃ó, ô ∈ M0 ∀α, â ∈ M0 ¬Qî ′ ,ç′ ,ϑ′ (ó, ô, α, â, c0 , d0 ). Recalling the definition of Qî,ç,ϑ (ó, ô, α, â, c, d), we see that if {ój }, {ôk } enumerate the strategies in M0 , {αℓ }, {âm } the reals in M0 , {ãn }, {äp } the reals ≤ c0 , d0 , resp., and we define ñi1 (α, â, ó, ô, ã, ä, r, ç, ϑ) ⇔ øi (α, ô ∗ α, ã, ä, r, ç, ϑ) ñ12 (α, â, ó, ô, ã, ä, r, ç, ϑ) ⇔ ¬÷i (α, ô ∗ α, ã, ä, r, ç, ϑ) ñi3 (α, â, ó, ô, ã, ä, r, ç, ϑ) ⇔ øi (ó ∗ â, â, ã, ä, r, ç, ϑ) ñi4 (α, â, ó, ô, ã, ä, r, ç, ϑ) ⇔ ¬÷i (ó ∗ â, â, ã, ä, r, ç, ϑ), then there is no î ′ < î, ç ′ , ϑ′ < î ′ such that: for all 1 ≤ t ≤ 4, j, k, ℓ, m, n, p ∈ ù and i ≤ k0 , we have Lî (R) |= ñit (αℓ , âm , ój , ôk , ãn , äp , r1 , ç0 , ϑ0 ) ⇔ Lî ′ (R) |= ñit (αℓ , âm , ój , ôk , ãn , äp , r1 , ç ′ , ϑ′ ). By suitably renumbering and coding, we conclude that there is a sequence ó0 , ó1 , . . . of Σn formulas, there exist ordinals ç0 , ϑ0 < î and a real r0 ∈ R such that for no ordinals î ′ < î, ç ′ , ϑ′ < î ′ we have ∀i[Lî (R) |= ói (r0 , ç0 , ϑ0 ) ⇔ Lî ′ (R) |= ói (r0 , ç ′ , ϑ′ )]. Thus by Steel’s Theorem 4.3 the pointclass Σ of sets of reals which are Σn e every set of reals whicheis over Lî (R) has the scale property, and moreover

THE EQUIVALENCE OF PARTITION PROPERTIES AND DETERMINACY

377

Πn over Lî (R) is in the class ∀R Σ. So we have that our undetermined set A e in ∃R ∀R Σ. Now Σ is closed under e continuous substitutions, ∧, ∨, ∃R , so by is e e the Second Periodicity Theorem, since Σ has the scale property, so does ∀R Σ, e granting Det(Σ), which is given to us bye the fact that every set in Σ is Suslin. e e So, by Second Periodicity again, ∃R ∀R Σ has the scale property, so A is Suslin, e proof of subcase 2 and Theorem 1.2 so determined, a contradiction. Thus the is complete. §5. Addendum: Update on an open problem. The conjecture mentioned in the last paragraph of Section 1 has now been proved by Neeman and Woodin. It is a matter of combining two theorems. Fix k > 0. Neeman [Nee95] proved that if for each real x, M#2k (x) exists and is countably iterable, where M#2k is the sharp of the Mitchell-Steel inner model with 2k Woodin cardinals, then for each n, all games in a2k (Mn ) are determined. Woodin (unpublished) proved that if all Π12k+1 games are determined, then for all reals x, M#2k (x) exists and e iterable. is countably REFERENCES

Leo A. Harrington [Har78] Analytic determinacy and 0# , The Journal of Symbolic Logic, vol. 43, pp. 685–693. Alexander S. Kechris, Eugene M. Kleinberg, Yiannis N. Moschovakis, and W. Hugh Woodin [KKMW81] The axiom of determinacy, strong partition properties and nonsingular measures, this volume, originally published in Kechris et al. [Cabal ii], pp. 75–100. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer. [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer. Alexander S. Kechris and Robert M. Solovay [KS85] On the relative consistency strength of determinacy hypotheses, Transactions of the American Mathematical Society, vol. 290, no. 1, pp. 179–211. Donald A. Martin [MarA] Borel and projective games, to appear. [Mar78] Infinite games, Proceedings of the international congress of mathematicatians, Helsinki 1978 (Olli Lehto, editor), Finnish Academy of Sciences, pp. 269–273. [Mar83A] The largest countable this, that, and the other, this volume, originally published in Kechris et al. [Cabal iii], pp. 97–106. Yiannis N. Moschovakis [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam. Itay Neeman [Nee95] Optimal proofs of determinacy, The Bulletin of Symbolic Logic, vol. 1, pp. 327–339.

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John R. Steel [Ste80] More measures from AD, mimeographed notes. [Ste83A] Scales in L(R), this volume, originally published in Kechris et al. [Cabal iii], pp. 107– 156. DEPARTMENT OF MATHEMATICS CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CA 91125, USA

E-mail: [email protected] DEPARTMENT OF MATHEMA TICS UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720, USA

E-mail: [email protected]

GENERIC CODES FOR UNCOUNTABLE ORDINALS, PARTITION PROPERTIES, AND ELEMENTARY EMBEDDINGS

ALEXANDER S. KECHRIS AND W. HUGH WOODIN

Historical Remark. This paper was circulated in handwritten form in December 1980 and contained Sections 1–7 below. There are two additional Sections 8 and 9 here that contain further material and comments. §1. A simple lemma. Let R = ù ù, the Baire space. Adopting (a variant of) a definition of [Bec79], let us call an ordinal ë reliable if there is a scale hϕi i on a set W ⊆ R such that ϕ0 : W ։ ë, ϕi : W → ë and the two relations x, y ∈ W ∧ ϕ0 (x) ≤ ϕ0 (y) x, y ∈ W ∧ ϕ0 (x) < ϕ0 (y) admit scales. [Note that if AD+V=L(R) holds, then every reliable ordinal is ≤ ä 21 , and in that case ë is reliable iff ∃hϕi i, W as above such that ϕ0 : W ։ e : W → ë (so that last condition is not required).] If hϕ i, W are as above, ë∧ϕ i i we will say that hϕi i, W witness the reliability of ë or simply are witnesses for ë. We call a countable subset of ë, S, î-honest, where î ∈ S, if there is a code w ∈ W of î, i.e., a w ∈ W such that ϕ0 (w) = î, with ϕi (w) ∈ S, ∀i. We call such an S honest if it is î-honest for all î ∈ S (this is of course all relative to hϕi i, W ). Note that for each î < ë there is some countable S0 containing î such that S ⊇ S0 ⇒ S is î-honest (however such an S0 cannot in general be canonically attached to each î). Note also that {S ∈ ℘ù1 (ë) : S is honest} is a strongly closed unbounded, in short scub, subset of ℘ù1 (ë) [Bec79]. Lemma 1.1 (AD). Let ë be reliable with witnesses hϕi i, W . (i) There is a Lipschitz function F0 : ù ë → R (i.e., F0 (f)↾n depends only on f↾n) such that ran(F0 ) ⊆ W and for any f ∈ ù ë: {f(0), f(1), . . . } is f(0)-honest ⇒ ϕ0 (F0 (f)) = f(0). Preparation of this paper is partially supported by NSF Grants DMS-0455285 (A.S.K.) and DMS-0355334 (W.H.W.). The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

379

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ALEXANDER S. KECHRIS AND W. HUGH WOODIN

(ii) There is a Lipschitz function F : ù ë → R such that ran(F ) ⊆ {w : ∀n((w)n ∈ W )} and for any f ∈ ù ë: {f(0), f(1), . . . } is honest ⇒ ∀n, ϕ0 ((F (f))n ) = f(n). Proof. We prove (i), the proof of (ii) being similar. Let T be the tree on ù × ë coming from the scale hϕi i on W . For each î < ë, let T(î) = {(s, u) ∈ T : u(0) = î}, be the subtree of T whose ordinal sequences start with î. Now consider the following game on ë: I f(0)

II w(0), h(0)

f(1) w(1), h(1) .. . f

w h

where f(i) < ë; w(i) ∈ ù; h(i) < ë; player II wins iff (w, h) ∈ [T(f(0)) ] ∧ ∀v[v ∈ p[T(f(0)) ↾{f(0), f(1), . . . }] ⇒ ϕ0 (v) ≤ ϕ0 (w)]. (For any tree J on ù × ë, and any S ⊆ ë, J ↾S is the restriction of J to ù × S.) It is enough to show that this game is determined. [If so, then since player I clearly can’t have a winning strategy (when player II sees f(0), he plays a w such that ϕ0 (x) = f(0) and h(i) = ϕi (w)), player II must have a winning strategy ô. Let then F (f) = w iff for some h, (f, w, h) is a run of the game in which player II follows ô.] To prove now the determinacy of the game, it is enough, by [Mos81], to verify that the ë-pointset R(f, w, h) ⇔ the run f, w, h is a win for player I and its negation, admit scales. This is clear from the closure properties of ë-pointsets carrying scales (see [Mos81]), noting that v ∈ p[Tf(0) ↾{f(0), f(1), . . . }] ⇔ ∃α ∈

ù

ù((v, f ◦ α) ∈ [Tf(0) ]).

Remark. (1) Note that the game I î0

II player II wins iff {î0 , î1 , . . . } is honest î1

î2 .. .

î3

⊣

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GENERIC CODES

is not determined, since otherwise for each î < ë we could explicitly find a î-honest Sî , violating a result of [Bec79] (or, by the preceding result, to each î = ë we could explicitly assign wî , a code of î). (2) For ë = ù1 , there are witnesses hϕi i, W such that the honest S are precisely the proper initial segments of ù1 . In the general case of arbitrary ë the honest S play in many respects a role similar to that of the proper initial segments in ù1 . §2. An ordinal determinacy result. For each P ⊆ consider the associated game, also denoted by P: I f(0)

ù

ë (ë reliable as before)

II f(i) < ë; player II wins iff P(f). f(1)

f(2) f(3) .. . Let, also assuming AD, U be a supercompactness measure on ℘ù1 (ë). Put ∀∗U S(. . . S . . . ) ⇔ ∀∗ S(. . . S . . . ) def

⇐⇒ {S : . . . S . . . } ∈ U. For each S ∈ ℘ù1 (ë), let P↾S be the game P restricted to S, i.e. I f(0)

II f(i) ∈ S; player II wins iff P(f). f(1) .. .

We now have player I (player II) has a winning strategy in P ⇔ ∀∗ S(player I (player II) has a winning strategy in P↾S) ⇔ ∃F : ℘ù1 (S) → strategies for games on S such that ∀∗ S(F (S) is a winning strategy for player I (player II) in P↾S). Clearly, by AD, for any S ∈ ℘ù1 (ë) player I or player II has a winning strategy in P↾S and thus either ∀∗ S (player I has a winning strategy in P↾S) or ∀∗ S (player II has a winning strategy in P↾S); we describe this situation by saying that P is weakly determined. If however P itself is not determined, then by the above equivalences, even if say ∀∗ S (player I has a winning strategy in P↾S), we can’t find a strategy for player I in P↾S explicitly from S (for ∀∗ S). We show that one can do the next best thing, i.e., have such a strategy depend explicitly (and in fact continuously) on any enumeration of S (for ∀∗ S).

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Theorem 2.1 (AD+V=L(R)). Let ë < ä 21 be reliable. Let P ⊆ ù ë. Then e (i) ∃H : <ù ë × <ù ë → ë such that for some scub subset C of ℘ù1 (ë), if f ∈ ù ë and S = {f(0), f(1), . . . ) ∈ C , then Hf is a winning strategy for player I in P↾S, where Hf (s) = H (f↾ lh(s), s). or (ii) Similarly for player II. Proof. Let hϕi i, W in ∆21 witness the reliability of ë. Let e P ∗ (w) ⇔∀n((w)n ∈ W ) ∧ hϕ0 ((w)n )i∞ n=1 ∈ P, be the coded version of P. Now consider the following game, where F is as in 1.1: I î0 , ç0

II ç1 , î1

î2 , ç2 .. .

ç3 , î3

where çi < ëi , îi < ë; player II wins iff ∃α{∀n(çn = îα(n) )∧∃w[∀n(ϕ0 ((w)n ) = ~ α(n) ) ∧ P ∗ (w)]}. ϕ0 ((F (î)) Assume that this game is determined. To find out who wins this game, note that if U is a supercompactness measure on ℘ù1 (ë), then ∀∗ S (player I has a winning strategy in P↾S) or ∀∗ S (player II has a winning strategy in P↾S). Say the second case occurs. We shall see then that player II has a winning strategy in the above game. Indeed if player I won by ó, fix S honest, closed under ó and such that player II has a winning strategy in P↾S. Let hî1 , î3 , . . . i enumerate S. Let î0 , ç0 , î2 , ç2 , . . . be determined by ó and ç1 , ç3 , . . . by following player II’s strategy in P↾S against ç0 , ç2 , . . . Then {ç0 , ç1 , . . . } ⊆ {î0 , î1 , . . . } = S, so let α be such that çn = îα(n) . As {î0 , î1 , . . . } enumerates an honest S, ϕ0 (F (hîn i∞ n=1 )k ) = îk for all k, so if v = F (hîn i), we have ϕ0 ((v)α(n) ) = îα(n) = çn . Let w be such that (w)n = (v)α(n) . Then ϕ0 ((w)n ) = çn , and as hçn i ∈ P, we have w ∈ P ∗ , so player II won against ó, a contradiction. So player II has a winning strategy, say ô. Let H be ô, forgetting about the î1 , î3 , . . . If S is honest and closed under ô, then, if f enumerates S, we claim that Hf is a winning strategy for player II in P↾S. Indeed assume player I played ç0 , ç2 , . . . in P↾S and player II produced ç1 , ç3 , . . . Let î1 , î3 , . . . be such that if î2i = f(i), then î0 , ç0 , ç1 , î1 , î2 , ç2 , . . . is a run of the above game

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GENERIC CODES

according to ô. Then {ç0 , ç1 , . . . } ⊆ {î0 , î1 . . . } and for some α, w, çn = îα(n) ~ α(n) ), and P ∗ (w). But {î0 , î1 , . . . } enumerates also and ϕ0 ((w)n ) = ϕ0 (F (î) S, therefore ~ α(n) ) = îα(n) = çn , ϕ0 (F (î) so P(hçn i, i.e., player II won P↾S and we are done. Now clearly the statement “The above game is determined” is a ∆21 property of P ∗ . So if it fails for some P ∗ , then by Solovay’s Basis Theoreme it fails for some P ∗ ∈ ∆21 . But, by the Martin-Moschovakis-Steel Theorem, if P ∗ ∈ ∆21 e then both Pe∗ and ¬P ∗ carry scales, therefore the above game carries scales, so is determined, and we are done. ⊣ Note. It is not immediately clear that the above argument works for ë = ä 21 , e although the result ought to be true in that case as well. §3. Generic codes for ordinals. Given P ⊆ game: I s0

ù

ë consider the Banach-Mazur

II s1

where si ∈ <ù ë \ {∅}; player II wins iff P(s0 as1 as2a. . . ).

s2 .. .

s3

def

Write ∀∗ f(P(f)) ⇐⇒ ∀s0 ∃s1 ∀s2 ∃s3 . . . P(s0 as1a. . . ). Also, for any given s ∈ <ù ë, ∀∗ f ⊇ sP(f) ⇔ ∀s0 ⊇ s∃s1 ∀s2 . . . P(s0 as1a. . . ), where the formula on the right asserts as usual the existence of a winning strategy for player II. Fact 3.1 (AD). Let ë be reliable, with witnesses hϕi i, W and let F0 be the function of 1.1. Then for all î < ë: ∀∗ f ⊇ (î)[ϕ0 (F0 (f)) = î]. Proof. In the Banach-Mazur game I s0

II player II wins iff ϕ0 (F0 (s0 as1a. . . )) = î, s1

s2 .. .

s3

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if player I starts with s0 ⊇ (î), then player II just enumerates any î-honest S0 . So player I cannot have a winning strategy By [Mos81] this game is determined, so player II has a winning strategy ⊣ For convenience, put for w ∈ W |w| = ϕ0 (w). The preceding fact allows us to define category notions on each Wî = {w ∈ W : |w| = î}. Indeed, for each A ⊆ Wî , say that A is comeager ⇔ ∀∗ f ⊇ (î)(F0 (f) ∈ A), and A is meager ⇔ Cî \ A is comeager. Then note that: (1) Wî is comeager. T (2) If An is comeager for each n, n An is comeager. (3) If A is comeager, A 6= ∅. (4) For each s ⊇ (î), s ∈ <ù ë say that A is comeager on s iff ∀∗ f ⊇ s(F0 (f) ∈ A). Then, if ë < ä 21 and V=L(R) holds, we have that every A ⊆ Wî has the e i.e., either A is comeager or W \ A is comeager on some property of Baire, î ∗ s, i.e., ∀ f ⊇ (î)(F0 (f) ∈ A) or ∃s ⊇ (î)∀∗ f ⊇ (s)(F0 (f) 6∈ A). This is because the Banach-Mazur game “F0 (f) ∈ A” is determined by an argument similar to that in § 2 (recall that F0 is continuous). It is now convenient to introduce the following notations (where A ⊆ R, s ∈ <ù ë, s ⊇ (î)): def

s |=î A ⇔ ∀∗s |w| = îA(w) ⇐⇒ ∀∗ f ⊇ s(F0 (f) ∈ A) ⇔ A is comeager on s, def

0 |=î A ⇔ (î) |=î A ⇔ ∀∗ |w| = îA(w) ⇐⇒ ∀∗ f ⊇ (î)F0 (f) ∈ A) ⇔ A is comeager. Then again for all ë < ä 21 , assuming AD+V=L(R), we have the usual rules e 1. s |=î T A ⇒ A 6= ∅, 2. s |=î n An ⇔ ∀n, s |=î An , 3. s |=î ¬A ⇔ ∀t ⊇ s(t 6|=î A), 4. Moreover we also have the following unfolding formula, where A ⊆ R × R, s |=î ∀α(0)∃α(1)∀α(2)∃α(3) . . . A(·, α) if and only if ∀s0 ⊇ s∃s1 ∀s2 ∃s3 . . . ∀α(0)∃α(1)∀α(2) . . . A(F0 (s0 as1a. . . ), α)

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if and only if ∀s0 ⊇ s∀α(0)∃s1 ∃α(1)∀s2 ∀α(2)∃s3 ∃α(3) . . . A(F0 (s0 as1a. . . ), α). (∗) Proof. Let U be a supercompactness measure on ℘ù1 (ë). As in §2, we can assume that the game in (∗) is determined, so we have ∀s0 ⊇ s∀α(0)∃s1 ∃α(1) . . . A(F0 (s0 as1 a. . . ), α) ⇔ ∀∗ S∀s0 ⊇ s, s0 ∈ <ù S∀α(0)∃s1 ∈ <ù S∃α(1) . . . A(F0 (s0 as1a. . . ), α) so (by the standard unfolding formula, see [Kec78B]) ⇔ ∀∗ S∀s0 ⊇ s, s0 ∈

<ù

S∃s1 ∈

<ù

S . . . ∀α(0)∃α(1) . . . A(F0 (s0 as1 a. . . ), α)

⇔ ∀s0 ⊇ s∃s1 ∀s2 . . . ∀α(0)∃α(1) . . . A(F0 (s0 as1 . . . ), α).

⊣

§4. Some new partition properties. Theorem 4.1 (AD). For each n ≥ 1, ä 12n+1 → (ä 12n+1) )ëç ; ∀ë < ä 12n , ∀ç < ä 12n+1 . e e e e Corollary 4.2 (AD). For each n ≥ 1, ä 12n+3 ≥ ä 12n+1 + ℵù 3 +1 , e e so ä 12n+3 ≥ ℵù 3 ·n+1 . e Proof. By imitating [MarC].

⊣

Corollary 4.3 (AD). Let ä 1ù = sup ä 1n . For each projective P ⊆ R and e e T on ù × ë, for some ë < ä 1 , tree each κ < ä 1ù , there is a homogeneous ù e such that P = p[T ] and the measures witnessing the homogeneity of P eare κ-additive. Proof. As in [Kec81A].

⊣

[Ste80] has first established the above result, without the extra additivity property. Proof of Theorem 4.1. We verify the Martin Criterion as formulated in [Kec78A, Lemma 11.1]: From [HK81] it follows that if hϕi i is a Π12n−1 -scale on W ∈ Π12n−1 , ϕi : W ։ ä 12n−1 , and ø : V ։ ä 12n+1 a Π12n+1 -norm, then there is a total ∆12n -function e e : R2 → R such thatefor every f : ä 1 1 G 2n−1 → ä 2n+1 there is ε ∈ R such that e e (i) w ∈ W ⇒ G(ε, w) ∈ V , (ii) ∀w ∈ W [ø(G(ε, w)) = f(ϕ0 (w))]. Fix now t : ù · ë ։ ä 12n−1 and define for î < ù · ë, letting |w| ≡ ϕ0 (w), e Cî = {ε : ∀∗ |w| = t(î)(G(ε, w) ∈ V )};

386

ALEXANDER S. KECHRIS AND W. HUGH WOODIN

then, for ε ∈ Cî , let f î (ε) = min {ø(G(ε, w) : |w| = t(î)}. We now verify 1), 2), 3), of [Kec78A, Lemma 11.1]. 1) is obvious by the above remarks. Let for î < ù · ë, ϑ < ä 12n+1 : e Cî,ϑ = {ε : ∀î ′ ≤ î∀∗ |w| = t(î ′ ) [G(ε, w) ∈ V ∧ ø(G(ε, w)) ≤ ϑ]}. T

Clearly Cî,ϑ ⊆ î ′ ≤î Cî and 3) is trivially satisfied. So it is enough to verify T 2). Thus let ó be continuous such that ó[ î ′ ⊆î Cî ′ ] ⊆ Cî , so that in particular A = ó[Cî,ϑ ] ⊆ Cî . We need now the following key lemma: Lemma 4.4. If R(w, x) is Σ12n+1 , so is P(x) ⇔ ∀∗ |w| = ñR(w, x), for each e ñ < ä 12n−1 . e Granting this lemma, we have, using also [HK81], that Cî,ϑ ∈ Σ12n+1 , thus e also A ∈ Σ12n+1 . Put e ó G(î,ϑ) = sup{f î (ó(ε)) + 1 : ε ∈ Cî,ϑ } = sup{f î (ε ′ ) + 1 : ε ′ ∈ A}. ó ó We have to show G(î,ϑ) < ä 12n+1 . Assume not, i.e., G(î,ϑ) = ä 12n+1 , towards a e e contradiction. Then we have

y ∈ V ⇔ ∃ε ′ ∈ A∀∗ |w| = t(î)(y ≤Σ12n+1 G(ε ′ , w)),

(∗∗)

where ≤Σ12n+1 ∈ Σ12n+1 and y ∈ V ⇒ [x ∈ V ∧ ø(x) ≤ ø(y) ⇔ x ≤Σ12n+1 y]. Proof of (∗∗). ⇒: If y ∈ V , let ε ′ ∈ A be such that ø(y) ≤ f î (ε ′ ). Then, since ∀∗ |w| = t(î)(G(ε ′ , w) ∈ V ) and ∀∗ |w| = t(î)(ø(y) ≤ ø(G(ε ′ , w))), we have ∀∗ |w| = t(î)(ø(y) ≤ ø(G(ε ′ , w)) ∧ G(ε ′ , w) ∈ V ), so ∀∗ |w| = t(î)(y ≤Σ12n+1 G(ε ′ , w)). ⇐: Let ε ′ satisfy the right-hand side of (∗∗). Then ∀∗ |w| = t(î)(y ≤Σ12n+1 G(ε ′ , w)). But also ∀∗ |w| = t(î)(G(ε ′ , w) ∈ V ), so ∀∗ |w| = t(î)(y ≤Σ12n+1 G(ε ′ , w) ∧ G(ε ′ , w) ∈ V, thus y ∈ V . ⊣ (∗∗) Now (∗∗) and Lemma 4.4 imply that V ∈ Σ12n+1 , a contradiction. e

GENERIC CODES

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Proof of Lemma 4.4. Let R(w, x) ⇔ ∃αQ(w, x, α), where Q ∈ Π12n . Then P(x) ⇔ ∀∗ |w| = ñ∃αQ(w, x, α) ⇔ ∀s0 ⊇ (ñ)∃α(0)∃s1 ∀s2 ∃α(1)∃s3 . . . Q(F0 (s0 as1a. . . ), x, α), (ä 12n−1 ), by the formula 4. in § 3. Now e Q(w, x, α) ⇔ ∀f ∈ ù (ä 12n−1 )∃n(w↾n, x↾n, α↾n, f↾n) 6∈ S, e for some tree S on ù × ù × ù × ä 12n−1 , so e P(x) ⇔ ∀s0 ⊇ (ñ)∃α(0)∃s1 ∀s2 ∃α(1) . . . ∀f ∈ ù (ä 12n−1 )∃n e (F0 (s0 as1 a. . . )↾n, x↾n, α↾n, f↾n) 6∈ S. where si ∈

<ù

Since F0 : ù (ä 12n−1 ) → ù ù is Lipschitz, it follows easily that P(x) is Σ11 in the e , <, Ai, for some A ⊆ ä 1 . (Here A encodes S and eF .) But structure hä 12n−1 0 2n−1 e ⊣ (Lemma 4.4) then by [HK81], P(x) is Σ12n+1 and weeare done. e ⊣ (Theorem 4.1) Actually a direct application of [Kec78A, Lemma 11.1] only shows that ä 12n+1 → (ä 12n+1 )ë , ∀ë < ä 12n . To prove that ä 12n+1 → (ä 12n+1 )ëç , ∀ç < ä 12n+1 , we e e modification e as in [KKMW81, e Lemma e1.2]. e need a minor §5. The weak Baire theory for ù ë. For each set X , ù X is the space of infinite sequences from X with the product topology, X taken to be discrete. The category notions for ù X are defined in the standard way. Abbreviate ∀∗ f ∈

ù

XP(f) ⇔ {f ∈

ù

X : P(f)} is comeager.

Call now P ⊆ ù X weakly comeager if there is C ⊆ ℘ù1 (X ) strongly closed unbounded such that ∀S ∈ C ∀∗ f ∈

ù

SP(f).

(Clearly comeager sets are weakly comeager.) The notion of weakly meager is defined in the obvious way, as the complement of a weakly comeager set. We now show that the weak Baire notions on ù ë work as the Baire notions on ù ù. Theorem 5.1 (AD + V=L(R)). Assume ë < ä 21 is reliable. e (i) If A ⊆ ù ë, then either A is weakly meager or else there is s ∈ <ù ë such that Ns(ë) \ A is weakly meager, where Ns(ë) = {f ∈ ù ë : f ⊇ s} (i.e, A is weakly comeager on Ns(ë) ). (ii) Let R ⊆ ù ë × V be a relation such that ∀f ∈ ù ë∃xR(f, x). Then there is G : ù ë → V such that R(f, G(f)) for a weakly comeager set of f’s.

388

ALEXANDER S. KECHRIS AND W. HUGH WOODIN

Proof. (i) Consider the Banach-Mazur game on ù ë defined as follows, where we fix hϕi i, W witnesses for ë, F0 the function in 1.1, and we let A∗ (w) ⇔ ∀n((w)n ∈ W ) ∧ hϕ0 (w)n i ∈ A: I s0

II s1

si ∈ <ù ë; player II wins iff F0 (s0 as1a. . . ) 6∈ A∗ .

s2 .. .

s3

By the usual arguments we have that this game is determined. If player II has a winning strategy, clearly A is weakly meager, and if player I has a winning strategy A is comeager on some Ns(ë) . (ii) Since V=L(R), we can assume that R ⊆ ù ë × R. Let hϕi i, W witness the reliability of ë, and let F0 be as in Lemma 1.1. Consider now the following game: I s0

II si ∈ s1 , α(0)

s2

ë; s0 as1a· · · = f; player II wins iff R∗ (F0 (f), α), <ù

s3 , α(1) .. . where R∗ (w, α) ⇔ ∀n((w)n ∈ W ) ∧ (hϕ0 (wn )i, α) ∈ R. By the usual argument, we can assume that this game is determined. Claim 5.2. Player I can’t have a winning strategy Because, if he had one, say ó, and S ∈ ℘ù1 (ë) is honest and closed under ó, we would have ∃s0 ∈

<ù

S∀s1 ∈

<ù

S . . . ∀α¬R(s0 as1a. . . , α),

<ù

S . . . ∀α¬R(s0 as1a. . . , α),

so by the usual game formula, ∃s0 ∈ thus ∃f ∈

ù

<ù

S∀s1 ∈

ë∀α¬R(f, α), a contradiction.

So player II has a winning strategy. Call f ∈ ù ë consistent with ô if there is a run of the game s0 , s1 , α(0), s2 , s3 , α(1), . . . in which player II follows ô and s0 as1a· · · = f. If f is consistent with ô, define a canonical run s¯0 , s¯1 , α(0), ¯ ... with s¯0as¯1 a· · · = f and player II following ô, as follows: • s¯0 is the shortest initial segment s0 of f such that there is a run s0 , s1 , α(0), . . . with s0as1 a· · · = f and player II following ô, • s¯1 , α(0) ¯ = ô(s¯0 ) (thus s¯0as¯1 ⊆ f),

GENERIC CODES

389

• s¯2 is the shortest sequence s2 such that s¯0 as¯1as2 ⊆ f and there is a run s¯0 , s¯1 , α(0), ¯ s2 , s, α(1), . . . with s¯0 as¯1as2 as a· · · = f and player II following ô, • etc. Now define G : ù ë → R by ( the α¯ produced as above, if f is consistent with ô, G(f) = ët · 0, otherwise. Note now that if S ∈ ℘ù1 (ë) is closed under ô and honest, then since ∀∗ f ∈ ù S (f is onto) and ∀∗ f ∈ ù S (f is consistent with ô), we have ∀∗ f ∈ ù S(f, G(f)) ∈ R, and we are done. ⊣ §6. Generic elementary embeddings for the L´evy collapse. We shall now use the results in § 5 to translate Woodin’s theory of generic elementary embeddings generated by the L´evy collapse (i.e., generic enumeration) of R in the framework of ADR (see [WooB]), to the context of the L´evy collapse of a reliable ë < ä 21 in the framework of AD+V=L(R). From noweon, in this section, assume ë < ä 21 is reliable and AD+V=L(R) e holds. Denote by I the ó-ideal of weakly meager subsets of ù ë. For A ⊆ ù ë, let A/I be the equivalence class of A modulo I. Let also ℘( ù ë)/I be the quotient Boolean algebra. We view this as a notion of forcing C = h℘( ù ë)/I, ≤i, where A/I ≤ B/I ⇔ A ⊆ B (modulo I). Note now that if Cë = h <ù ë, ≤i, where t ≤ s ⇔ t ⊇ s, is the notion of forcing for collapsing ë to ù, then Cë is canonically isomorphic to a dense subset of C. Indeed, for each s ∈ <ù ë, let Ns(ë) = {f ∈

ù

ë : f ⊇ s}.

Then if ð(s) = Ns(ë) /I, ð is an embedding of Cë onto a dense subset ð[Cë ] of C. This follows immediately from Theorem 5.1 (i). Then if G is a generic (over V) subset of Cë , G gives rise canonically to an ultrafilter H ∈ V[G] for the Boolean algebra C and thus to an ultrafilter U ∈ V[G] on (℘( ù ë))V such that {A ∈ V : A is a weakly comeager subset of ( ù ë)V } ⊆ U. Let now J = ( ù ë)V and consider the ultrapower (taken in V[G]): VJ ∩ V/U;

390

ALEXANDER S. KECHRIS AND W. HUGH WOODIN

we want first to check that the Łos Theorem goes through in this case. As usual this comes down to verifying (working in V), that if R(f, x) is a relation, A ⊆ ù ë is in U and ∀f ∈ A∃xR(f, x), then there is G : ù ë → V such that for some B ∈ U, (f, G(f)) ∈ R, for all f ∈ B. This however is obvious from Theorem 5.1 (ii) and the fact that U contains all weakly comeager sets. To show the wellfoundedness of this ultrapower and study some of its further properties, we digress briefly to discuss the supercompactness measure on ℘ù1 (ë), working until further notice in V. By [HK81], there is a supercompactness measure on ℘ù1 (ë), and by [Bec79] it is unique, and coincides with the strongly closed unbounded filter on ℘ù1 (ë). Denote it by U ë . Clearly each F : ℘ù1 (ë) → V gives rise to a map F ∗ : ù ë → V defined by F ∗ (f) = F (ran(f)). Lemma 6.1. The map (in V[G]) [F ]U ë 7→ [F ∗ ]U , is an isomorphism between Ord℘ù1 (ë) /Uë and OrdJ ∩V/U. In particular, VJ ∩ V/U is wellfounded. Proof. It is routine to verify that this is indeed an embedding. For example, if [F ]U ë ∈ [G]U ë , then for some C ⊆ ℘ù1 (ë) scub we have S ∈ C ⇒ F (S) ∈ G(S), so for S ∈ C , ∀∗ f ∈

ù

S(F (ran(f)) ∈ G(ran(f)),

so ∀S ∈ C ∀∗ f ∈

ù

S(F ∗ (f) ∈ G ∗ (f)),

so {f : F ∗ (f) ∈ G ∗ (f)} is weakly comeager, thus [F ∗ ]U ∈ [G ∗ ]U . We now show that this map is onto OrdJ ∩V/U. So let H : ù ë → V be in V. We shall show that: (∗∗∗) For any A ⊆ ù ë not weakly meager, there is B ⊆ ù ë not weakly meager, B ⊆ A such that for f, g ∈ B: ran(f) = ran(g) ⇒ H (f) = H (g). By genericity this implies that there is C ∈ U such that f, g ∈ C ∧ ran(f) = ran(g) ⇒ H (f) = H (g). Define then F : ℘ù1 (ë) → V in V by ( H (f), where f ∈ C and S = ran(f), F (S) = 0, Clearly [F ∗ ]U = [H ]U and we are done.

if such exists otherwise.

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391

Proof of (∗∗∗). Fix A ⊆ ù ë not weakly meager. Then A is weakly comeager on some Ns(ë) ; say A is scub such that S ∈ A ∧ s0 ∈ <ù S ⇒ ∀∗ f ∈ 0 (ë) Ns0 ∩ ù S(f ∈ A). For each such S, we have by Theorem 5.1 that there is some s ∈ <ù S so that s ⊇ s0 and H (f) is constant on a comeager in Ns(ë) ∩ ù S set of f’s. By normality this s is independent of S for S ∈ D, where D is scub in ℘ù1 (ë), call it s1 (clearly s1 ⊇ s0 ). Thus for S ∈ D, s1 ∈ <ù S and for comeager many f ∈ Ns(ë) ∩ ù S, H (f) = fixed = αS . Let 0 B = {f ∈

ù

ë : f ∈ A ∧ ran(f) ∈ D ∧ f ⊇ s1 ∧ H (f) = αran(f) }.

Clearly for S ∈ D ∧ s1 ∈ <ù S, ∀∗ f ∈ Ns(ë) ∩ ù S(f ∈ B), thus B is not weakly 1 meager. Moreover if f, g ∈ B and range(f) = range(g) = S ∈ D, then H (f) = H (g) = αs and we are done. ⊣ (∗∗∗) ⊣ (Lemma 6.1) Let us now denote by M ⊆ V[G] the transitive collapse of VJ ∩ V/U, and by j ≡ jG : V → M ∼ = VJ ∩ V/U the associated elementary embedding. We identify also each [H ]U with its image under the isomorphism of VJ ∩ V/U with M . Working now in V, let HWO = {a : tcl(a) is wellorderable} and consider the ultrapower Ult(HWO, U) = {F : ℘ù1 (ë) → V : ran(F ) ∈ HWO}/U ë It is easy to verify that the Łos Theorem goes through for ∆0 formulas, and thus this ultrapower is wellfounded and extensional, so it can be identified with its transitive collapse N . Let then jë be the associated ∆0 -elementary embedding jë : HWO → N ∼ = Ult(HWO, U ë ). The argument used in Lemma 6.1 clearly also establishes that the map ñ([F ]Uë ) = [F ∗ ]U is an isomorphism between Ult(HWO, U ë ) and {H :

ù

ë → V : ran(H ) ∈ HWO}/U

and clearly we have jG ↾HWO = jë ◦ ñ. In particular, for ϑ ∈ Ord, jë (ϑ) = jG (ϑ)

392

ALEXANDER S. KECHRIS AND W. HUGH WOODIN

and also for X ⊆ Ord jë (X ) = jG (X ). As jG : V → M is elementary and V |= V=L(R), clearly M |= V=L(R). We actually show now that M = L(R)V[G] . For that we have to show that, if x ∈ RV[G] , then x ∈ M . Let ô be a term in the forcing language denoting x. Let p0 ∈ G be such that p0 |= ô ∈ R. Note ù then that Np(ë) ë → V in V such that [F ]U = x. 0 ∈ U. We shall find some F : We have p0 |= ∀n∃m(ô(n) = m), so ∀n∀p ≤ p0 ∃q ≤ p∃m(q |= ô(n) = m) (conditions are in Cë = <ù ë). Thus ∀∗ f ⊇ p0 , there is a unique α ∈ ù ù such that for all n there is k with f↾k |= ô(n) = α(n). Denote this α by ô(f). This is only defined for f ∈ C ⊆ ù ë, where ∀∗ f ⊇ p0 (f ∈ C ). Put ( ô(f), if f ∈ C, F (f) = ët · 0, otherwise. Clearly [F ]U ∈ RV[G] . Fix n. We show that [F ]U (n) = x(n). Let [F ]U (n) = m. Then for some A ∈ U, f ∈ A ⇒ F (f)(n) = m. If now x(n) 6= m, there is p1 ≤ p0 , p1 ∈ G such that p1 |= ô(n) 6= m. Thus ∀∗ f ⊇ p1 (F (f)(n) 6= m). But Np(ë) ∈ U, so {f ⊇ p1 : F (f)(n) 6= m} ∩ A 6= ∅, which is a 1 contradiction. We can summarize what we have proved until now as follows: Theorem 6.2 (AD+V=L(R)). Let ë < ä 21 be reliable. Let Cë = <ù ë be the e is a generic subset of C (over V), notion of forcing for collapsing ë to ù. If G ë then in V[G] we can define an elementary embedding jG : V → M = L[R]V[G] Moreover, if HWO = {a ∈ V : tcl(a) is wellorderable}, U ë is the scub measure on pù1 (ë) and jë : HWO → N ∼ = {F : pù1 (ë) → V : ran(F ) ∈ HWO}/U ë is the ∆0 -embedding generated by U ë , then jG = jë ◦ñ, where ñ is the inclusion map from N to M , in particular jë = jG ↾HWO. We conclude this section by pointing out an equivalence between forcing and Banach-Mazur games in the preceding context. Let ô be a term (always for forcing with Cë ) and p ∈ Cë be a condition such that p |= ô ∈ R. Let ϕ be a formula, x ∈ V. We claim that the following equivalence holds: For any q ≤ p, q |= ϕ L(R) (ô, jG (x)) ⇔ ∀∗ f ⊇ q(ϕ(ô(f), x)).

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Indeed, if ∀∗ f ⊇ q(ϕ(ô(f)), x) and G is generic containing q, therefore Nq(ë) ∈ U, we have that {f ∈ ù ë : ϕ(ô(f), x)} ∈ U, as this set is the intersection of a comeager (therefore weakly comeager) set in ù ë and Nq(ë) . So, since [f 7→ ô(f)]U = the real represented by ô in V[G] (= say, a) we have, by Łos, M = L(R)V[G] |= ϕ(a, jG (x)), i.e., ϕ L(R) (a, jG (x)). Assume now ¬∀∗ f ⊇ q(ϕ(ô(f), x)). As in §2, we can assume that {f ⊇ q : ϕ(ô(f), x)) is a determined Banach-Mazur game, therefore ∃r ≤ q such that ∀∗ f ⊇ r¬(ϕ(ô(f), x)), so as before r |= ¬ϕ L(R) (ô, jG (x)), thus q 6|= ϕ L(R) (ô, jG (x)) and we are done. §7. Some applications of the generic elementary embeddings. Theorem 7.1 (AD). For each n ≥ 1 and α < ù2 : ä 12n ≤ ùα+1 ⇒ cf(ùα+1 ) ≥ ä 12n . e e In particular, cf(ùα+1 ) ≥ ùù+2 (= ä 14 ), for all α ≥ ù + 1. e Proof. Assume ä 12n ≤ ùα+1 , with α < ù1 . Let ë = ä 12n−1 ; ë is reliable. e e embeddings as in § 6. Let ë′ be the predecessor Let jë , jG be the two of ë (ë′ ′ is semireliable), let jë be the embedding generated by the supercompactness measure on ℘ù1 (ë′ ). By [Bec79], jë′ (ë) < ë+ = ä 12n . But if ìù is the e check that j (ϑ) ≤ ù-closed unbounded measure on ë, then it is easy to ë ++ + jìù (jë′ (ϑ)), so jë (ë) < ë . Similarly, jë′ (ë ) = ë+ , thus jë (ë′ ) ≤ ë++ , so, since jë (ϑ) ≥ jìù (ϑ) and jìù (ë+ ) = ë++ , we have jë (ë+ ) = ë++ , and more generally jë (ë+(n) ) = ë+(n+1) , for n ≥ 1. Since however jë = jG , and V → V[G] does not collapse cardinals ≥ ë+ , we also have jë (ùâ ) = ùâ , for all â < ù1 , with ùâ ≥ ë+(ù). Consider now ùα+1 as above. If ùα+1 = ë+(k) for some k ≥ 1, we are done. Also, by the above, jë (ùα+1 ) = ùα+1 , in particular jë (ùα+1 ) = sup{jë (î) : î < ùα+1 } is a continuity point of jë . Now assume, towards a contradiction, that cf(ùα+1 ) = κ ≤ ë. Let ð : ℘ù1 (ë) → κ be given by ð(S) = sup(S ∩ κ). Let ð∗ (U ë ) = V be the induced measure on κ. Clearly V is uniform and jV (ϑ) ≤ jë (ϑ), so ùα+1 is a continuity point of jV as well, a contradiction, since if f : κ → ùα+1 is cofinal and nondecreasing, then sup{jV (î) : î < ùα+1 } < [f]v < jV (ùα+1 ). ⊣ 2 In [Bec79], he calculates that if ë < ä 1 is reliable (and AD+V=L(R) holds), e then jë (w1 ) = ë+ . Theorem 7.2 (AD+V=L(R)). Let ë ≤ ä 21 be reliable and let uα(ë) = αth e uniform indiscernible for subsets of ë (u1(ë) = ë+ ). Then jë (ùn ) = un(ë) . Proof. Since jë = jG , jë (ùn ) = jG (ùn ) = jG (un ) (where un ≡ un(ù) ) = (un )V[G] . But subsets of ë in V are reals in V[G], while reals in V[G] are

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represented by terms which are essentially subsets of ë in V, so are generic over subsets of ë in V. Thus (un )V(G) = un(ë) . ⊣ It is clear that (granting AD again), un(ùm ) = um+n+1 . For ùù we have however un(ùù ) < ùù+2 and cf(un(ùù ) ) = ùù+1 (= ä 13 ). e Proof. From [Bec79], jùù (ùn ) < ùù+2 , so un(ùù ) = jùù (ùn ) < ùù+2 . Now assume cf(u2(ùù ) ) < ùù+1 , towards a contradiction. Then u2(ùù ) is a continuity point for jìù , where ìù is the ù-cub measure on ùù+1 . Thus jìù (u2(ùù ) ) = ≤ = ≤

sup{jìù (α) : α < u2(ùù ) } u2(ùù+1 ) jùù+1 (ù2 ) jìù (jùù (ù2 ))

= jìù (u2(ùù ) ),

therefore u2(ùù+1 ) has also cofinality < ùù+1 , so it can’t be a continuity point for jùù , therefore it can’t be a fixed point for jùù = jG , where G is generic for Cùù . (ä 13 )

(ä 13 )

Let M = L(R)V[G] . Then (u2e )M ≤ (u2e )V , since every A ⊆ (ä 13 )M , A ∈ M , e is a subset of (ä 13 )M = jG ((ä 13 )V ) < (ä 14 )V , so is generic over a subset of (ä 13 )V e e e e in V. But also (ä 13 )

(ä 13 )

(ä 13 )

thus u2e is a fixed point of jG , a contradiction. Similarly

(ä 13 )

(u2e )V ≤ jG ((u2e )V ) ≤ (u2e )M ,

un(ùù+1 )

⊣

< ùù+3 , ∀n, although the obvious conjecture that

cf(un(ùù+1 ) ) = ùù+2 = ä 14 e has not been proven yet. Similar results hold about the higher ä 12n+1 and their e predecessors.

A final observation: Granting AD, it is known that the only reliable cardinals above ùù+1 and below ùù 3 can be ùù·2 or ùù 2 , although it ought to be true that neither of them actually is. We can eliminate ùù·2 as follows: If ë = ùù·2 was reliable, let jë be the associated supercompactness measure and jG be the associated generic embedding. As no cardinal above ë+ is collapsed in the extension V → V[G], it is easy to see that for α < ù1 , jG (ùα ) ≤ ùù·2+α , so if α ≥ ù 2 , jG (ùα ) = ùα , thus as before α ≥ ù 2 ⇒ cf(ùα+1) ≥ ë = ùù·2 . But by [MarC], there are unboundedly many cardinals in ùù 3 of cofinality ùù+2 , a contradiction. §8. Addendum; non-existence of strong codes. (The nonexistence of strong codes was proved by Woodin and a simplified argument was given afterwards by Kechris but the proofs were not included in the original note. We would like to thank John Steel for writing up this addendum.)

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We shall show that, assuming AD, there is no coding of the ordinals below ù2 which admits a notion of category with the properties described in §3, and such that the ideals of meager sets are ù2 -additive, and have quotients with uniformly wellordered dense sets. More precisely, suppose hCα : α < ù2 i is a sequence of non-empty, pairwise disjoint sets of reals. Suppose also we have a sequence h(Iα , Pα , <α ) : α < ù2 i such that (a) Iα is a nontrivial ideal on Cα which is closed under wellordered unions of length ≤ ù1 , that is, Iα ⊆ S P(Cα ), Cα 6∈ Iα , A ⊆ B ∈ Iα ⇒ A ∈ Iα , and ∀î < ù1 (Aî ∈ Iα )) ⇒ î<ù1 Aî ∈ Iα ., (b) Pα is dense in the Iα -positive sets under almost inclusion, that is, letting Iα+ = P(Cα ) \ Iα , we have Pα ⊆ Iα+ , and for all X ∈ Iα+ , there is a p ∈ Pα such that p \ X ∈ Iα , and (c) <α is a wellorder of Pα . We then call h(Cα , Iα , Pα , <α ) : α < ù2 i a strong coding system for ù2 . It doesn’t matter too much how apt this term is, because we can show Theorem 8.1. Assume AD; then there is no strong coding system for ù2 . Proof. AssumeSthat h(Cα , Iα , Pα , <α ) : α < ù2 i is a strong coding system for ù2 . Let C = ù1 ≤α<ù2 Cα , and for w ∈ C , let |w| be the unique α such that w ∈ Cα . By the Coding Lemma, there is a function w 7→ f w with domain C such that f w is a bijection from ù1 onto |w| , for all w ∈ C . Claim 8.2. For any α ∈ [ù1 , ù2 ), there are club many ó ∈ ℘ù1 (α) such that ∃p ∈ Pα ∃î < ù1 ∀∗ w ∈ p(f w [î] = ó). Proof. Here and below, whenever Y ⊆ Iâ+ , then ∀∗ w ∈ Yϕ(w) means that {w ∈ Y : ¬ϕ(w)} ∈ Iâ . Fix α. It is enough to show there are stationary many such ó, since the club filter on ℘ù1 (α) is an ultrafilter. For that, it is enough to show that whenever g : ù1 → α is a bijection, then ∃î < ù1 ∃p ∈ Pα ∀∗ w ∈ p(f w [î] = g[î]). So fix a g. Clearly ∀∗ w ∈ Cα ∃î < ù1 (f w [î] = g[î]). But Iα is ù2 -additive, so ∃î({w : f w [î] = g[î]} ∈ Iα+ ). Fixing such a î, by clause (b) in the definition of strong coding we have a p ∈ Pα such that ∀∗ w ∈ p(f w [î] = g[î]), as desired. ⊣ (Claim 8.2) ∗ If p ∈ Pα , î < ù1 , and ó ∈ ℘ù1 (α) are such that ∀ w ∈ p(f w [î] = ó), then we set Bp,î = ó. So Claim 1 tells us that club many ó ∈ powù1 (α) are of the form Bp,î . Note that, because of the <α ’s, the family of all Bp,î is wellordered.

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Claim 8.3. For ù-club many ã < ù2 , ∃p∃î(Bp,î is cofinal in ã). Proof. Otherwise, we have an ù-club D ⊆ ù2 such that for no ã ∈ D are there such p, î. However, let α be a limit point of D of cofinality ù1 , and let U = {ó ∈ powù1 (α) : sup(ó) ∈ D}. Since U is club in powù1 (α), we have by Claim 1 some p, î such that Bp,î ∈ U . But then sup(Bp,î ) = ã ∈ D, a contradiction. ⊣ (Claim 8.3) Now set h(ã) = least Bp,î ⊂ ã which is cofinal in ã, whenever there is one. Here “least” refers to some fixed wellorder of the family of all Bp,î . By Claim 2, h is defined on an ù-club in ù2 . It is easy to see using the ù2 -additivity of the ù-club ultrafilter on ù2 that h is constant on an ù-club. This is impossible, as h(ã) is cofinal in ã. ⊣ §9. Addendum; Ordinal games and reliable cardinals. The ordinal games used in the proofs of both Lemma 1.1 and Theorem 2.1 are special cases of the following class of ordinal games: Suppose ë < Θ and that ð : ëù → ù ù is a continuous function, where the topology on ëù is the product topology induced by the discrete topology. Suppose A ⊆ ù ù . Then associated to the pair (ð, A) is the ordinal game on ë given by ð−1 [A], the preimage of A under ð. By [KKMW81, Theorem 2.7], assuming AD, if A is Suslin and co-Suslin, then this ordinal game is determined. As corollary, by the Solovay Basis Theorem and the Martin-Steel Theorem for Scale(Σ21 ) in L(R), assuming AD holds in L(R), then all the ordinal games given by a pair (ð, A) as above, are determined. This is ordinal determinacy. Ordinal determinacy in this form has emerged has a fundamental concept; it is one of the axioms of AD+ which is a structural generalization of AD. A major open question is whether AD implies AD+ . There are a number of partial results, for example, assuming ZF+DC, ADR implies AD+ . In analyzing the Suslin cardinals of L(R), Steel [Ste83A] obtained as a corollary (assuming AD+V=L(R)) that every reliable cardinal is a Suslin cardinal. For this equivalence one need only assume that the cardinal κ is weakly reliable. This is the assertion that there exists a scale on a set B with norms ñi such that for each i, ñi : B → κ and such that ñ0 is a surjection. Applying the theory of AD+ , one obtains this equivalence of weakly reliable cardinals and Suslin cardinals just assuming AD+ (so for example, as a consequence of ZF+DC+ADR ).

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REFERENCES

Howard S. Becker [Bec79] Some applications of ordinal games, Ph.D. thesis, UCLA, 1979. Leo A. Harrington and Alexander S. Kechris [HK81] On the determinacy of games on ordinals, Annals of Mathematical Logic, vol. 20 (1981), pp. 109–154. Alexander S. Kechris [Kec78A] AD and projective ordinals, In Kechris and Moschovakis [Cabal i], pp. 91–132. [Kec78B] Forcing in analysis, Higher set theory. Proceedings of a conference held at the Mathema¨ tisches Forschungsinstitut, Oberwolfach, April 13–23, 1977 (Gert H. Muller and Dana Scott, editors), Lecture Notes in Mathematics, vol. 669, Springer, 1978, pp. 277–302. [Kec81A] Homogeneous trees and projective scales, In Kechris et al. [Cabal ii], pp. 33–74. Alexander S. Kechris, Eugene M. Kleinberg, Yiannis N. Moschovakis, and W. Hugh Woodin [KKMW81] The axiom of determinacy, strong partition properties and nonsingular measures, this volume, originally published in Kechris et al. [Cabal ii], pp. 75–100. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer, 1981. [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. Donald A. Martin [MarC] On Victoria Delfino problem number 1, notes. Yiannis N. Moschovakis [Mos81] Ordinal games and playful models, In Kechris et al. [Cabal ii], pp. 169–201. John R. Steel [Ste80] More measures from AD, mimeographed notes, 1980. [Ste83A] Scales in L(R), this volume, originally published in Kechris et al. [Cabal iii], pp. 107– 156. W. Hugh Woodin [WooB] ℵ1 -dense ideals, to appear. DEPARTMENT OF MATHEMATICS CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CA 91125, USA

E-mail: [email protected] DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720, USA

E-mail: [email protected]

A CODING THEOREM FOR MEASURES

ALEXANDER S. KECHRIS

§1. Introduction. Assuming ZF+DC+AD, Moschovakis (see [Mos80, §7D]) has shown that if there is a surjection ð : R → ë from the reals (R = ù ù in this paper) onto an ordinal ë, then there is a surjection ð∗ : R → ℘(ë) from the reals onto the power set of ë. Let us denote by â(ë) the set of ultrafilters on ë. The question was raised whether there is an analog of Moschovakis’ Theorem for â(ë), i.e., if there is a surjection from R onto ë, is there one from R onto â(ë)? Martin showed that this cannot be proved in ZF+DC+AD alone, because if V = L(R) and ë = ä 21 , there is no surjection of R onto â(ë). We prove in this paper that if one estrengthens the determinacy hypothesis from 1/2 AD to ADR (see [Kec88A]), then this question has a positive answer. Our main theorem is then: 1/2

Theorem 1.1. Assume ZF+DC+ADR . If there is a surjection from R onto an ordinal ë, then there is a surjection from R onto â(ë). This result can be rephrased as follows. By a result of Kunen (for a proof see [Kec85]) â(ë) is wellorderable, so it has a definite cardinality, which we denote also by â(ë). Since the sup of the ordinals onto which we can map the continuum is denoted by Θ, the Theorem 1.1 says: 1/2

Corollary 1.2. Assuming ZF+DC+ADR , if ë < Θ, then â(ë) < Θ. Combining the above result with Moschovakis’ Theorem, one can actually obtain a stronger statement which implies both. Recall first the standard fact that in ZF+DC+AD every ultrafilter is countably complete. Denote then, for each ordinal ë, by q(ë) the set of countably complete filters on ë. This contains â(ë) but also contains ℘(ë) by the natural identification of A ⊆ ë with the filter Aˆ = {X ⊆ ë : A ⊆ X }. We now have the following: 1/2

Corollary 1.3. Assume ZF+DC+ADR . Then if ë < Θ, there is a surjection of R onto q(ë). Research partially supported by NSF Grants MCS-8117804 and DMS-8416349. The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

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Actually in Moschovakis’ Theorem one obtains, for each ð : R → ë, estimates of the complexity of Að = {x ∈ R : ð(x) ∈ A}, for A ⊆ ë, in terms of ≤ð , where x ≤ð y ⇔ ð(x) ≤ ð(y), which are very useful in various applications. Similarly we obtain definability estimates for the complexity of ultrafilters. As an immediate consequence we have various local versions of the main theorem, of which we mention as an example the following: Corollary 1.4. Assume ZF+DC+AD (this is all we need here). Then (i) If ë < ä 1ù = sup{ä 1n : n < ù}, then â(ë) < ä 1ù . e (ii) If ë <eκ KL (the eKleene ordinal of the continuum), then â(ë) < κ KL . R KL R Similarly for κ (for the definition of κ , κ , see, for instance [Kec85]). (iii) If ë < (ä 21 )L(R) , â(ë) < (ä 11 )L(R) . e e Finally let us mention another corollary concerning the absoluteness of ultrafilters for certain inner models of AD. Corollary 1.5. Assume ZF+DC+AD and let M be an inner model of 1/2 ZF+DC+ADR containing R. Let ë < ΘM . Then every ultrafilter on ë is in M . In particular, if ë is measurable, then M |= “ë is measurable”. Actually a finer version of this result is possible, which implies the following 1/2 (note that L(R) |= ¬ADR ). Corollary 1.6. Assume ZF+DC+AD. If ë < ΘL(R) , then every ultrafilter on ë belongs to L(R), thus if ë is measurable, L(R) |= “ë is measurable”. §2. A game for coding ultrafilters. Let C ⊆ R, ð : C → ℘(ë), ë an ordinal, and write for simplicity Xα ≡ ð(α). Consider then the following game G ≡ Gð , which is a “coded” version of a “cut-and-choose” game on ë: I

α0

αj ∈ R

α1 ...

II

i0

i1

ij ∈ {0, 1}

Player I wins iff: (i) ∀n(αn ∈ C ) and (ii) \ \ {Xαn : in = 0} ∩ {¬Xαn : in = 1} 6= ∅. We claim first that, assuming ZF+DC+AD, player I has no winning strategy in G. Indeed if ô was such a strategy, towards a contradiction, let, for every x ∈ ù 2, α0x , α1x , . . . be player I’s moves following ô, when player II plays x(0), x(1), . . . . Put n \ f(x) = min î : î ∈ {Xαnx : x(n) = 0} ∩ o \ {¬Xαnx : x(n) = 1} .

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Then f : ù 2 → ë is an injection, i.e., there is a wellordering of ù 2, which violates AD. A strategy F for player II in this game is a map F : <ù R → (0, 1). We now have the key lemma. Lemma 2.1. If F is a winning strategy for player II in G, and U ⊆ ℘(ë) is an ultrafilter, then there are α0 , α1 , . . . , αn−1 ∈ C , such that for all α ∈ C , Xα ∈ U ⇔ F (α0 , . . . , αn−1 , α) = 1. In particular each such U is completely determined by F ↾Rn+1 . Proof. Call a sequence (α0 , i0 , α1 , i1 , . . . , am−1 , im−1 ) U-good if it is a finite run of the game G in which player II follows F, α0 , . . . , αm−1 ∈ C and ∀j ≤ m − 1(Xαj ∈ U ⇔ ij = 0). The empty sequence is U-good by convention. If every U-good sequence has a U-good proper extension, we can obtain an infinite run (α0 , i0 , α1 , . . . ) of G in which playerTII followed F, α0 , αT 1, · · · ∈ C and Xαj ∈ U ⇔ ij = 0, for all j. Then clearly {Xαn : in = 0} ∩ {¬Xαn : in = 1} ∈ U (recall that U is countably complete), so this intersection is non-∅, and player I won, a contradiction. So let (α0 , i0 , . . . , αn−1 , in−1 ) be a maximal U-good sequence. Let α ∈ C . Then (α0 , i0 , . . . , αn−1 , α, F (α0 , . . . , αn−1 , α)) is not U-good, i.e., Xα ∈ U ⇔ F (α0 , . . . , αn−1 , α) = 1 and we are done.

⊣

§3. On real-integer games. Note that the preceding game, Gð , is a game in which one player plays reals and the other integers, so it is not necessarily determined using AD alone. We denote the determinacy of such games by 1/2 1/2 ADR . (In [Kec88A] it is shown that in ZF+DC, ADR is equivalent to Unif.) So we have immediately 1/2

Theorem 3.1. Assume ZF+DC+ADR . If ë < Θ, then â(ë) < Θ. Proof. Using the notation of §2, with C = R, ð : R → ë, map each (n + 1)~ = 1}. Then tuple α ~ = (α0 , . . . , αn−1 , αn ) in Rn+1 to Vα~ = {Xαn : F (α) â(ë) ⊆ {Vα~ : α ~ ∈ C n+1 , n = 0, 1, 2, . . . }. So there is a surjection of R onto â(ë). We also have easily the result about countably complete filters.

⊣

1/2

Corollary 3.2. Assume ZF+DC+ADR . If ë < Θ, then there is a surjection from R onto q(ë), the set of countably complete filters on ë. Proof. By a result of Kunen, if I is a proper countably complete filter on ë, then there is an ultrafilter U on ë extending I. (For a proof T see [Sol78A, p. 148]) Thus if I is a proper countably complete filter, then I = {U : U in an

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ultrafilter on ë containing I}. Thus ℘(â(ë)) can be mapped onto q(ë) and, since â(ë) < Θ, R can be mapped onto q(ë). ⊣ §4. The complexity of ultrafilters. In the notation of §2 again, let for each ultrafilter U on ë: U ∗ ≡ Uð∗ = {α ∈ C : Xα ∈ U}. It follows that for some α0 , . . . , αn−1 ∈ C , and all α ∈ C , α ∈ U ∗ ⇔ F (α0 , . . . , αn−1 , α) = 1, for any winning strategy F for player II in Gð . Thus the complexity of U ∗ depends (beyond that of C, ð) on the complexity of F ↾Rm , m = 1, 2, . . . (each of which is essentially a set of reals), for any winning strategy F for player II. There are certain real-integer games which can be proved determined in AD only. In those instances we get reasonably good estimates for the winning strategies, which give us corresponding estimates for ultrafilters. Here is a relevant result. Theorem 4.1 (Woodin, unpublished). Assume ZF+DC+AD. If A ⊆ ù R × ù is co-Suslin, then the (real-integer) game corresponding to A is determined. ù

Here A ⊆ ù R × ù ù is co-Suslin iff A′ ⊆ ù ù × ù ù given by A′ (α, â) ⇔ A({(α)n }, â) is co-Suslin, where α 7→ {(α)n } is a recursive 1-1 correspondence between ù ù and ù R. We give below an alternative version and proof (motivated by the ideas of [Kec88A]) of that result, which also gives the definability estimates we want. Theorem 4.2. Assume ZF+DC+AD. Let A ⊆ ù R × ù ù and suppose ¬A (viewed as a subset of ù ù × ù ù as above) carries a scale hϕn i such that each corresponding relation ≤∗ϕn , <∗ϕn belongs to a pointclass Γn , where Γ0 ⊆ Γ1 ⊆ · · · ⊆ Γn ⊆ . . . and Γn is closed under ∧, ∨, ∃m ≤ k, ∀m ≤ k and recursive substitutions. Consider the game G I II

α0

α1 i0

i1

... ...

αj ∈ R ij ∈ ù

Player II wins iff (α, ~ ~i) 6∈ A. If player I has no winning strategy in G, then player II has a winning strategy F : <ù R → ù such that for each n, F ↾Rn ∈ a2 Γn . (Here a is the game quantifier, see [Mos80], and a2 = aa). Proof. Fix a Turing degree d. Then clearly player I does not have a winning strategy in the game where he plays reals αi ≤T d. So, by the Third Periodicity

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ALEXANDER S. KECHRIS

Theorem of [Mos80, 6E.1], player II has a winning strategy Fd in this restricted game, such that for each n the relation αi ≤T d ∧ Fd (α0 , . . . , αn−1 ) = i is in aΓn , uniformly on d, i.e., the relation Rn (x, α0 . . . αn−1 , i) ⇔ αi ≤T x ∧ F[x]T (α0 , . . . , αn−1 ) = i is in aΓn . Define now F (α0 , . . . , αn−1 ) = i ⇔ for a cone of d’s, Fd (α0 , . . . , αn−1 ) = i. Clearly F ↾Rn ∈ a(aΓn ) = a2 Γn and F is a winning strategy for player II in the game G. ⊣ Let us now mention some specific applications. We assume ZF+DC+AD below. First let ë < ä 1ù be a projective ordinal, let ϕ : R → ë be a projective norm e and, using the Moschovakis’ Coding Lemma (see [Mos80, 7D.5]), let in the notation of §2, C ⊆ R, ð : C → ℘(ë) be also projective. (For ð this means that the relation “ϕ(x) ∈ ð(y)” is projective.) Then for any ultrafilter U on ë, U ∗ is projective (actually in some fixed level of the projective hierarchy) and â(ë) < ä 1ù . e of course state finer level-by-level versions of this result. However One can the recent work of Steve Jackson (see for example [Jac88]) provides a fairly complete analysis of ultrafilters on projective ordinals, which provides much more accurate estimates for the definability of ultrafilters. Similarly, if ë < κ KL , every ultrafilter on ë is Kleene recursive in 3 E and a real (in the codes), and â(ë) < κ KL . If ë < κ R , then every ultrafilter on ë is hyperprojective in the codes and â(ë) < κ R . For ë = κ R itself, we have that each ultrafilter on κ R is Σ∗m for some m (depending on U). For the definition of the classes Σ∗m see [Mos83]. So â(κ R ) ≤ ä ∗ù . Martin has showed that also e â(κ R ) ≥ ä ∗ù , so in fact â(κ R ) = ä ∗ù . e e Finally, if ë < (ä 21 )L(R) , then every ultrafilter on ë is (∆21 )L(R) in the codes, e e and â(ë) < (ä 21 )L(R) . e §5. Absoluteness of ultrafilters. The following fact is immediate from the analysis in §2.

Corollary 5.1. Assume ZF+DC+AD. Let M be an inner model of ZF+DC +AD containing R, let ë < ΘM , and let C, ð (as in §2) be in M . If F is a winning strategy for player II in Gð such that F ↾Rn ∈ M for all n ∈ ù, then every ultrafilter on ë is in M . In particular we have Corollary 5.2. Assume ZF+DC+AD. Let M be an inner model of ZF+DC 1/2 +ADR containing R, and let ë < ΘM . Then every ultrafilter on ë is in M , and thus, if ë is measurable, M |= “ë is measurable”.

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Finally we can obtain the result for L(R) itself. Corollary 5.3. Assume ZF+DC+AD. Let ë < ΘL(R) . Then every ultrafilter on ë is in L(R), and if ë is measurable, L(R) |= “ë is measurable”. Proof. If V = L(R), there is nothing to prove. If V 6= L(R) then by [SVW82], R# exists. Then by a result of Solovay (unpublished—see however [MS83, p. 93]), every A ∈ L(R) admits a scale hϕn i with ≤∗ϕn , <∗ϕn in L(R), thus, by Theorem 4.2, the hypothesis of Corollary 5.1 is satisfied and we are done. ⊣ REFERENCES

Stephen Jackson [Jac88] AD and the projective ordinals, In Kechris et al. [Cabal iv], pp. 117–220. Alexander S. Kechris [Kec85] Determinacy and the structure of L(R), Proceedings of symposia in pure mathematics, vol. 42, American Mathematical Society, pp. 271–283. 1 [Kec88A] AD + Unif is equivalent to AD/R2 , In Kechris et al. [Cabal iv], pp. 98–102. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer. Alexander S. Kechris, Donald A. Martin, and John R. Steel [Cabal iv] Cabal seminar 81–85, Lecture Notes in Mathematics, no. 1333, Berlin, Springer. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer. Donald A. Martin and John R. Steel [MS83] The extent of scales in L(R), this volume, originally published in Kechris et al. [Cabal iii], pp. 86–96. Yiannis N. Moschovakis [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam. [Mos83] Scales on coinductive sets, this volume, originally published in Kechris et al. [Cabal iii], pp. 77–85. Robert M. Solovay [Sol78A] A ∆13 coding of the subsets of ù ù, In Kechris and Moschovakis [Cabal i], pp. 133–150. e John R. Steel and Robert Van Wesep [SVW82] Two consequences of determinacy consistent with choice, Transactions of the American Mathematical Society, no. 272, pp. 67–85. DEPARTMENT OF MATHEMATICS CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CA 91125, USA

E-mail: [email protected]

THE TREE OF A MOSCHOVAKIS SCALE IS HOMOGENEOUS

DONALD A. MARTIN AND JOHN R. STEEL

§0. We work in the theory ZF+AD+DCR throughout this paper. Let T be a tree on ù × κ, for some ordinal κ. The projection of T is the set of reals p[T ] = {x ∈

ù

ù : ∃f ∈

ù

κ∀n(x↾n, f↾n) ∈ T }.

We say that p[T ] is Suslin via T . A homogeneity system for T is a system hìs : s ∈ <ù ùi of countably additive, 0-1 valued measures1 such that for all s, t ∈ <ù ù and x ∈ ù ù ìs (Ts ) = 1, and s ⊆ t ⇒ ìt projects to ìs , and x ∈ p[T ] ⇒ hìx↾n : n < ùi is countably complete. Here, Ts = {u : (s, u) ∈ T }, and ìt projects to ìs iff whenever ìs (A) = 1, then ìt ({u : u↾ dom(s) ∈ A}) = 1, and the tower of measures hìx↾n : n < ùi is called countably complete just in case whenever hAn : n < ùi is a sequence of sets such that ìx↾n (An ) = 1 for all n, then ∃f ∈ ù κ∀n(f↾n ∈ An ).2 If ì ~ is a homogeneity system for T , then we shall say p[T ] is homogeneously Suslin via T and ì ~. Which sets of reals are homogeneously Suslin? One can use partition cardinals to propagate homogeneity, and thereby show that all projective sets are homogeneously Suslin. (This is a result of Kunen and Martin; cf. [Kec81A, §6].) This method bogs down a bit past the projective sets, however, and leaves open, for example, whether all hyperprojective sets of reals are homogeneously Suslin. In the other direction, every homogeneously Suslin set is Suslin, and 1 That

is, countably complete ultrafilters. An = Tx↾n , we see that the countable completeness of the tower hìx↾n : n < ùi implies x ∈ p[T ]. 2 Taking

The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

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405

by [MS69], its complement in R is Suslin as well. Thus if V = L(R), every homogeneously Suslin set is ∆21 . In this paper we shall showethat the homogeneously Suslin sets are precisely those Suslin sets whose complements are also Suslin. Thus if V = L(R), a set of reals is homogeneously Suslin if and only if it is ∆21 . The homogeneous e trees in question are not generated by the partition cardinal construction; rather they are trees associated to scales produced by Moschovakis’ second periodicity construction. This result is due to the first author. We also include a result of the second author concerning the additivity of the homogeneity measures produced by this construction. §1. Our main result is Theorem 1.1. Assume ZF+AD+DCR ; then for any A ⊆ R, the following are equivalent: (1) A is homogeneously Suslin, (2) A and R \ A are Suslin. Proof. (1) ⇒ (2) follows at once from [MS69]. Now let A be Suslin and co-Suslin. Since ∅ is homogeneously Suslin, we may assume A is nonempty. Let hui : i < ùi be an enumeration without repetitions of <ù ù. We write n(u) for the unique n ∈ ù such that u = un . Let us choose our enumeration so that (dom(u) ⊆ dom(v) ∧ ∀i ∈ dom(u)(u(i) ≤ v(i))) ⇒ n(u) ≤ n(v)). This implies that proper initial segments of u are enumerated before u.3 This enumeration lets us identify Lipschitz continuous functions on R (i.e. strategies for player II in games on ù) with reals. Namely, for any ñ ∈ <ù ù, let ñ ∗ : <ù ù → <ù ù with domain {u : n(u) ∈ dom(ñ)} be given by ñ ∗ (u)(k) = ñ(n(u↾(k + 1)), whenever k ∈ dom(u). Let ñ ∗ (∅) = ∅. For any ó ∈ [ ó∗ = (ó↾n)∗ ,

ù

ù, we let

n<ù ∗

and we extend ó to a Lipschitz continuous function on R, which we also call ó ∗ , by setting [ ó ∗ (x↾i). ó ∗ (x) = i<ù 3 For

example, we can let n(u) ≤ n(v) iff the ith prime.

Q i ∈dom(u)

piu(i )+1 ≤

Q i ∈dom(v)

piv(i )+1 , where pi is

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So ñ ∗ is a finite neighborhood condition in the space of all Lipschitz continuous functions on R, and ó ∗ is the set of all neighborhood conditions coded into ó, as well as the Lipschitz continuous function meeting all these neighborhood conditions. We shall tend to use Greek letters ñ, ó, . . . for finite or infinite sequences of natural numbers when we are thinking of them as codes in this fashion, and Latin letters s, x, . . . otherwise. Let A∗ = {ó ∈

ù

ù : ∀x ∈

ù

ù(ó ∗ (x) ∈ A)}

be the set of codes for Lipschitz functions mapping into A. It is easy to see that A is Lipschitz-reducible to A∗ ; simply send x to the code for the constantly x function. It will therefore suffice to show that A∗ is homogeneously Suslin. Let hϕn : n < ùi be a very good scale on A. Moschovakis’ second periodicity construction [Mos80] produces a very good scale on A∗ . It is most natural to index the norms in the Moschovakis scale with finite sequences, so we shall do that. Let u ∈ <ù ù and let ó1 , ó2 ∈ A∗ . Consider the game G(u, ó1 , ó2 ): Player I plays out x1 ∈ R, and player II plays out x2 ∈ R, the players alternating moves, with player I going first, as usual. Let u ax ∈ R be the result of appending x to u, that is, (u ax)(n) = u(n) for n ∈ dom(u), and (u ax)(n) = x(n − dom(u)) otherwise. Since ói ∈ A∗ , we have ói∗ (u axi ) ∈ A for i = 1, 2. We say (x1 , x2 ) wins G(u, ó1 , ó2 ) for player II ⇔ ϕdom(u) (ó1∗ (u ax1 )) ≤ ϕdom(u) (ó2∗ (u ax2 )). Putting ó1 ≤u ó2 ⇔ player II has a winning strategy in G(u, ó1 , ó2 ), Moschovakis shows that each ≤u is a prewellorder of A∗ . For ó ∈ A∗ and u ∈ <ù ù, we let ϑu (ó) = rank of ó in ≤u , and we have by Moschovakis’ arguments that hϑu : u ∈ <ù ùi is a scale on A∗ .4 It is clear that ϑu (ó) only depends on ó↾{i : u ⊆ ui ∨ ui ⊆ u}. It is also clear that ϑu (ó) ≤ ϑu (ô) ⇔ ∀i∃j(ϑuahii (ó) ≤ ϑuahji (ô). ~ is given by: The tree of the scale ϑ T = {(ó↾n, hϑu0 (ó), . . . , ϑun−1 (ó)i) : ó ∈ A∗ }. 4 To see the limit property: let ó → ó (mod ϑ). ~ Let x ∈ R; we must show ó ∗ (x) ∈ A. We i can thin out the ói ’s so that ón+1 ≤x↾n ón for all n; let Σn be a winning strategy for player II in G(x↾n, ón+1 , ón ) witnessing this. Playing the Σn against each other simultaneously, we obtain zn so that zn → x and ón∗ (zn ) → ó ∗ (x) (mod ϕ). ~ (Here zn is defined by: x↾n ⊆ zn , and for k ≥ n, zn (k) = Σn (hx(n), zn+1 (n + 1), . . . , zn+1 (k)i).) Since ϕ ~ is a scale on A, ó ∗ (x) ∈ A.

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407

By the limit property of scales, p[T ] = A∗ . We shall show that T is homogeneous. The homogeneity measures ìñ on Tñ come from a game in which the two players interact to produce a potential member of Tñ . We need a Suslin representation of the complement of A in order to define our game, so fix a tree S on some ù × κ such that p[S] = R \ A. From S we get a Suslin representation of R \ A∗ : for ó ∈ R \ A∗ , we define the leftmost witness that ó 6∈ A∗ to be the pair (x, f), where x ∈ R and f ∈ ù κ are defined inductively by x(n) = min{k : ∃y∃g((x↾n)ahki ⊆ y ∧ f↾n ⊆ g ∧ (ó ∗ (y), g) ∈ [S])}, and f(n) = min{α : ∃y∃g(x↾(n + 1) ⊆ y ∧ (f↾n)ahαi ⊆ g ∧ (ó ∗ (y), g) ∈ [S])}. Now let ñ ∈ <ù ù and X ⊆ Tñ . We shall say that ìñ (X ) = 1 if and only if player II has a winning strategy in the game Gñ (X ), which is played as follows. Players I and II alternate playing natural numbers, with player I moving first, as usual. At the end of a run player I has produced ô1 ∈ R and player II has produced ô2 . Set ói = ñaôi . Player II wins Gñ (X ) if and only if either (1) ó1 6∈ A∗ and ó2 ∈ A∗ , or (2) ó1 ∈ 6 A∗ and ó2 6∈ A∗ , and letting (x1 , f1 ) and (x2 , f2 ) be the leftmost witnesses respectively, and ki least such that xi ↾ki 6∈ dom(ñ ∗ ), we have (x1 ↾k1 , f1 ↾k1 ) ≤lex (x2 ↾k2 , f2 ↾k2 ), or (3) ó1 ∈ A∗ and ó2 ∈ A∗ , and for all u ∈ dom(ñ ∗ ), ϑu (ó1 ) ≤ ϑu (ó2 ), and hϑui (ó2 ) : i ∈ dom(ñ)i ∈ X . Here the lexicographic order ≤lex is defined in a way which parallels our definition of the leftmost witnesses: (s, t) ≤lex (u, v) iff (s, t) = (u, v), or there is an i ∈ dom(s) such that s↾i = u↾i, t↾i = v↾i, and either s(i) < u(i), or s(i) = u(i) and t(i) < v(i). So ≤lex is a partial, not total, order on <ù ù × <ù κ. We also define ≤lex on ù ù × ù κ in the obvious way: (x, f) ≤lex (y, g) ⇔ ∀k(x↾k, f↾k) ≤lex (y↾k, g↾k). Here the order is total. It is clear that ìñ (Tñ ) = 1, since player II can win Gñ (Tñ ) by simply copying player I. It is equally clear that ìñ (∅) = 0, and that X ⊆ Y ∧ ìñ (X ) = 1 ⇒ ìñ (Y ) = 1. For the rest of the proof that the ìñ constitute a homogeneity system, we need some terminology concerning embeddings of one Lipschitz continuous function in another. Let ó, ô, ð ∈ ù ù. We say ð embeds ó in ô iff ó ∗ = ô ∗ ◦ ð∗ . We say that ð is an embedding over ñ, for ñ ∈ <ù ù, iff ð∗ (u) = u for all u ∈ dom(ñ ∗ ). For any ð ∈ ù ù, we say ð is II-safe ⇔ ∀u ∈

<ù

ù(u ≤lex ð∗ (u)),

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and if ð is an embedding over ñ, ð is I-safe over ñ ⇔ ∀u ∈

<ù

ù \ dom(ñ ∗ )(u
We say ô ∈ ù ù is the amalgamation of hói : i ∈ X i via hði : i ∈ X i iff each ði embeds ói into ô for all i, and {ran(ði ) : i ∈ X } is a partition of ù ù. We say the amalgamation is II-safe if all the ði are II-safe, and over ñ if all the ði are embeddings over ñ. Remark. The following two facts help explain our interest in safe amalgamations: (a) If ô is an amalgamation of hói : i ∈ X i and each ói ∈ A∗ , then ô ∈ A∗ . If the amalgamation is over ñ, then ϑu (ói ) ≤ ϑu (ô) for all i and all u ∈ dom(ñ ∗ ). (b) If the amalgamation is via II-safe embeddings ði , and ô 6∈ A∗ , then letting (x, f) be the leftmost witness that ô 6∈ A∗ , and x = ði (y), then ói∗ (y) = ô ∗ (x), so (y, f) is a witness that ói 6∈ A∗ such that (y, f) ≤lex (x, f). Claim 1.2. For any ñ ∈ X ) = 1.

<ù

ù and X ⊆ Tñ , either ìñ (X ) = 1 or ìñ (Tñ \

Proof. Suppose ìñ (X ) 6= 1, and fix a winning strategy Σ for player I in Gñ (X ). Player II cannot quite afford to simply use Σ as his own strategy in Gñ (Tñ \ X ), because of the demand that ϑu (ó1 ) ≤ ϑu (ó2 ) for all u ∈ dom(ñ ∗ ) in player II’s payoff condition (3). So player II will also embed player I’s play into his own as he proceeds. In order not to lose because of payoff condition (2), he will use a II-safe embedding. More precisely, player II’s winning strategy in Gñ (Tñ \ X ) is to insure that if player I plays out x ∈ R, then player II’s response y is such that ô = ñay is a II-safe amalgamation over ñ of ó = ñax and ø = ñaΣ(y). We show first that such a play (x, y) is a win for player II in Gñ (Tñ \ X ). Suppose first that ô ∈ A∗ . Then both ó and ø are in A∗ , since they embed in ô. Since the embedding is over ñ, ϑu (ó) ≤ ϑu (ô) and ϑu (ø) ≤ ϑu (ô) for all u ∈ dom(ñ ∗ ). Now (ø, ô) comes from a play of Gñ (X ) according to player I’s winning strategy Σ, and the second set of inequalities displayed therefore imply hϑu0 (ô), . . . , ϑudom(ñ)−1 (ô)i ∈ Tñ \ X. But then the first set of inequalities show that (x, y), which gives rise to (ó, ô), is a win for player II under payoff clause (3). Suppose next that ô 6∈ A∗ , and let (z, f) be the leftmost witness of this fact. Since the amalgamation yielding ô is II-safe, we can find (w, g) ≤lex (z, f) such that either (w, g) is the leftmost witness that ó 6∈ A∗ or (w, g) is the

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leftmost witness that ø 6∈ A∗ . (See Remark 1.) The latter is impossible, since it implies that (Σ(y), y) is a defeat for Σ under payoff condition (2) of Gñ (X ). The former implies that (x, y) is a win for player II in Gñ (Tñ \ X ) under payoff condition (2), as desired. Finally, we must see that player II can play his y in response to x so that ñay is the desired amalgamation. For each u ∈ dom(ñ ∗ ), let Iuñ = {i : u ahii 6∈ dom(ñ ∗ )}, and hu1 , hu2 : Iuñ → Iuñ be order-preserving functions such that ran(hu1 ) and ran(hu2 ) partition Iu , and ñ

k ≤ hui (k) for all k ∈ Iuñ and all i ∈ {0, 1}. We define our embeddings ð1 and ð2 by i ði∗ (v)(k) = hv↾k (v(k))

if v↾k ∈ dom(ñ ∗ ) but v↾(k + 1) 6∈ dom(ñ ∗ ), and ði∗ (v)(k) = v(k) otherwise. Notice that dom(ði∗ (v)) = dom(ði (v)), and that ði∗ (v) differs from v at at most one k, and for that k, v(k) ≤ ði∗ (v)(k). It follows that ∀v(n(v) ≤ n(ði∗ (v))). Player II can now determine y(k), granted x↾(k + 1), as follows: let u be such that n(u) = dom(ñ) + k. Then u = ði∗ (v) for exactly one pair i, v; moreover, n(v) = dom(ñ) + j for some j ≤ k. We let y(k) = x(j) if i = 0 and y(k) = Σ(y↾k)(j) if i = 1. We leave any further detail to the reader. T Claim 1.3. If ìñ (Xi ) = 1 for all i < ù, then ìñ ( i Xi ) = 1.

⊣ (Claim 1.2)

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DONALD A. MARTIN AND JOHN R. STEEL

Proof. Let Σi be a winning strategy for player II in Gñ (Xi ). For each u ∈ dom(ñ ∗ ), let hu1 , hu2 : Iuñ → Iuñ be as in the proof of Claim 1.2, but with k < hu2 (k) for all k ∈ Iuñ in addition. Let ð1 and ð2 be the II-safe embeddings over ñ determined from hu1 and hu2 as in Claim 1.2. Our additional property of hu2 guarantees that ð2 is in fact I-safe over ñ. Our winning strategy Σ for player II in Gñ (X ) will behave as follows: suppose player I plays out x ∈ ù ù, and let ó = ñax. Let y be player II’s response as dictated by Σ, and ô = ñay. Let ϕ be the amalgamation of ó and ô via ð1 and ð2 . (Here ð2 is used to embed ô into ϕ.) Since ó and ô extend ñ and the amalgamation is over ñ, ϕ extends ñ. Let ϕ = ñaz, let wi = Σi (z), and let øi = ñawi . Player II’s strategy Σ will be to arrange that ô is a II-safe amalgamation over ñ of høi : i < ùi. We first show that there is a strategy Σ such that plays (x, y) according to Σ have the property that there are z and wi as above. Let gui : Iuñ → Iuñ for i ∈ ù be nondecreasing functions whose ranges partition Iuñ , for each u ∈ dom(ñ ∗ ). Let hçi : i ∈ ùi be the II-safe amalgamating functions these gui determine (as in Claim 1.2). It is easy to see that there is a Lipschitz function which, given x, will produce y, z, and wi so that the conditions above hold. Note first that z↾(k + 1), and hence wi ↾(k + 1), are determined by x↾k and y↾k. For to determine z(k), let n(v) = dom(ñ)+k. If v = ð1∗ (r), then set z(k) = x(n(r)− dom(ñ)), noting that n(r) ≤ n(v), so that n(r) − dom(ñ) ≤ k. If v = ð2∗ (r), then n(r) < n(v) because of our additional requirement guaranteeing the I-safety of ð2 . Thus it makes sense to set z(k) = y(n(r) − dom(ñ)). It is enough, then, to determine y(k) from x↾(k + 1) together with z↾(k + 1) and wi ↾(k + 1). Let n(v) = dom(ñ) + k, and let v = çi∗ (r), and note that n(v) ≤ n(r). We then set y(k) = w Ti (n(r) − dom(ñ)). We now verify that Σ wins Gñ ( i<ù Xi ) for player II. Let (x, y) be a play by Σ, with z and wi for i ∈ ù associated as above. Let ó = ñax, ô = ñay, ϕ = ñaz, and øi = ñawi for all i. Suppose first that ô 6∈ A∗ , and let (p, f) be the leftmost witness to this fact. Let p = ði∗ (q); then (q, f) is a witness that øi 6∈ A∗ , and (q, f) ≤lex (p, f) because ði is II-safe. Now (z, wi ) is a play by Σi , which is winning for player II, and hence we have a witness (r, g) that ϕ 6∈ A∗ such that (r↾k1 , g↾k1 ) ≤lex (q↾k2 , f↾k2 ), where k1 , k2 are least such that r↾k1 6∈ dom(ñ ∗ ) and q↾k2 6∈ dom(ñ ∗ ). We cannot have r ∈ ran(ð2∗ ), for if r = ð2∗ (s), then because ð2 is I-safe, we have s↾k1
THE TREE OF A MOSCHOVAKIS SCALE IS HOMOGENEOUS

411

that ó 6∈ A∗ , and (s↾k1 , g↾k1 ) ≤lex (r↾k1 , g↾k1 ) ≤lex (p↾k2 , f↾k2 ), and since all embeddings above are over ñ, we have k1 is least such that ∗ s↾k1 6∈ dom(ñ ∗ ) and k2 is least such T that p↾k2 6∈ dom(ñ ). It follows that (x, y) is a win for player II in Gñ ( Xi ) under payoff condition (2). Next, suppose ô ∈ A∗ . We may assume that ó ∈ A∗ , as otherwise we are done in virtue of payoff condition (1). But then ϕ ∈ A∗ , as it is an amalgamation of ó and ô, and øi ∈ A∗ for all i, as ô is an amalgamation of høi : i ∈ ùi. Further, for u ∈ dom(ñ ∗ ) we have ϑu (ó) ≤ ϑu (ϕ) ≤ ϑu (øi ) ≤ ϑu (ô) for all i. (Here the first and last inequalities come from the properties of amalgamations, and the middle one holds because Σi wins for player II.) This shows player II has met his norm-inequality obligations. We also have, for u ∈ dom(ñ ∗ ), ϑu (ô) ≤ ϑu (ϕ ≤ ϑu (øi ) ≤ ϑu (ô), for similar reasons. Thus hϑun (ô) : n ∈ dom(ñ)i = hϑun (øi ) : n ∈ dom(ñ)i ∈ Xi , T for all i. This implies that (x, y) is a win for player II in Gñ ( i<ù Xi ) under payoff condition (3). ⊣ (Claim 1.3) So ìñ is a countably complete ultrafilter concentrating on Tñ . We now verify compatibility. Claim 1.4. If ñ, ϕ ∈ X }, then ìϕ (Y ) = 1.

<ù

ù, ñ ⊆ ϕ, ìñ (X ) = 1, and Y = {w : w↾ dom(ñ) ∈

Proof. Fix a winning strategy Σ for player II in Gñ (X ). Our strategy for player II in Gϕ (Y ) is roughly the following. If player I plays ó extending ϕ, and ø is the extension of ñ which Σ would play when ó is regarded as player I’s play in Gñ (X ), then player II will play a safe amalgamation ô of ø and the parts of ó below u’s in dom(ϕ ∗ ) \ dom(ñ ∗ ). More precisely, for u ∈ dom(ñ ∗ ), let hu : Iuñ ↔ Iuϕ be an order-preserving bijection, where Iuñ = {i : u ai 6∈ dom(ñ ∗ )}, and Iu is defined similarly. Note that i ≤ hu (i) for all i, as Iuϕ ⊆ Iuñ . We define a II-safe embedding over ñ by: ϕ

ð∗ (u ahiiav) = u ahhu (i)iav whenever u ∈ dom(ñ ∗ ) and i ∈ Iuñ , and of course ð∗ (u) = u for all u ∈ dom(ñ ∗ ). We can now define ô from ó and ø by letting ô ∗ (ð∗ (u)) = ø ∗ (u),

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and ô ∗ (v) = ó ∗ (v) for v 6∈ ran(ð∗ ). We leave it to the reader to check that ô(n) is determined by ó↾n (via ø↾n), for all n, so that we have legitimately described a strategy for player II in Gϕ (Y ). (Notice here that ô does extend ϕ, because ran(ð∗ ) is disjoint from dom(ϕ ∗ ) \ dom(ñ ∗ ).) In order to see that this strategy wins for player II, suppose ó, ô, and ø are as above. Suppose first ô 6∈ A∗ , and let (x, f) be the leftmost witness to this fact. If x 6∈ ran(ð∗ ), then (x, f) is a witness that ó 6∈ A∗ , so player II has won under payoff condition (2). Now suppose x = ð∗ (y), and let u be the longest initial segment of x, or equivalently y, such that u ∈ dom(ñ ∗ ). Since (y, f) is a witness that ø 6∈ A∗ , and (ó, ø) is a play by Σ, we have a leftmost witness (z, g) that ó 6∈ A∗ such that (z↾(j + 1), g↾(j + 1)) ≤lex (y↾(k + 1), f↾(k + 1)), where j is largest such that z↾j ∈ dom(ñ ∗ ) and k is largest such that y↾k ∈ dom(ñ ∗ ). Note that k = dom(u) since x↾(k + 1) 6∈ dom(ñ ∗ ) and ð∗ (y) = x. If the inequality just displayed is strict, then we have by the II-safety of ð∗ (z↾ℓ, g↾ℓ)
Iuϕ ,

(z↾ℓ, g↾ℓ)
THE TREE OF A MOSCHOVAKIS SCALE IS HOMOGENEOUS

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find a one-term extension r of u such that ϑv (ô) ≤ ϑr (ø). Again, r = v works if v ∈ dom(ñ ∗ ). If v 6∈ dom(ϕ ∗ ), then r = (ð∗ )−1 (v) works, because ð∗ is an “isomorphism” between ø ∗ below r and ô ∗ below v. Finally, if v ∈ dom(ϕ ∗ ) \ dom(ñ ∗ ), then ô ∗ agrees with ó ∗ below v. But ϑu (ó) ≤ ϑu (ø) because Σ is winning, so there is a one-term extension r of u such that ϑv (ô) = ϑv (ó) ≤ ϑr (ø). This completes the proof of Claim 1.4. ⊣ (Claim 1.4) Finally, we must show the countable completeness of towers associated to ó ∈ A∗ . Claim 1.5. Let ó ∈ A∗ , and suppose ìó↾n (Xn ) = 1 for all n; then there is an infinite sequence f such that f↾n ∈ Xn for all n. Proof. By Claim 1.4, we may as well assume that Xn+1 projects into Xn , for all n. Let Σn be a winning strategy for player II in Gó↾n (Xn ). By playing all the Σn together in the usual way, we obtain ôn for n < ù such that for each n, ó↾n ⊆ ôn and (ôn+1 , ôn ) arises from a play by Σn in which player I is responsible for ôn+1 and player II is responsible for ôn .5 We claim that ôn ∈ A∗ for all n. For if ôn 6∈ A∗ , then since the Σ’s win for player II, ôk 6∈ A∗ for all k ≥ n, by induction on k. Letting (xk , fk ) be the leftmost witness that ôk 6∈ A∗ , payoff condition (2) implies that (xk , fk ) converges to some (x, f). It is clear that (x, f) is a witness that ó 6∈ A∗ , a contradiction. Thus (ôn+1 , ôn ) falls under player II’s payoff condition (3), for all n. But this implies that for each u ∈ <ù ù, ϑu (ôk ) ≥ ϑu (ôk+1 ) for all k > n(u), so that we can let f(n) = eventual value of ϑun (ôk ) as k → ù. It is easy to see that f↾n ∈ Xn for all n. ⊣ (Claim 1.5) These claims complete the proof of Theorem 1.1. ⊣ (Theorem 1.1) We conclude with two further observations. First, the tree T constructed in the proof of Theorem 1.1 has the genericcodes property isolated by Kechris and Woodin [KW07]. ~ and let (r, v) ∈ T . Consider the BanachLet T be the tree of a scale ϑ, Mazur game G∗∗ (r, v) on T in which at round n, player I plays (r2n , v2n ) ∈ T , then player II plays (r2n+1 , v2n+1 ) ∈ T , and we Srequire (r, v) ⊆ S (r0 , v0 ) and (rn , vn ) ⊆ (rn+1 , vn+1 ) for all n. Letting x = n rn and f = n vn , we say player II wins G∗∗ (r, v) iff for all k ∈ dom(r), ϑuk (x) = v(k). (By the lower 5 Formally,

ôn = (ó↾n)axn , where xn (k) = Σn (hó(n)ia(xn+1 ↾k)).

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~ ϑu (x) ≤ v(k).) Kechris and Woodin say that semi-continuity of the scale ϑ, k T has the generic codes property if for all (r, v) ∈ T , player II has a winning strategy in G∗∗ (r, v). For any Suslin set A, they construct a T with the generic codes property projecting to A. ~ constructed in the proof of 1.1 It is quite easy to show that the T and ϑ ∗∗ have the generic codes property. To win G (r, v), player II simply picks at the outset some ó ∈ A∗ such that r ⊆ ó and ϑuk (ó) = v(k) for all k ∈ dom(r). He then plays so that if (ô, f) is the output of the game, then ó is embedded into ô. Since player II can only make finite extensions, he can never prevent player II from further extending the embedding he is building. Our second observation is that in certain natural situations, the homogeneity measures we have constructed are more than countably complete. Theorem 1.6. Let Γ be an ù-parametrized pointclass which is closed under recursive substitution, number quantification, and universal real quantification, and suppose that every set in Γ is Suslin and co-Suslin. Let ä be the e ˘ then every set supremum of the order types of prewellorders in Γ ∩ Γ; in Γ is ä -homogeneously Suslin. e Proof. We illustrate the proof in the special case Γ = Π13 . Let ä = ä 13 . We shall define a complete Π13 set A∗ such that the homogeneity measuresewe get from the proof of 1.1 are ä-complete. In this connection, it will help if we are more careful in our choice of the ingredients A (which will be Σ12 ), the scale ϕ ~ on A, and the Suslin representation S of R \ A. Let U ⊆ ù × R be a universal Σ12 set. Fix a map H : ù ։ HF, where HF is the set of all hereditarily finite sets. For x ∈

ù

ù, put x ∈ A¯ iff

(a) for all n, H (x(n)) = hrn , vn i ∈ ( <ù ù ×

<ù

ù),

where dom(rn ) = dom(vn ) > 0, (b) rn ⊆ rn+1 , and rn 6= rn+1 , and (c) letting eix be the common value of rn (i) for all i > n, and defining (x)i ∈ ù ù by (x)i (n) = vp+n (i), where p = min{k : i ∈ dom(vk )}, we have ∃iU (eix , (x)i ). Note that the set of x ∈ ù ù satisfying (a) and (b) in the definition of A is ¯ or x 6∈ [W0 ]. closed; let W0 be the tree of this closed set. Put x ∈ A iff x ∈ A,

THE TREE OF A MOSCHOVAKIS SCALE IS HOMOGENEOUS

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Let hϕi∗ : i < ùi be a very good Σ12 -scale on U . We get a Σ12 -scale on A in a routine way.6 Let x ∈ A. If x 6∈ [W0 ], then set ϕn (x) = h0, ki, where k is least s.t. x↾k 6∈ W0 . ¯ and we have U (eix , (x)i ) for some i. Let i0 be such Otherwise, x ∈ A, ∗ x that ϕ0 (ei0 , (x)i0 ) = α is as small as possible, and i0 is the least i such that ϕ0∗ (eix , (x)i ) = α, and set ϕn (x) = h1, ϕ0∗ (i0 , (x)i0 ), i0 , ϕ1∗ (i0 , (x)i0 ), . . . , ϕn∗ (i0 , (x)i0 )i for all n. Here we use the lexicographic order to identify the range of ϕn with an ordinal. It is clear that ϕ ~ is a Σ12 -scale on A. For a technical reason, we have not tried to make ϕ ~ very good; it could happen that ϕn (xi ) is eventually constant for each n, but the xi do not converge. However, it is true that n ≤ m and ϕn (x) < ϕn (y) imply ϕm (x) < ϕm (y). This refinement property ~ on A∗ , defined as in is enough to guarantee that the Moschovakis norms ϑ ∗ the proof of Theorem 1.1, constitute a scale on A . It is easy to check that if ó ∈ A∗ and ó ∗ (u) 6∈ W0 , then since ϕdom(u) is constant on the neighborhood determined by ó ∗ (u), ϑu (ó ∗ ) = 0. This property, which might have been lost if we had tried to make ϕ ~ very good, will be of use later. Let S0 be a tree on ù × κ such that p[S0 ] = (ù × R) \ U . We construct a tree S projecting to R \ A. For s ∈ W0 , let hrns , vns i = H (s(n)), and letting n be largest in dom(s), set k(s) = dom(rns ) and eis = rns (i) for all i < k(s), and s (s)i (j) = vp+j (i), where p = least ℓ s.t. i ∈ dom(vℓs ).

(The domain of (s)i is (n − p) + 1.) That is, eis and (s)i , for i < k(s), are the common initial segments of all eix and (x)i for s ⊆ x determined by s. Now fix a bijection c : ë → <ù κ, for some ë, with the property that dom(s) < dom(t) ⇒ c −1 (s) < c −1 (t). For s ∈ <ù ù and t ∈ <ù ë, put (s, t) in S iff (i) s ∈ W0 , dom(s) = dom(t), and ∀n ∈ dom(s), dom(c(t(n)) = dom(vns ), and (ii) letting (t)i (j) = c(t(p + j))(i), where p = least ℓ s.t. i ∈ dom(vℓs ), (where j is such that p + j < dom(s), we have (eis , (s)i , (t)i ) ∈ S0 for all i < k(s). 6 The definability of the scale ϕ ~ on A∗ it induces, are actually ~ , and of the Moschovakis scale ϑ not relevant to Theorems 1.1 or 1.6.

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That is, S is the tree of attempts to build an (x, f) such that x satisfies (a) ¯ and for each i < ù, (f)i (defined as above) and (b) of the definition of A, x witnesses that ¬U (ei , (x)i ). Having defined the ingredients A, ϕ, ~ and S in the proof of Theorem 1.1, we ~ be the Moschovakis scale on A∗ , T be the tree of ϑ, ~ and hìñ : ñ ∈ <ù ùi let ϑ be the homogeneity system for T defined there. We also carry over all the notation regarding Lipschitz maps from the proof of 1.1. Fix ñ ∈ <ù ù, and suppose T that ìñ (Xα ) = 1 for all α < ã0 , where ã0 < ä. We must show that ìñ ( α<ã0 Xα ) = 1, that is, we must describe a winning strategy for player II in T Gñ ( α<ã0 Xα ). This strategy will be, roughly, to amalgamate plays according to strategies for player II in the games Gñ (Xα ), as was done in the proof that ìñ is countably complete. The new problem, that there are uncountably many such strategies, makes the amalgamation somewhat more delicate, but we can use a rather direct transcription of an argument of Kunen.7 That argument rests on the Coding Lemma. Strategies for player II in games of the form Gñ (Y ) are Lipschitz functions on ù ù, and so can be coded by reals Σ ∈ ù ù as we have done above. By the Coding Lemma, there is a Σ13 set B such that ∀Σ(B(Σ) ⇒ ∃α < ã0 (Σ is a winning strategy in Gñ (Xα ))), and ∀α < ã0 ∃Σ(B(Σ) ∧ Σ is a winning strategy in Gñ (Xα )). Since B is Σ13 , we can fix a real z0 and an a ∈ ù such that for all Σ e B(Σ) ⇔ ∃y¬U (a, [Σ, y, z0 ]). For use below, we spell out our coding of tuples here: if s0 , . . . , sk−1 ∈ <ù ù are sequences of the same length n, then [s0 , . . . , sk−1 ] is the sequence of length n such that for i < n, [s0 , . . . , sk−1 ](i)S= n(hs0 (i), . . . , sk−1 (i)i).) If x0 , . . . , xk−1 ∈ ù ù, then [x0 , . . . , xk−1 ] = n [x0 ↾n, . . . , xk−1 ↾n]. Finally, if t = [s0 , . . . , sk−1 ], then we call t a k-code, and write (t)i = si . (The more common (t)i notation has already been dedicated to something slightly different.) Let ð1 , ð2 be the II-safe embeddings over ñ defined in the proof of Claim 1.2 (induced by picking for each u ∈ dom(ñ ∗ ) increasing maps hu1 , hu2 from Iuñ to itself, whose ranges partition Iuñ ). If ó, ô ∈ ù ù extend ñ, then we write Tó ⊕ ô for their amalgamation via (ð1 , ð2 ). Our strategy for player II in Gñ ( α Xα ) will behave as follows: as player I produces ó extending ñ, we produce a ô extending ñ whose membership in A∗ would code the truth of the statement ∀Σ(B(Σ) ⇒ Σ(ó ⊕ ô) ∈ A∗ ). 7 From

his proof that all ä 1n are measurable; see [Kec78A]. e

THE TREE OF A MOSCHOVAKIS SCALE IS HOMOGENEOUS

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Notice that this statement is Π13 , and A∗ is complete Π13 , so it is reasonable to try to construct such a ô. Of course, the statement coded by ô is self-referential. It is possible to bring in the recursion theorem at this point, but instead we shall simply give a direct construction of the desired ô. More precisely, for each u ∈ dom(ñ ∗ ), we want the restriction of ô ∗ to sequences extending u to code the truth of the statement ∀Σ∀y∀x ⊇ u(U (a, [Σ, y, z0 ]) ∨ Σ(ó ⊕ ô)(x) ∈ A). This is achieved by having ô ∗ regard the s in u as as a 3-code for an initial segment of a possible Σ, y, and x, then compute an appropriate fragment of Σ(ó ⊕ ô)(x), then give an output whose membership in A codes up the disjunction in the matrix of the formula just displayed. To make this work, we have to let (s)0 give information about Σ quickly enough, so we make the convention: for any s ∈ <ù ù, se = us(0)a. . .aus(dom(s)−1) , and for x ∈ ù ù, S g It will be se0 which is considered by ô ∗ as a possible initial xe = n x↾n. segment of some Σ in B. We turn to the details of the construction of ô from ó. Suppose we are given ô↾n and ó↾(n + 1); we must define ô(n). We may assume dom(ñ) ≤ n, as otherwise ô(n) = ñ(n). Our amalgamation process yields (ó ⊕ ô)↾(n + 1). Letting n = n(v), our job is to define ô ∗ (v). Let v¯ = v↾(dom(v) − 1). If ô ∗ (v) ¯ 6∈ W0 , then we just let ô ∗ (v) be any one-term extension of ô ∗ (v). ¯ Assume now that ô ∗ (v) ¯ ∈ W0 . Let us write v = u aw, where u is the longest initial segment of v in dom(ñ ∗ ). If w is not a 3-code, then again we just let ô ∗ (v) be any one-term extension of ô ∗ (v) ¯ which is not in W0 ; note we can always leave W0 in one step. So assume w is a 3-code, and let g0 . Γ = (w)

This is the approximation to Σ determined by w, the approximations to y and x being (w)1 and u a(w)2 respectively . If (ó ⊕ ô)↾(n + 1) 6∈ dom(Γ∗ ) or ñ 6⊆ Γ∗ ((ó⊕ô)↾(n+1)), then once again we let ô ∗ (v) be any one-term extension of ô ∗ (v) ¯ which is not in W0 . Assume then that (ó ⊕ ô)↾(n + 1) ∈ dom(Γ∗ ), and that Γ∗ ((ó ⊕ ô)↾(n + 1)) extends ñ. Clearly, n(u a(w)2 ) ≤ n(v), so we can set s = (Γ∗ ((ó ⊕ ô)↾(n + 1)))∗ (u a(w)2 ), and we have ñ ∗ (u) ⊂ s and dom(s) = dom(v). If s 6∈ W0 , then again we can let ô ∗ (v) be any one-term extension of ô ∗ (v) ¯ which is not in W0 . We are finally in what we shall call the interesting case in the definition of ô ∗ (v). Let s k = dom(u), ℓ = dom(v), ¯ and m = dom(rk−1 ).

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DONALD A. MARTIN AND JOHN R. STEEL

We must define ô ∗ (v)(ℓ), the earlier entries having been given by ô ∗ (v). ¯ We do this by setting H (ô ∗ (v)(ℓ)) = hr, vi, where r = hrℓs (0), . . . , rℓs (m − 1), a, rℓs (m), . . . , rℓs (dom(rℓs − 1))i, and v = hvℓs (0), . . . , vℓs (m − 1), [Γ∗ , (w)1 , z0 ](ℓ − k), vℓs (m), . . . , vℓs (dom(vℓs − 1))i. To be pedantic, we should have written [Γ∗ ↾(ℓ +1−k), (w)1 ↾(ℓ +1−k), z0 ↾(ℓ + 1 − k)] above. We leave it to the reader to check that ô ∗ (v) ∈ W0 . This completes the definition of ô ∗ (v), T and thereby our description of player II’s putative winning strategy in G ( Tñ α Xα ). Let (ó, ô) be a run of Gñ ( α Xα ) according to this strategy. We must see player II wins. Suppose first that ô 6∈ A∗ , and let (p, f) be the leftmost witness to this fact. Note that ô ∗ (p↾i) is defined using the interesting case, for all i. Let k be largest such that p↾k ∈ dom(ñ ∗ ), and p = u az, where u = p↾k. Then z = [Σ, y, x] ¯ must be a 3-code. Inspecting the tree S and our definition of ô ∗ (p), we see that ô (p) (a, [Σ, y, z0 ], (f)m ) ∈ [S0 ], where m = dom(rk−1 ), ∗

which implies that B(Σ) holds. Let g ∈

ù

ë be given as follows: if

c(f(j)) = hα0 , . . . , αn i, then g(j) = f(j) if n < m, and otherwise c(g(j)) = hα0 , . . . , αm−1 , αm+1 , . . . , αn i. Letting x = u ax¯ it is easy to see that our construction guarantees that (x, g) is a witness that Σ∗ (ó ⊕ ô) 6∈ A∗ , and moreover, (x, g) ≤lex (p, f). (Note here that g(j) = f(j) for j < k, and g(j) < f(j) for j ≥ k because c(g(j)) is a sequence of length one less than that of c(f(j)).) The remainder of the argument goes as in the countable additivity proof: since Σ is winning, there is a witness (q, h) that ó ⊕ ô 6∈ A∗ such that (q↾k1 , h↾k1 ) ≤lex (x↾k2 , g↾k2 ) for the appropriate k1 , k2 . The nature of the amalgamation and the fact that (p, f) is leftmost implies that (q, h) is a witness that ó 6∈ A∗ , and that therefore (ó, ô) is a win for player II under payoff clause (2).

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Next, suppose ô ∈ A∗ . If ó 6∈ A∗ , we are done, so suppose ó ∈ A∗ . Notice that if u ∈ dom(ñ ∗ ) and B(Σ), then sup(ϑu (ó), ϑu (ô)) ≤ ϑu (ó ⊕ ô) ≤ ϑu (Σ∗ (ó ⊕ ô)), because Σ is winning for player II. Claim. For u ∈ dom(ñ ∗ ) and Σ such that B(Σ), ϑu (Σ∗ (ó ⊕ ô)) ≤ ϑu (ô). Proof of Claim. We describe a winning strategy Γ for player II in the Moschovakis comparison game G(u, Σ∗ (ó ⊕ ô), ô). Fix a witness y to B(Σ), so that ¬U (a, [Σ, y, z0 ]). Suppose that player I plays out x extending u in this game (his first move being x(dom(u))). Let uˆ be the longest initial segment of x which is in dom(ñ ∗ ), and x = uˆ ax. ¯ Then player II will respond with e Γ(x) = uˆ a[w, y, z0 ], where Σ = w,

and w codes Σ sufficiently rapidly.8 It is clear that this defines a strategy Γ for player II. To see that it wins, suppose player I has played x extending u. If Σ∗ (ó ⊕ ô)(x) is not a branch of W0 , then by our construction, neither is ô ∗ (Γ(x)), and moreover both reals leave W0 at the same time, so that ϕn (Σ∗ (ó ⊕ ô)(x)) = ϕn (ô ∗ (Γ(x))) = h0, ki for all n, where k is this common departure time. Thus player II has won the Moschovakis comparison game. If p = Σ∗ (ó ⊕ ô)(x) is a branch of W0 , then by construction so is q = ô ∗ (Γ(x)). p By construction, setting k = dom(u) ˆ and m = dom(rk−1 ), we have  p  when i < m ei eiq = a when i = m   p ei−1 when i > m and

  (p)i (q)i = [Σ, y, z0 ]   (p)i−1

when i < m when i = m when i > m.

Using these facts, and inspecting the definition of the ϕn , it is easy to see that ϕn (p) ≤ ϕn (q) for all n. The key point here is just that U (a, (q)m ) is false, so it cannot lower the norm values. ⊣ (Claim) 8 Namely,

Σ↾n(ó ⊕ ô↾(k + 5)) ⊆ w↾(k ^ − dom(u)) ˆ for all k ≥ dom(u). ˆ

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We can now complete the proof of Theorem 1.6. We have that whenever B(Σ) and u ∈ dom(ñ ∗ ), then ϑu (ô) = ϑu (Σ∗ (ó ⊕ ô)). Fixing α < ã0 and Σ a winning strategy for player II in Gñ (Xα ) such that B(Σ), this implies hϑu0 (ô), . . . , ϑuk (ô)i ∈ Xα , where k = dom(ñ) − 1. Since this T is true for all α < ã0 , (ó, ô) is a win for player II under clause (3) in Gñ ( α<ã0 Xα ), as desired. ⊣ REFERENCES

Alexander S. Kechris [Kec78A] AD and projective ordinals, In Kechris and Moschovakis [Cabal i], pp. 91–132. [Kec81A] Homogeneous trees and projective scales, In Kechris et al. [Cabal ii], pp. 33–74. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer, 1981. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. Alexander S. Kechris and W. Hugh Woodin [KW07] Generic codes for uncountable ordinals, this volume, originally circulated manuscript, 2007. Donald A. Martin and Robert M. Solovay [MS69] A basis theorem for Σ13 sets of reals, Annals of Mathematics, vol. 89 (1969), pp. 138–160. Yiannis N. Moschovakis [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES, CA 90095, USA

E-mail: [email protected] DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720, USA

E-mail: [email protected]

WEAKLY HOMOGENEOUS TREES

DONALD A. MARTIN AND W. HUGH WOODIN

§1. Introduction and background. We work in ZF set theory, i.e., in ZFC without the axiom of Choice. We will give some sufficient conditions for the truth of “For all ordinals ë, every tree on ù × ë is weakly homogeneous,” and we will discuss the extent to which the projection of a weakly homogeneous tree must be the projection of a homogeneous tree.1 If ë is an ordinal number, then a tree on ù × ë is a set T with the following properties. (1) The members of T are pairs (s, t) with s ∈ <ù ù, t ∈ <ù ë, and lh(s) = lh(t). (2) If (s, t) ∈ T and n < lh(s), then (s↾n, t↾n) ∈ T . In this paper, we shall mean by a tree a tree on ù×ë for some ordinal number ë. We shall treat the word “measure” as synonymous with “ultrafilter.” We shall use “measure” almost exclusively for ultrafilters on sets n ë where n < ù and ë is a non-zero ordinal number. This includes the trivial ultrafilters on one-point sets 0 ë. By a cardinal number, we mean an initial ordinal, i.e., an ordinal number that cannot be injected into any smaller ordinal number. If n < ù, κ is an infinite cardinal number and ë is a non-zero ordinal n number, let MEASκ,ë n be the set of all κ-complete measures on ë. m n If m < n < ù and X ⊆ ë, then define extn (X ) = {t ∈ ë : x↾m ∈ X }. If ì is a measure on n ë, define projm (ì) = {X ⊆

m

ë : extn (X ) ∈ ì}.

It is easy to see that projm (ì) is a measure on m ë and that it is as complete (i.e., as additive) as ì is. For non-zero ordinal numbers ë, a ë-tower of measures is a sequence hìn : n < ùi such that 1 Theorem 3.1 is due to Martin. Theorem 3.2 is a variation due to Woodin as are the results of the last section, except for Theorem 4.8 which is due to Steel. The proof given here of Theorem 3.1 is based on the proof of Theorem 3.2. Martin’s original proof used partition cardinals. Theorem 3.1 and Theorem 3.2 were proved in the early 1980’s.

The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

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(i) for each n, ìn ∈ MEASℵn 1 ,ë ; (ii) for m < n, ìm = projm (ìn ). We say that a tower hìn : n < ùi is countably complete if, for every sequence hXn : n < ùi with Xn ∈ ìn for all n, there is an f : ù → ë such that f(n) ∈ Xn for all n. Let T be a tree on some ù × ë. Define [T ] = {(x, f) : (∀n < ù)(x↾n, f↾n) ∈ T }; p[T ] = {x ∈ Ts = {t ∈

ù

ù : (∃f)(x, f) ∈ [T ]};

lh(s)

ë : (s, t) ∈ T } (for s ∈

The tree T is homogeneous if there is a system hìs : s ∈ ℵ1 ,ë MEASlh(s)

<ù

ù).

<ù

ùi such that

and ìs (Ts ) = 1. (i) for each s, ìs ∈ (ii) if s1 ⊆ s2 then ìs1 = projlh(s1 ) (ìs2 ). (iii) for all x ∈ p[T ] the ë-tower of measures hìx↾n : n < ùi is countably complete. The tree T is weakly homogeneous if there is a system hMs : s ∈ <ù ùi such that 1 ,ë (i) for each s, Ms is a countable subset of MEASℵlh(s) and Ts belongs to every member of Ms ; (ii) for all x ∈ p[T ] there is a countably complete ë-tower of measures hìn : n < ùi such that each ìn belongs to Mx↾n . In this paper we show that every tree is weakly homogeneous if either of the following holds: (1) ADR , the Axiom of Determinacy for real games (i.e., for infinitely long games whose moves are real numbers); (2) the universe comes from an inner model M of ZFC by taking the M (R) of the L´evy collapse to ù of all ordinals less than some supercompact cardinal. The proofs of these two theorems have a common core, to which §2 will be devoted. The two theorems will be deduced in §3. In §4 results related to the second theorem will be proved, including a weakening of its hypothesis. These results will be applied to the question of when a projection of a weakly homogeneous tree must be homogeneous. If κ is a cardinal number and A is a set, then ℘κ (A) is the set of all subsets of A of size < κ. Let κ and ä be cardinal numbers, let A be a nonempty set, and let U be an ultrafilter on ℘κ (A). U is: (a) fine if, for every x ∈ A, {a : x ∈ a} ∈ U; (b) normal if, for every f : ℘κ (A) → ℘(A), if {a : a ∩ f(a) 6= ∅} ∈ U then there is an x ∈ A such that {a : x ∈ f(a)} ∈ U.

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It is easy to prove that U is normal just in case it is closed under diagonal intersections, i.e., just in case, for any hYx : x ∈ Ai, if Yx ∈ U for each x ∈ A, then {a : (∀x ∈ a) a ∈ Yx } ∈ U. In the presence of the Axiom of Choice, it is usual to define U to be normal just in case, for every f : ℘κ (A) → A, if {a : f(a) ∈ a} ∈ U, then f is constant on a set ∈ U. Clearly U is normal in this sense if it is normal in our sense. Moreover, it is easy to prove that the converse holds if A can be wellordered. If κ and ë ≥ κ are infinite cardinal numbers, then κ is ë-supercompact if there is a κ-complete, fine, normal ultrafilter on ℘κ (ë). A different sort of example of a fine, normal ultrafilter exists under the hypothesis ADR . We define a set U of subsets of ℘ℵ1 (R) as follows. For X ⊆ ℘ℵ1 (R), consider the real game GX a play hxi : i < ùi of which is a win for player II just in case {xi : i < ù} ∈ X . We define X ∈ U ↔ II has a winning strategy for GX . Lemma 1.1 (Solovay, [Sol78B]). Assume ADR . Then U is a countably complete, fine, normal ultrafilter on ℘ℵ1 (R). Proof. Obviously ℘ℵ1 (R) ∈ U, and obviously if X ⊆ Y and X ∈ U then Y ∈ U. Suppose that strategies ô1 and ô2 witness X1 ∈ U and X2 ∈ U respectively. We get a strategy witnessing X1 ∩ X2 ∈ U as follows. Let E1 = {4n : n < ù} and let E2 = {4n + 2 : n < ù}. Let II’s moves on Ei be gotten by playing ôi delayed against an opponent whose moves are the xj for j < ù \ Ei . An argument similar that just given for closure under intersections could be used to show directly that U is countably complete. But this is unnecessary, since AD implies—and so ADR implies—that every ultrafilter is countably complete. (See Proposition 28.1 of [Kan94].) We thus can finish the proof that U is a countably complete ultrafilter by demonstrating that (∀X )(X ∈ U ↔ (℘ℵ1 (R) \ X ) ∈ / U). If ó is a strategy for I witnessing that (℘ℵ1 (R) \ X ) ∈ / U, then a strategy for II witnessing that X ∈ U is ó itself, delayed by a move. If ô is a strategy for II witnessing that X ∈ U, then a strategy for I witnessing that (℘ℵ1 (R)\X ) ∈ / U is gotten by choosing an arbitrary first move for I and then letting I get the x2i+1 by playing ô against an opponent who makes the moves x0 , x1 , x3 , x5 , . . . . Fineness is trivial, since player II can choose any given real as, for example, his first move. For normality, let f : ℘ℵ1 (R) → ℘(R). Assume that (∀x ∈ R){a : x ∈ / f(a)} ∈ U.

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Let X = {a : a ∩ f(a) = ∅}. We must show that II has a winning strategy for GX . Consider the following ¯ A play hxi : i < ùi of G¯ is a win for II just in case real game G. x0 ∈ / f({xi : i < ù}). Our assumption implies that, for any fixed x, II has a winning strategy for G¯ when I is constrained to play x0 = x. Thus I does not have a winning strategy ¯ and therefore II has a winning strategy ô. We can use ô to get a winning in G, strategy for II in GX . To do this, break the even natural numbers into infinite pieces Ei , i < ù, in such a way that the least element of Ei is greater than i. Let II’s moves on Ei be gotten by playing ô against an opponent who plays xi followed by, in order, all the remaining xj for j < ù \ Ei . ⊣ Let Θ be the least non-zero ordinal number that is not a surjective image of R. Clearly Θ is a cardinal number. Let 0 < ë < Θ. By [Mos70A], AD implies that ℘(ë) is a surjective image of R, and hence that ℘( n ë) is such a surjective image for each n < ù. By [Kec88B], ADR implies that MEASën is a surjective image of R for each S n < ù. It follows that ADR implies that n<ù (℘( n ë) ∪ MEASën ) is a surjective image of R. Lemma 1.2. Assume ADR and let 0 < ë < Θ. There is a countably complete, fine, normal ultrafilter on [ ℘ℵ1 ( (℘( n ë) ∪ MEASën )). n<ù

S

Proof. Let f : R → n<ù (℘( n ë) ∪ MEASën ) be a surjection. S Let U be the ultrafilter given by Lemma 1.1. Define an ultrafilter V on ℘ℵ1 ( n<ù (℘( n ë) ∪ MEASën )) by setting X ∈ V ↔ f −1 [X ] ∈ U. It is easy to check that V inherits the properties of countable completeness, fineness, and normality from U. ⊣ §2. The main lemmas. Throughout this section, (A) let κ be an uncountable cardinal number; (B) let ë be an ordinal number with κ ≤ ë; S (C) let V be a κ-complete, fine, normal ultrafilter on ℘κ (℘(ë)∪ n<ù MEASκ,ë n ). (D) Assume that the Axiom of Dependent Choice holds for relations on ℘(ë). S (E) assume that n<ù MEASκ,ë n can be wellordered;

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Remarks. (a) Under ADR , Lemma 1.2 gives examples satisfying our assumptions with κ = ℵ1 . As long as ë < Θ, Condition (D) follows from Uniformization, a consequence of ADR . Condition (E) follows from (D) and Corollary 28.21 of [Kan94], a result of Kenneth Kunen. (b) Under the Axiom of Choice, there are examples for all κ and ë such that ë κ ≤ ë and κ is 22 -supercompact. S We use “a” as a variable ranging over ℘κ ( n<ù (℘( n ë) ∪ MEASκ,ë n )). When we say “for almost all a” or “for almost every a,” we mean “for a set of a that belongs to V.” For n < ù, let measn (a) = MEASκ,ë n ∩ a. For n < ù and ì ∈ MEASκ,ë n , let Yì (a) =

\

X.

X ∈ì∩a

Lemma 2.1. Let n < ù. For almost all a, (i) if ì1 and ì2 are distinct members of measn (a), then Yì1 and Yì2 are disjoint; S (ii) ì∈measn (a) Yì (a) = n ë. Proof. Assume that (i) fails for almost all a. Since MEASën is wellordered, let a 7→ ì1 (a), ì2 (a) be such that, for almost all a: (a) ì1 (a) ∈ measn (a) ∧ ì2 (a) ∈ measn (a); (b) ì1 6= ì2 ; (c) Yì1 (a) (a) ∩ Yì2 (a) (a) 6= ∅. By normality there are ì1 and ì2 such that, for almost all a, ì1 (a) = ì1 and ì2 (a) = ì2 . Let X ⊆ n ë be such that X ∈ ì1 and X ∈ / ì2 . Fineness implies that, for almost all a, both X and n ë \ X belong to a, and so Yì1 (a) ⊆ X and Yì2 (a) ∩ X = ∅. This is a contradiction. Now assume that (ii) fails for almost all a. Define [ Z(a) = n ë \ Yì (a). ì∈measn (a)

Since n ë is wellordered, there is an f such that f(a) ∈ Z(a) for almost all a. Define an element ì of MEASκ,ë n by X ∈ ì ↔ f(a) ∈ X for almost all a. Fineness implies that ì belongs to almost all a. Since Z(a) and Yì (a) are disjoint whenever ì ∈ a, it follows that f(a) ∈ / Yì (a) for almost all a. Thus if X (a) = {X ∈ ì : f(a) ∈ / X },

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then, for almost all a, a ∩ X (a) 6= ∅. By normality, there is an X such that X ∈ X (a) for almost all a. By the definition of X , X ∈ ì, and f(a) ∈ / X for almost all a. This contradicts the definition of ì. ⊣ If ì = hìi : i < ùi is a ë-tower of κ-complete measures, then for each a we define Yì (a) = {f ∈ ù ë : (∀i < ù) f↾i ∈ Yìi (a)}. Lemma 2.2. For almost all a, if ì = hìi : i < ùi is a ë-tower of κ-complete measures and each ìi ∈ a, then ì is countably complete if and only if Yì (a) 6= ∅.

(†)

Proof. Observe first that if we replace “if and only if ” by “only if ” then (†) holds for every a, by the definition of countable completeness. To prove the lemma we shall first show that there is a function assigning to each a for which (†) fails a tower witnessing that failure. Then we shall apply normality to show that if the lemma is false then there is a single tower that witnesses failure for almost all a, and there can be no such tower. Fix an a such that (†) fails for a. Suppose that ì = hìi : i < ùi witnesses this failure. Then there is an f : ù → ë such that (1) ì is a ë-tower of κ-complete measures and each ìi ∈ a; (2) f ∈ Yì (a); (3) ì is not countably complete. By (3), there are sets Xi , i < ù, such that Xi ∈ ìi for each i but there is no g : ù → ë such that g↾i ∈ Xi for all i. For any such hXi : i < ùi, there are functions h¯ i : Xi → ë+ such that   (4) (∀i)(∀t ∈ Xi+1 ) (∀j ≤ i) t↾j ∈ Xj → h¯ i+1 (t) < h¯ i (t↾i) . The functions h¯ i can be extended to functions hi : i ë → ë+ . By the definition of a tower, the hi satisfy (5) (∀i)({t ∈ i+1 ë : hi+1 (t) < hi (t ↾ i)} ∈ ìi+1 ). Moreover any sequence hhi : i < ùi satisfying (5) determines sequences hh¯ i : i < ùi and hXi : i < ùi satisfying (4). Just set X0 = 0 ë; Xi+1 = {t ∈ i+1 ë : hi+1 (t) < hi (t↾i)}; h¯ i = hi ↾Xi . Note also that whether or not (5) is satisfied depends only on h[hi ]ìi : i < ùi and not on the full hhi : i < ùi. (Our assumption (D) implies that the ordinals [hi ]ìi exist.)

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Now forget our particular ì. We define a triple (ì, f, g) as follows. Suppose that hìi : i < ni, f↾n, and g↾n have been defined. For i < n, let hi be such that [hi ]ìi = g(i). Assume inductively that (hìi : i < ni, f↾n, hhi : i < ni) can be extended to at least one triple (ì, f, hhi : i < ùi) satisfying (1), (2), and (5). Note that whether this assumption holds depends only on g↾n and not on the representative functions hi . Let (ìn , f(n), g(n)) be lexicographically least preserving the extendability. Here the ordering of measures is one given by (E), and the orderings of ë and the ordinals are the standard ones. Using (D) to pick functions hi representating the ordinals g(i), we see that our triple extends to a triple satisfying (1), (2), and (5). Thus ì witnesses the failure of (†) for a. Our inductive definition depends on a, so let us call its components ì(a), f(a), and g(a). These objects are defined for each a that does not satisfy (†), and so ì(a) is a witness to the failure of (†) for every a for which it fails. Assume that the lemma is false. By normality and countable completeness of V, there is a tower ì such that ì = ì(a) for almost all a. Thus ì is not countably complete. Let hXi : i < ùi witness this. By fineness, there is an a for which (†) fails such that each Xi ∈ a. For this a, Yìi (a) ⊆ Xi for each i. But this is a contradiction. ⊣ For the rest of this section, let T be a tree on ù × ë. Lemma 2.3. For almost all a, the following five statements are true. (1) For every n < ù, a satisfies (i) and (ii) of Lemma 2.1. (2) a satisfies (†) of Lemma 2.2. (3) Both Ts and lh(s) ë \ Ts belong to a for every s ∈ <ù ù. (4) If m < n < ù, X ⊆ m ë, and X ∈ a, then extn (X ) ∈ a. (5) If m < n < ù and ì ∈ measn (a), then projm (ì) ∈ a. Proof. The lemma follows easily from Lemma 2.1, Lemma 2.2, and the fact that V is countably complete, fine, and normal. ⊣ For the rest of this section, fix an a for which (1)–(5) are true. We shall write measn for measn (a), Yì for Yì (a), and Yì for Yì (a). It follows immediately from (1) that for all n < ù and t ∈ n ë there is a unique ì ∈ measn such that t ∈ Yì . Let this unique ì be ìt . Lemma 2.4. Let n < ù and let t ∈ n ë. Let m < n and let s ∈ s ⊆ t. Then ìs = projm (ìt ).

m

ë with

Proof. By (5), it is enough to show that s ∈ Yprojm (ìt ) . Let X ∈ a ∩ projm (ìt ). We must show that s ∈ X . By (4) and the definition of projm (ìt ), extn (X ) ∈ a ∩ ìt . Hence Yìt ⊆ extn (X ). But this implies that t ∈ extn (X ), which in turn implies that s ∈ X . ⊣

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For each f : ù → ë, let ì f = hìf↾n : n < ùi. By Lemma 2.4, ì f is a tower of measures. Lemma 2.5. If (x, f) ∈ [T ], then Tx↾n ∈ ìf↾n for each n. Proof. Since a satisfies (3), both Tx↾n and n ë \ Tx↾n belong to a. By (1) it follows that either Yìf↾n ⊆ Tx↾n or Yìf↾n ∩ Tx↾n = ∅. Since fx↾n belongs to Yìf↾n ∩ Tx↾n , the latter possibility is ruled out. ⊣ Lemma 2.6. Let (x, f) ∈ [T ]. Then (a) ì f is a tower of measures; (b) ìf↾n ∈ a for all n < ù; (c) Tx↾n ∈ ìf↾n for all n < ù; (d) ì f is countably complete. Proof. (a) follows from Lemma 2.4. (b) holds because ìt ∈ a for all t ∈ ë. (c) is an instance of Lemma 2.5. Assume that (d) does not hold. By Lemma 2.2, Yì is empty. But that is impossible, since f ∈ Yì . ⊣ <ù

For s ∈

<ù

ù, let Ms = {ì ∈ a : Ts ∈ ì}.

Lemma 2.7. If κ = ℵ1 , then the system hMs : s ∈ is weakly homogeneous.

<ù

ùi witnesses that T

Proof. Lemma 2.6 implies that hMs : s ∈ <ù ùi meets all requirements for witnessing weak homogeneity of T except perhaps the requirement that the sets Ms be countable. If κ = ℵ1 , then a is countable, and the Ms are countable. ⊣ §3. The main theorems. Theorem 3.1. Assume ADR . Every tree is weakly homogeneous. Proof. Let T be a tree. We first show that we may assume that T is on ù × ë for some ë < Θ. For each x ∈ p[T ], let fx be lexicographically least such that (x, f) ∈ [T ]. The set {fx (n) : x ∈ p[T ] ∧ n < ù} is a surjective image of R. Let ð be a bijection from this set to an ordinal number ë < Θ. Define a tree T¯ on ù × ë by (s, t) ∈ T¯ ↔ (s, ð−1 (t)) ∈ T. Clearly p[T¯ ] = p[T ], and any system witnessing that T¯ is weakly homogeneous is essentially a system witnessing that T is weakly homogeneous. We may then assume T¯ = T without loss of generality.

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429

Let V be the ultrafilter given by Lemma 1.2. Let a satisfy (1)–(5) of Lemma 2.3. By Lemma 2.7, the system hMs : s ∈ <ù ùi defined as for Lemma 2.7 witnesses that T is weakly homogeneous. ⊣ Theorem 3.2. Assume the Axiom of Choice. Let κ be supercompact. Let B VB come by the L´evy collapse of all ordinals < κ to ù. Let N = V(RV ). In N , every tree is weakly homogeneous. Proof. It suffices to prove the weakened version of the theorem in which “every tree” is replaced by “every tree in V.” For suppose that we have proved this weakened version. Let κ, B, and N be as in the statement of the theorem. Let T ∈ N . For some α < κ, T belongs to an inner model of N of the form VBα , where VBα comes from V by the collapse of α to ù. Now κ is supercompact in VBα , and VB comes from VBα by a L´evy collapse B of all ordinals < κ to ù. Moreover N = VBα (RV ). By our supposition, the theorem holds in VBα . Hence T is weakly homogeneous in N . Let κ, B, and N be as in the statement of the theorem and let T be a tree in V. Let ë be such that T is on ù × ë. If ë < κ, then T is countable in N and so is trivially weakly homogeneous. Assume then that κ ≤ ë. For the moment, let us work in V. Let V be a κ-complete, fine, normal, S ultrafilter on ℘κ (℘(ë) ∪ n<ù MEASκ,ë n ). The assumptions (A)–(E) hold. • Let a satisfy (1)–(5) of Lemma 2.3; • For t ∈ <ù ù, let ìt be defined as for Lemma 2.4; • Let hMs : s ∈ <ù ùi be as defined as for Lemma 2.7. Since a satisfies (2) of Lemma 2.3, it satisfies the condition (†) of Lemma 2.2. By the proof of Lemma 2.2 and the definition of a ë-complete tower of measures, it follows that the following statement is true of a. (∗) There is no triple (hìi : i < ùi, f, g) such that ì and f satisfy the conditions (1) and (2) of the proof of Lemma 2.2 and such that condition (5) from the same proof holds for sequences hhi : i < ùi for which [hi ]ìi = g(i) for every i < ù. For the rest of the proof of the theorem, we work in N . V For hìi : i < ùi such that each ì ∈ (MEASκ,ë n ) , we extend a previous definition by setting Yì = {f : (∀i ∈ ù)f↾i ∈ Yìi }, where Yìi is the Yìi (a) defined in V. V + n For n < ù and ì ∈ (MEASκ,ë n ) , define ì ⊆ ë by X ∈ ì+ ↔ (∃Y ∈ ì)Y ⊆ X. Standard arguments show that (1) ì+ is a κ-complete (i.e., countably complete) measure on n ë; (2) for every f : n ë → V there is a g : n ë → V such that g ∈ V and {t : f(t) = g(t)} ∈ ì+ .

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It follows that
(3) (∀h) (h ∈ V ∧ h : n ë → Ord) → [h]ì+ = ([h]ì )V . Here is a version of the condition (†) of Lemma 2.2 that a satisfies with respect to measures ì+ . V Lemma 3.3. Let ì = hìi : i < ùi, and assume that each ìi ∈ (MEASκ,ë n ) .

If ì + = hì+ i : i < ùi is a ë-tower of κ-complete measures and each ìi ∈ a, then ì + is countably complete if and only if Yì (a) 6= ∅.

(†)+

+ Proof. Since Yìi ∈ ì+ i for each i, we need only prove the “if ” part of (†) . By the definition of a ë-tower of measures, the fact that (∗) holds in V implies that there is in V no triple (hìi : i < ùi, f, g) such that:

(i) (ii) (iii) (iv) (v) (vi)

V for i < ù, ìi ∈ (MEASκ,ë n ) ;
V for j < i < ù, ìj = projj (ìi ) ; f : ù → ë; for i < ù, f(i) ∈ Yìi ; g : ù → Ord; i i+1 for i < ù, for h ∈ ( ë ë+ )V , and for h ′ ∈ ( ë ë+ )V ,


([h]ìi )V = g(i) ∧ ([h ′ ]ìi+1 )V = g(i + 1) → {t ∈ i+1 ë : h ′ (t) < h(t↾i)} ∈ ìi+1 .

By the absoluteness of wellfoundedness, there is no triple in N satisfying (i)– (vi). Assume that Yì 6= ∅, and let f witness this. Assume, to derive a contradiction, that ì + f is not countably complete. Let hhi : i < ùi witness this. By property (2) of ì 7→ ì+ , we may assume that each hi ∈ V. Let g : ù → Ord be defined by g(i) = [hi ]ì+i . By property (3) of ì 7→ ì+ , g(i) = ([hi ]ìi )V for each i. But then (hìi : i < ùi, f, g) has properties (i)–(vi), and we know that no triple can have these properties. ⊣ + For t ∈ <ù ë, let ì+ t = (ìt ) . For f : ù → ë, let ì f = hìf↾n : n < ùi. The following Lemma does the work of Lemma 2.6.

Lemma 3.4. Let (x, f) ∈ [T ]. Then (a) ì + f is a tower of measures; (b) ìf↾n ∈ a for all n < ù; (c) Tx↾n ∈ ìf↾n for all n < ù; (d) ì + f is countably complete. V Proof. Since clause (a) of Lemma 2.6 holds in V, ìf↾i ∈ (MEASκ,ë i ) for
V i < ù, and ìf↾j = projj (ìf↾i ) for j < i < ù. κ,ë From property (1) of ì 7→ ì+ , it follows that ì+ f↾i ∈ MEASi .

WEAKLY HOMOGENEOUS TREES



From the definition of ì 7→ ì+ , it follows that ì+ f↾j

431 V = projj (ì+ for f↾i )

j < i < ù. Hence clause (a) of the lemma holds. Clauses (b) and (c) are the same as the corresponding clauses of Lemma 2.6. Clause (d) follows from Lemma 3.3 in the same way as clause (d) of Lemma 2.6 followed from Lemma 2.2. ⊣ For s ∈

<ù

ù, define

Ms+ = {ì+ : (ì ∈ measn (a) ∧ ì ∈ a ∧ Ts ∈ ì}. The theorem follows from Lemma 3.4 and the fact that κ = ℵ1 in N .

⊣

§4. Further results and open problems. The basic open problems concern to what extent it is true that the projection p[T ] of a weakly homogeneous tree must be representable as the projection of a homogeneous tree. To avoid fairly simple counterexamples one must (and we do) assume that the collection {p[T ] : T is a weakly homogeneous tree} is reasonably closed, for example that the collection is closed under complements. This question in turn involves variations on Theorem 3.2 where the large cardinal hypothesis on κ is reduced below that of the hypothesis that κ be supercompact. A weakly homogeneous tree, T , is κ-weakly homogeneous if there is a witness to the weak homogeneity of T such that all the measures are κ-complete. Similarly, a homogeneous tree, T , is κ-homogeneous if there is a witness to the homogeneity of T such that all the measures are κ-complete. A cardinal κ is Vα -strong if there exists an elementary embedding j: V → M with critical point κ such that Vα ⊆ j(Vκ ). A cardinal κ is a strong cardinal if κ is Vα -strong for all ordinals α. A cardinal κ is Vα -1-strong if there exists an elementary embedding j: V → M with critical point κ such that Vα ⊆ j(Vκ ) and such that for all ä < α, ä is a strong cardinal in V if and only if ä is a strong cardinal in M . κ is a 1-strong cardinal if κ is Vα -1-strong for all ordinals α. Theorem 4.1 (ZFC). Suppose that κ is a regular cardinal which is a limit of 1-strong cardinals and that there is a strong cardinal above κ. Let VB come B by the L´evy collapse of all ordinals < κ to ù. Let N = V(RV ). In N , for every tree T the following are equivalent. 1. There is a weakly homogeneous tree, T¯ , such that p[T ] = p[T¯ ]. 2. There is a tree S such that p[T ] = ù ù\ p[S].

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Proof. By standard arguments (1) implies (2), and so we have only to prove that in N , (2) implies (1). As in the proof of Theorem 3.2, it suffices to prove the weakened version of the theorem in which the trees S and T belong to V. Clearly it suffices to produce the tree, T¯ , such that T¯ is weakly homogeneous in VB . Clearly we may suppose that T is a tree on κ and by reshaping the tree T if necessary we may suppose that that for all ã < κ, if ã is strongly inaccessible then for all complete Boolean algebras, B0 ∈ Vã , VB0 |= “p[T ] = p[T ↾ã]”. Fix κ0 < κ < κ1 such that κ0 is a 1-strong cardinal and such that κ1 is a strong cardinal. Let j0 : V → M0 be an elementary embedding with critical point κ0 such that κ1 < j0 (κ0 ), Vκ1 +ù ⊆ M0 , and such that κ1 is a strong cardinal in M0 . Let j1 : M0 → M1 be an elementary embedding in M0 with critical point κ1 such that j0 (κ0 ) < j1 (κ1 ) and such that M0 ∩ Vj0 (κ0 )+ù ⊆ M1 . For each n < ù, let ðn : MEASnκ0 ,κ0 → MEASnκ1 ,κ1 be the (unique) map such that for all ì ∈ MEASnκ0 ,κ0 , for all í ∈ MEASnκ1 ,κ1 , ðn (ì) = í if for all A ⊆ κ1 , A ∈ í if and only if j1 (A) ∩ n j0 (κ0 ) ∈ j0 (ì). For each t ∈ n j0 (κ0 ), let ìt ∈ MEASnκ0 ,κ0 be such that for all A ⊆ A ∈ ìt if and only if t ∈ j0 (A). For each s ∈ <ù ù, let n

n

κ0 ,

Ms = {ðn (ìt ) : t ∈ n j0 (κ0 ), n = lh(s), (s, t) ∈ j0 (T0 )}. A key point is that for each s ∈ <ù ù, |Ms | ≤ |Vκ0 +2 | < κ1 . Thus there is a tree T¯ on ù ×κ1 (with the tree T¯ in V) such that the sequence hMs : s ∈

<ù

ùi

witnesses that T¯ is weakly homogeneous in VB where VB comes by the L´evy collapse of all ordinals < κ to ù. We next show that VB |= “p[T ] = p[T¯ ]”. This we do in two steps. First we show that VB |= “p[T¯ ] ⊆ p[T ]”.

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433

If not then by absoluteness there exists x ∈ p[S] ∩ p[T¯ ] such that x ∈ V. Let hín : n < ùi be a countably complete tower of measures such that for each n < ù, ìn ∈ Mx↾n . Let hìn : n < ùi be such that for each n < ù, ín = ðn (ìn ). It follows that hìn : n < ùi is a countably complete tower at κ0 and moreover that for each n < ù, {t ∈ n κ0 : (x↾n, t) ∈ T0 } ∈ ìn . This implies that x ∈ p[T ] which contradicts that p[T ] ∩ p[S] = ∅. Next we show that VB |= “p[T ] ⊆ p[T¯ ]”. Suppose x ∈ VB and VB |= “x ∈ p[T ]”. Then we have VB |= “x ∈ p[j0 (T0 )↾κ1 ]”, by the reshaping of T . Let f ∈ VB be such that f ∈ ù κ1 and such that in VB , (x, f) is an infinite branch of j0 (T0 )↾κ1 . For each n < ù, let ìn = ìt where t = f↾n and let ín = ðn (ìn ). Thus hín : n < ùi is a tower of measures at κ1 and for each n < ù, ín ∈ Mx↾n . Therefore it suffices to show that in VB , the tower, hín : n < ùi, is countably complete. If not then again by absoluteness it follows that we can suppose that f ∈ V. But then hìn : n < ùi ∈ V and in V, hìn : n < ùi is a countably complete tower of measures at κ0 . This implies that hj0 (ìn ) : n < ùi ∈ M0 and that in M0 , hj0 (ìn ) : n < ùi is a countably complete tower of measures at j0 (κ0 ). But this in turn implies hín : n < ùi ∈ M0 and that in M0 , hín : n < ùi is a countably complete tower of measures at κ1 . Since Vκ1 +ù ⊆ M0 , this implies that in V, hín : n < ùi is a countably complete tower of measures at κ1 , which is a contradiction.

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This proves the claim that VB |= “p[T ] = p[T¯ ]”. Since T¯ is weakly homogeneous in VB , T¯ is as required. ⊣ The proof of Theorem 4.1 is actually an adaptation of the proof of the following theorem, for a proof see (Theorem 1.5.12) of [Lar04]. Theorem 4.2 (ZFC). Suppose that κ is a Woodin cardinal and T is a tree. There exists ã < κ such that if VB comes by the L´evy collapse of ã to ù, then in VB , T is α-weakly homogeneous for all α < κ. One corollary of Theorem 4.2 is that the large cardinal hypothesis of Theorem 3.2 can be reduced significantly. Theorem 4.3 (ZFC). Suppose that κ is a measurable Woodin cardinal. Let B V come by the L´evy collapse of of all ordinals < κ to ù. Let N = V(RV ). In N , every tree is weakly homogeneous. B

Proof. It suffices to show that if κ is a measurable cardinal and if T is a tree which is α-weakly homogeneous for all α < κ, then T is κ-weakly homogeneous. This is essentially immediate. Fix the tree T and fix ã such that T is a tree on ù × ã. Let í be a κ-complete uniform measure on κ. For each α < κ, let hMsα : s ∈ <ù ùi witness that T is α-weakly homogeneous. For each α < κ and for each s ∈ <ù ù, let fsα : ù → Msα be a surjection. For each pair (i, s) ∈ ù × <ù ù, define a measure ìsi on lh(s) ã by A ∈ ìsi if {α : A ∈ fsα (i)} ∈ í. Clearly ìsi is a κ-complete measure on For each s ∈ <ù ù, let

lh(s)

ã such that Ts ∈ ìsi .

Ms = {ìsi : i < ù}. We finish by showing that hMs : s ∈

<ù

ùi

witnesses that T is κ-weakly homogeneous. Suppose that x ∈ p[T ]. Then for each α < κ there exists a countably complete tower, hìαi : i < ùi , α . such that for each i < ù, ìαi ∈ Mx↾i Since í is countably complete, for each i < ù there exists i ∗ < ù such that α {α < κ : ìαi = fx↾i (i ∗ )} ∈ í.

WEAKLY HOMOGENEOUS TREES

Therefore

D

ìx↾i i∗ : i < ù

435

E

is a countably complete tower such that for all i < ù, ìx↾i i ∗ ∈ Mx↾i .

⊣

A proof of the following theorem can also be found in [Lar04], the theorem appears there in a slightly different form as (Theorem 3.3.8). Theorem 4.4 (ZFC). Suppose that ä is a Woodin cardinal and suppose that S and T are trees such that if VB comes by the L´evy collapse of ä to ù then in VB , p[S] = ù ù\ p[T ]. Then in V, for all α < ä, S and T are α-weakly homogeneous. As a corollary to Theorem 4.2 and Theorem 4.4 one obtains the following theorem. Theorem 4.5 (ZFC). Suppose that κ < ä are Woodin cardinals and κ is a limit of strong cardinals. Let VB come by the L´evy collapse of of all ordinals B < κ to ù. Let N = V(RV ). In N , for every tree T there exists a tree T¯ such that p[T ] = p[T¯ ] and such that T¯ is weakly homogeneous. The large cardinal hypothesis of Theorem 4.1 is significantly weaker than the large cardinal hypothesis of Theorem 4.5. But the conclusions do not seem that different; Theorem 4.1 obtains a symmetric L´evy extension of V in which every Suslin, co-Suslin, set A ⊆ ù ù is the projection of a weakly homogeneous tree, and Theorem 4.5 obtains obtains a symmetric L´evy extension of V in which every Suslin set A ⊆ ù ù is the projection of a weakly homogeneous tree. Does the latter really require the additional large cardinal assumptions? We note that by our main theorem, Theorem 3.1, the consistency strength of the theory, ZF+DC+“Every tree is weakly homogeneous”, is at most that of the theory, ZF+DC+ADR , and the consistency of the latter theory is well below that of the large cardinal hypothesis of Theorem 4.5. Similar considerations apply to Theorem 4.3, and so one can reasonably ask whether the large cardinal hypothesis for that theorem can also be reduced. For each cardinal κ, let ΓκwHom be the set of all A ⊆ ù ù such that A is the projection of a κ-weakly homogeneous tree, an let ΓκHom be the set of all A ⊆ ù ù such that A is the projection of a κ-homogeneous tree. It is straightforward to verify that ΓκwHom is closed under countable unions, countable intersections, and closed under both images preimages by continuous functions, f : ù ù → ù ù,

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and moreover every set A ∈ ΓκwHom is the image of a set B ∈ ΓκHom by a continuous function, f : ù ù → ù ù. Therefore by consideration of Wadge games, one can easily prove the following lemma. Lemma 4.6 (ZFC). Suppose that κ is a cardinal such that ΓκwHom is closed under complements. Then the following are equivalent. 1. ΓκHom = ΓκwHom . 2. Every set in ΓκwHom is determined. Theorem 4.1 can be reformulated quite easily as follows. Theorem 4.7 (ZFC). Suppose that κ is a regular cardinal which is a limit of 1-strong cardinals and that there is a strong cardinal above κ. Let VB come by the L´evy collapse of all ordinals < κ to ù. Then in VB , ΓκwHom is closed under complements. Theorem 4.7 shows that one cannot hope to prove that ΓκHom = ΓκwHom if one assumes just ΓκwHom is closed under complements (and so is a ó-algebra). In fact assuming ΓκwHom is closed under complements one cannot hope to prove even that ΓκHom contains all ∆12 sets. On the other hand there is the following theorem of Steel. Theorem 4.8 (Steel). (ZFC) Suppose that ℘( ù ù) ∩ L( ù ù) ⊆ ΓκwHom . Then ℘( ù ù) ∩ L( ù ù) ⊆ ΓκHom . The proof of Theorem 4.8 makes essential use of the smallness of ℘( ù ù) ∩ L( ù ù), in particular the proof exploits the relationship of the determinacy hypothesis that all sets in L( ù ù) are determined and the existence of (iterable) inner models with infinitely many Woodin cardinals. The following theorem can be proved by the methods of [Lar04], details will appear in [WooC]. Theorem 4.9 (ZFC). Suppose that κ is strongly compact. Then ΓκwHom has the scale property and for each A ∈ ΓκwHom , ℘( ù ù) ∩ L(A, ù ù) ⊆ ΓwHom . Building on this, and adapting the core model methods used to prove Theorem 4.8, one can very likely obtain the following theorem, [WooC]. Theorem 4.10 (ZFC). Suppose that κ is strongly compact. Then ΓκHom =

ΓκwHom .

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437

This suggests the following question. Fix a cardinal κ and suppose A ⊆ ù ù is such that ℘( ù ù) ∩ L(A, ù ù) ⊆ ΓκwHom . ù ù Must ℘( ù) ∩ L(A, ù) ⊆ ΓκHom ? If the answer is yes then this would provide a different, and possibly more direct, proof of Theorem 4.10. Finally, as noted above, the assumption that ΓκwHom is closed under complements cannot imply (modulo inconsistency) that ℘( ù ù) ∩ Lù1 ( ù ù) ⊆ ΓκHom . Of course if ΓκwHom is closed under complements then necessarily ℘( ù ù) ∩ Lù1 ( ù ù) ⊆ ΓκwHom . Suppose that ℘( ù ù) ∩ Lù1 ( ù ù) ⊆ ΓκwHom . Does this imply that ℘( ù) ∩ Lù1 ( ù ù) has the scale property? This may seem unlikely, but there is no known counterexample. For the natural models in which ℘( ù ù) ∩ Lù1 ( ù ù) ⊆ ΓκwHom , these are the L´evy collapse extensions given by Theorem 4.7 where the ground model, V, is a reasonably closed extender model; it is true that ℘( ù ù) ∩ Lù1 ( ù ù) has the scale property. The proof of this remarkable fact divides into two cases. If Lù1 ( ù ù) |= AD then the scale property holds by the Moschovakis periodicity theorems. If ù

Lù1 ( ù ù) 6|= AD then again ℘( ù ù) ∩ Lù1 ( ù ù) has the scale property by an analysis due to Steel. It is for this case that the assumption that the ground model be an extender model seems essential. If one could eliminate this assumption then very likely the answer to the question above is yes. REFERENCES

Akihiro Kanamori [Kan94] The higher infinite, Springer-Verlag, Berlin, 1994. Alexander S. Kechris [Kec88B] A coding theorem for measures, this volume, originally published in Kechris et al. [Cabal iv], pp. 103–109. Alexander S. Kechris, Donald A. Martin, and John R. Steel [Cabal iv] Cabal seminar 81–85, Lecture Notes in Mathematics, no. 1333, Berlin, Springer, 1988. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978.

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DONALD A. MARTIN AND W. HUGH WOODIN

Paul B. Larson [Lar04] The stationary tower: Notes on a course by W. Hugh Woodin, University Lecture Series (AMS), vol. 32, Oxford University Press, Providence, RI, 2004. Yiannis N. Moschovakis [Mos70A] Determinacy and prewellorderings of the continuum, Mathematical logic and foundations of set theory. Proceedings of an international colloquium held under the auspices of the Israel Academy of Sciences and Humanities, Jerusalem, 11–14 November 1968 (Y. Bar-Hillel, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam-London, 1970, pp. 24–62. Robert M. Solovay [Sol78B] The independence of DC from AD, In Kechris and Moschovakis [Cabal i], pp. 171–184. W. Hugh Woodin [WooC] Category, scales and determinacy, In preparation. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES, CA 90095, USA

E-mail: [email protected] DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720, USA

E-mail: [email protected]

BIBLIOGRAPHY

John W. Addison [Add59] Some consequences of the axiom of constructibility, Fundamenta Mathematicae, vol. 46 (1959), pp. 123–135. John W. Addison and Yiannis N. Moschovakis [AM68] Some consequences of the axiom of definable determinateness, Proceedings of the National Academy of Sciences of the United States of America, no. 59, 1968, pp. 708–712. James Baumgartner, Donald A. Martin, and Saharon Shelah [BMS84] Axiomatic set theory. Proceedings of the AMS-IMS-SIAM joint summer research conference held in Boulder, Colo., June 19–25, 1983, Contemporary Mathematics, vol. 31, Amer. Math. Soc., Providence, RI, 1984. Howard S. Becker [Bec78] Partially playful universes, In Kechris and Moschovakis [Cabal i], pp. 55–90. [Bec79] Some applications of ordinal games, Ph.D. thesis, UCLA, 1979. [Bec80] Thin collections of sets of projective ordinals and analogs of L, Annals of Mathematical Logic, vol. 19 (1980), pp. 205–241. [Bec81] Determinacy implies that ℵ2 is supercompact, Israel Journal of Mathematics, vol. 40 (1981), no. 3– 4, pp. 229–234. [Bec85] A property equivalent to the existence of scales, Transactions of the AMS, vol. 287 (1985), pp. 591–612. Howard S. Becker and Alexander S. Kechris [BK84] Sets of ordinals constructible from trees and the third Victoria Delfino problem, In Baumgartner et al. [BMS84], pp. 13–29. Andreas Blass [Bla75] Equivalence of two strong forms of determinacy, Proceedings of the American Mathematical Society, vol. 52 (1975), pp. 373–376. Douglas R. Busch [Bus76] ë-Scales, κ-Souslin sets and a new definition of analytic sets, The Journal of Symbolic Logic, vol. 41 (1976), p. 373. Chen-Lian Chuang [Chu82] The propagation of scales by game quantifiers, Ph.D. thesis, UCLA, 1982. Daniel Cunningham [Cun90] The real core model, Ph.D. thesis, UCLA, 1990. Morton Davis [Dav64] Infinite games of perfect information, Advances in game theory (Melvin Dresher, Lloyd S. Shapley, and Alan W. Tucker, editors), Annals of Mathematical Studies, vol. 52, 1964, pp. 85– 101. Qi Feng, Menachem Magidor, and W. Hugh Woodin [FMW92] Universally Baire sets of reals, In Judah et al. [JJW92], pp. 203–242. The Cabal Seminar. Volume I: Games, Scales and Suslin Cardinals ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 31 c 2008, Association for Symbolic Logic

439

440

BIBLIOGRAPHY

Leo A. Harrington [Har73] Contributions to recursion theory in higher types, Ph.D. thesis, MIT, 1973. [Har78] Analytic determinacy and 0# , The Journal of Symbolic Logic, vol. 43 (1978), pp. 685–693. Leo A. Harrington and Alexander S. Kechris [HK77] Ordinal quantification and the models L[T 2n+1 ], Mimeographed note, January 1977. [HK81] On the determinacy of games on ordinals, Annals of Mathematical Logic, vol. 20 (1981), pp. 109–154. Gregory Hjorth [Hjo96] Two applications of inner model theory to the study of Σ12 sets, The Bulletin of Symbolic e Logic, vol. 2 (1996), no. 1, pp. 94–107. [Hjo97] Some applications of coarse inner model theory, The Journal of Symbolic Logic, vol. 62 (1997), no. 2, pp. 337–365. [Hjo01] A boundedness lemma for iterations, The Journal of Symbolic Logic, vol. 66 (2001), no. 3, pp. 1058–1072. Stephen Jackson [Jac] Non-partition results in the projective hierarchy, to appear. [Jac88] AD and the projective ordinals, In Kechris et al. [Cabal iv], pp. 117–220. [Jac90] A new proof of the strong partition relation on ù1 , Transactions of the American Mathematical Society, vol. 320 (1990), no. 2, pp. 737–745. [Jac91] Admissible Suslin cardinals in L(R), The Journal of Symbolic Logic, vol. 56 (1991), no. 1, pp. 260–275. [Jac99] A computation of ä 15 , vol. 140, Memoirs of the AMS, no. 670, American Mathematical e Society, July 1999. [Jac07] Suslin cardinals, partition properties, homogeneity. Introduction to Part II, this volume, 2007. [Jac08] Structural consequences of AD, In Kanamori and Foreman [KF08]. Stephen Jackson and Donald A. Martin [JM83] Pointclasses and wellordered unions, In Kechris et al. [Cabal iii], pp. 55–66. Stephen Jackson and Russell May [JM04] The strong partition relation on ù1 revisited, Mathematical Logic Quarterly, vol. 50 (2004), no. 1, pp. 33– 40. Thomas Jech [Jec02] Set theory, second ed., Springer Monographs in Mathematics, Springer, 2002. Ronald B. Jensen [Jen72] The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308. H. Judah, W. Just, and W. Hugh Woodin [JJW92] Set theory of the continuum, MSRI publications, vol. 26, Springer-Verlag, 1992. Akihiro Kanamori [Kan94] The higher infinite, Springer-Verlag, Berlin, 1994. Akihiro Kanamori and Matthew Foreman [KF08] Handbook of set theory into the 21st century, Springer, 2008. Alexander S. Kechris [Kec75] The theory of countable analytical sets, Transactions of the American Mathematical Society, vol. 202 (1975), pp. 259–297.

BIBLIOGRAPHY

441

[Kec77] AD and infinite exponent partition relations, Circulated manuscript, 1977. [Kec78A] AD and projective ordinals, In Kechris and Moschovakis [Cabal i], pp. 91–132. [Kec78B] Forcing in analysis, Higher set theory. Proceedings of a conference held at the Mathema¨ tisches Forschungsinstitut, Oberwolfach, April 13–23, 1977 (Gert H. Muller and Dana Scott, editors), Lecture Notes in Mathematics, vol. 669, Springer, 1978, pp. 277–302. [Kec78C] On transfinite sequences of projective sets with an application to Σ12 equivalence relations, e North-Holland, 1978, Logic colloquium ’77 (A. Macintyre, L. Pacholski, and J. Paris, editors), pp. 155–160. [Kec81A] Homogeneous trees and projective scales, In Kechris et al. [Cabal ii], pp. 33–74. [Kec81B] Suslin cardinals, κ-Suslin sets, and the scale property in the hyperprojective hierarchy, this volume, originally published in Kechris et al. [Cabal ii], pp. 127–146. [Kec85] Determinacy and the structure of L(R), Proceedings of symposia in pure mathematics, vol. 42, American Mathematical Society, 1985, pp. 271–283. 1 [Kec88A] AD + Unif is equivalent to AD/R2 , In Kechris et al. [Cabal iv], pp. 98–102. [Kec88B] A coding theorem for measures, this volume, originally published in Kechris et al. [Cabal iv], pp. 103–109. Alexander S. Kechris, Eugene M. Kleinberg, Yiannis N. Moschovakis, and W. Hugh Woodin [KKMW81] The axiom of determinacy, strong partition properties and nonsingular measures, this volume, originally published in Kechris et al. [Cabal ii], pp. 75–100. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer, 1981. [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Alexander S. Kechris, Donald A. Martin, and Robert M. Solovay [KMS83] Introduction to Q-theory, In Kechris et al. [Cabal iii], pp. 199–282. Alexander S. Kechris, Donald A. Martin, and John R. Steel [Cabal iv] Cabal seminar 81–85, Lecture Notes in Mathematics, no. 1333, Berlin, Springer, 1988. Alexander S. Kechris and Yiannis N. Moschovakis [KM77] Recursion in higher types, Handbook of mathematical logic (K. J. Barwise, editor), NorthHolland, 1977, pp. 681–737. [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. [KM78B] Notes on the theory of scales, this volume, originally published in Cabal Seminar 76–77 [Cabal i], pp. 1–53. Alexander S. Kechris and Robert M. Solovay [KS85] On the relative consistency strength of determinacy hypotheses, Transactions of the American Mathematical Society, vol. 290 (1985), no. 1, pp. 179–211. Alexander S. Kechris, Robert M. Solovay, and John R. Steel [KSS81] The axiom of determinacy and the prewellordering property, In Kechris et al. [Cabal ii], pp. 101–125. Alexander S. Kechris and W. Hugh Woodin [KW83] Equivalence of determinacy and partition properties, Proceedings of the National Academy of Sciences of the United States of America, vol. 80 (1983), no. 6 i., pp. 1783–1786. [KW07] Generic codes for uncountable ordinals, this volume, originally circulated manuscript, 2007.

442

BIBLIOGRAPHY

John Kelley [Kel55] General topology, The University series in higher mathematics, Van Nostrand, Princeton, NJ, 1955. Richard O. Ketchersid [Ket00] Toward ADR from the continuum hypothesis and an ù1 -dense ideal, Ph.D. thesis, Berkeley, 2000. Stephen C. Kleene [Kle55] Arithmetical predicates and function quantifiers, Transactions of the American Mathematical Society, vol. 79 (1955), pp. 312–340. Eugene M. Kleinberg [Kle77] Infinitary combinatorics and the axiom of determinacy, Lecture Notes in Mathematics, vol. 612, Springer-Verlag, 1977. [Kle82] A measure representation theorem for strong partition cardinals, The Journal of Symbolic Logic, vol. 47 (1982), no. 1, pp. 161–168. Motokiti Kondoˆ [Kon38] Sur l’uniformization des complementaires analytiques et les ensembles projectifs de la seconde classe, Japanese Journal of Mathematics, vol. 15 (1938), pp. 197–230. Kazimierz Kuratowski [Kur66] Topology, vol. 1, Academic Press, New York and London, 1966. Paul B. Larson [Lar04] The stationary tower: Notes on a course by W. Hugh Woodin, University Lecture Series (AMS), vol. 32, American Mathematical Society, Providence, RI, 2004. Azriel L´evy [L´ev66] Definability in axiomatic set theory, Logic, methodology and philosophy of science. Proceedings of the 1964 international congress. (Amsterdam) (Yehoshua Bar-Hillel, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, 1966, pp. 127–151. N. N. Luzin and P. S. Novikov [LN35] Choix effectif d’un point dans un complemetaire analytique arbitraire, donne par un crible, Fundamenta Mathematicae, vol. 25 (1935), pp. 559–560. Richard Mansfield [Man69] The theory of Σ12 sets, Ph.D. thesis, Stanford University, 1969. [Man70] Perfect subsets of definable sets of real numbers, Pacific Journal of Mathematics, vol. 35 (1970), no. 2, pp. 451– 457. [Man71] A Souslin operation on Π12 , Israel Journal of Mathematics, vol. 9 (1971), no. 3, pp. 367– 379. Donald A. Martin [MarA] Borel and projective games, to appear. [MarB] Games of countable length, to appear. [MarC] On Victoria Delfino problem number 1, notes. [MarD] Weakly homogeneous trees, Circulated manuscript. [Mar68] The axiom of determinateness and reduction principles in the analytical hierarchy, Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 687–689. [Mar70A] Measurable cardinals and analytic games, Fundamenta Mathematicae, (1970), no. LXVI, pp. 287–291. [Mar70B] Pleasant and unpleasant consequences of determinateness, March 1970, unpublished manuscript.

BIBLIOGRAPHY

443

[Mar73] Countable Σ12n+1 sets, 1973, circulated note. [Mar75] Borel determinacy, Annals of Mathematics, vol. 102 (1975), no. 2, pp. 363–371. [Mar78] Infinite games, Proceedings of the international congress of mathematicatians, Helsinki 1978 (Olli Lehto, editor), Finnish Academy of Sciences, 1978, pp. 269–273. [Mar83A] The largest countable this, that, and the other, this volume, originally published in Kechris et al. [Cabal iii], pp. 97–106. [Mar83B] The real game quantifier propagates scales, this volume, originally published in Kechris et al. [Cabal iii], pp. 157–171. Donald A. Martin, Yiannis N. Moschovakis, and John R. Steel [MMS82] The extent of definable scales, Bulletin of the American Mathematical Society, vol. 6 (1982), pp. 435– 440. Donald A. Martin and Robert M. Solovay [MS] Basis theorems for Π12k sets of reals, unpublished. [MS69] A basis theorem for Σ13 sets of reals, Annals of Mathematics, vol. 89 (1969), pp. 138–160. Donald A. Martin and John R. Steel [MS83] The extent of scales in L(R), this volume, originally published in Kechris et al. [Cabal iii], pp. 86–96. [MS89] A proof of projective determinacy, Journal of the American Mathematical Society, vol. 2 (1989), pp. 71–125. [MS94] Iteration trees, Journal of the American Mathematical Society, vol. 7 (1994), pp. 1–73. [MS07] The tree of a Moschovakis scale is homogeneous, this volume, 2007. Donald A. Martin and W. Hugh Woodin [MW07] Weakly homogeneous trees, this volume, 2007. William J. Mitchell and John R. Steel [MS94] Fine structure and iteration trees, Lecture Notes in Logic, vol. 3, Springer-Verlag, Berlin, 1994. Yiannis N. Moschovakis [Mos67] Hyperanalytic predicates, Transactions of the American Mathematical Society, vol. 129 (1967), pp. 249–282. [Mos70A] Determinacy and prewellorderings of the continuum, Mathematical logic and foundations of set theory. Proceedings of an international colloquium held under the auspices of the Israel Academy of Sciences and Humanities, Jerusalem, 11–14 November 1968 (Y. Bar-Hillel, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam-London, 1970, pp. 24–62. [Mos70B] The Suslin-Kleene theorem for countable structures, Duke Mathematical Journal, vol. 37 (1970), no. 2, pp. 341–352. [Mos71A] Uniformization in a playful universe, Bulletin of the American Mathematical Society, vol. 77 (1971), pp. 731–736. [Mos71B] Uniformization in a playful universe, Bulletin of the American Mathematical Society, vol. 77 (1971), pp. 731–736. [Mos73] Analytical definability in a playful universe, Logic, methodology, and philosophy of science IV (Patrick Suppes, Leon Henkin, Athanase Joja, and Gr. C. Moisil, editors), North-Holland, 1973, pp. 77–83. [Mos74A] Elementary induction on abstract structures, North-Holland, 1974. [Mos74B] Structural characterizations of classes of relations, Generalized recursion theory. Proceedings of the 1972 Oslo symposium (Jens Erik Fenstad and Peter G. Hinman, editors), Studies in Logic and the Foundations of Mathematics, North-Holland, 1974, pp. 53–79.

444

BIBLIOGRAPHY

[Mos78] Inductive scales on inductive sets, this volume, originally published in Kechris and Moschovakis [Cabal i], pp. 185–192. [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. [Mos81] Ordinal games and playful models, In Kechris et al. [Cabal ii], pp. 169–201. [Mos83] Scales on coinductive sets, this volume, originally published in Kechris et al. [Cabal iii], pp. 77–85. Jan Mycielski [Myc64] On the axiom of determinateness, Fundamenta Mathematicae, vol. 53 (1964), pp. 205– 224. [Myc66] On the axiom of determinateness II, Fundamenta Mathematicae, vol. 59 (1966), pp. 203– 212. Jan Mycielski and Stanislaw Swierczkowski [MS64] On the Lebesgue measurability and the axiom of determinateness, Fundamenta Mathematicae, vol. 54 (1964), pp. 67–71. Itay Neeman [Nee] Games of length ù1 , Journal of Mathematical Logic, to appear. [Nee95] Optimal proofs of determinacy, The Bulletin of Symbolic Logic, vol. 1 (1995), pp. 327–339. [Nee02] Optimal proofs of determinacy II, Journal of Mathematical Logic, vol. 2 (2002), pp. 227– 258. [Nee04] The determinacy of long games, de Gruyter Series in Logic and its Applications, vol. 7, Walter de Gruyter, Berlin, 2004. [Nee06] Determinacy for games ending at the first admissible relative to the play, The Journal of Symbolic Logic, vol. 71 (2006), no. 2, pp. 425– 459. [Nee07] Propagation of the scale property using games, this volume, 2007. Mitchell Rudominer [Rud99] The largest countable inductive set is a mouse set, The Journal of Symbolic Logic, vol. 64 (1999), pp. 443– 459. Joseph R. Schoenfield [Sch61] The problem of predicativity, Essays on the foundations of mathematics (Y. Bar-Hillel et al., editors), Magnes Press, Jerusalem, 1961, pp. 132–139. Jack H. Silver [Sil71] Measurable cardinals and ∆13 wellorderings, Annals of Mathematics, vol. 94 (1971), no. 2, pp. 141– 446. Stephen G. Simpson [Sim78] A short course in admissible recursion theory, Generalized recursion theory II, Studies in Logic, vol. 94, North Holland, Amsterdam, 1978. Robert M. Solovay [Sol66] On the cardinality of Σ12 set of reals, Foundations of Mathematics: Symposium papers commemorating the 60th birthday of Kurt G¨odel (Jack J. Bulloff, Thomas C. Holyoke, and S. W. Hahn, editors), Springer-Verlag, 1966, pp. 58–73. [Sol67] Measurable cardinals and the axiom of determinateness, Lecture notes prepared in connection with the Summer Institute of Axiomatic Set Theory held at UCLA, Summer 1967. [Sol70] A model of set theory in which every set is Lebesgue measurable, Annals of Mathematics, vol. 92 (1970), pp. 1–56. [Sol78A] A ∆13 coding of the subsets of ù ù, In Kechris and Moschovakis [Cabal i], pp. 133–150. [Sol78B] Thee independence of DC from AD, In Kechris and Moschovakis [Cabal i], pp. 171–184.

BIBLIOGRAPHY

445

John R. Steel [SteA] The derived model theorem, Available at http://www.math.berkeley.edu/∼ steel. [SteB] Scales in K(R) at the end of a weak gap, Preprint, available at http://www.math. berkeley.edu/∼ steel. [Ste80] More measures from AD, mimeographed notes, 1980. [Ste81A] Closure properties of pointclasses, In Kechris et al. [Cabal ii], pp. 147–163. [Ste81B] Determinateness and the separation property, The Journal of Symbolic Logic, vol. 46 (1981), no. 1, pp. 41– 44. [Ste83A] Scales in L(R), this volume, originally published in Kechris et al. [Cabal iii], pp. 107– 156. [Ste83B] Scales on Σ11 -sets, this volume, originally published in Kechris et al. [Cabal iii], pp. 72– 76. [Ste88] Long games, this volume, originally published in Kechris et al. [Cabal iv], pp. 56–97. [Ste95A] HODL(R) is a core model below Θ, The Bulletin of Symbolic Logic, vol. 1 (1995), pp. 75– 84. [Ste95B] Projectively wellordered inner models, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 77–104. [Ste96] The core model iterability problem, Lecture Notes in Logic, no. 8, Springer-Verlag, Berlin, 1996. [Ste05] PFA implies ADL(R) , The Journal of Symbolic Logic, vol. 70 (2005), no. 4, pp. 1255–1296. [Ste07A] The length-ù1 open game quantifier propagates scales, this volume, 2007. [Ste07B] Scales in K(R), this volume, 2007. [Ste08] An outline of inner model theory, In Kanamori and Foreman [KF08]. John R. Steel and Robert Van Wesep [SVW82] Two consequences of determinacy consistent with choice, Transactions of the American Mathematical Society, (1982), no. 272, pp. 67–85. M. Y. Suslin [Sus17] Sur une d´efinition des ensembles mesurables B sans nombres transfinis, Comptes Rendus Hebdomadaires des S´eances de l’Acad´emie des Sciences, vol. 164 (1917), pp. 88–91. Robert Van Wesep [Van78A] Separation principles and the axiom of determinateness, The Journal of Symbolic Logic, vol. 43 (1978), pp. 77–81. [Van78B] Wadge degrees and descriptive set theory, In Kechris and Moschovakis [Cabal i], pp. 151–170. John von Neumann [vN49] On rings of operators, reduction theory, Annals of Mathematics, vol. 50 (1949), pp. 448– 451. W. Hugh Woodin [WooA] unpublished. [WooB] ℵ1 -dense ideals, to appear. [WooC] Category, scales and determinacy, In preparation. [Woo99] The axiom of determinacy, forcing axioms, and the nonstationary ideal, De Gruyter Series in Logic and its Applications, Walter de Gruyter, Berlin, 1999. A. Stuart Zoble [Zob00] Stationary reflection and the determinacy of inductive games, Ph.D. thesis, U.C. Berkeley, 2000.

Lecture Notes in Logic 1. Recursion Theory. J. R. Shoenfield. (1993, reprinted 2001; 84 pp.) 2. Logic Colloquium ’90; Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic, Held in Helsinki, Finland, July 15–22, 1990. Eds. J. Oikkonen and J. V¨aa¨ n¨anen. (1993, reprinted 2001; 305 pp.) 3. Fine Structure and Iteration Trees. W. Mitchell and J. Steel. (1994; 130 pp.) 4. Descriptive Set Theory and Forcing: How to Prove Theorems about Borel Sets the Hard Way. A. W. Miller. (1995; 130 pp.) 5. Model Theory of Fields. D. Marker, M. Messmer, and A. Pillay. (First edition, 1996; 154 pp. Second edition, 2006; 155 pp.) 6. G¨odel ’96; Logical Foundations of Mathematics, Computer Science and Physics; Kurt G¨odel’s Legacy. Brno, Czech Republic, August 1996, Proceedings. Ed. P. Hajek. (1996, reprinted 2001; 322 pp.) 7. A General Algebraic Semantics for Sentential Objects. J. M. Font and R. Jansana. (1996; 135 pp.) 8. The Core Model Iterability Problem. J. Steel. (1997; 112 pp.) 9. Bounded Variable Logics and Counting. M. Otto. (1997; 183 pp.) 10. Aspects of Incompleteness. P. Lindstrom. (First edition, 1997; 133 pp. Second edition, 2003; 163 pp.) 11. Logic Colloquium ’95; Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic, Held in Haifa, Israel, August 9–18, 1995. Eds. J. A. Makowsky and E. V. Ravve. (1998; 364 pp.) 12. Logic Colloquium ’96; Proceedings of the Colloquium held in San Sebastian, Spain, July 9–15, 1996. Eds. J. M. Larrazabal, D. Lascar, and G. Mints. (1998; 268 pp.) 13. Logic Colloquium ’98; Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic, Held in Prague, Czech Republic, August 9–15, 1998. Eds. S. R. Buss, P. H´ajek, and P. Pudl´ak. (2000; 541 pp.) 14. Model Theory of Stochastic Processes. S. Fajardo and H. J. Keisler. (2002; 136 pp.) 15. Reflections on the Foundations of Mathematics; Essays in Honor of Solomon Feferman. Eds. W. Seig, R. Sommer, and C. Talcott. (2002; 444 pp.) 16. Inexhaustibility; A Non-Exhaustive Treatment. T. Franz´en. (2004; 255 pp.) 17. Logic Colloquium ’99; Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic, Held in Utrecht, Netherlands, August 1–6, 1999. Eds. J. van Eijck, V. van Oostrom, and A. Visser. (2004; 208 pp.) 18. The Notre Dame Lectures. Ed. P. Cholak. (2005; 185 pp.)

19. Logic Colloquium 2000; Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic, Held in Paris, France, July 23–31, 2000. Eds. R. Cori, A. Razborov, S. Todorˇcevi´c, and C. Wood. (2005; 408 pp.) 20. Logic Colloquium ’01; Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic, Held in Vienna, Austria, August 1–6, 2001. Eds. M. Baaz, S. Friedman, and J. Kraj´ıcˇ ek. (2005; 486 pp.) 21. Reverse Mathematics 2001. Ed. S. Simpson. (2005; 401 pp.) 22. Intensionality. Ed. R. Kahle. (2005; 265 pp.) 23. Logicism Renewed: Logical Foundations for Mathematics and Computer Science. P. Gilmore. (2005; 230 pp.) 24. Logic Colloquium ’03: Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic, Held in Helsinki, Finland, August 14–20, 2003. Eds. V. Stoltenberg-Hansen and J. V¨aa¨ n¨anen. (2006; 407 pp.) 25. Nonstandard Methods and Applications in Mathematics. Eds. N. J. Cutland, M. Di Nasso, and D. Ross. (2006; 248 pp.) 26. Logic in Tehran: Proceedings of the Workshop and Conference on Logic, Algebra, and Arithmetic, Held October 18–22, 2003. Eds. A. Enayat, I. Kalantari, and M. Moniri. (2006; 341 pp.) 27. Logic Colloquium ’02: Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic and the Colloquium Logicum, Held in M¨unster, Germany, August 3–11, 2002. Eds. Z. Chatzidakis, P. Koepke, and W. Pohlers. (2006; 359 pp.) 28. Logic Colloquium ’05: Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic, Held in Athens, Greece, July 28–August 3, 2005. Eds. C. Dimitracopoulos, L. Newelski, D. Normann, and J. Steel. (2007; 272 pp.) 29. Logic Colloquium ’04: Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic, Held in Torino, Italy, July 25– 31, 2004. Eds. A. Andretta, K. Kearnes, and D. Zambella. (2007; 220 pp.) 30. Stable Domination and Independence in Algebraically Closed Valued Fields. D. Haskell, E. Hrushovski, and D. Macpherson. (2007; 190 pp.) 31. Games, Scales, and Suslin Cardinals: The Cabal Seminar, Volume I. Eds. ¨ A. S. Kechris, B. Lowe, and J. R. Steel. (2008; 445 pp.)