Selected logic papers

SELECTED LOGIC PAPERS

World Scientific Series in 20th Century Mathematics

Published Vol. 1

The Neumann Compendium edited by F. Brddy and T. Vamos

Vol. 2

40 Years in Mathematical Physics by L D. Faddeev

Vol. 3

Selected Papers of Yu I Manin by Y. Manin

Vol. 4

A Mathematician and His Mathematical Work — Selected Papers of S. S. Chem edited by S. Y. Cheng, P. Li and G. Tian

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Vol. 6

Selected Logic Papers by G. E. Sacks

World Scientific Series in 20th Century Mathematics - Vol. 6

SELECTED LOGIC PAPERS

Gerald E. Sacks Department of Mathematics Harvard University and Massachusetts Institute of Technology USA

World Scientific Singapore 'New Jersey •London ■ Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Sacks, Gerald E. Selected logic papers / Gerald E. Sacks. p. cm. -- (World Scientific series in 20th century mathematics; vol. 6) Includes bibliographical references. ISBN 9810232675 (alk. paper) 1. Logic, symbolic and mathematical. I. Title. II. Series. QA9.2.S23 1999 511.3-dc21 99-22812 CIP

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

The author and publisher would like to thank the following publishers of the various journals and books for their assistance and permission to include the selected reprints found in this volume: Academic Press American Mathematical Society Association for Symbolic Logic Elsevier Science Publishers Gauthier-Villars Johns Hopkins University Press Polish Academy of Sciences Princeton University Press While every effort has been made to contact the publishers of reprinted papers prior to publication, we have not been successful in some cases. Where we could not contact the publishers, we have acknowledged the source of the material. Proper credit will be accorded to these publishers in future editions of this work after permission is granted.

Copyright © 1999 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic oo mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Printed in Singapore by Uto-Print.

In Memory Of My Mother Ethel Sacks

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vil

Contents

General Introduction

ix

[I]

On the Degrees Less Than Q\Ann. Math. 77 (1963) 211-231

1

[2]

Recursive Enumerability and the Jump Operator, Trans. Amer. Math. Soc. 108 (1963) 223-239

22

The Recursively Enumerable Degrees are Dense, Ann. Math. 80 (1964)300-312

39

A Simple Set Which is Not Effectively Simple, Proc. Amer. Math. Soc. 15(1964)51-55

52

Metarecursive Sets (with Georg Kreisel), J. Symbolic Logic 30(1965)318-338

57

Post's Problem, Admissible Ordinals, and Regularity, Trans. Amer. Math. Soc. 124 (1966) 1-23

78

[3]

[4]

[5]

[6]

[7]

[8]

[9]

On a Theorem of Lachlan and Martin, Proc. Amer. Math. Soc. 18(1967)140-141

101

A Minimal Hyperdegree (with Robin O. Gandy)) Fundamenta Math. 61(1967)215-223

103

Measure-Theoretic Uniformity in Recursion Theory and Set Theory, Trans. Amer. Math. Soc. .42 21969) )31-420

[10] Recursion in Objects of Finite Type, Proc. Int. Congress Math. (1970), Vol. 1,251-254 [II]

Forcing with Perfect Closed Sets, Proc. Symp. Pure Math.. Vol. 13, Part 1, Axiomatic Set Theory (1971) 331-355

[ 12] The a-Finite Injury Method (with Stephen G. Simpson), Ann. Math. Logic 4 (1972)343-367

111

152

156

181

Vlll

[ 13] The 1 -Sectton of a Type n Objectt Generalized Recursion Theory (edited by J. E. Fenstad and P. G. Hinman) (1974) 81-93

206

[14] Remarks Against Foundational Activity, Historia Math. 2 (1975) 523-528

219

[15] Countable Admissible Ordinals and Hyperdegrees, Adv. Math. 20(1976)213-262

225

[ 16] Thefc-Secttonof a Type n Object, Amer. J. Math. 99 (1977) 901-917

275

[17] Effective Bounds on Morley Rank, Fundamenta Math. 53 (1979) 111-121

292

[18] Post's Problem, Absoluteness and Recursion in Finite Types, The Kleene Symposium (edited by J. Barwise, H. J. Keisler and K. Kunen) (1980) 201 -222

303

[19] On the Number of Countable Models, Southeast Asian Conference on Logic (edited by C.-T Chong and M. J. Wicks) (1983)185-195

325

[20] Post's Problem in ^-Recursion, Proc. Symp. Pure Math., Vol. 42 (1985)177-193

336

[21 ] The Limits of ^-Recursive Enumerability, Ann. Pure Appl. Logic 31(1986)87-120

353

[22] Inadmissible Forcing (with T A. Slaman), Adv. Math. 66 (1987) 1-30

[23] Effective Forcing Versus Proper Forcing, Ann. Pure Appl. Logic 81(1996)171-185

387

417

ix

General Introduction

Nothing

is forgotten,

fortunately

or not. -H. M.

Looking backward I see a problem solver more than a system builder and still do. I always felt more comfortable working on someone else's problem. Avoidance of some sort I suppose. Two exceptions come to mind immediately, splitting an re set into incomparable re s e t s [ l ] and solving Post's problem invariantly. The first was my initial foray into re sets and I am still working on the second. T h e problem with problem solving is being unable to let go of an intractable problem. This did not happen to me, fortunately, until fairly late in the game, about fifteen years ago I think. The papers I selected for reprinting in this volume are more interesting to me t h a n the ones I left out. I do not know if there is a common theme. For me logic is about definability, but t h a t could be a recursion theorist talking. On the other hand the principal methods of recursion theory, priority arguments and forcing, do apply in other areas of mathematical logic. a a

T . Slaman put it as follows. "There is a particular perspective obtained by consistently including an understanding of 'definability' in the solution to a mathematical problem. It is the assumption of that vantage point which I identify with recursion theory. Of course not only 'recursion theorists', but also computer scientists, descriptive set theorists and others have claim to it; many of us ascend but we do not all look in the same direction. We are all trying to understand the interaction between mathematical objects and the means needed to speak .about, them.!'., , . , F

Copyngnted Material

X

On the degrees less than 0' [1] appeared in 1963. T h e paper presents two results from my Ph.D. thesis (Cornell 1961) directed by J. B. Rosser. T h e first is a splitting theorem for re sets with several corollaries. My favorite then and now is: each degree strictly between 0 and 0' is incomparable with some re degree. The second is the existence of a minimal degree below 0'. I was intrigued by the idea of using the priority method to construct a non-re degree, an approach taught to me by Shoenfield [rl]. T h e paper is devoid of intuitive remarks save for the final paragraph. It is tempting to blame the lack on Kleene's influence, but the t r u t h is that in 1961 I greatly distrusted intuitive remarks and regarded them as intellectually corrupting. These days formal reasoning seems too weak to stand alone, like a skeleton without a body. Recursive enumerability and the jump operator [2] was written in the fall of 1961. The main result, a determination of the range of the j u m p restricted to re degrees, was inspired by Friedberg's determination of the range of the j u m p [r2], and by Shoenfield [rl] on the j u m p restricted to degrees below 0'. This paper in harmony with its predecessor was free of intuitive remarks. The search for the proof led me to a simple instance of the infinite injury method. Phrases such as "finite injury" or "infinite injury" did not occur. The recursiveyy enumerable degrees are dense [3] needed something more than infinite injury, an insight that was painfully long in coming. Density says t h a t if B and C are re sets such t h a t B
xi

were tractable as long as they retained some of the combinatoric properties of finite computations essential to ordinary recursion theory. We delighted in the term, "metafinite". I began to say good-bye to finite computations. Post's problem, admissible ordinals, and regularity [6] continued the study of metarecursion and pushed it upward to Kripke-Platek admissi­ ble ordinals. Most of the questions in this paper were raised by Kreisel. Some of the answers given are as follows. Let a be a Si admissible ordinal. There exists a non-a-recursive, a-re, subgeneric set. (A subgeneric means (L(a),A) is Si admissible.) There exists a countable a such that no a-re set is maximal. (A complete understanding of maximal a-re sets was found by Lerman [r3] and Simpson.) For every a there is an intermediate a-re degree. (A better solution to Post's problem was offered in [12].) In retrospect the result I am most fond of has to do with regularity. A subset A of a is regular if (A(i3) G L(a) for every j3 < a. Regularity is the same as Jensen's notion of amenability. In other words the intersection of A with any a-finite set is a-finite. Each a-re set has the same a-degree as some regular a-re set by an argument that goes back to Dekker's deficiency set construction in classical recursion theory. On a theorem of Lachlan and Martin [7] reproves their result that there exists an re degree d such that Vn[0W 0} a standard enumeration of all sets re in A. According to Kleene there is a fixed point c such that TjA

_ T>A

The twist, which enlivened the paper, is that c is independent of A. A minimal hyperdegree [8] (with R. O. Gandy) was an application of perfect forcing to hyperdegrees. Gandy and I worked independently on the problem of finding a minimal hyperdegree, and after conferring with him at his cottage in Wales, I wrote it up in a style more ponderous than I would choose now. This was my first excursion into class forcing over a countable structure with severely restricted closure properties, a setting where considerations of effectiveness matter more than they do in'models of set theory. The experience gained here was helpful in the 1-section [13] and k-section [16] papers.

xii

Measure-theoretic uniformity in recursion theory and set theory [9] started with an interest in the question: what is the measure of {X | uf = ufK}? The answer turned out to be 1, although every authority I consulted beforehand thought it should be 0. Along the way I noticed that a Il{ set of positive measure had to have a hyperarithmetic member. The argument looked at definable aspects of adjoining an arbitrary real T to the hyper­ arithmetic hierarchy. Soon after I could not resist looking further at L(a,T) when L(a) is a countable model of ZF & V = L. Let T be a sentence of in the language of set theory with parameters from L(a,T). Define p{T), the probability that F is true in L(a,T), to be the Lebesgue measure of {T | L(a,T) N T}. The L(a)-definability of p{F) readily shows that L(a,T) is a model of ZFC & V ^ L with probability 1. Thus Cohen's historic results were obtainable by familiar facts about the constructive nature of the measurability of Borel sets and prenex normal form. I later learned that the above naive measuretheoretic point of view was equivalent to Solovay's forcing with closed sets of positive measure. I spoke on p{F) at a logic seminar at Stanford in 1965. The small audience had the highest density of distinguished logicians I ever faced. There sat Cohen, Kreisel, Solovay, Barwise, Feferman and Platek. Recursion in objects of finite type [10] begins with an elaborate joke that went unnoticed when the paper was delivered in Nice (ICM) in 1970. My mistake was thinking that France is a Catholic country. The paper announced some results about l-sections building on Kleene's work. A type n object F is normal according to Kleene if nE, the equality predicate for objects of type less than n, is recursive in F. (The objects of type 0 are non-negative integers.) SXF, the 1-section of F, is the set of all reals, i.e. objects of type 1, recursive in F. The main theorem says (i) is equivalent to (ii). Let Ai(SiF) be the transitive set whose elements are coded by the reals in SiF. (i) A is a countable Si admissible set that satisfies Si dependent choice. (ii) There exists a normal type 2 F such that A = A^F). The proof that (i) implies (ii) is a class forcing argument over a Si admissible set, a technique I learned from [8] and extended to ^-closed, inadmissible structures in [21]. The corollary of most interest to me was: if F is normal of type > 2, then there is a normal G of type 2 such that S^ = 5iG. Forcing with perfect closed sets [11] looks at applications of perfect forc­ ing to set theory, hyperarithmetic theory, and Turing degrees. My inspi­ ration was Spector's construction of a minimal Turing degree via recursive

XIii

Cantor sets. I added the sequential (or fusion) lemma: let Ft {i < w) be a sequence of sentences; then

VP(3QC P){\/i Ri Ih {-&$. In set theory minimal sidewise extensions of countable initial segments of L were obtained. In the hyperarithmetic case there was a wrinkle. Forcing was defined to be identical with truth for all ranked sentences in order to make it immediate that the forcing relation, restricted to ranked sentences, was U\ in V, hence Si over L(LJ^K). In recursion theory a real X was found such that (a) X^ =T 0 ^ and (b) 0
Vx[x

€P^P
(P ambiguously denotes the perfect condition P and its canonical real code.) Hyperarithmetic pointedness proved useful in the study of hyperdegrees as in [15] below. The a-finite injury method [12] (with S. Simpson) lifts the FriedbergMuchnik (FM) solution to Post's problem from u to every S x admissible a by means of condensation (i.e. downward Skolem-Lowenheim) arguments in admissible L(a)'s. Simpson and I focused on this problem only after Shore demolished an earlier approach based on ordinal arithmetic suggested by Stillwell. The FM solution works straightforwardly if L(a) satisfies £2 replacement. As in [5] the FM solution is readily adapted when a*, the S? projectum of a, is less than a. The most interesting case is when a* = a and there is a largest a-cardinal in the sense of L(a), call it H. Then L{a) is the limit of S i substructures (hence of a-stable ordinals), requirements can be arranged in blocks of length H, and injuries can be bounded by a-stable ordinals. Later Lerman's notion of tame and Shore's notion of block clarified the dynamics of the construction. A simple combinatoric lemma turned out to be useful. If p is an infinite regular a-cardinal, then fi is regular with respect to IT? functions. (A regular a-cardinal is by definition regular with respect to a-finite functions.) The 1-section of a type n object [13] gave the details of the proof of the main result of [10]. I tried for a long time with a notable lack of success to stretch the argument to show the fc-section (jfc > 1) of a normal type

xiv

n object (n > k) was the fc-section of a normal type (k + 1) object. T h e missing ingredient is described below [16]. In Remarks against foundational activity [14] I was out of place, an amateur among professionals, but I enjoyed speaking against foundational activity, dismissing as groundless the fears of contradiction in mathematics, and pointing out the dangers of excessive precision and clarity. Countable admissible ordinals and hyperdegrees [15] began as t h e writeup of "each countable admissible ordinal greater t h a n w is of the form w f for some X C u", a result announced in 1967, but gradually evolved into an obsession with class forcing over a countable admissible set. I doubt if it was worth the effort expended to show: if a is a countable admissible > u, then there is an X C w such t h a t w f = a and u\ < a for all Y of fower hyperdegree than X. T h e main result of the paper is: if A is a countable S i admissible set t h a t satisfies S i dependent choice, the set of hyperdegrees of reals in A has a minimal upper bound. The notion of pointedness [11] was extended to the hyperarithmetic case. P, a perfect condition, is pointed if v r [ T e P ^ P t f t

T].

( P ambiguously denotes a condition and its canonical real code.) To cope with the reality that A may fall far short of resembling an L{a), a triple (so-called) forcing relation, \H-A, was introduced. Ilh^combined with S i DC made it possible to carry out in A the sort of fusion argument with hyperarithmetically pointed P ' s more readily done in an admissible L{a). I spent a lot of time trying to eliminate the use of S i DC above until F. Abramson showed it was impossible to do so. He constructed a countable admissible A such that no G generic over A is a minimal upper bound for the hyperdegrees contained in A He assumed that the forcing relation underlying G, restricted to E x sentences, was S i over A. The k-section of a type n object [16] completed the proof of the PlusOne theorem: if n > k > 1 and F is a normal type n object, then there is a normal type (k + 1) object G whose k-section equals t h a t of F. The ingredient absent from [13] was reflection. I noticed one day, I forget why, t h a t Grilliot selection for sets of reals re in 3 £ led to a simple reflection principle in the setting of recursion in normal objects t h a t stretched the forcing argument of [13] from k = l t o c > . . I asked Harrington to extend and apply reflection further in the above setting in his thesis. His success inspired much of my work on ^-closed initial segments of L [20, 21, 22]. Effective bounds on Morley rank [17] attempted to pin down the absolute nature of Morley rank with the help of hyperarithmetic theory and model

XV

theory on admissible sets. I had long been intrigued by the thought t h a t the central notions of model theory were absolute. A typical result was as follows. Let A be a countable substructure of a model of a complete, substructure complete, countable theory T, and p G Si A. If p has a Morley rank, then p is hyperarithmetic in A and its Morley rank is an ordinal recursive in A. (Earlier Lachlan had shown t h a t Morley rank is always countable.) It follows t h a t the Morley rank of a totally transcendental theory T is an ordinal recursive in T. The impetus for this paper goes back to Kreisel's effective analysis of the Cantor-Bendixson theorem [r5]. Deeper absoluteness results on the Morley derivative were obtained by Calmer [r6]. Post's problem, absoluteness and recursion in finite types [18] worked directly with Kleene's theory of recursion in finite types and tried to reach some definite conclusions about Post's problem. In retrospect it served as a of the class warm for [20] and [21]. The main result in the case {R | R C 2" A 3 6 G 2U(R recursive

in

3

E,b)}

is not re in 3E. It was proved by a forcing argument that relied on a regularity assumption introduced by Normann: there exists a recursively regular well-ordering -< of 2W such that -< is recursive in 3£; recursively regular means there is no function / recursive in 3E, a (for some a C u) such that dom(f) is a bounded initial segment of -< and range(f) is unbounded in -<. Two projecta related to 3E, n and p, were defined and proved equal with the aid of Normann's regularity assumption. In a footnote a claim was made t h a t a positive solution of an absolute version of Post's problem for 3E followed from T) = p. T h e proof of the claim is extended in [20] to E-closed initial segments of L, where 77 and p reappear, and without any regularity assumption lead to a positive solution to Post's problem. On the number of countable models [19] was my first foray into the wilder­ ness of Vaught's conjecture. I have been planning to rewrite, clarify and extend this paper for some time. T h e main result slightly updated is: if r{M) < u>!M for every M a model of T, then r(T) < wjf. A brief explanation may help. T is a countable theory. M is a countable model of T. r(M) is the Morley rank of M, the ordinal height of CM, a canonical fragment of CWljUi. M is the atomic model of Th(M), the complete theory of M in £M. r(M) is at most w^ + 1. T h e differences between (1) r{M) = LJf

and

(2) r(M)

= wf + 1

xvi

are noteworthy. (1) implies M is the atomic model of its complete theory TM in Cwuw restricted to L(cjf f ,M), but does not imply TM is u;-categorical. (2) means that M realizes a non-principal type of TMr(T) is the least ordinal where a witness to the truth of Vaught's conjecture is constructible from T. The witness has two forms: a perfect kernel of models of T, or a countable family of Scott sentences that characterize all countable models of T. uj is the least 7 such that £(7, T) satisfies S2 replacement. A similar result holds for PC^^ classes. Post's problem in E-recursion [20] completes the attempt on Post's problem begun in [18]. Suppose L(K) is .E-closed: if / is a partial ^-recursive function, b € L(K), and f(b) I, then f(b) G L(K). The partial .E-recursive functions, discovered by Normann [r7], and independently and subsequently by Moschovakis, are a set-theoretic version of Kleene recursion in normal objects of finite type. The main result of the paper is a positive solution to Post's problem: there exist two subsets of L(K), each E-recursively enumerable on L(K), such that neither is ^-reducible to the other. The proof has some curious features. None of the Friedberg-Muchnik requirements is injured. Each requirement is attempted at most once. A form of reflection is key to the argument, in particular a Harringtonesque connection between reflection and Moschovakis divergence witnesses. The presence of divergence witnesses in L(K) makes it possible to avoid injuries. The construction is guided by two projecta, n and p, seen to be equal by a fine structure argument. One proposition, 3.2, is stated and proved in a way that to this day I do not understand. The sole purpose of 3.2 is to assist in the proof of 3.3, a theorem whose statement seems intelligible. The limits of E-recursive enumerability [21] continues a theme of [18]: certain naturally enumerable sets are not "recursively enumerable". [21] assumes L(K) is ^-closed but not Si admissible and answers the following question. Let gc{n) be the greatest cardinal in the sense of L{K). Is 2gc^K' n L(K) ^-recursively enumerable in some a £ L(K)? The answer breaks into 3 cases. (1) if L(K) II- [#c(/c) is regular], then no. (2) if L(K) lh [ui < cofinality(gc{K)) <
XV11

Case (2) is an application of combinatoric and fine structural ideas of Silver [r8] and S. Friedman [r9]. Friedman showed that the um-degrees above 0' are well-ordered if V = L. His approach shows there is a d € L{K) such that 0(x) is ^-recursive in x,d (uniformly) for all x 6 29CW n L(K). (O(x) is the least ordinal by which x is constructible.) The computability of 0(x) yields the ^-recursive enumerability of 29C(I) n L(K). Case (3) is an application of infmitary model theory. The principal inspiration is Green's compactness theorem [rlO]: R(u + u)+, the least Ei admissible set with R(u + u) as a member, is Ei compact. Inadmissible farcing [22] (with T. Slaman) applies pointed perfect forc­ ing [15] to solve a problem in E-recursion. The result is: suppose L{K) is countable, E-closed, and not Ei admissible; assume L(K) \= [cofinality(gc(K))

> w];

(9C(K) is the greatest cardinal in the sense of !(«)); then there exists a T C gc(k) such that L{K,T) — E(T), the least E-closed set with T as a member. A forcing condition p is a suitable uncountable tree in L(K). p is Epointed in the sense that the standard code for p is E-recursive in any path through p. Fusion and reflection combine to show L(«, T) is E-closed for generic T. Effective versus proper forcing [23] tries to characterize those notions of set forcing that preserve E-closure. Recall: if A is a Ei admissible set, V € A is a forcing relation, and G is P-generic over A, then A(G) is Ei admissible. The situation is more complex for an E-closed L(K). Consider E(ui), the E-closure of wi. E(ui) = L{K) for some K. If G collapses wi to w, then L(K, G) is not E-closed. The main result says that P G % ) preserves E-closure iff P is locally proper, a variant of Shelah's notion of proper. Let L(K) be E-closed. / € L{K) is a co-re substructure of L(K) if / is E-closed and for all m,ae I and e < CJ, Bx[x€mA{e}(x,a)t] —>3x[se(mn/)A{c}(x,a)f]. / is pointed if 3 6[6 € / A Kbr ^ /]. Kbr is the greatest reflecting ordinal relative to b. V is locally proper if for all p £ V and a € L(/e), there is a countable pointed co-re substructure / such that p,a € I and 3?[9ePAP>gA?

generic over/].

xviii

The last means: if D is a dense set of conditions, suitably denned over /, then q C U(D D / ) . The genericity of q implies p cannot force {e}(a) to converge via a computation not in / . [23] ends with a sentence that fits here: Farewell to higher recursion theory, but not to recursion theory; there is no way to say good-bye to recursion theory. Cambridge, Massachusetts February 1999

References [1] to [23] refer to the list of Selected Logic Papers. [rl] Shoenfield J. R., On degrees of unsolvability, Ann. Math. 69 (1959) 644-655. [r2] FriedbergR., A criterion for completensss of degrees of unsolvability, J. Symb. Logic 22 (1957) 159-160. [r3] Lerman M., Maximal a - re sets, Trans. Amer. Math. Soc. 260 (1980) 363-378. [r4] Abramson F., Sacks forcing does not always produce a minimal upper bound, Adv. Math. 31 (1979) 110-130. [r5] Kreisel G., Set theoretic methods suggested by the notion of potential totality, in Infinitistic Methods (Pergamon, 1961), pp. 325-369. [r6] Calmer C , Ph.D. Thesis, Stanford University (1978). [r7] Normann D., Set recursion, in J. E. Fenstad et al. eds., Generalized Recursion Theory II (North-Holland, 1978), pp. 303-320. [r8] Silver J., On the singular cardinals problem, in Proc. Int. Cong. Math. Vancouver, 1974. [r9] Friedman S., Uncountable admissibles II: compactness, Israel J. Math. 40 (1984) 129-149. [rlO] Green J., E x compactness for next admissbble sets, J. Symb. Logic 39 (1974) 105-116.

1

ANNALS OF

MATHEMATICS

Vol. 77, No. 2, March, 1963

ON THE DEGREES LESS THAN 0' BY GERALD E. SACKS*

(Received October 2, 1961)

By degree, we mean degree of recursive unsolvability as denned by Kleene and Post in [6]. Two of the principal results of this paper are: (1) there exists a degree h such that 0 < h< 0', and such that 0 < d < h for no degree d; (2) if c is a non-zero degree less than 0', then there exists a degree b which is the degree of a recursively enumerable set, and which is in­ comparable with c. The first result answers in the affirmative a question raised by Shoenfield in [10]: "Does there exist a minimal degree a such that a^O'V The second result has as a consequence: a set is recursive if and only if it is recursive in every non-recursive, recursively enumerable set. Our notation, unless otherwise specified, is that of [4], [6] and [11]; we assume a considerable amount of familiarity with [4] and [6]. If e is a natural number and h(n) is a function, then we employ {e}h{n) to denote the partial function U{nzT\{h{z),e,n,zj). h

This use of {e} (n) is consistent with the definition of it given by Kleene in [4]. Recall that a non-empty set D is recursively enumerable in a set B if there is a function recursive in B whose range is D. THEOREM 1. Let B, C and D be sets such that C is recursive in D, D is recursively enumerable in B, and C is not recursive in B. Then there exist disjoint sets D„ and A such that D = A U A , and such that for each i < 2, A is recursively enumerable in B, and C is not recursive in B, APROOF. Let B* = {2n\ne B) and D* = {2n + 1 1 n e D ) . Then C ii recursive in D*, D* is recursively enumerable in B*, and C is not re­ cursive in B*. Suppose there are disjoint sets A* and D* such that A* u A* =D*, and such that for each i < 2, D? is recursively enumer­ able in 5*, and C is not recursive in B*,D*. Then the disjoint sets D0 = {n|2n + 1e A*} and A = {n 12n + 1 e A*} are such that A U A = A, and sucn that for each i < 2, A is recursively enumerable in B, and C is not recursive in B, A- Thus we may assume without any loss of generality that every member of B is even, and every member of D is odd. " * The author is a National Sgte^gfo/H&WafePK&Aoctoral 211

Fellow.

2 212

GERALD E. SACKS

Let b(n) be the representing function of B: n 6 B^b(n) = = 0nd d $ 0 — 6(n) - 11 Lee c(n) and d(») be the representtng functions of C and D respectively. Then b(n)-d(n) is the representing functions of B\jD. Since there exists a set, namely B, in which D is not recursive, it must be that D is infinite. Let/(n) be a function recursive in 5 which enumer­ ates D without repetitions: D = {/(«) | « ^ 0} and (n, m)(» =£ m —/(») # / ( » ) ) . Let d(s, n) = 0 if there is a /c g s such that /(&) = n, and let d(s, n) = 1 otherwise. It is clear that the function d(s, n) is recursive in B, and that for each n, lim.d(«, ») exists and is equal to d(n). Since C is re­ cursive in D, there exists a Gbdel numberffwith the property that c(n) = {g}\n) for all ». We make use of g to define a function e(s, «) which is recursive in B:

, C(S %

' )

\U{tiyT\{Jl.<^^\g,n,y)) =

|s + 1

if (Ey)vS,Tl{Jl, >\ g, n,y) otherwise.

Since lims d(s, n) = d(«) for all n, and since c(») = {g}\n) for all » , it follows that lim, c(s, n) exists and is equal to c(n) for all n. A very useful property of the Gbdel numbering devised by Kleene in [4] to arithmetize his formalism for recursive functions is: the Gbdel number of a deduction d is greater than the integer denoted by any formal numeral occurring in d.1 This property was utilized by Friedberg in [1]. It follows from GND that c(s, n) = s + 1 whenever n ^ s. It is our intention to define twelve functions simultaneously by induc­ tion: dfa n), Us), yt(s, n, e), P{(s, n, e), m4(«, e) and K^s, e) (i = 0, 1). Each of these functions will be recursive in B. At stage s (s ^ 0) of the induction, we will define dfa, n), Us), y^s, n, e), P4(s, n, e), m{(s, e) and Ki(s, e) for all i < 2, all n and all e. The sole purpose of stage s is to put the natural number/(s) (the stb member of D) in D0 or in Dlt but not in both. Thus at stage s, we set either d0(s,f(s)) or (£,(«,/(«)) equal to 0, but not both. Stage s = 0. We set ^(0, n) = 1 for all n ^ /(0), and we set ^(0, /(0)) = 0. We set P,(0, n, e) - 2, d0(0, n) = tt(0) = ^(0, n, e) = 1 and m,(0, e) = Ki(0, e) = 0 for all i < 2, all e and all n. Stage s > 0. For each i < 2, let (S)

\Mf(s) < Us - 1- e)) \s + 1

1 We will denote this fact by GND.

if (^).s.(/(«) < KA* ~ 1, e)) otherwise.

3 DEGREES LESS THAN 0'

213

Let z(s) = 1 if Us) ^ t0(s), and let z(s) = 0 otherwise. Thus *„.,(«) ^ «!_,,„(*). We set d,(.,(s, n) = d2(e)(s - 1, n) for all % * /(*), and we set dzM(s,f(s)) = 0. We set <*,_.„(*, n) = (*,_,«„(« - 1, «) for all n. We now define ^ ( s , n, e) and P4(g, n, e) for all i < 2, all % and all e:

s + 1 P«(*,n,e) =

otherwise;

(CT(v«(«,»,e)) s + 2

if *<(•,»»•)£« otherwise.

Before we define m{{s, e), we note that (i)i<2(e)(Et)tSs(c(s, t) * A(s, t, e)) . This last is clear, since it is immediate from GND that c(s, s) = s + 1, and that P0(s, s, e) = A(s, s, e) = s + 2 for all e. We complete stage s > 0 by defining m^s, e) and K^s, e) for all i < 2 and all e: m£s, e) = fd{c(s, t) * A(s, t, e)) ; Ai(S

' C) - U « ^ „ U y ^ . *. e> ^ *)

otherwise.

We list two remarks which will be needed in Lemmas 1, 3 and 5: (Rl) (i) i<2 (s)(e)(^(s + 1, e) 2= JBT«(*. e)); (R2) (i)4<,(s)(e)(»).<m4(I.f)(»4(sf », e) ^ Z,-(s, e)). It can easily be verified by the methods of [6] that each of the twelve functions we have defined is recursive in B. Such a verification is possible because of two facts: each of the functions/(s), b(s) and c(s, n) is recursive in B; only bounded quantifiers occur in the case hypotheses of the definitions of the functions Us), yt(s, n, e) and Z,(s, e) (i = 0,1). It is clear that (i)i<2(s)(n){0 ^ d((s + l,n)£

dfa n)^l)

.

It follows that for each i < 2 and each n, lim.d4(s, n) exists and is less than 2. For each i < 2, let d{{n) = lim.d«(«, n), and let A be the set whose representing function is d4(n). Then A n A = 0 , and D = A U A . since at stage s,f{s) was put in A or in A , but not in both. We now show by means of a reductio ad absurdurn argument that, for each i < 2, C is not recursive in B, A . Suppose then that C is recur­ sive in B, A<- where i' < 2. It is our intention to prove that C is recur­ sive in B, contrary to the hypothesis of our theorem. Recall that B con­ tains only even numbers, and D only odd. This means that B and Dt.

4 214

GERALD E. SACKS

are each recursive in B U A-, since n e B = (n is even & % e 5 U A ) n 6 A- = (« is odd & % e 5 u A ) . The representing function of 5 U A< is 6(»)-di((n). It follows that there is a Godel number e' such that c(») = {e'Y^'in) for all n. 1. The set {K^s, e')\s^ 0} is infinite. Suppose, for the sake of a reductio ad absurdum, that {KAs, e')\s^ 0} is finite; let m be its largest member. Let s' be such that c(s, ft) = c(n) for all s and % such that s^s'andn^ m. Let y(n) denote the partial function LEMMA PROOF.

^ r HII 3 '<^ < J '" a i ' < y , . e '. % .2/)Since c(») = {e'}'-"''(») for all n, y(n) is defined and c(n) = I7(y(»)) for all n. Let «' be the largest member of {y(n) \n^m}. Let s" be such that s" ^ s', s" ^ i/' and cZ*,(s, n) = d,.,(») for all n and s such that n^y' and s ^ s". It follows that y{n) = 2/;.(s", ft, e') for all n ^ m. Then P,-,(s", «, e') = i7(i/(ft)) = c(«) = e(g", ft) for all « ^ m, since y{n) ^y'^ s" for all n ^ m, and since s" S s'. This means that mv{s", e') > m. But then by remark (R2), y,.(«", m, e') g ^-(s", e') g m. This last is absurd, since by GND,1 we have m < y(m) = y^s", m, e'). Let e* be the least e such that {K{(a, e) \ i < 2 & s ^ 0} is infinite; by Lemma 1, e* ^ e'. Let i* be the least i such that {^(s, e*) | s 2: 0} is infinite. LEMMA 2. There is an s' such that for all s ^ s', z(s) = 1 - i* or e* < t^s). PROOF. Suppose, for the sake of a reductio ad absurdum, that there are infinitely many s with the property that z(s) = i* and e* ^ t{.(s). Let S be an infinite set such that for each seS,s^e*, z{s) = i* and e* ^ Us). Recall that t„.,(s) 2= k-.«.,(*). Then for each s e S, e* ^ ^.(s)^ *,_,.(«). As S is infinite, there must be an infinite subset of S, call it T, such that for each r and s e T, t^.(r) = *,_„(«). Thus there is an e** such that for each s e T, we have s ^ e* ^ e** = ^ . ( s ) . It follows from the definition of the function *,_«.(») that, for each se T, /(s) < K^.(s - 1, «**). Since T is infinite, and since the function /(«) enumerates Z> without repetitions, it must be that {K^.{s, e**) | s £ 0} is infinite. It follows from the definition of e* thate* = e"; thus {iTw.(s, «*) | s g 0} is infinite. This means that 1 - i* > t* = 0. But then for any s e T, we have e* = t,.(s) = t,_,.(s) = e** and z(s) = i* = 0. This last is impossible because z(s) = 1 if t,(«) = t0(«) and s > 0.

5 DEGREES LESS THAN 0'

215

Let s* be the least s' such that s' ^ e*, and such that for all s ^ s\ z{s) = 1 - i* or e* < t{.(s). LEMMA 3. If s ^ s* a?td ifn < m^s, e*), then d^s', j) = d^s, j) for all j and s' such that j < y^s, n, e*) and s' ^ s. PROOF. Let s and n be such that s ^ s* and n< m^s, e*). We prove the lemma by induction on s' ^ s. Suppose s' ^ s and <*«.(«', j) = d^s, j) for all j < y^s, n, e*). Since s' ^ s*, we know that (a) z(s' + 1) = 1 - i*, or (b) e* < tt.(s' + 1) is the case. If (a) holds, then a\.(s' + 1, j) = <Ms', j) for all j . Suppose now that (b) is true. Note that e* ^ s*. Then f(s' + 1) ^ X,-.(s',e*), since otherwise (b) is false. By remark (Rl), K^s', e*) 2= X«,(s, «*), since s' 2: s. By remark (R2), Kt.(s, e*) ^ yt.{s, n, e*), since n < TO,.(S, «*). But then d,.(s'+ 1, j) = d(.(s', i) for all j < y^s, n, e*) ^ f(s' + 1). LEMMA

4. If $ ^ s* and if TO < m;.(s, e*), t^en ^.(s, TO, e*) = y(.(8\ n, e*) ^ s

/ o r aZZ s' ^ s .

PROOF. Let s andTObe such that s ^ s* and n< mt~.(s, e*). Let s' ^ s. Since n < m^s, e*), it must be that c(s, n) = P^s, n, e*). Since (by GND) c(s, n) ^ s + 1, we have P<.(s, n, e*) = £%j»(s,TO,«*)) and &.(s,TO,«*) ^ s. But then

M«,

n, e*) = MTKli^p^-^,

e*,n, y) .

r

By Lemma 3, d{.(s , j) = d,.(s, i) for all j < »«.(«, n, e*). This means that (y)„£V(..»,.,(^l(^y<^rJ',•ai*<,y^e^n,2/)=^K^y<^(i,•ai,l''•y^e^n,2/)). It follows that ^.(s, n, e*) = ^.(s',TO,e*). 5. TAe sei {m{.(8, e*)\s^ 0} is infinite. Suppose, for the purpose of a reductio ad absurdum, that the set {m(.(s, e*) \s ^ 0} is finite; let m' be its greatest member. Let s' be such that s'^s*, and such that c(s, n) = lim.c(«,TO)= C(TO) for all s and TO such that s ^ s' andTO^ TO'. Let TO" be the greatest member of {m^s, e*) | s ^ s'}. Let s" be such that s" ^ s' and TO,.(S", e*) = TO" ^ TO'. We now show by induction on s ^ s" that TO;.(S, «*) = TO" and ^ . ( s , «*) - Kds", e*) for all s ^ s". Suppose s S s", m^s, «*) = m", and #,..(s, «*) = iT,.(s", e*). By Lemma 4, ^.(s + 1,TO,e*) = y^s, TO, e*) ^ s for all TO < TO^S, e*), since s ^ s" ^ s' ^ s*. This means that Pt*(s + 1,TO,e*) = P(.(s,TO,e*) = c(s,TO)for allTO< m^«. e*) = m"- But c(s + 1, *) = c(s, TO) = C(TO) for allTO^ TO" ^ TO', since s ^ s'. Then P,.(s + 1, n, e*) = LEMMA PROOF.

6 216

GERALD E. SACKS

c(s + 1, n) for all n < mAs, e*) = m"; consequently, mAs + 1, e*) ^ m". It follows from the definition of m" that m,(s + 1, e*) = m". By remark (R2) and Lemma 4, y(.(s + 1, n, e*) = yAs, n, e*) ^ KAs, e*) for all n < mAs, e*) = m" = mAs + 1, e*). From this last, it is clear that KAs, e*) = KAs + 1, e*). Thus we have shown that KAs, e*) = ^ . ( s " , e*) for all s ^ s"; but this is absurd, since the set {KAs, e*)\s^ 0} is infinite. 6. If s ^ s* and if n < mAs, e*), then c(n) = PAs, n, e*). Let n < mAs, e*) and s^s*. By Lemma 4, yAs, n, e*) = yAs', n, e*) ^ s for all s' ^ s. By Lemma 5, there is an s" ^ s such that m;.(s", e*) ^ m ^ s , e*) > n and such that c(s", n) = lim,c(s, %) = c(n). But then c(s", %) = PAs", n, e*) = U(yAs", n, e*)), since n < mAs", e*) and s" ^ s. It follows that c(n) = c(s", n) = U(yAs, n, e*)) = P<.(s, n, e*), since y;.(s", ?i, e*) = yAs, %, e*) ^ s. LEMMA PROOF.

Let *(») S / < S ( S ^ S * & K < mAs,

e*)) ;

by Lemma 5, t(n) is denned for all n. By Lemma 6, c(n) = P,-.(i(w), %, e*) for all %. Since each of the functions mAs, e) and PAs, n, e) is recursive in B, it must be that C is recursive in B. But this last is contrary to the hypothesis of our theorem. Thus, by a reductio ad absurdum argument, we have shown that C is not recursive in B, A for any i < 2. It follows that A # 0, because otherwise A = A and C is recursive in B, A- We know from stage 0 that A * 0 - For each i < 2, we define a function &(») which is recursive in P and which enumerates A : f(«)i

ft (/Y)\ — J

if ^(Mo, (n\) = 0

^ U(«eA) otherwise. We will use the following elementary lemma to obtain some corollaries to Theorem 1. LEMMA 7. / / A, and A, are disjoint sets, and if for each i < 2, A, is recursively enumerable in A0 U Au then the degree of A0 U A, is equal to the least upper bound of the degrees of A0 and A,. PROOF. Let a„, a, and a be the degrees of A0, A, and A0 U A, respec­ tively. Clearly a g a0\Jttl. We show a, ^ a. Suppose we wish to know if n 6 A9. We first ask if ne A0 U Ax. If the answer is no, then n € A0. If the answer is yes, we simultaneously enumerate A0 and A,; eventually n will turn up in A„ or in Au but not in both, and our original question

7 DEGREES LESS THAN 0'

217

will be answered. Each of the enumeration functions is recursive in A0 U A,. Thus a0 g a, and similarly, a, ^ a. Following Shoenfield [10], we say a degree b is recursively enumerable in a degree c if there is a set of degree b which is recursively enumerable in a set of degree c, and we say a degree is recursively enumerable if it is the degree of a recursively enumerable set. Thus b' is recursively enumerable in b, and a recursively enumerable degree is a degree which is recursively enumerable in 0. COROLLARY 1. Let b and c be degrees such that b < c and c is recur­ sively enumerable in b. Then there exist degrees c0 and c, such that CoUci = c, c0I Cl, and such that for each i < 2, 6 < ct < c and Ci is re­ cursively enumerable in b. PROOF. Let B and C be sets of degrees b and c respectively, and let C be recursively enumerable in B. We set D equal to C and apply Theorem 1. Thus we obtain disjoint sets A and A such that C = A U A , and such that for each i < 2, C is not recursive in B, A> and A is recursively enumerable in B. For each i < 2, letdf
8 218 COROLLARY

GERALD E. SACKS

3. The set of all degrees less than 0' has no maximal

member. PROOF. Let 6 be a degree less than 0'. Then 0' is recursively enu­ merable in 6, and we apply Corollary 1 with c = 0' to obtain a degree c0 such that b < c0 < 0'. Shoenfield [10] proved that the set of all degrees less than 0' has no greatest member. COROLLARY 4. / / b, c and d are degrees such that b < c < d and d is recursively enumerable in b, then there exists a degree ft such that b< h
in

COROLLARY 7. If c is a degree which is recursively enumerable in some degree less than itself, then c is the least upper bound of the set of all degrees less than c. PROOF. Let c be recursively enumerable in b, where b < c, and apply Corollary 1. Then c = c0 U
9 DEGREES LESS THAN 0'

219

less than itself; however, it seems likely that such a b exists. (Note added in proof: We now can show there exist degrees 6, c, and d, each greater than 0 and less than 0', such that the only degrees less than 6 are c, d, and 0, and such that c and d are incomparable. Then 6 is the least upper bound of all degrees less than b, but by Corollaries 1 and 4, 6 is not recursively enumerable in any degree less than itself. This result will appear in [9].) COROLLARY 8. If c is a degree which is recursively enumerable in some degree less than itself, then there exists an infinite, ascending sequence of degrees b0 < b, < 62 < •■• •uch that b0 < b, < b, < ■• ■ ■ < and such that b0 < b^ < b, < ■ ■ ■ • < g < for no degree g. PROOF. Let c be recursively enumerable in b, where b < c. Let B = {d\b^d 0, let r(n) be the least k such that tk > *,.,„_„. Let 6„ = tr(n) for all n. Then b0 < b, < b2 < • ■ • < c. II there were a g such that b0 < &i < b2 < ■•• < g < c, then n would db ereater rhan nvery member of T, and T would not be maximal. In [11] Spector left open the following question: does there exist an infinite, ascending sequence of degrees with a minimal upper bound? Corollary 8 answers this question in the affirmative. In particular, there is an infinite, ascending sequence of degrees which has 0' as a minimal upper bound. From Corollary 1 of Theorem 3 of [6], we know that the infinite, ascending sequence 0 < 0' < 0" < • • • does noo have 0''" as a minimal upper bound; however, by [2] 0M is complete, and so by Corollary 8, there exists an infinite, ascending sequence of degrees which has 0("" as a minimal upper bound. (Note added in proof: We now can show that every countable set of degrees has a minimal upper bound. We can also show that the set of all arithmetical degrees has a minimal upper bound less than 0(
10 220

GERALD E. SACKS

In general it appears that a problem P requires the priority method for its solution S if and only if the nature of P is such that it severely re­ stricts the number of quantifiers that may appear in S. We discuss this matter further in [8], where, insofar as is possible, we isolate the priority method from recursion theory in order to expose its combinatorial nature. Following Kleene [5], we represent the sequence of natural numbers /(0), / ( l ) , • • • • / ( % - 1) by the sequence numbee

and we write Seq(a) when a is a sequence number, i.e., Seq(a) = a > 0 & (t}t 0) . We will need two recursive predicates defined in [11]: Cpt(o, 6) s ( i ^ w f t a ) , = 0 V (6)< = 0 V (a), = (6),) ) Ext(i>, a) = Seq(a) & Seq(6) & Cpt(a, b) & b ^ a . We will need an additional recursive predicate and a recursive function: In(6, a) = ~Ext(o, 6) & ~Ext(6, a) , Note that if f(n) is a function, then Cv(/(»)) = / ( » ) for all w. Let {a0, olf a2, • • •• }b e aequence eo fequence eumbers such htat tor rach i, Ext(o i+1 , o() & ai+l > a;. Then there is a unique _function / ( » ) with the property that for each t there is an n such that / ( » t = a,. If A is a set of sequence numbers such that {a0, olf a2, • • •• S A, ,hen we sayjthe function /(») is a member of the closure of A, or in symbols, / ( » ) e A. In [11] Spector defined a contracting sequence F p F ^ F p - • o sets of sequence numbers and then obtained a function of minimal degree as the unique member of D , " ^ - ^ e defined his sets one at a time; that is, he defined a function G such that for all n, Fn+1 = G(Fn). Our construction of a minimal degree has his as an ancestor, but it differs from his in that we define our sets of sequence numbers simultaneously. They nonetheless manage to have properties similar to those of his sets because of our use of the priority method. Possibly the most apparent difference between the two proofs is that each of his sets of sequence numbers is recursive, whereas each of ours is merely recursively enu­ merable. We define a recursive predicate which is central to our construction of a minimal degree less than 0':

11 DEGREES LESS THAN 0'

221

H(c, t, e, x, m, b, d) s (i)i<2{Ext{(x)it t) & Tx{c, (m),.0, (m)iA) & (x), = C/((m),,) & r!(Cv((a00, e, 6, (d)«) & (d)t g lh((a)t)) & tf((d)0) * t/((d),) . We define a partial recursive function: y(c, t, e) = fixH(c, t, e, (x\, (x)u (x)2, (»),) . It is important to observe that if Y(c, t, e) is defined, then We prove this observation with the help of GND.1 Let c, t and e be such that Y(c, t, e) is defined, and let y = Y(c, t, e). Then for each i < 2, we have ri(Cv((»U, e, (?/)2, MM) & (y%A ^ lh((y)M) . Suppose Ert((y)0.„ (?/)„.,). It follows from GND that ri(Cv((y)0.0), e, (y)„ (y)3a) , since (j/)3.1 g lh((y)sa) ^ lh((i/)M); this means that (y) w = (y)8iJ, since ri(Cv((y)0.,), e, (?/)2, 2) is true for at most one z. But £/((2/)3.„) * U((y)3J. Thus ~Ex(((3/)0.„ (!/)„.!), and similarly, ~Ex(((y) 0il , (y)0..)We define a recursively enumerable set of sequence numbers denoted by W(c, t, «): if Seq(t), then t e W(c, t, e); if u e W(c, t, e) and if r(c, u, e) is defined, then (Y(c, u, e))0it. e W(c, i, e) for all i < 2; the only members of W(c, i, e) are those given by the last two clauses. If Wis a recursively enumerable set, we say e is a Gbdel number of W if W is the range of the partial recursive function UiftzTfc, n, a)). Note that if F(c, i, e) is defined, then both (Y(c, t, e))0.„ and (F(c, t, e)) w are members of the unique recursively enumerable set which has c as a Gbdel number. It is clear that there exists an effective procedure P for obtaining a Gbdel number of W(c, t, e) from the triple (c, t, e). We define the recursive function V(c, t, e) to be that, function whose value for the triple (c, t, e) is equal to the result of applying procedure P to (c, t, e). 2. There exists a degree h such that 0 < h< 0' and such h for no degree d. PROOF. We will define three functions, u(s), v(s) and t(s), and two partial functions, v\ and Q(i, s), simultaneously by induction; neither of the partial functions will be defined when i > s. For each s, we will have Ext(w(* + 1),u(s))&u(s + 1) > u(s); our function of minimal degree will be the unique member of the closure of {«(«) | s ^ 0}. THEOREM

that 0
12 222

GERALD E. SACKS

Let «(0) = 2, and let g be a Gddel number of the recursively enu­ merable set of sequence numbers {t | Ext(t, u(0))}. We define a recursive function w(t): W(t)

if

= (° \Mln((x\,

(x),) & (»)t<1 Ext((aO„ t))

~SeqW

otherwise .

We set v(0) = w(tt(0)), i(0) = 0, «° = g and Q(0, 0) = 0. That completes stage 0 of our definition by induction. Before we proceed with stage s > 0, we make several assumptions concerning what took place at stage s - 1 of the definition: (Al) u(s - 1), v(s - 1) and t(s - 1) were defined; (A2) if h < t(s - 1), then vl'1 was defined; (A3) Q(i, s - 1) was defined if and only if i < s; (A4) In((tK» - 1))„, (v(s - 1)),); (A5) (i)i<2 Ext{(v(s - 1))(, «(s - 1)); (A6) t(s - 1) is less than or equal to the cardinality of the set {t | Q(i, s - 1) = 1}. Clearly these six assumptions are true if s = 1. Let 0(») be the function whose value is 0 for all n. From (A4) and (A5) we know that (v(s - 1))„ and (v(s - 1)\ are sequence numbers such that neither one extends the other. If {s}°(i) is defined for all i < v(s - 1), then the sequence number TT

pi+MHi)

must fail to extend at least one of the sequence numbers (v(s - 1))0 and (v(s - 1)X. With this last remark in mind,"we define u(s): (0

if (Ei)i
m(s)

1

- l^(~Ext(n i <. 8 _ 1 ) P-" °"', (Ks - l))t)) u(s) = (v(s - !))„,„ .

otherwise ,

It follows from (A4) and (A5) that Ext(u(s), u(s - 1)) & u(s) > u(s - 1). Let {t | Q(i, s - 1) = 1} U {s} = {i[, tj, * ■ •, i'rM}, where i\ < i\ < ■■■ < ij,„ = s. By (A6), t(s - 1 ) < r(«). We define i7j for all k ^ r(s): *

[vV 1 V(vU, u(a), ii)

if k g t(s - 1) if t(s - 1 ) < k g r(s) .

It follows from (A2) that v\ is well-defined for all k g r(s). Note that <<«, =

V(t*U»-l,

M(S), S).

Let i(s) be the greatest ft such that for all j < k, Y{v), u(s), i'j+1) is

13 223

DEGREES LESS THAN 0'

defined; clearly 0 ^ t(s) g r(s). We define v(s) and Q(i, s) for all i S s: t;(s) = ■

w(u(s))

if t(s) = 0

(Y(v'H^uU{s),i'lw))o ■Q(i, s - 1)

ift(s)>0;

1

Q(z, s)

if i < s) if £(s) = r(s) if

t

-

8)

G(», • - 1 ) 0

if ♦ < • : „ « if i = tj u ) + ,

) _ h if i(s) < r(s) .

1

if i;„)+1 < i ^ sj

That completes stage s. It is readily seen that (Al) - (A6) remain true when s + 1 is replaced by s throughout. For example, we have In((v(s))0l («(«)X) as a consequence either of the definition of w(t) or of the observation made immediately after the definition of Y(c, t, e). Note that Q(0, s) = 0 and V0 = g for all s. LEMMA 8. For each i there is an s' such that for all s>s*, Q(i, s - 1) = Q(i, s). PROOF. By induction on i. Let i ^ 0; suppose s** has the property that Q(j, s - 1) = QU, s) for all j < i and s>s**. Suppose, for the sake of a reductio ad absurdum, that there is no s* such that Q(i, s - 1) = Q(i, s) for all s>s*. Since Q(i, s) equals either 0 or 1 for all s s t, there must be an s' such that Q(i, s' - 1 ) = 0, Q(i, s') = 1 and s' > s**. It follows that ti;..,+1 < i < s'. But then Q(i;;,.)+1I s' - 1) = 1 and Q(i;;.-)+lf s') = 0; however, this is impossible since i't'{B,i+1 < i and s' > s**. For each i, let s(i) be the least s* such that Q(i, s - 1) = Q(i, s) for all s > s*. Thus Q(i, s) = Q(t, s(i)) = Iim.Q(», s) for all s ^ s(i). In addition, if Q(i, s(i) - 1) is defined, then Q(i, s(i) - 1) =£ Q(i, «(i)). Let s*(i) be the largest member of {s(i) | i ^ i).

9. If lim,Q(t, s) = 0, i/ien s(y) < s(i) /or all j < i. Suppose lim,Q(t, s) = 0, s(j) ^ s(i) and j < i. Since Q(i, «(i)) is defined, we have i S s(i). Then Q(j, s(j) - 1) is defined, since j ^ i - l S s(i) - 1 ^ s(i) - 1. It follows that »;{&,,„ S i < » , since LEMMA PROOF.

Q(j, s(i) - l) * 0 ( i , s ( i ) ) . But then Q(i, s(j)) = 1, which is impossible, since s(j) ^ s(t) and lim.Q(i, s) = 0. It is an immediate consequence of Lemma 9 that if lim.Q(i, s) = 0, then s*{i) = s(i) > s*(j) for all j < i. Let S = {i\ \im.Q(i, a) = 1}. We define a partial function ift by in­ duction:

14 224

GERALD E. SACKS

it

£ [ii(i G S)

ik+1 = fti(ieS&i>ik)

(k > 0 ) .

Thus if S is infinite, S = {ilr i2, i3, ■ ■ ■}, where t, < i, < ^ < •• -. At this point in our argument we do not know anything about the cardi­ nality of S; however, we will eventually see that ik is denned for all k >0. 10. If ik is defined and if s > s*(ik), then i\ = ik. Let ik be denned, and let s > s*(ik). Then Q{j, s - 1) is defined for all j ^ik, since s - 1 §: s*(is) ^ »*. But then Q(j, s - 1) = lim.Q(i, s) for all j g *„, since s > s*(ik) ^ s(j) for all i ^ ik. It follows that ij = %i for all 3 such that U i g i . LEMMA PROOF.

11. / / ik is defined and if s > s*(ik), then k ^ t(s). Let ik be defined and let s > s*(ik). Suppose t(s) < k. Then by Lemma 10, i;(I1+1 = im+1, since s > s*(ik) ^ 8*(i<(„+1). This means that Q(t„. )+ll s) = Q(i;„)+1, s) = 0. But Q(i((„+1, s) = 1, since s > s(itM+1). For each & such that ik is denned, let s'{ik) be the least s* such that fe ^ t(s) for all s 2: s*; by Lemma 11, «'(tt) ^ s*(i) + 1. It is immediate from the definition of v\ that v'k+1 = v\ for all s such that k ^ t(«). Thus if ik is defined, we have v'k = <«*> for all s ^ «'(**)• For each k such that t» is defined, let vk = <«*>. Recall that vi = v°0 = q for all s; let v„ = g. Observe that if ♦, is denned, then vk = V(v£P, u(s'(ik)), tj'«*>), since t{s'{ik) - 1) < k. LEMMA PROOF.

12. If ik is defined and if s ^ s'(Q, tfeen i\ = ik. Let ik be denned. First we show by induction on s that %% = *:'«»' for all s ^ s'(i t ). Let s be such that *i = i|'«*> and s 2 s'(i*). Then by definition of s'(ik), k g i(s). Recall that Q(i, s - 1) = Q(j, s) for all j < i\M. This means that (a) Q(j, s - 1) = Q(j, s) for all j < ti = ik'^ ^ »JW. Since fc ^ t(s), we have (b) Q(ii, s) = 1. It follows from (a) and (b) that ij+1 - tj for all j such that l g y g f c . It is clear that s'{ik) S «'(»,) for all j such that U j g i . Thus we have that ij = if**1 for all j and s such that s 2 s«(%) and 1 g / ^ k. It follows that Q(i, s'(ik)) = \im,Q(j, s) for all j ^ ij'W; consequently, i*/(t'*! = i, for all i such that U i g J s . LEMMA PROOF.

13. J/% is denned, Men vk = V(vk.u u(s'(ik)), ik). Let ik be defined. If k = 1, then *£$»' = <"■' = g = v0 = i>t_„ since vl = q for all s. If k > 1, then wj!"*' = <«•-'» = *;._„ since LEMMA PROOF.

15 DEGREES LESS THAN 0'

225

«'(**) ^ *'(»»-»). Thus we have established that = V{v'k'l\^, u(s'(ik)), i'kH^) = 7 ( I I W I u(s'(ik)), ih). Let (c)* be the unique recursively enumerable set which has c as a Gbdel number; thus (c)* is the range of the partial recursive function U(pLzTx{c, n, 2)). Note that (V(c, i, e))* = W(c, t, e). We list four notes which will be needed in Lemmas 14, 15, 17 and 18: (Nl) (V(c,t,e))* -{£}£(c)*; (N2) if ~Ext(m, t), then m $(V(c, t, e))*; (N3) if t, e (V(c, t, «))* and if t2 = (F(c, i,, e))0ii for some i < 2, then

t2e(V(c,t,e))*; (N4) if F(e, t, e) is denned, then (Y(c, t, e))oi>t and Ext(( Y(c, t, e))0,„ i) for all t < 2. Note (N4) is a consequence of the observation made immediately after the definition of Y(c, t, e). Suppose k and s are such that v\ is denned and k > 0. Then there [is a unique s* ^ s such that < = F « _ t , u(s*), if) for all s' such that s* ^ s' ^ s; the existence of s* is evident from the definition of Vk. For each k and s such that vl is defined and k > 0, we define y(fc, s) to be the unique s* described above. We define 2/(0, s) = 0 for all s. If «; is defined and if k> 0, then y(ft - 1, s) is defined and y(k - 1, s) g y(fc, s) ^ s. This last follows from the definition of *;. LEMMA

14. If v°k is defined and i/fcgl,

tfe« (v°k)* - {u(y(k, «))} S

PROOF. Let v'k be defined and k 2: 1. We know that i>'t = V(i^«, «(3/(fc, »)), il"-"). ^ follows from note (Nl) that (vl)* ~ {u(y(k, «))} g WV>)*. But »l«i" = i*Lf"> = vU, since y(fe - 1, s) ^ y(fe, s) g s. Thus we have that («i)* - {u(y(fc, »))} E M-,)*. We need only show that u(y(k - 1, s)) $ (vl)* - {u(y(k, «))}; if y(fc - 1, s) = irffc, a), there is nothing to prove. Suppose y(k - 1, s ) < 2/(fc, s). Recall that Ext(»(a), u(« - 1)) and u(s)>u(s-l) for all s > 0. This means that ~Ext(u(y(k1, «)), tt(»(fc, »))). It follows from note (N2) that u(y(k - 1, s)) g (vj)«.

15. 1/ ik is defined and if s ^ s'(ik), then u(s) e (vk)*. PROOF. Let i, be defined; then fc>0. We prove the lemma by induction on s. By Lemma 13, vk = V(vk-lr u(s'(ik)), ik); consequently, u(s'(ik)) e (vk)*. Suppose «(*) e {vk)\ where s £ s'{ik). Then from the definition of s'(ik), we have 0 < k £ t(s). It must then be that LEMMA

U(s

+ i) = (»oo)< = (rw,,,.,,

u(S), i«(.,))M

for some i < 2; furthermore, u(« + 1) € («{,„_,)*. In order to show

16 226

GERALD E. SACKS

u(s + 1) e (%)*, we consider two cases: (a) k = t{s); (b) k 1. Thus we have %(s + l) = (rK_ 1 ,u(s),i t )) 0 . i for some i < 2. It follows from raoie (N3) that M(S + 1) e (v*)*, since u(s)e(0*. Now suppose (b) is the case. We have that u(s + 1) e (Vti.^)*. We know u(s + 1) > u(y(t(s) - 1, s)), because s ^ i/(i(s) - 1, s). It follows that u(s + 1) e (^,,,-J* - {u(y(t(s) - 1, «))}. Let r = i(s) - 1 - k; then r ^ 0. We apply Lemma 14 r times in succession in order to obtain K . , - 0 * - {u(y(t(s) - 1, s))} E (vl)* - {u(y(k, s))} . Then u(s + 1) e 04)* = (vk)*, since s ^ s'(i t ). Let h(n) be the unique function which is a member of the closure of {u(0), u(l), u(2), ■■■}. Thus for all s, u(s) = h(\h(u(s))). By Lemma 15, if ik is defined, then h(n) is a member of the closure of (v*)*. We will show h{n) has minimal degree less than 0'. To see that h(n) is not recursive, it is merely necessary to recollect the definition of u(s) at stage s > 0. If s > 0, there is an i < v(s - 1) such that either {s}°(i) is not defined or {s}\i) is defined but not equal to h(i). Thus h{n) is not recursive with Gbdel number s in 0(n) for any s > 0, and consequently, h(n) is not re­ cursive. To see that the function u(s), and thereby the function h(n), has degree less than or equal to 0', we study stage s > 0 (the induction step) of our definition by induction and apply the method of Kleene and Post [6]. We first define

F1(s) = JIisspV^

,

F(S) = j I U w ^ + 1

if

(2F(s) = 2ulB)-3vM-5tls)-7FlM.IF2'8'

if

00 s

= 0,

.

Our simultaneous definition by induction of the functions u(s), v(s) and t(s) and the partial functions v\ and Q(i, s) can be regarded as a definition by induction of the function F(s). There exists a function L(n) such that F(0) = 2 2 -3"" 2 '-5°-7 20 .ir +I F(s) = L{F(s - 1))

(S > 0) .

17 227

DEGREES LESS THAN 0'

If L(n) has degree less than or equal to 0', then so do F(s) and u(s). It is not necessary for us to exhibit the function L(n) in order to indicate why its degree is less than or equal to 0'. It is sufficient to examine stage s >0. Let g be a Gbdel number of the partial recursive function Y(c, t, e); i.e., Y(c, t, e) s U(pyT3(g, c, t, e, y)). We claim that the function L(n) is recursive in the predicates {Ey)T\(0(y), s, i, y), (Ey)Tz(g, c, t, e, y). The first predicate is needed to determine which case holds in the definition of m(s) (s > 0). Thus m(s) = 0 if (but not only if) {Ei)^,^

~{Ey)T\${y),s,i,y)

.

The second predicate is needed in the definition of t{s) (s > 0). Thus t(s) = 0 if and only if ~(Ey)Ts{gfvl,u(s),H,y). Since both the first and the second predicates have degree less than or equal to 0', it follows that L(n), F{s) and h(ri) also have degree less than or equal to 0'. LEMMA 16. If e > 0, if limsQ(e, s) = 0, and if {e}"(b) is defined for all b, then the function {e}k(b) is recursive. PROOF. By Lemma 9, s(e) = s*(e) > s(j) for all j < e. This means that Q(j, s(e) - 1) = Q(j, s(e)) = lim.Q(i, s) for all j < e. Since Q(e, s(e) - 1) is either undefined or equal to 1, there must be a A; such that i'kw = e. Then Q(ip\ s(e) - 1) )s either undefined or equal to 1 and Q(iy\ s(e)) = 0. It follows that t(s(e)) + 1 = k and that Y(vk% «(«(«)), ij(") is undefined. We show that Vk'l{ = vk.,. If k = 1, then t#_1 = Q sup = v0 = % _ j . Suppose k k > 1; then ij is defined and is equal to i}« for all j such that i ^ j < k, since Q(j, s(e) - 1) = Q(j, s(e)) = limeQ(j, s) for all j <e = i'k[e); in par­ ticular, ij£J = w Recall lhat t'(iit_1) ^ ss(i,_,) + 11 Then s(e) = s*(e) = s*(^(c)) > B*{i£Q = «*(**-,) ^ «'(**_,) - 1 . Since s(e) ^ s'fe-J, we have »£:i = vk.,. Thus we have shown that Y(vk^, u(s(e)), e) is undefined. It is clear that u(s) 6 (tvO* for all s ^ s(e), since if k = 1, then t^-i = g, and since if fe > 1, then t*-, is defined, s(e) ^ s'(**-»), and Lemma 15 applies. We define a recursive predicate L(sc, 6):

L(a, 6) = Ext((a;)„ it(s(e))) & r ^

( 4 f c («)«)

& (»)„ = t/((«)w) & Ti(Cv((a;).), «. 6, (»),) & (x), g lh((a;)„) .

18 228

GERALD E. SACKS

We claim that for each b, U((fixL(x, b))3) is defined and is equal to {e}h(b). Since u(s) e (vk^)* for all s ^ s(e), and since {e}h(b) is defined for all b, it follows that (b)(Ex)(L(x, b) & {e}"(b) = U((x)3)) . Thus the partial recursive function U((ftxL(x, b))3) is defined for all b and hence is recursive. It remains only to see that U(([ixL(x, 6))3) = {e}h(b) for all b. If this last were not the case, then there would exist y, z and b such that L(y, b) & L{z, b) & U((y)3) * U{(z)3) . But then we would have H{vk^, u(s(e)), e, 2<">°-3,"°, 2,»)i-3")', b, 2<>"^3(z)3) , and Y{vk.„ u(s(e)), e) would be defined, which it is not. LEMMA 17. Ifue W(c, t,e)&we W(c, t,e)&w>u& Ext(w, u), then Y(c, u, e) is defined and Ext(w>, (Y(c, u, e))M) for some i < 2. PROOF. Since u e W(c, t, e), there must exist um uu---,un and ylt y„ • • •, yn such that u0 = t uk = (Y(c, uk_x, e))0,Vlc (0
It is of course possible that u = t, in which case we take n = 0. Similarly, there exist w„, wlt ■ ■ •, wm and *,, z2, ■ ■ •, zm such that w0 = t wk^(Y(c,wk-1,e))0,,1i wm = w .

(0
Let d be the largest k such that u, = Wj for all j ^ k. Clearly d^n. In addition, d < m by ?ioie (N4), since w > u. For the sake of a reductio ad absurdum, suppose d < n. Then % +1 = (Y(c, wd, e))0.v<(+l and wi+1 = (F(c, «;„, «))Mg+1. But yd+l # zd+1, since w„+, =£ w„+I. It follows from the observation made immediately after the definition of Y(c, t, e) that ln(ud+1, Wm). This means that ~Ext(w, u d+1 ), since Ext(w, wa+1). But then ~Ext(w, u), since Ext(u,ud+1). This last contradicts the hypothesis of the lemma, so it must be that d = n. Thus u = un = ud = wd< w. It follows with the help of note (N4) that Y(c, u, e) is defined and that Ext(w,(Y(c,u,e))0ld ).

19 DEGREES LESS THAN 0'

229

LEMMA 18. If lim, Q(e, s) = 1 and if {e}*(6) is defined for all b, then the function h(b) is recursive in the function {e}\b). PROOF. Let k be such that ik = e. By Lemma 15, u(s) e (vk)* for all s ^ s'(e). This last fact enables us to define a function p(r):

p(0) = u(s'(e)) p(r + 1) = fix(x e (vt)* &x> p(r) & Ext(z, p(r)) & (Es) Ext(tt(s), x)) . It is clear that for each r there is an n such that h(n) = p(r). Thus it is sufficient for us to show that the function p(r) is recursive in the function

mb). By Lemma 13, vk = V(vt.lt u(s'(e)), e). Thus for each r, p{r)e W(vk.lt u(s'{e)), e) & p(r + 1) e W(vt.u u(s'(e)), e) & p(r + 1) > p(r) & Ext(p(r + 1), p(r)). It follows from Lemma 17 that for each r, Y(vk^,p(r), e) is defined and Ext(p(r + 1), {Y{vk^, p(r), e)\.t) for some i < 2. For each r, let 2(r)

= Y{vk_u p(r), e) , t(r) = fiiExt{p(r + 1), (2(r))0,,) .

We now show for all r that (a) p{r + 1) = (z(r))0.((r); (b) t{r) = (ii(U{{z{r)\,{) = {e}"((z(r))2)). Let r be fixed. First we prove (a). By note (N3), (z(r))0.((r, 6 (vk)*, since p(r) e (v4)*. By wote (N4), (z(r))0Mr) > p(r) and Ext((z(r))0.t(r), p{r)). There is an s such that Ext(u(s), (z(r))0Mr)), since Ext(p(r + l j , (z(r))0.((r)), and since there is an s such that Ext(u(s), p(r + 1)). Clearly p(r + 1) ^ (z(r))0.1(r); (a) readily follows. From the definition of the partial recursive function Y(c, t, e) it is evident that Ti(Cv((z(r))0.4(r)), e, (z(r))2, (z(r))lMr)) & {z(r))3,tM ^ lh((z(r))0.((r)) . This last means that U((z(r))3.tlT)) = {e}h((z(r)),), since (z(r))0.t(r) = p ( r + 1) = h(n) for some n. But then (b) follows, since t/((z(r))3.0) * £7((z(?"))3.1). It is immediate from (a) and (b) that the following system of equations defines the function p(r) recursively from the function {e}h(b): p(0) = z{r) = t(r) = p(r + 1) =

u{s'(e)) , Y(vk_u p(r), e) , fii{U{(z(r))3A) = {e}"((z(r))2)) , (z(r))0,((r) .

We conclude from Lemmas 16 and 18 that for each e > 0, either {e}"(6)

20 230

GERALD E. SACKS

is not denned for some b, or the function {e}n(b) is secursivee ,o rhe functioo h(b) is recursive in the function {e}h(b). (It is now clear that i„ is denned for all k > 0, since the function h(b) is recursive in the function {e}n(b) for infinitely many e.) Thus h(b) has minimal degree less than 0', since 0' was shown to non-minimal by Kleene and Post in [6]. THEOREM 3. For each degree b, there is a degree csuch that b
Hb(c, t, e, x, m, b, d) = (i)«,(Ext((a;)i, t) & TifSftm),.,), c, (m),.„ (m),-.,) & (»), = U{tm)J) & Tl^biid),), Cvftafc), e, b, (d)t) & (d)( g lh((aO,)) & U((d)0) * U{(d)J ■ In this manner we obtain a function h(n) which is not recursive in the function b(n) and whose degree is less than or equal to b'. It has the property that for each e > 0, either {e)hk(n) is undefined for some n or the function {e}hh(n) is recursive in the function b(n) or the function h(n) is recursive in the functions {e}bk(n), b(n). The desired degree c is the least upper bound of the degrees of b(n) and h(n). We end with a word on the role of the priority method in the proof of Theorem 2. Let v(k, s) = v\ if v\ is defined, and let v(k, s) = 0 otherwise. Then for each k, limXfc, s) = l i m X = vk, since for each k, v\ is defined for all sufficiently large s. It is readily seen that the degree of the function v(k, s) is less than or equal to 0'. It follows that the degree of the function vk is less than or equal to 0". Spector [11] worked directly with a function resembling vk, and that is why the best upper bound he could supply for his minimal degree was 0". By working indirectly with the function v(k, s), we were able to obtain 0' as an upper bound. It is generally true of the priority method that one works with a convergent sequence of functions rather than with their limit in order to avoid an unwanted quantifier. INSTITUTE FOE ADVANCED STUDY

REFEEENCES

1. Richard M. FEIEDBERG, TWO recursively enumerable sets of incomparable degrees oj unsolvability, Proc. Nat. Acad. Sci U.S.A., 43 (1957), 236-238. 2. , A criterion for completeness oj degrees of unsolvability, J. Symb. Logic, 22 (1957), 159-160.

21 DEGREES LESS THAN 0' 3. 4. 5. 6. 7. 8. 9. 10. 11.

231

, Three theorems on recursive enumeraiion, J. Symb. Logic, 23 (1958), 309316. S. C. KLEENE, Introduction to Metamathematics, New York, Toronto, Amsterdam and Grbningen, 1952. , On the forms of the predicates in the theory of constructive ordinals (second paper), Amer. J. Math., 77 (1955), 405-428. and Emil L. POST, The upper semi-lattice of degrees of recursive unsolvability, Ann. of Math., 59 (1954), 379-407. A. A. MUCHNIK, Negative answer to the problem of reducibility of the theory of algorithms (in Russian), Dokl. Akad. Nauk SSSR, 108 (1956), 194-197. Gerald E. SACKS, On the priority method of Friedberg and Muchnik, in preparation. , Degrees of Unsolvability, Annals of Mathematics Studies, Princeton, forthcoming. J. R. SHOENFIELD, On degrees of unsolvability, Ann. of Math., 69 (1959), 644-653. Clifford SPECTOR, On degrees of recursive unsolvability, Ann. of Math., 64 (1956), 581592.

22

RECURSIVE ENUMERABILITY AND THE JUMP OPERATOR BY GERALD E. SACKS(')

By degree we mean degree of recursive unsolvability as denned by Kleene and Post in [4]. Following Shoenfield [7], we say a degree c is recursively enumerable in a degree b if there is a set of degree c which is the range of a function of degree less than or equal to b, and we call a degree recursively enumerable if it is re­ cursively enumerable in 0 (i.e., if it is the degree of a recursively enumerable set). The jump operator, which takes the degree d to the degree d' (the completion of d), was defined in [4] and has the following properties: if h is recursively enumerable in d, then h ^ d'; d' > d; and d' is recursively enumerable in d. In [4] a degree c is said to be complete if there exists a degree d such tl at d' = c. Friedberg [1] showed that a degree c is complete if and only if c ^ 0'. For any degree b, if b ^ d ^ b', then b ' ^ d ' g b and d' is recursively enu­ merable in b'. Shoenfield [7] proved that if b' ^ c ^ b" and c is recursively enumerable in b', then there is a degree d such that b g d g b' and d' = c. Thus the degrees which lie between b' and b" and are recursively enumerable in b' can be viewed as the completions of the degrees which lie between b and b'. He also showed there is a degree greater than b and less than b' which is not recursively enumerable in b. Our main result below is that the degrees which lie between b' and b" and are recursively enumerable in b' can be viewed as the completions of the degrees which lie between b and b' and are recursively enumerable in b. Our notation is that of [3]. THEOREM 1. Let a, b and c be degrees such that a ^ b , a ^ b ' ^ c and c is recursively enumerable in b'. Then there exists a degree d such that a $ d , b ^ d, d' = c and d is recursively enumerable in b.

Proof. We first prove the theorem when b = 0, and then indicate the changes needed when b > 0. Thus we have degrees a and c such that a > 0, a ^ 0' g c and c is recursively enumerable in 0', and we wish find a recursively enumerable degree d such that a ^ d and d' = c. L e t / be a function of degree less than or equal to 0' whose range is a set C of Presented to the Society, April 19, 1962; received by the editors May 4, 1962. (') The author is a National Science Foundation Postdoctoral Fellow. 223

23 224

G. E. SACKS

[August

degree c. Let c be the representing function of C. Let g be a recursive function whose range is a set J of degree 0'. Let j be the representing function of J. We define r0 if (Ek)k<s(g(k) = n), j(s,n)= i [1 otherwise. It is clear that j(s, n) is a recursive function, and that for each n, lims j(s,n) exists and is equal to j(n). Since/is recursive in j , there is a Godel number zx such that fin) = {z.Yin) =

U(wTl(J(y),Zl,n,y))

for all n. We define a recursive function/(s, n) of supreme importance to our argument; pi(s-n)!zun,y^

'utnyT\(j\

HK&yjySs1

.s + 1

l l 1 1 Pi

JZJJBJJI,

otherwise.

We claim that lim s /(s, n) exists and is equal to/(n) for all n. Our claim is a con­ sequence of the fact that/(n) = {z^Kn) and lims j(s, n) =j(n) for all n. Let a be an everywhere positive function of degree a, and let z2 be a Godel number such that {z2}\n) = a(«) for all n. We define U^yTl^Up{^\z2,n,y^ a{s,n)= ■ 1

otherwise.

The function a(s, n) is recursive; for each n, lims a(s, n) exists and is equal to a(n). A useful property of the Godel numbering devised by Kleene in [3] to arithmetize his formalism for recursive functions is: the Godel number of a deduction is greater than the intuitive counterpart of any formal numeral occurring in the deduction. We will denote this fact by GND. It follows from GND that a(s, n) = 1 whenever n^s. We define two recursive functions, t(s, n) and h(s, n), by means of an induction on s: t(s,n) = nmm<s(j(s,m)=n);

h(fl,n) = 0; h(s + 1, n) = h(s, n) + sg(| t(s + 1, n) - t(s, n) |). Recall that the bounded least number operator is defined in such a way that t(s, 77) = s if and only if there is no m < s such that/(s, m) = n.

24 1963]

RECURSIVE ENUMERABILITY AND THE JUMP OPERATOR

225

We now proceed to define four recursive functions, y(s,n,e), m(s,e), r(s,n,e) and d(s, n), simultaneously by induction on s. The function d(s, n) will be such that 0£d(s+,,n)g((s,n)gl for all s and n. Thus for each n, lira, d(s,n) will exist; furthermore, lims d(s,n) will be the representing function of a recursively enumerable set D. The degree of D will be the desired degree d. At stage s of the construction we put finitely or infinitely many natural numbers in D; our main objective is to see that c g d ' ; however, with the aid of a system of priorities, we exercise restraint when we add members to D in order to insure that a ^ d and d ' g c . Stage s = 0. We set y(0,n,e) = r(0,n,e) = 0, m(0,e) = e + 1 and d(0,n) = 1 for all n and e. Stage s > 0. We define y(s, n,e) for all n and e:

^ri(n^ (s ~ 1 ' i) ><w) .0

otherwise.

It follows from GND that y(s,n,e) = 0 whenever n^:s We define m(s,e) for all e; there are three mutually exclusive cases. Case 1. y(s, e, e) = 0- We set m{s, e) = e + 1. Case 2. y(s, e, e) > 0 and there is an n such that e
# y{s - 1,n,e)&a(s,n) * U(y(s,n,e)).

m(«,e) = ime<„[y(s, n, e) # y(s - 1, n,e) &a(s, n) * l/(y(j; n,
&a(s,t) ¥= U(y(s,t,e))~\.

Note that Case 3 of the definition of m{s, e), the least number operator is bounded. We define r(s,n,e) and d(s,pne) for all n and e by means of a simultaneous induction on e. Let e ^ 0 and suppose r(s,n,i) and d^p^) have been defined for all i < e and all «; we define r(s, n, e) and d(s, p") for all n as follows: 0 if (£i) (£m) (£t) [i < e gf gn &p™ < y(ss,,e) r(s,n,e)

&d(s,P7)*d(s-l,p?)l

=z:

.1

otherwise;

25 226

G. E. SACKS

rd(s-UpS d(s-l, d{S

if )^)

[August

n^h(s,e),

if (£0 (Em) [i ^ e&i g m < m(s,i)

P e) =

' "

& r(s, m, i) = 1 &p] < y(s, m, i)], ■0

otherwise.

We conclude the construction by setting d(s,n) = d(s-l1n) for all n not a power of a prime. It is readily verified by the method of [4] that each of the four functions just defined is recursive. Such a verification is possible for two reasons: each of the functions a(s, n) and h(s,n) is recursive; at stage s > 0, all quantifiers, as well as all applications of the least number operator, are bounded. For each n, let d(n) =

limsd(s,n);

it is clear that d(n) = 0 if and only if there is an s such that d(s, n) = 0. Thus d is the representing function of a recursively enumerable set. Let d be the degree of d. We list some remarks which will be needed in vital parts of the body of our argument: (Rl) (s)(e)[m(s,e)>e]; (R2) (s)(n)(e)[Ks,n,e) = 0-+ r(s,n + l,e) = 0 ] ; (R3) (s)(n)(e)[(y(s,n,e) = O&n > e) -> m(0, e) ^ n]. Let s be such that s > 0 and (n)(e)[(Xs - hn,e) = O&n >
and«>e.

Then fl(s,«) # U(y(s, n, e)), since a(s,n)^1 and 17(0) = 0. First we suppose n < m(s -lle). Then y(s- l,n,e) > 0 as a consequence of the inductton hypo­ thesis. But then either Case 1 or Case 2 of the definition of m(s,e) holds, and so m(s,e) ^ n. Now we suppose m(s - lle) ^ n. If eiiher Case 1 or Case 2 of the definition of m(s,e) holds, then m(s,e) g m ( s - l,e) g n by remark (RI). If Case 3 holds and n < 2m(s - 11 ,) + ss then m(s, e) g «. If Case 3 holds and n ^ 2m(s -l,e) + s, then m(s,e) ^ 2m(s -,,e) + sgn. (Note that if Case 3 holds and (0[e < t < 2m(s - 1, c) + s -• a(s, f) = U(^(5, r, e))], then m(s,e) = m(s - l,e) + s; this last is a consequence of the definition of the bounded least number operator.)

26 1963]

RECURSIVE ENUMERABILITY AND THE JUMP OPERATOR

227

We introduce two predicates: ^(e): if the set {m(s,e))s^0} is infinite, then there is an n ^ e such that ]im s y(s,n,e) either does not exist or is equal to 0. B(e): lim„ d(p") exists and is equal to 1 - c(e). We will prove (e)A(e) and (e)B(e) by means of a simultaneous induction on e. From (e)A(e) it will follow that a ^ d . From (e)B(e) it will follow that c ^ d'. Fix e* ^ 0 and suppose A(e) and 5(e) are true for all e < e*. We proceed to prove A(e*) and B(e*). LEMMA 1. Let y(s, n,e*) > 0 and m(s,e*) >n^e*. Let d(s,pT) = d(s - l,pT) for all i, m and t such that i<e*-gt^nand n p?
Since y(s,n,e*)>

0, we have pf-s~ui\e*,n,y\.

y{s,n,e*) = [iyT\\Y\

We suppose y(s + 1,n,e*)¥= y(s, n, e*) and then show there is an i, an m and a t such that i < e* g t£ n &jT < y(s, tfe*)& d(s,PT)

*dd"-\,p").

Since y(s + l,n,e*) # y(s,H,e*), there must be a 7 < y(s,n,e*) such that d(s,j)¥=d(s - 1,j). Recall that d(s,w) = d(s - l,w) for all w not a power of a prime. Thus there is an V and an m' such that

d(s,Pr.')*d(s-l,P?) and p? < y(s,n,e*). But then by the hypothesis of the lemma, e* g V. Thus we have e*gi'8ce*gn< m(s,e*) &pf < y(s,n,e*)&d(s,pT.') ± d(s l,flf). It follows from the definition of d(s,p?) that r(s,n,e*) = 0. But this last means the desired i, m and * exist. LEMMA 2. Let y{s,n,e*) > 0 and m(s,e*) >n>e*. foralli,mandtsuchthati<e*^t^nandp,;n>e*, the definition of m(s,e) that

Let d(s,p?) = d(s - l,pf) + l,e*)>n. Then

it follows from remark (R3) and Case 1 of y(s,t,e*)>0

for all t such that e*^fg

n. But then by Lemma 1, y(s,t,e*) =

y(s+l,t,e*)

for all * such that e* g t £ n. Suppose m(s + 1,e*) & n. Then m(s+1, e*) < m(s,e*),

27

228

G. E. SACKS

[August

and consequently, Case 2 of the definition of m(s + l,e*) holds. This means there is a t (namely, m(s + l,e*)) such that e*< t ^n&y(s,t,e*) LEMMA 3.

* y(s,+

l,t,e*).

A(e*).

Proof. By the hypothesis of our theorem the function a is nonrecursive. We suppose A(e*) is false and show a is recursive. Thus the set {m(s,e*)\s ^ 0} is infinite, and for each n ^ e*, lims y(s, n, e*) exists and is positive. Let R(n,s) denote the predicate m(s,e*) >n&(e)(m)(t)[(p?
<e*gt^n)

-»J(s-l,p™

) = dQ£)]1

We know B(e) is true for all e < e*. This means limm dip ?) exists for all e < e*. For each e < e*, let £(e) be such that (m)[m £ g(e) -* dfjtf) = d(p,»<«>)]. We define a recursive function z(n) as follows: first we require that z(n) = 1 for all n not a power of a prime; then we specify d(pl(e)) Z(pf) = . d(p?) 1

ife>e*&m^g(e), if e > e* & m
The predicate R(B,s) can now be rewritten as m(s,e*)>n&(e)(m)(/)

[fj»; < j;fo *, **)& e < e* g t g> n) -»i(s-l,p^)

= z(jJ«)].

It is clear that R(n,s) is recursive, since the functions m,y and z are recursive. Now we show (n)(Es)R(n,s). Fix n. Since Kms y(s,n,e*) exists for all n ^ e* there is a y such that y£y(j,f.e*) for all t and s such that e* ^ ! | n. Let s' be so large that

d(.-l,w)-d(w) for all s and w such that 5 ^ s' and w < y. Since the set {m(s,e*) \ s ^ 0} is infinite, there is an s ^ s' such that m(s,e*) > n. But then R(n,s). Let w(«) denote the recursive function fisR(n,s). Note that w(n + 1)^w(n) for all n.

28 1963]

RECURSIVE ENUMERABILITY AND THE JUMP OPERATOR

229

Next we prove y(w(n),n,e*) = limsy(s,w,e*) for all n > e*. Fix n > e*. We show by induction on s that y(w(n), n,e*) = y(s,n,e*) for all s ^ w(n). Let s be such that s ^ w(n) and y(w(n), e*) = y(s, n, e*) &R(n,s). Since m(s,e*) >n>e*, it follows from remark (R3) and Case 1 of the definition of m(s,e*) that y(s,t,e*) > 0 for all t such that e* ^ t£ n. By the definition of R(n,s), we have d(s-l,p") = d(p™) for all e, m and t such that e < e* ^ l ^ n and p™ < y(s,t,e*). Recall that tf d(s-1,w) = d(w) then d(s', w) = d(w) for all s' ^ 5. It follows from Lemma 1 that y(s,t,e*) =

y(s+l,t,e*)

for all t such that e* g i ^ n. It follows from Lemma 2 that m(s +

l,e*)>n.

But then Xw(n), «, e*) = j(s + 1,n, e*) & R(n, s + 1). Thus y{w(n),n,e*) = y(s,n, e*) for all s ^ w(n), andlim s y(s,n,e*) = y(w(n), n,e*). Finally, we show by means of a reductio ad absurdum that a(n)=U(y(w(n),n,e*)) for all n > e*. It will then follow that a is recursive, since w is recursive. Fix n > e* and suppose a(n) # £/<>0(n), n, e*)). Since y(w(n),n,e*) = lim s Xs,«,e*), and since a(n) = limsa(s, n), there is an s* such that for all s ^ s*, a{n) = a(s,n) =£ U(y(s»n,c*)) = U(y(w(n), n, e*)). Let s > s* and suppose m(s - l,e*) ^ m{s*,e*) + n + e* + 1. If either Case 1 or Case 2 of the of the definition of m(s, e*) holds, then m(s,e*) g max(e* + l,m(s - l,e*)) ^ m(s*,e*) + « + e* + 1. If Case 3 holds and n < 2m(s - 1, e*) + s, ,hen m{s,e*) ^ n. If Case 3 holds and 2m(s - l , e * ) + s^ n, ,hen m((se*) g n. Thus we have shown by inductton on s that «j(s,e*)gm(s*,e*) + n + e* + 1 for all s ^ s*. But this last is absurd, since the set {w(s, e*) \ s ^ 0} is infinite. For each e ^ 0, we say e is stable if for all n^e, lim.Xs,",*) exists and is positive. Note that if e is not the Godel number of a system of equations, then y(s,n,e) = 0 for all s and n, and consequently, e is not stable. It follows that there are infinitely many e which are not stable, since there are infinitely many e which are not Godel numbers of systems of equations. We define

29 230

G. E. SACKS

[August

e0 = ne(e is not stable); eJ+1

= ne (e > e} and e is not stable).

Thus e0 < ex < e2 < ••• ii s aisting oo fll lhe e which are eot ttablee For rach j ^ 0, let nj be the least n ^ e- such that lims y(s, n, e,) either does not exist or is equal to 0. The most important part of our argument is contained in Lemma 4. If the proof of our theorem is a heavy meal, then the proof of Lemma 4 is the main course; furthermore, it is there that the combinatorial flavor of our reasoning is strongest. LEMMA

4. For each k and v, there is ans^v

such that

(j)j
(s^v); M\_i 0] ]

S(f+1)

=

fis(Em)[_s^S(t)&.m
Clearly S(0) is well defined and greater than or equal to t;. Suppose t ^ 0 and S(0 is well defined and greater than or equal to v. Then M(t) < k, since we have supposed the lemma to be false. Thus y(S(t),nMit),eMit))>0 and lim s ^(s,n M(r) ,e M(r) ) does not exist or is equal to 0. Then there must be an s > S(t) such that y(s,nuw,eMm)

#

y(S(t),nmpeMm);

note that S(f) > 0, since ^(0, n, e) = 0 for all n and e; this means there is an s > S(t) and an m such that m < y{S{t),nuw,eMi,))&.d{s

- 1,m) #

d(S(t)-1,m).

Then S(t + 1) is well defined and greater than or equal to v. For each t ^ 0, let «(r) = fim[_d(S(t + 1), m) # d(S(t) - 1, m)].

30 1963]

RECURSIVE ENUMERABILITY AND THE JUMP OPERATOR

231

Now we show u(t) < u(t - 1) for all r > 0. Fix t > 0. Since we have u(t)
u(t - 1).

Since d(w) = 1 for all w not a power of a prime, there must exist i and m such that u(t - 1) = p?. Note that d(S(t), u(t - 1)) # d(S(t) - 1, i/(t - 1)); this last follows from the definitions of S(t) and u(t - 1)) Let e = emty, s = S(t) and n = n M ( ( ) . First we suppose i < e. This means j < e g n &d(s,pT) * d(s - \,P?)&<s,n,e)

= 1,

since M(t) < k. But then it follows from the definition of r{s, n,e) that y(s, n,e) g p™. Now we suppose i ^ e. This means e^i&e£n<

m{s,e)8cr{s, n,e) = l &d(s,p?) / d(s - l,p"),

since M(t) < k. But then it follows from the definition of d(s,pT) that y(s,n,e)^PT = u(i - 1). LEMMA 5. 7/c(e*) = 0, ffcen limndQ)^) exzsfs and is equal to 1. Proof. Let t be the least m such that/(m) = e*. Let s' be so large that s'>t and / ( s , m) = / ( m ) for all s and m such that s ^ s ' a n d m ^ f. Then f(s,e*) = t for all s ^ s', and consequently, ft(s,e*) = h(s\e*) for all s ^ s'. But then

for all s and n such that 5 > 0 and « ^ fc(s',e*), since /z(s,e*) ^ fc(s',e*) for all s ^ ' . . It follows that lim,d(s,^.) = l for all n^h(s',e*), since d(0,w) = l for all w. Then ^Cp".) = 1 for all n ^ h(s',e*), and limnd(pen.) = 1. LEMMA

6. / / c(e*) = 1, then limnd(p%) exists and is equal to 0.

Proof. First we show that the set {r(s,e*)|s ^ 0} is infinite. Suppose t(s, e*) ^ t for all s. Let s' be so large that s'> t a n d / ( s , m) = / ( m ) for all s and m such that s ^ s' and m ^ f. Then / ( s ' , *(*',**)) = e*, since t(s',e*) < s'. But f(s',t(s-,e*))=f(t(s',e*)), since f(s',e*)^f. But then/(t(s>*)) = e*; this last is impossible because C is the range o f / a n d c(e*) = 1. Since the set {t(s,e*)\s2 0} is infinite, it is clear that the set {h(s,e*))|^ 0} is infinite. By Lemma 3, we know A(e) holds for all e ^ e*. This means that if e ^ e* and e

31 232

G. E. SACKS

[August

is stable, then the set {m(s,e*)|sS 0} is finite. If e ^ e* and e is stable, let m(e) be the greatest member of {m(s,e*)\s ^ 0}; if e g e* and e is not stable, let m(e) = np where / is such that e = e,. If e < e* and e g m < m(e), then lim s y(s, m,e) exists. Let y be so large that y ^ y(jj, m , e)

for all s, m and e such that e g e* and e ^ m < m(e). Fix n > y. We show dQ£.) = 0. It will suffice to find an s such that d(p£, s) = 0. Let v be such that h(v,e*) > n. Let k be such that if e g e* and e is not stable, then e = e} for some j < fc. By Lemma 4 there is an s ^ i> such that (j)Jn;

(e) (in) [(e g e* & e g m < w(s, e)) - ►r(s, m, ee » 0 V p" ^ y(s, m, •))]. It will then follow from the definition of d(s,pne.) that: d(s,^.) = 0. We have h(s,e*) > n, since s 1 p and fc(s,e*) is a nondecreasing function of s. Fix e and m so that e g e* and e g m < m(s,e). Suppose e is stable. Then m < m(e), since m(s, e) g m(e). But then y ^ y(s, m,e), and consequently fe. ^ y(s, w, e), since n > y. Now suppose e is not stable. Then e = ep wherej < k, and m(e) = «,, If m < np then m<m(e) and p". £ y(s, m, e). Suppose m ^ ^,, Then m(s,e£>np This last means that either r(s,n,-,e) = 0 or y(s,n;,e) = 0. If r(s,npe) = 0, then by remark (R2), r(s, m, e) = 0, since m ^ n,. Suppose y{s, np e) = 0. Since nj g m < m(s, e), it follows from remark (R3) that rij = e. But then y(s, e, e) = 0, and Case 1 of the definition of m(s,e) holds. It follows that m(s,e) = e+1,m = e and y(s, m, e) = 0. Thus d(p"e.) = 0 for all n> y, and lim„dQ>en.) exists and is equal to 0. Lemmas 5 and 6 constitute a proof of B{e*). That concludes our proof by induction of(e)A{e) and (e)B(e). It is now easily seen that c g d'. Observe that

(e)(Et)[[m)mkt(d(p7) = D V(m) mg MpD = 0)] is an immediate consequence of (e)B(e). We define k{e) = ^([(m)mi,(d(P:)

= 1) V (n) B S ,(i(j£) = Q)].

The function fc has degree less than or equal to d', and by (e)B(e), c(e) = 1 -

d(pke(e))

for all e. LEMMA

7.

a^d.

Proof. We suppose there is a Godel number e such that

32 1963]

RECURSIVE ENUMERABILITY AND THE JUMP OPERATOR

233

a{ri) = {e} (n) for all n, and then show A(e) is false. First we show that limsy(s,n,e) is positive for all n ^ e. Fix n ^ e; let W

=

exists and

wT}(d(y),e,n,y).

Let s' be so large that d(s, m) = d{m) whenever s £ s' and m < w. Then y(s,n,e) = w&U(w) = a(n) for all s ^ s' + w; w > 0, since 0 is not the Godel number of a deduction. Now we show the set {m(s, e) \ s ^ 0} is infinite. We fix m > t and obtain an s' such that m(s',e) > m. Let s be so large that 5 > m and a(s,t) = a(t) = {ef(t) = U(y(s, t,e)) for all t such that e^t^m. If m(s - l,e) > m, then s - 1 1i sth eesired d's Suppose m(s - l,e) ^ m. This means fl(S,0=t/(Xs,r,e))

for all t such that e^f^ m(s - l,e); in addition, Xs,e,e) > 0, since a(s,e) ^ 1 and £7(0) = 0. But then Case 3 of the definition of m(s,e) holds. It follows that m(s, e)>m, since s>m. LEMMA 8. d' ^ c.

Proof. We will define two functions, E{e,n) and L(e), simultaneously by induction on e so that each is recursive in the function c(n). We will combine the definition of £ and L with a proof by induction on e of the following: (e)(n) {£(e,«)

= 0 <- [n ^ e &(w)(Es)(s> w&m(s,e)> >« &(m)(» ^ m

(e)(m)[m ^ L(e) -

^e^{E^)T\{d(y.,e,mjy))1};

W

d(pD = <*0£ )]-

It follows immediately from the above and remark (Rl) that (e)[E(e,e) =

0~(Ey)T\(d{y),e,e,y)-];

but then if £ is recursive in c, we have d ' ^ c . Fix e ^ 0. Our induction hypothesis has two parts: (1, e) for each i < e and each n, E(i,n) has been defined and E(i,n) = 0<-+[n ^ i&(w))£s))( >w&m(s,i)

> n«

&(m)(fi ^ m ^ i -»(£>)T(Jd(y),i,m,y))]; (2, e) for each i < e, L(i) has been defined and (m) [m ^ L(z) -»d(p?) = dQ>,lw)].

33

234

G. E. SACKS

[August

We proceed to define E(e,n) for all n, verify (1, e + 1), define L(e) and verify (2, e + 1). Let (1, e + 1, n) denote the following predicate: for each t < n, E(e, t) has been defined and E(e, t) = 0 ~ [ t £ e & (w) (£s) (s > w & m(s, e) > t) &(m)(t ^ m ^ e ^ C E ^ r K ^ e . m , > - ) ) ] . To verify (1, e + 1), it suffices to prove (1, e + 1, n) for all n. We define E(e,n) and prove (1, e + 1, n) for all « by means of an induction on n. First we set E(e, t) = 1 for all t < e. Then it is clear that (1, e + 1, r) holds for all t g e. Now we fix n ^ e and suppose (1, e + 1, n) holds. We define £(e,n) and then prove (1, e + 1, n + 1). The definition of £(e, n) has two cases. Case 1. (Et) (e^t m ^ e —*

(Ey)T\(3(y),e,m,y)).

For each m such that n > m ^ e, let

Let y* be the largest member of {y(m) \ n > m ^ e} U {0}. Let s* =

AU[(0(J

< y* -»<*0 - 1 , 0 - <*(0) &s > >*].

Recall that for each s > 0 and i, if d(s - 1, i) = d(i), then d(*'>0 = d(i) for all s' ^ s. It follows from the definition of y(s,m,e) and the fact that 0 is not the Godel number of a deduction that (sHm)[(s ^s*&n>m^e)->

y(s, m, e) = y{m) > 0].

We define '0{

if (Es){y(s,n,e)>0&n<

m(s,e)&s>

s* &

(0i<«[( m )( m < W) -* d(s — l,pT) = d(p™)) &{tn)(y{s,n,e)>

E(e,n) —

m ^ L(i)-*d(s

l,pl )

= d(pfw))]}. L1

otherwise.

To verify (1, e + 1, n + l),it suffices to prove £(e,n) =

0~[ii£e&(w)(£s)(s>w&in(s,e)>ii) &(m)(n ^ m ^

e-(Ej»)rJ(*y).e,m,y))].

Suppose £(e, n) = 0. Then Case 2 of the definition of E(e, n) must hold. Let s be the natural number whose existence is required by the fact E(e, n) = 0; thus

34 1963]

RECURSIVE ENUMERABILITY AND THE JUMP OPERATOR

y(s, n,e)>0&n<

m(s, e)&s>

235

s*.

It is our aim now to prove y(s,n,e) = y(s',n,e)&n<

m(s',e)

for all s' ^ s by means of an induction on s'. Fix s' > s and suppose y(s,n,e) = y(s' -l,n,e)&n<

m(s' -

l,e).

Suppose for the sake of a reductio ad absurdum that there is a w such that d(s' - 1, w) # d(s' - 2, w) )an w < y(s' - l,n,e). Then there eusu be aa n and an m such that (0) and (1) are true: (0)

d($'-l,pT)

# d(s'-2,p?)&PT
(1)

(i%
- l,pf)

= y(s,n,e);

= d(s' - 2,pT')WpT' ^ y(s' -

l,n,e)].

If j < e, then it follows from the second half of (0), the definition of 5 and (2, e) that d(s - l,p7)=d(p?); but this last contradicts the first half of (0), since s'> s. Thus i ^ e. Since e ^ n < m(s' - l,e), ii follows from (0) and the definition of d(s' - \,p?) that r{s' - l,n,e) = 0. This last means there is an i', an m' and a / such that i' < e g f £ n & / # ' < y(*' - l,f,e) &d(s' - l,p?') * d(s' - 2 , ^ ' ) . If t = n, this last contradicts (1), since V < e ^ i. Thus t < n, and

since $' > s> s* and e g t < n. But this is absurd, since d(s' - UpT')= d(s' - 2lPf)

= d(s - l,p?) = d(p?')

is a consequence of the fact that s' > s> s* and p$' < y*. Since d(s' - l,w) = d(s'- 2,w) for all w < y(ss-lln,e)) and s i n c e X s ' - l , H , e ) = y ( s , n , e ) > 0 , it must be that y(s', n,e) = y(s'-1,«,

e) = y(s, n, e).

Then we have (m)[n ^m^e-+

y(s', n, e) = y(s' - 1, n, e) > 0],

since s' > s > s*. It follows that either Case 2 or Case 3 of the definition of m(s', e) holds. If Case 2 of the definition of m(s',e) holds, then it is clear n < m(s',e). If Case 3 holds, then n < m(s',e) because n < m(s' - l,e). Thus we have shown that y(s', n,e) = y(s, n,e)>0&n<

m(s\ e)

for all s' ^ s. It follows immediately that (Ey)T\(d(y), e, n, y) and

35 236

G. E. SACKS

[August

(w)(Es)(s>w&m(s,e)>n). Note that (m) (n > m = e - (EjOritfOO, e, m, y)) is a consequence of the fact that Case 2 of the definition of E(e,n) holds. That completes the first half of the verification of (1, e + 1, n + 1); in order to verify the second half, we suppose (w) (Es) (s>w& m(s, e) > n) & (m) (n = m = e -+ (Ey)T\(3(y), e, m, y)), and then show E(e, n) = 0. It follows from (1, e + 1, n) that Case 2 of the definition of E(e,n) holds. Let v be so large that v > L(i) for all i < e. Let z = iiyT\{d{y), e, n, v). Let w be so large that w> z and

d(w-l,t)

= d(t)

for all f < pl + v. Let s be such that s> w + s* and m(s,e) > n. It follows easily from (2, e) that s has the properties required to conclude E(e,n) = 0; note that y(s, n, e) = z > 0. The definition of L(e) has two cases: Case 1. c(e) = 0. Then by B(e), \immd(p") = 1. We set L(e) = nt(s)(m)[m

=

f -+ d(s,p?) = 1].

Case 2. c(e) = 1. It is a consequence of (1, e + 1) and of the definition of y(s, n, i) that for each i ^ e and each n, E(i, n) = 0 ^ [(£s)(m(s, i) > n) & lim.y(s, n,f) exists and is positive]. For each i g e, it follows from A(i) that there is a f such that t = i and £(i, 0 = 1. For each i = e, let t, = fit(E(i,t) = l&t^i). It follows from (1, e + 1) that for each i = e, tt satisfies either (2) or (3): (2) (£w) )5) (s > w -+ m(s, i) g tt);

(3) lim^Cs, tt, i) does not exist or is equal to 0. Note that since £(i, 0 = 0 whenever tt > t = i, it follows from (1, e + 1) that ( m ) ( t , > B i £ i - » l i m ) < s , i n , 0 exists and is positive). But then (Ey) (0 (m) (s)[(i = e & i, > m = i & s = 0) -► y(s, m, i) ^ y]; let L(e) be the least such y. We now verify (2, e + 1). What follows is similar to the proof of Lemma 6. Fix » ^ L(e). We must show d(>") = limmd(>™). If Case 1 of the definition of L(e) holds, there is nothing to prove. Suppose Case 2 of the definition of L(e) holds. Then c(e) = 1, and by B{e), limmdO™) = 0. In order to

36 1963]

RECURSIVE ENUMERABILITY AND THE JUMP OPERATOR

237

show d(p"e) = 0, it suffices to find an s such that d(s,p$ = 0. Let k be such that if i ^ e and i is not stable, then i = es for some j < k. Let w be so large that for all i£e,if (2) holds, then (s)(s>w-+m(s,i)^ti). By the same argument as in Lemma 6, there is a v > w such that (s)(s^t>->/j(s,ee>n). By Lemma 4, there is an s ^ p such that U)j i) = 0 V rf ^ J, i). Suppose (2) holds. Then m(s, i) g r„ since s jg f > w. But then (( > m 5 i, and consequently, P:>n^L(e)^y(s,m,i).

Now suppose (3) holds. Then i is not stable, and there is a j < k such that e,- = i; in addition, n; = tt If w < n; = r„ then pj ^ v(s, ™> 0. since n ^ L(e). Suppose m ^ «j. Then m(«, i) > R,-. This last means either r(s,nj,ej) = 0

or

j>(s, n,-, e_,) = 0.

If Ks,«/.0 = 0, then by remark (R2), r(s,m,i) = 0, since m^nr Suppose yfort^i) = 0. Since n, g m < m(s,0, it follows from remark (R3) that rij = i. But then v(s, i, i) = 0, m(s, t) = i + I, w = i and y(s, m, ,) = 0 < />". Inspection of the definitions of E and L readily reveals they are recursive in c. We make some informal remarks to indicate how to write equations defining E and L recursively in c. Fix e £ 0 and n^e, and consider the definition of E(e, n). The choice between Case 1 and Case 2 can be made effectively once the values of E(e,t)(t< n) are known. Suppose Case 2 holds. The values of v* and s* are defined by means of a predicate recursive in d. Then the value of £(e, n) is found from the values of d(p'") (m ^ L(i) and i < e) by means of a predicate of degree less than or equal to 0'. Thus the value of E(e, n) can be expressed in terms of the values of E(e, t)(t < n) and L(i') (i < e) with the aid of a predicate of degree less than or equal to d U 0 ' . Similarly, the value of L(e) can be expressed in terms of the values of E(i n) (i ^ c and n ^ 0) with the aid of a predicate of degree c u 0'. Then d' ^ c since d'is7he degree ofthe function £(& e) and since d ^~J 0' ^J c = c When b > 0, the changes needed in the above argument are largely notational. The notion of recursiveness is replaced throughout by the notion of recursiveness in a function of degree b. The arguments contained in Lemmas 1-8 are retained

37

238

G. E. SACKS

[August

unaltered save for relativization to a function of degree b. The functions a(s,n), f(s,)) and d(s,n) are now recursive in a function of degree b. We set d(s,2■ 3") = b(n) for all s and all n > 0, where b has degree b, in order to insure b ^; d. The value of E(e, n) is now obtained from the values of E(e, t) (t < n) and L(i)(i < e) with the aid of a predicate of degree less than or equal t o b ' U d . COROLLARY 1. If b and c are degrees, then the following conditions are equivalent: (i) b' ^ c g b* and c is recursively enumerable in b ' ; (ii) there is a d such that b g d ^ b' and d' = c; (iii) there is a d such that b g d ^ b', d is recursively enumerable in b and d' = c.

For each degree b, let Rb denote the set of all degrees greater than or equal to b, recursively enumerable in b and less than or equal to b'. L e t j denote the jump operator. Then Corollary 1 tells us that the order-preserving map j:Rb

-* Rb-

is onto. It also follows from Theorem 1 that any element of Rb. greater than b' is the image of more than one element of Rb; Friedberg(result unpublished)has shown that b ' does not have a unique pre-image in Rb. We do not know if Rb and Rb. are order-isomorphic, but we conjecture that they are. We can show (announced in [6] for b = 0): for any degree b, Rb is a universal, countable partial ordering. COROLLARY 2. There exists a recursively enumerable degree d such that d < 0 ' < 0" = d'.

Proof. Let b = 0, c = 0" and a = 0', and apply Theorem 1 to obtain d. Then d is recursively enumerable, 0 $ d and d' = 0". Note that Corollary 2 provides still another solution to Post's problem. COROLLARY 3. For each degree b and each natural number n, there is a degree d recursively enumerable in b such that

b < d < b' < d' < b" < ••• < b
and

hj-,.-x.

Let d = h„. Then d is recursively enumerable in b, and for all i ^ n,

38 1963]

RECURSIVE ENUMERABILITY A N D THE J U M P OPERATOR

239

Corollary 3 improves a result of Shoenfield [7]; he showed that for each degree b, there is a degree d such that b < d < b' < d < b". We do not know if for any degree b there exists a degree d such that for all n ^ 0, b(n)< d(n)< -(n i));

if for some b, such a d exists, then by Theorem 1, d can be given the additional property of recursive enumerability in b. Theorem 1 can be extended without any radical alteration of its proof. For example, we can show: if g is a recursively enumerable degree such that g' < 0", then there is a recursively enumerable degree d such that g < d < 0' < 0" = d'. We say a sequence a 0 ,a t ,a 2 , ••• of degrees is simultaneously recursively enumer­ able if there is a sequence A0,AUA2,~of simultaneously recursively enumerable sets such that a, is the degree of Ai for all i. Using the method underlying the proof of Theorem 1, we can show: if a 0 < a t < a 2 < ••• is an infinite, ascending sequence of simultaneously recursively enumerable degrees, then there exists a recursively enumerable degree d such that a 0 < aj < a 2 < ••• < d < 0'. We end with a conjecture: the upper semi-lattice of recursively enumerable degrees is dense (i.e., if b and c are recursively enumerable degrees such that b < c, then there exists a recursively enumerable degree d such that b < d < c). The only evidence we have to offer in favor of this conjecture is contained in the results we announced above and the result of Muchnik [5] that there is no minimal, nonzero, recursively enumerable degree(2). REFERENCES 1. R. M. Friedberg, A criterion for completeness of degrees of unsolvability, J. Symbolic Logic 22 (1957), 159-160. 2. , Two recursively enumerable sets of incomparable degrees of unsolvability, Proc. Nat. Acad.^Sci. U.S.A. 43 (1957), 236-238. 3. S. C. Kleene, Introduction to metamathematics, Van Nostrand, New York, 1952. 4. S. C. Kleene a n d E . L. Post, The upper semi-lattice of degrees of recursive unsolvability, Ann. of Math. (2) 59 (1954), 379^07. 5. A. A. Muchnik, Negative answer to the problem of reducibility of algorithms, Dokl. Akad. Nauk SSSR 108 (1956), 194-197. (Russian) 6. Gerald E. Sacks, The universality of the recursively enumerable degrees, Abstract No. 60T-14, Notices Amer. Math. Soc. 7 (1960), 996. 7. J. R. Shoenfield, On degrees of unsolvability, Ann. of Math. (2) 69 (1959), 644-653. 8. Clifford Spector, On degrees of recursive unsolvability, Ann. of Math. (2) 64 (1956), 581-592INSTITUTE FOR ADVANCED STUDY, PRINCETON, N E W JERSEY

(2) Added in proof. Our conjucture is true; we will give a proof elsewhere.

39

T h e Recursively Enumerable Degrees are Dense* By GERALD E. SACKS Our main result is: if b and c are recursively enumerable degrees such that b < c, then there exists a recursively enumerable degree d such that b < d < c. By degree we mean degree of recursive unsolvability as denned by Kleene and Post [2]. A degree is called recursively enumerable if it is the degree of a recursively enumerable set. The upper semi-lattice of recursively enumerable degrees has a least member, 0, the degree of all recursive sets, and a greatest member, 0', the degree of all complete sets. Post [4] asked: does there exist a recursively enumerable degree d such that 0 < d < 0'? Friedberg [1] and Muchnik [3] answered Post's question in the affirmative. Sacks [5] showed that every countable, partially ordered set is imbeddable in the upper semi-lattice of recursively enumerable degrees. Muchnik [3] an­ nounced that if c is a non-zero, recursively enumerable degree, then there exists a recursively enumerable degree d such that 0 < d < c. The arguments of Friedberg [1], Muchnik [3] and Sacks [5] have a great deal in common. All three authors make use of a method which may be roughly described as follows (a precise abstract description is given in Sacks [5]): One or more sets are to be recursively enumerated. Certain requirements are to be met. A typical requirement is: the set B is not recursive in the set A with Gbdel number e. Unfortunately, the requirements tend to conflict. Thus we may wish to add n to B to insure that B # {e}A, but the addiiton of n to B may ruin what we did earlier in the enumeration to insure A =£ {/}*. We resolve all conflicts by appealing to a system of priorities assigned to the requirements before the enumeration begins. Then it can be shown that each requirement is "ruined" only finitely often, and that, consequently, each re­ quirement is eventually met. In the argument below we assign priorities to requirements, but we are forced to permit each requirement to be "ruined" infinitely often. Nonethe­ less, we are still able to meet each requirement. Thus there is an important combinatorial difference between the method of Friedberg [1] and Muchnik [3] and the method we use below. The difference is a consequence of the fact t h a t Friedberg and Muchnik deal only with finite, initial segments of func­ tions while we are forced to deal indirectly with entire functions. Almost all * This * research was

40 RECURSIVELY ENUMERABLE DEGREES

301

of the complications we encounter below arise from the non-recursiveness of B. We are given a non-recursive, recursively enumerable set B, and we wish to find a recursively enumerable set D such that B is recursive in D but D is not recursive in B. Since B is to be recursive in D, we are forced to add many members to D in complete disregard of all priorities. Thus even the require­ ment of highest priority may be "ruined" infinitely often. If B were recursive, we could put all of B in D in one step, and then the arguments of [1] and [3] would suffice. The basic combinatorial principle of this paper appears in primitive form in Sacks [6]. Our basic source of notation is [2]. We conclude with intuitive remarks and an open problem. If b and c are recursively enumerable degrees such that b < c, then there exists a recursively enumerable degree d such that b < d < c. PROOF. L e t / , g respectively, be a one-one recursive function whose range is a set B, C respectively, of degree b, c respectively. We define: THEOREM.

,

(0

if (Et)t<s(f(t)

(1

otherwise ;

= n)

, (2 if (Et)t<Jg(t) = n) v v c(s, n) = ' (1 otherwise . Then lim, b(s, n) exists for each n; in addition, lim, b(s, n) is the representing function of B. Similarly, 2 - lim, c(s, n) is the representing function of C. Let k(s) be a recursive function such that, for each s, s < k(s) < k(s + 1). We define three recursive functions: yb(s, n, e) =

v

3
otherwise-

h(0,n,e) = 0 ; S ° Ms + 1, n, e) = 0(s, n, e) + sg(\ yb(s + 1, n, e) - yds, n, e) \) ; We define four recursive functions, y(s, n, e), m(s, e), r(s, n, e) and d(s, n), simultaneously by induction on s. The desired degree d will be the degree of the predicate {Es)(d{s, n) = 0). Stage 8 = 0. We set y(0, n, e) = m(0, e) = 0 and r(0, n, e) = d(0, n) = 1 for all n and e, except that d(0, p(/(0), 1)) = 0. Stage s > 0. We set y(s,n,e)

__ {PyTl(U}
^°-"J,

e, n, y)

if {Ey)v<,Tl(\l]
pf*-> «, e, n, y)

41 302

GERALD E. SACKS

The definition of m(s, e) has two cases: Case m l . There exists an n < m(s - 1, e) such t h a t c(s, n) ± u(y(s, n, e)) & y(s, n, e) * y(s - 1, », e) . We set m(s, e) equal to the least such n. Case m2. Otherwise. We set m(s, e) equal to the least n such that ra(s - 1, e) < n < 2m(s - 1, e) + s & (Et)ts„(c(s, t)) * v(y(s, t, e)) . If no such n exists, then m(s, e) = m(s - 1, e) + s; this last is in accord with the definition of the bounded, least number operator. Let p(i, m) = $+m and d(e, i, m) = p(e, p(i, m)). We eefine e(s, n, e) and d(s, p(e, i, m)) for all e, n, i and m by a simultaneous induction on e. First we set d(s, p(f(s), 1)) = 0.

r($, n, e)

(0 j

if {Ei)i<e{Et)lin{Em)[p{i, m) < y(s, t, e) & d(s, p(i, m)) =fc d(s - 1, p(i, m))] ,

J0 (l

if (^)1£„b(/(s), otherwise .

1) < y(s, t, e)] ,

The definition of d(s, p(e, i, m)) has three cases. Case d l . t(s, i, e) > m. We set (d(s - 1, p(e, i, m))

if (Eu)uie(Ev)l<w,.,ul[r(s,

d{s, p(e, i, m)) =

v,w) = 1

& p(e, i, m) < y(s, v, u)] , (o

otherwise .

Case d2. t(s, i, e) = m. We set /0

if c(s, ») = 2 &

d(s, p(e, t,m))=•

(j)j
& ~ (JBM)u£e(JEv)„<m,„u)[r(s, v,») = 1 & p(e, i, m) < »(«, t», u)] , ,d{$ - 1, P(e, i, m)) otherwise .

Case d3. t(s, i, e) < m. We set d(s, p(e, i, m)) = d(s - 1, p(e, i, m))) The construction is concluded by setting d(s, w) = d(s - 1, w) for all w not equal to p(f(s), 1) or p(e, i, m) for some e, t and m. Let d(n) = lim, d(s, w) for each n. Let Z> be the set whose representing function is d; D is recursively enumerable since neD if and only if (Es)(d(s, n) = 0). We list two notes which will be needed later: (Nl) (s)(n)(e)(r(s, n, e) = 0 — r(s, » + 1,e) = 0);

42 RECURSIVELY ENUMERABLE DEGREES

303

(N2) (s)(n)(e){y(s, n, e) = 0 - n ^ m(s, e)). We prove (N2) by induction on s. Certainly, {n)(e){y(0, n, e) = 0 - n ^ m(0, e)) . Fix s > 0, n and e. Suppose v(s, n, e) = 0, and (n)(e)(v(s - 1, n, e) = 0 — n Jg m(* - 1, e)e . Suppose Case ml of the definition of m(s, e) holds. If y(s - 1, w, e) = 0, then n^m(s1, e) > m(s, e)) Suppose y(s - 1, », e) > 0. Then y(s, n, e) =jfc y(s - 1, %, e) & 0 = u(y(s, », e)) qt e(a, ») > 0 ; Consequently, n g m(s, e). Now suppose Case m2 of the definition of m(s, e) holds. Once again, v(y(s, n, e)) =£ c(s, n); it follows n ^ ra(s, e). LEMMA 1. Lei y(s, n, e) > 0, m(s, e) > » and p(f(s),

1) ^ y(s, £, e) /or

all

t g » . Let <2(s, p(i, m)) = d(s - 1, p(t, m)) for all i, t and m suc/i tfeot i < e, t^n andp(i, m)n. PROOF. Since y(s, n, e) > 0, we have y(s, n, e) = j u y l Y d l ^ , Pp-1*", e, » , y) . If d(s, j) = d(s - 1, 3) for all i < y(s, n, The hypothesis of our lemma tells us that from the definition of d(s, j) t h a t d(s, j) since m(s, e) > n. Note (Nl) makes clear that the above Thus we have (t)tin{y(s,

e), then y(s + 1, «, e) = |/(s» », e). r(s, n, e) = 1. But then it follows = d(s - 1, j) for all j < y(s, n, e), argument also works for any t < n.

t, e) = y(s + 1, t, e)) .

Suppose m(s + 1, e) ^ w. Then m(s + 1, e) < m(s, e), and Case m l of the definition of m(s + 1, e) holds. But then we have a t (namely, m(s + 1, e)) such that t ^ n, and y(s, t, e) ^ y(s + 1, t, e). For each e, we say e is stable if lim„ y(s, n, e) exists and is positive for all ii. Since there are infinitely many e which are not Gbdel numbers of systems of equations, there are infinitely many non-stable e. Let {e0 < ex < e2 < • • •• be the set of all non-stable e. For each j , let % be the least n such that lim, y(s, n, e3) either does not exist or is zero. Lemma 2 is our basic com­ binatorial principle (cf. [6, Lemma 4]). LEMMA

2. For each k and v, there is an s g v such that

(i)i<*[«*(e, e}) ^ »i V »■(*, »/, «i) = 0 V !/(«, n„ •,) = 0] . PROOF.

Fix A: and v. We suppose there is no s with the desired propeertes

43 304

GERALD E. SACKS

and then define an infinite descending sequence of natural numbers. We propose the following system of equations as a means of defining two functions, S(t) and M(t), simultaneously by induction: S{0) = v ■ M(t) = ftji 0] ; S(t + 1) = fi8(Em)[s ^ S(t) & TO < y(S(t), nMU), emt)) & d(s, TO) * d{S(t) - 1, m)] . Clearly S(0) ^ v. Suppose t ^ 0, S(t) is well-defined and S(t) 5: v. M(t) < k, since we have supposed the lemma to be false. Thus

Then

y(S(t), nM(„, emt)) > 0 , and lim. y(s, nxm, eMW) does not exist or does equal 0. Then there must be an s > S{t) such that TO < y{S(t), nMil), eMW) & d(s - 1,TO)gfe d(S(t) - 1,TO). But then S(* + 1) is well-defined and S(t + 1 ) § 1 ) . For each t ^ 0, let u(t + 1) = fim[d(S(t + 1),TO)# d(S(tj - 1,TO)]. Fix « > 0. We show u(t + 1 ) < u(t). Since we know «(t + 1 ) < y(S(t), nMm, emt)) , it suffices to show y(S(t), nmtu

emt)) ^ u(t) .

It follows from the definition of S that d(w,

TO)

= d(S(t - 1) - 1,

TO)

whenever S(t) > w ^ S(t - 1) and TO < y(S(t - 1), nmt~», eMil^). quently,

Conse­

d(S(t), u{t)) # d(S(t) - 1, u(t)) . First suppose u(t) = p(f(S(t)), 1). Then y(S(t), »*,,,, a,,,,) ^ u(«), since r(,S(£), TO*,,,, e,,,,) = 1. Now suppose u(t) = p(i,f, TO) for some i,f and TO. Let s = S(t), n = nMW and e = e*,,,. If i < e, then the definition of r tells us that y(s, n, e) g u(t), since r(s, n, e) = 1. If i ^ e, then the definition of d tells us that y(s, n, e) 5S u{t), since w
44 RECURSIVELY ENUMERABLE DEGREES

305

We introduce two predicates: A{e): if e is stable, then the set {m(s, e) | s ^ 0} is finite. B(e): the set {m | d(p?) = 0} is recursive in B. We will prove (e)A(e) and (e)B(e) by means of a simultaneous induction on e. From (e)A(e) it will follow that c £ d. From (e)JS(e) it will follow that d ^ b. LEMMA 3. (i)i<eB(i) — A(e). PROOF. We know c | b . We suppose that B(i) is true for each i < e and that A(e) is false, and we show c ^ b. Thus the set {m(s, e) | s ^ 0} is infinite, and for eachTO,lira, y(s, n, e) exists and is positive. Let R(n, s) denote the predicate

m(s, e)>n & (i)(m)(t)[p(i, m) < y(s, t, e) & i < e & t ^ n - d(s - 1, p(i, m)) = d{p(i, m))] & (z)(£)[z eB & p(z, 1) < y(s, t,e) & t ^ n - d(s - 1, p(z, 1)) = 0] . (Recall that ze B if and only if d(p(z, 1)) = 0.) Since B(i) is true for all i < e, it follows that R is recursive in B. Since the set {m(s, e) | s ^ 0} is infinite, and since lims y(s,TO,e) exists and is positive for each TO, it follows that (n){Es)R{n, s). Let w(n) = fxsR(n, s) . Then w is recursive in B, and for eachTO,w(n + 1) ^ w(n). Fix TO. We show Km, y(s, TO, e) = 2/(W(TO),TO,e). Let s be such that s ^ and 2/(W(TO), TO, e) = y(s, n, e) & R(n,

W(TO)

s) .

Since m(s, e) > n, it follows from note (N2) that y(s, £, e) > 0 for all t ^ TO. We know / enumerates B without repetitions. It follows from the definition of R that p(f(s), 1) ^ i/(s, t, e) for all t g TO. But then Lemma 1 tells us that y(s + 1, t, e) = y(s, t, e) for all t ^ n, and that m(s + 1, e) > TO. Thus y{w(w), w, e) = y(s + 1, n, e) It follows that lim, y(s, n, e) = Finally, we show

I/(W(TO), TO,

and

R(«, s + 1) .

e).

lim, c(s, n) = U(y(w(n),TO,e)) for all n. If this last is true, then C is recursive in B, since 2 - lim. c(s,TO)is the representing function of C; and since w is recursive in B. Fix TO and

45 306

GERALD E. SACKS

suppose lim, c(s, re) * v(y(w(n), n, «)). There must be an s* such that for all s ^ s*, c(s, re) = lim, e(a, re) * v{y(w(n), re, »)) = v(y(s, re, e)) . Fix s > s* and suppose m(s - 1, e) S m(fl*, e) + » . If Case m l of the defini­ tion of m(s, e) holds, then m(s, e) ^ m(s*, e) + re. If Case m2 holds and ■re < 2m(s - 1, e) + s, then m(s, e) ^ m(s*, e) + », since c(s, re) ^ u(y(s, re, e)). If re ^ 2m(s - 1, e) + s, then m(s, e) g re. Thus w(s, e) ^ m(s*, e) + re for all s ^ s*. This last is impossible, since the set {ra(s, e) | s ^ 0} is infinite. Suppose {e}\n) is denned for all re. Then lim, yb(s, re, e) exists for each re, and lim, *(*, i, e) exists for each i. In addition, lim, t(«, i, e) (regarded as a function of i) is recursive in 5 . All this is clear from the definition of yb and t. LEMMA 4. 1 / {e}\n) is defined for all re and (u)uSeA(u), then d(p(e, i, lim, t(s, i, e))) = 0 for only finitely many i. PROOF. We suppose the lemma is false and show C is recursive in B. Our first claim is:

d = {e}b .

Fix j . Let s be so large that d{j) = d(w - 1, j) & v(yb(w, j , e)) = {e}b(j) for all w^s. Let t(i) = lim, £(s, i, e) for all i. Let to and i be such that i < t, to ^ s, c(w, i) = 2, 0 = d(w, p(e, i, t(*))) * d(w _ 1, p(e, i, t(i))) , and i(i) = t(w, i, e). It follows from Case d2 of the definition of d that d(w - 1, J) = U(lfc(8, J , 6)) ,

since j < i. But then d(j) = {e}<>(j), since w ^ s, and our first claim is proved. Our second claim is: for all sufficiently large i, d{p{e, i, t(i))) = lim, c(s, i) . Our first and second claims, together with the fact that t is recursive in B, imply that C is recursive in B. Our second claim is a consequence of Lemma 2 and (w)„£,A(w). If u ^ e and w is stable, then A(u) tells us that the set {m(s, re) | s ^ 0} is finite. If re £ e and u is stable, let m(re) be the greatest member of {m(s, u) | s 2 0}. If re g e and re is non-stable, then re = e, for some i; let m(re) = n{. Recall that ref is the least witness to the fact that e< is nonstable. Thus lim, re(s, B, M) exists if re ^ e and v < m(re). Let y be so large that y(s, v,u)^y if s ^ 0. re g e and v < rei(re). Fix i S w. We show

46 RECURSIVELY ENUMERABLE DEGREES

307

d(p(e, i, t{i))) = lira, c(s, i). If lira, c(s, i) = 1, then c(s, i) = 1 for all s, and d(s, p(e, i, t(i))) = 1 for all s. Suppose lim, c(s, i) = 2. Let w be so large that c(s, i) = 2, i(s, t, e) = i(t), and (3)j
r(-, n„ M) = 0 V ¥<*, *«, w) = 0 .

We show d(s, p(e, t, t(i))) = 0. Case d2 is such that we need only show ~(Eu)uie(Ev)v<m{S:U)[r(s,

v, u) = 1 & p{e, i, t(i)) < y(s, v, u)] .

Fix u g. e and v < m(s, u). Suppose u is stable. Then y(s, v,u)gygig p (e, i, t{i)). Suppose u is non-stable. Let u = e„ If v < %, then v < m(u) and 2/(s, «, B ) ^ ^ p(e, i, *(t)). Suppose v ^ nt. Then m(s, it) > »«, and conse­ quently, r(s, n„ u) = 0V y(s, »„ u) = 0 . If r(s, n<( w) = 0, then r(s, t>, M) = 0 by -Note (Nl), since nt ^ u. Suppose y(s, ni} u) = 0. Then n, ^ m(s, w) by JVoie (N2). But m(s, u) > nf since we have supposed v > nt. LEMMA 5. (u)ui,A(M) — 5(e). PROOF.

(L)

Our first claim is (L).

(Ey)(m)(x)(i)[m

< i(x, i, e) & p(e, i, m) ^ y — d(p(e, i, m)) = 0] .

The proof of our first claim is virtually identical with the proof of the second claim of Lemma 4. For each u ^ e, we define m(u) as in Lemma 4. Then lim, y(s, v, e) exists if u ^ e and v < m{u). Define y as in Lemma 4. Fix m, x and i so that m < i(£, », e) and p(e, i, m) ^ ?/. We show d(p(e, i, ra)) = 0. By Lemma 2 there is an s ^ a such that for all u g e, if u is non-stable (hence equal to e{ for some i), then m(s, et) g n, V r(», »,, e;) = 0 V *(s, n„ e;) = 0 . We show d(s, p(e, i, ro)) = 0. Since s ^; .r, we have m < t(x, i, e) ^ £(s, i, e) . Thus the definition of d(s, p(e, i, m)) is given by Case d l . We need only show ~(Eu)uSe(Ev)v<mU,ut[r(s,

v, u) = 1 & p(e, i, m) < y(s, «, M)] .

Fix u g e and v < m(s, w). If M is stable, then y(s, v, u) g y ^ p(e, i, m). Suppose M is non-stable. Then we repeat the argument at the end of Lemma 4

47 808

GERALD E. SACKS

to show r(s, v, u) = 0 or p(e, i, m) ^ y(s, v, u). We are ready to prove B(e). We will use only Lemma 4 and (L). First we suppose that {e}"(n) is undefined for some n; let N be the least such n. It follows from the definition of y„ that lim, t(s, t, e) = w for all t ^ JV. Let v have the property described by (L). Then (i)(m)[i ^N Ifi
& p(e, i, m) ^ y ^ d{p(e, i, m)) = 0] .

then lim, t(s, i, e) is finite; consequently, the set {p(e, t, m) | d{p(e, i, m)) = 0 & i < N}

is finite by Case d3 of the definition of d. Now d(p'.) = 0 only if p." = p(e, i, m) for some t and m or p*. = p(f(s), 1). It follows that the set {w \ d(p:) = 0} is recursive, hence recursive in B. Now suppose {e}b(n) is denned for all n. Then lim, t(s, i, e) is finite for all i. Let lim, t(s, i, e) = t(i). Recall that t(s + 1, i, e) ^ t(«, », e) for all s and i. It follows from (L) that (i)(m)[m < t(i) & p(e, i, m) g 2/ - d(p(e, », m)) = 00 ] By Lemma 4, the set {p(e, i, t(i)) ! d(p{e, i, t(i))) = 0} is finite. By Case d3, d(p(e, i, m)) = 1 if m > t(t). Since t{i) is recursive in B, it follows that the set {n | «f(pr) = 0} is recursive in £ . Lemmas 3 and 5 constitute a proof of (e)A(e) and (e)B{e). LEMMA 6. C is not recursive in D.

Suppose lim, c(s, n) = {e}a(n) for all n. Then e is stable, and by A(e), the set {m(s, e) \ s ^ 0} has a greatest member; call it M. Let w be so large that PROOF.

lim, c(s, n) = c(s, n) = u(»(a, n, «)) when s ^ w and % ^ M. lis^w, then nither (a) c(s, n) ^ u(y(s, n, e)) for some n g «(S, e), or (b) m(s, e) = m(S - 1, e) + s. If (a) holds, then M < m(s, e). Thus (b) must hold for infinitely many s. But then M < m(s, e) for some s. LEMMA 7. D is not recursive l

in B.

PROOF. Suppose d(n) = {e} (n) for all n, where b(n) = lim, 6(8, n) for all n. Then lim, t(s, i, e) is finite for each i. Let t(i) = lim, i(s, i, e). By Lemma 4, there is an n such that d(p(e, i, t(i))) = 1 for each i ^ n. We shall find an i such that i ^ u and d(p(e, i, t(i))) = 0. We proceed as in Lemma 4. Define y

48 RECURSIVELY ENUMERABLE DEGREES

309

as in Lemma 4. Fix i ^ y + n. Define w and s as in Lemma 4. Then the final argument of Lemma 4 tells us that d(s, p{e, i, t{i))) = 0. Let d be the degree of D. It follows from Lemmas 6 and 7 that c ^ d and d S b. We have b S d, since d(p(m, 1)) = 0 if and only if meB. It remains only to show that D is recursive in C. We give an intuitive description of a procedure E for computing D from C. Our description is such that the trans­ lation of E into a system of equations which define D recursively in C is not difficult. We need a recursive predicate Q(u, v, s, e, i, m): u S e & v < m{s, u) & r(s, v, u) = 1 & p(e, i, m) < y(s, v, u) . We also need a predicate R(s, e, n): (u)(i)(m)(t)[u <e & t£n & p(u, i, m) < y(s, t, e) - d{s - 1, p(u, i, « ) ) = =(p(u, i, m))] & (t)(i)[ie5

& tSB

& p(i, 1) < y(s, t, e)

- d(8 - 1, p(i, 1)) = d(p(i, 1))] . We need the next two lemmas to describe E. LEMMA 8. Q{u, v, s, e, i, m) & R(s, u, v) -> (w)„£,Q(u, v, w, e, i, m).

We proceed with an induction on w g s. Fix w ^ s and suppose Q(M, v, w, e, i, m) and B(w, w, v) hold. Recall that / , the recursive function whose range is B, is one-one. It follows from Lemma 1 that v < m(w + 1, u) and that PROOF.

y(w, t, u) = y(w + 1, £, u) for all i g v. But then R(w + 1, u, t>) holds and r(w + 1, v, u) = 1, and con­ sequently, Q(u, v, w + 1, e, i, m). LEMMA 9. (w)(Es)aSJu)(v)[Q(w,

v, s, e, i, m) -

R(s, u, v)}.

Fix w, e, i and m. For each u g e, define m(%) as in the second half of the proof of Lemma 4. Define y as in Lemma 4. Thus y(s, v,u)^y if s ^; 0, « g e and v < m(w). It is safe to suppose that to is so large that PROOF.

(s)(v)(s >w &v£y

— d(s-l,v)

= d(v)) .

By Lemma 1 there is an s > w such that for any non-stable u g e, we have m(s, u) :£ n, V r(s, n „ u) = 0 V »(s, % , u) = 0 ,

where u = »,; recall that if u is non-stable, then m(u) = nt. Fix « and v and suppose Q(u, «, s, e, t, m) holds. We show R(s, u, v). First we suppose u is stable. Then v < m(s, u) g m(u), since Q(«, v, s, e, i, m) holds; consequently, (n)(t)[t gv

& n< y(s, i, M) — d(s - 1, n) = d(n)] ,

49 310

GERALD B. SACKS

since s > w. But then R(s, u, v) holds. Now we suppose u is non-stable. Let u = e,. Then m(u) = n(i). If m(s, u) ^ m(tt), then v < m(w), and as above, fl(s, tt, «) holds. Thus r(s, m{u), M) = 0 V V(«. m(tt), u) = 0 . If r(s, m(u), u) = 0, then v < m(«), since r(s, v, u) = 1; this last follows from (Nl). Suppose y{s, m(u), u) = 0. Then it follows from (N2) that m(u) £ m{s, u). But then v < m(w). We return to the description of procedure E. Fix z. We compute d(z) with the help of C. If z is not of the form p(b, 1) or p(e, i, m), then d(z) = 1. If 2 = p(6, 1), then d(z) = 0 if and only if 6 6 B. Suppose 2 = p(e, i, m). The predicate (JS?s)(t(s, i, e) = i) is recursive in B, since t(8, i, e) g i(s + 1, %, e) for all s. We consider three cases: (a) (s)(t(s, i, e)< m); (b) (^s)(*(s, i, e) > m); (c) lim, t(s, i, e) = m. With the help of B, we can decide which one of (a), (b) and (c) holds; note that the monotonicity of t(s, i, e), regarded as a function of s, is vital. If (a) holds, then Case d3 tells us that d(z) = 1. Suppose (b) holds. Let w* be such that t(s, i, e) > m for all s ^ w*. We can determine w* with the help of B. Suppose d(s, z) = 1 for all s < w'. Then d(z) = 0 if and only if (Es)(u)(v)[s ^w*

& ~Q{u, v, s, e, i, m)] ,

since Case d l applies when s ^ w*. By Lemma 9, there is an s ^ w* such that (u)(v)[Q(u, v, s, e, i, m) — B(s, ,t, v)] . We can find s if we know the value of d(x) for some of the following x: x = p(u, j , k), where u < e; x = p(j, 1), where j e B. (Recall that d(p{j, 1)) = 0 if and only if j e B.) If ~Q(w, v, s, e, t, m) holds for all u and t>, then d(z) = 0; we only have to check u g, e and « < m(s, w). Suppose for some u and v, Q(u, v, s, e, i, m) holds. Then we have R(s, u, v) by definition of s. By Lemma 8, it follows that (wU.QK

v, w, e, i, m) .

But then d(z) = 0 if and only if d(w, z) = 0 for some w < s. Finally, we suppose (c) holds. Let w' be such that t(s, i, e) = m for all s S w'. We can determine w' with the help of B. Suppose d(s, z) = 1 for all s < w/. Then d(z) = 0 if and only if

50 RECUSIVELY ENUMERABLE DEGREES

311

(Es)[s ^ W' & c{s, i) = 2 & (u)(v) ~ Q(u, v, s, e, i, m) & tih«{d(s - 1, j) = U{yb(s, j , e)))] , since Case d2 applies when s ^ w \ If i < i, then 3/»(«, i, e) - lim, m(s, i, e) for all s ^ w', since t(w', i, e) = lim, t(s, i, e) = m. Let v* 2: w' be such that for all s^v*, c(s, i) = lim, c(s, i) & (i)J<;[d(s - 1, j) = d(i)] . We can determine v* with the help of C and the values of d(x) for x < z, since j < i implies that j < p(e, i, m). If lim, c(s, i) = 1 or d(i) ^ u(lim, yt(s, i, e)) for some i < i, then d(z) = 1. Suppose this last hypothesis is false. Suppose also that d(s, z) = 1 for all s; that is we set d(s vie i Us i e))) = 0. If for some w > s, t(w, i, e) > t(s, i, e), then Case dl or d2 may give us a chance of setting d(w, p{e, i, 1 + t(s, i, e») = 0; if this last happens, we have unplanted i. We have the opportunity to unplant i if and only if lim, Us, i e) > t(s i, e). This happens if and only if {e}b(i) is undefined or

51 312

GERALD E. SACKS

unequal to yb(s, i, e). If for some i, {e}\i) is undefined, there is no need to plant any member of C in D. If {e}b(i) is defined for all i, then Lemma 4 tells us that we do not permanently plant infinitely much of C in D. Of course the proof of Lemma 4 turns on the fact that C is not recursive in B. There are still many open questions concerning the recursively enumerable degrees; however, it may be possible to say something about the elementary theory of recursively enumerable degrees if the following question is answered: if b and c are non-zero, recursively enumerable degrees such that b < c, does there exist a recursively enumerable degree d such that d < c and d U b = c? In a talk at the Berkeley Model Theory Symposium (1963), J. R. Shoenfield conjectured that the answer is yes. We agree with him, but we are unable to prove it. INSTITUTE FOR D E F E N S E ANALYSES, PRINCETON CORNELL UNIVERSITY

REFERENCES

1. RICHARD M. FRIEDBERG, TWO recursiveyy enumerabee sets of incomparable degrees of unsolvability, Proc. Nat. Acad. Sci., U.S.A., 43 (1957), 236-238. 2. S. C. KLEENE and EMIL L. POST, The upper semi-lattice of degrees of recursive unsolvability, Ann. of Math., 59 (1954), 379-407. 3. A. A. MUCHNIK, Negative answer to the problem of reductibility of the theory of algorithms (in Russian), Dokl. Akad. Nauk SSSR, 108 (1956), 194-197. 4. EMIL L. POST, Recursively enumerable sets of positive integers and their decision problems, Bull. Amer. Math. S o c , 50 (1944), 284-316. 5. GERALD E. SACKS, Degrees of Unsolvability, Annals of Mathematics Studies Number 55, Princeton, 1963. 6. , Recursive enumerability and the jump operator, Trans. Amer. Math. Soc, in press. (Received August 6, 1963)

52

A SIMPLE SET WHICH IS NOT EFFECTIVELY SIMPLE GERALD E. SACKS1

For each e, let/„ be the partial recursive function UinyTxie, n, y)), and let We be the range of /,. Then W0, Wu Wt, • • ■ ii she Kllene enumeration of the recursively enumerable sets. Post [5] calls a recursively enumerable set simple if its complement is infinite but does not contain any infinite, recursively enumerable set. Raymond Smullyan calls a recursively enumerable set W effectively simple if its complement is infinite, and if there is a partial recursive function / such that for each e, if We is contained in the complement of W, then/(e) is defined and is greater than the cardinality of W,.1 Clearly, an effectively simple set is simple. The simple set 5 constructed by Post in [5] is effectively simple. This latter is no accident. In fact it is not unreasonable to claim that any direct attack on the problem of constructing a simple set must result in an effectively simple set. Our purpose here is to obtain a simple set which is not effectively simple. We will make strong use of the recursion theorem of Kleene [2j; however, we will use it in the informal manner of Myhill [4]. Our notation is that of [2]. We introduce a recursive function E: £(0) = nxTi((x)o, (x)h (*)»); E(s + 1) = nx\x > E(s) & Ti((x),, (x)h

(x)2)].

We will need E to simultaneously enumerate all the recursively enumerable sets in a fashion suitable for the proving of our theorem. It is a peculiarity of our proof that we cannot rely merely on the usual properties associated with any standard enumeration of the recursively enumerable sets; instead, we are forced to specify a par­ ticular enumeration. For each e and 5 we define a finite set Wt: for each m, m<E.Wse if and only if for some t g s , m = Z7((E(0)j) & e = (JEW)o. Then for each e, W?QW}QW?Q

• • • , and We = l) { W'e\ s^O}.

We

Presented to the Society January 24, 1963; received by the editors October 25, 1962. 1 The preparation of this paper was partially supported by N.S.F. Grant GP-124. 2 Smullyan actually requires that / be recursive, but it is easy to show the two definitions equivalent.

51

53 52

G. E. SACKS

[February

say 5 defines/.(w) \(e=(E(s))0 and n= (£(s))i. If 5 defines/,(»), then /.<») = £/((£(s)) 2 ). For each e and n, let 5(e, « ) ^ M *

(J defines/.(»)).

5 is partial recursive, and S(e, ») is defined if and only if /.(«) is defined. Let 0 denote the empty set. It is clear there exists a recursive func­ tion g such that for each e, i and z, we have

( {2*-3' | £(5(e,z)) < i ^ E(S(e,z)) +/.(«)} 10

if/.(s) isdefined,

otherwise.

The recursion theorem tells us that there exists a recursive function z such that for each e and t, we have Wz(e,i)

= Jr,(,,,',»(e,i))

' {2*-3*l-E(S(«,«(«,«"))) < * S £($(*,•(«,*)))+/,(i(«,*))} if/.(«(e,0) is defined, .0

otherwise.

We note some properties of z: (1) if /.(*(«, »)) is defined, then/.(«(«, 0 ) is equal to the cardinality of

W.(.,M

(2) if «Vj, then Wtleii)nW«.ti) = 0; (3) if/.(«(e, t)) is defined, then for all n, fff'W)n^,,o = 0; (4) if «Vj and both JMe, i)) and /.(*(•, j)) are defined, then z(e, i)*z(e,j). To prove (3), let * - $ ( « , *(e, ♦)) and let meWnnW,l.,t). Then m g £ ( s ) , since m = £7((£(*)),) for some * ^ 5, and since E is an increas­ ing function. (Recall that U(x)^x for all x.) But m>E(s), since m = 2i-3' for some t>E(s). THEOREM

1. There exists a simple set which is not effectively simple.

PROOF. We will define a sequence A, B, Qt, Q,, Q,, • • • of simul­ taneously recursively enumerable sets. A will be simple, but not effec­ tively simple. B will be such that if eEB, then W.r\A*0. We will see to it that if We is infinite, then eGB. Each Q, will be finite and will contain a set that will serve as a witness to the fact t h a t / , does not effectively bound the cardinalities of the finite subsets of the complement of A. Stage 5 = 0. We set X° = .B0 = Q? = 0for all *.

54 1964]

A SIMPLE SET WHICH IS NOT EFFECTIVELY SIMPLE

53

Stage 5 > 0 . Let e=(£(*))0 and n = (£(*))i. Thus 5 defines/,(»). We perform the following two operations in the indicated order: (a) We set QJ = 0 " 1 for all j>e. If there is no * such that i^e and n = z(e, i), we set QJ = QJ-1. If there is such an *, then by (4) it is unique. In addition, S(e, z(e, i)) is defined and WzU,{) = {2--3' | £(5(e, «(«, 0)) < t ^ E(S(e, z(e, »*))) + /.(«(«, 0 ) 1 WesetG-QrUWW,. (b) If e G 5 - » or if there is no m such that

then we set B' = B'~l and A' = A<~\ If e ^ ' " 1 and there is an m with the above property, let i be the least one. We set B* = B>-l\j\e\ and

A-= A'-^li}. Let 4 = U U ' | 5 ^ 0 } and £ = U{£«|5^0J. Since £ and z are re­ cursive, it follows A is recursively enumerable. For each e, let Qc = U{
Q.= V[W.ttM\

»^),

since
1. 7/ We is infinite, then

AC\We^Q.

PROOF. We know Q, is finite for every j . Let m be a member of W which is greater than every member of Q,- for all j£e. Let 5 be such that mGWl. First we suppose e£B*~\ Then there must be a / < s such that eQB'-1 and eEB'. At stage / we must have performed operation (b) in such a manner that Bt = B'-i\j{e) and A'^A''1 1 U { t | , where t g ^ . Now we suppose e^-B*- . We have

But then operation (b) at stage 5 forces us to put a member of Wt \nA>. LEMMA

2. IfmEWl-Q,

then m^Qi-

PROOF. Suppose for the sake of a reductio ad absurdum that fflGWI-g and w £ & . Since <2, = U{W. l i t i ) \ i£j], there must be an i^j such that mEWlU,i}. Since W,y,{, is nonempty, /y( 2 (j, *)) is defined. Let t**S(j, «(/'. *))• Then ' defines/ y (z(j, *}). and conse­ quently,

55 54

G. E. SACKS

[February

since t ^ j . Since «$
have

wGw:nw.ttM*o. But this last contradicts (3). LEMMA 3. If mEQil^A, then there exists an s and an e such that (E(s))0 = e
mEW'tk(j)js,

{mEQ'i).

It follows from Lemma 2 that (j)is. (mEQ,). mGQi.

But then e
4. TTse set QiC\A has at most i members.

LEMMA

PROOF. Suppose m and n are distinct members of Qtr\A. Lemma 3 guarantees the existence of s(m), e(m), s(n) and e{n) with properties as stated in the conclusion of Lemma 3. Thus

[m] = A'™ - A-™-1 & \n) = A>™ Since m^n,

A'^~\

it follows j(wt)^i(n). But then e(m)^e(n),

{e(w)}

since

s

= -B<™> - £ • ' - ) - ! & {«(n)} = B<M - B><»>-\

We also know from Lemma 3 that e(m)
W»(..o>, If,,,.!), • • • , W, are nonempty and disjoint. If each of them has a member in A, then their union has at least e + 1 members in A. But their union is Qt, and Lemma 4 tells us that Q. has at most e members in A. It follows from Lemma 5 that A is not effectively simple. It also

56 1964]

A SIMPLE SET WHICH IS NOT EFFECTIVELY SIMPLE

55

follows from Lemma 5 that the complement of A is infinite, since otherwise, the constant function fin) = 1 + cardinality of the complement of A would constitute a counterexample to Lemma 5. Finally, by Lemma 1, A is simple. Post [5] calls a recursively enumerable set W hyper-simple if its complement is infinite, and if there does not exist a recursively enumerable sequence of disjoint, finite sets, each one of which con­ tains a member of the complement of W. It can be shown with the help of Lemma 4 that A is not hyper-simple. The proof of Theorem 1 above is, as far as we know, the first proof in recursion theory to make simultaneous use of the recursion theo­ rem and the priority method of Friedberg [ l ] and Muchnik [3]. The priority method was needed to resolve the inevitable conflict between putting elements in A as required by Lemma 1 and keeping them out of A as required by Lemma 4. Thus in operation (b), we are not allowed to take m from W\ and add it to A' if for some j^e, w^?J. The recursion theorem was needed to prove that our system of pri­ orities does eventually resolve all conflicts happily; in particular, the recursion theorem made possible the proof of Lemma 2. REFERENCES 1. R. M. Friedberg, Two recursively enumerable sets of incomparable degrees of unsolvability, Proc. Nat. Acad. Sci. U. S. A. 43 (1957), 236-238. 2. S. C. Kleene, Introduciion to metamathematics, Van Nostrand, New York, 1952. 3. A. A. Muchnik, Negative answer to the problem of reducibility of the theory of algorithms, Dokl. Akad. Nauk SSSR 108 (1956), 194-197. (Russian) 4. John Myhill, Creative sets, Z. Math. Logik Grundlagen Math. 1 (1955), 97-108. 5. E. Post, Recursively enumerable sets of positive integers and their decision prob­ lems, Bull. Amer. Math. Soc. 50 (1944), 284-316. CORNELL UNIVERSITY

57

T H E JOURNAL or SYMBOLIC LOGIC

Volume 30, Number 3, Sept. 1965

METARECURSIVE SETS G. KREISEL AND GERALD E. SACKS*

1. Introduction. Our ultimate purpose is to give an axiomatic treat­ ment of recursion theory sufficient to develop the priority method. The direct or abstract approach is to keep in mind as clearly as possible the methods actually used in recursion theory, and then to formulate them explicitly. The indirect or experimental approach is to look first for other mathematical theories which seem similar to recursion theory, to formulate the analogies precisely, and then to search for an axiomatic treatment which covers not only recursion theory but also the analogous theories as particular cases. The first approach is more general because it does not depend on the existence of (familiar) analogues. A concrete mathematical theory, it seems, need have no such analogues and still be important, as e.g. classical number theory. In such a case, an axiomatic treatment may still be useful for exhibiting the mathematical structure of the theory considered and the assumptions on which it rests. However, it will lack one of the most heavily advertised advantages of the axiomatic method, namely, the "economy of thought" which results from an uniform theory for several different and interesting cases : we cannot hope for this if by hypothesis we know of only one particular case. In contrast, the second approach, if successful at all, is bound to achieve such "economy" because we start out with several interesting particular cases. Another possible virtue of the second approach is that of field work over insight: the abstract pattern that we are looking for and horring to formalize in axioms may not be evident in any one mathematical theory but may spring to the eye if one happens to look simultaneously at several theories which happen to realize the pattern The present paper is intended as a contribution to the second approach. But, for the time being, we play safe and stress its independent interest as a recursion theory on the recursive ordinals, or, at most, as an important special case of: recursion theory on initial segments of the ordinals. Since almost any generalization of recursion theory- that comes to mind, wiU preserve pre-Post [14] recursion theory, such as Kleene's enumeration theorem, we treat the elementary parts briefly, and concentrate on postFriedberg recursion theory. We generalize and discuss Friedberg's solution [2] to Post's problem and the maximal set constructions of Friedberg [3] Received October 21, 1964. * This research was partially supported by NSF Grant GP-124. The authors wish to thank Saul Knpke for many helpful conversations. Reprinted with permission from Association for Symbolic Logic. All rights reserved. This reproduction is by special permission for this publication only.

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and Yates [20]. We also present some applications to the theory of 11} sets of natural numbers. By use of unique notations for recursive ordinals, e.g. Gandy [4], the concepts of our metarecursion theory may be formulated in familiar terms: Let Q be a 11} set of natural numbers (the set of unique notations), ordered by the restriction of a recursive relation < to Q, and let \n\ be the ordinal of the predecessors of n neO Let the order type of 0 be Kleene's toi. The definitions below are invariant for all such ( such that X = Yi and 0-X = Y2 Metarecursive functions- k\n| \m\>|Cn m><X) (graph of the function) where X is a U} subset of Q2, and (^n€Q){E1meQ)X(n, m). Metarecursively enumerable sets: If non-empty, the range of a metarecursive function; it turns out: {\n\ | n<X}, where X is any 11} subset of Q. Bounded sets: X is bounded if I C {|«| | \n\ < a} for some a < coi; all metafinite sets are bounded; all bounded, metarecursive sets are metafinite, but some bounded, metarecursively enumerable sets are not metafinite. We discuss the notion of relative metarecursiveness in section 3. The definitions above in terms of notations are given only for comparison with the literature. One of the advantages of our treatment is that the theory is notation-free, instead of first being developed in terms of notations and then proved independent of the notation system used. (The use of notations, particularly non-unique ones, is a thoughtless hangover from a constructive theory of ordinals, where one wanted the usual ordinal functions to be represented by constructive functions For this one needs non-unique notations. But once the jump operator is introduced, constructivity is crone by the board, and the use of such notations becomes pointless. But once one has unique notations, it is natural to work with abstract ordinals instead.) The remainder of the paper can be read independently of the next section, which sets out the considerations which led to the correct definitions. 2. Analysis of the problem, (a) Language of recursion theory. The first step in any generalization of an informal theory is to pick out the primitive notions and the formal language. A simple-minded choice is this: A two-sorted language, with variables x, y, ... for individuals, f.g,h,... for monadic functions; = between individuals and applications of functions to individuals; two distinct individuals, say 0, 1. In ordinary recursion theory, the interpretation is this: x.y, ... range over o, f.g, ... over the recursive number-theoretic functions; 0, 1 have their usual meaning.

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The theory of non-empty, recursively enumerable sets c w can be inter­ preted, by defining the relation / e g (read: the range of / is contained in the range of g) explicitly by (x)(Ey)(f(x) = g(y)). And / enumerates a recursive set if g enumerates its complement, i.e., /?(/) - , (Eg)(x){y)(Ez)\J(x) ^ g(y) & (f{z) = x V g{z) = xj\, or if it has a characteristic function g, i.e., /?+(/) ~ (Eg)(x)[g{x) = 0 « (Ey)(/(y) = at)]. Fin{f) ~ (e)fes/ — R(g)) is a definition of finite set suggested by Lacombe, i.e., every r.e. set contained in the set (enumerated by) / is recursive, or if the range of every recursive function restricted to / is recursive, i.e., Fit) ~ (g)(EA)(x)[R(h) & (h{x) = 0 ~ (Ey)(g(/(y)) = x))}. The theory of recursive functions of finite sequences can be interpreted in the above theory by use of inverse pairing functions in the usual way, by assuming function constants j l t j 2 , such that (*, y)(Eiz)[j'i(2) = x & jt{z) = y & /i(0) = 0 & ja(0) = 0]] Evidently, relative recursiveness (degree theory) of r e . sets does not need an extension of the language. (b) Transfer principles or axiomatic theory! Keeping to the language (a), there are two kinds of generalizations. First, we may look for other inter­ pretations (i.e. different ranges for the variables, but requiring the second sort to range over Y g F if X is the range of the first) such that for a suitable class of formulae, the interpretations are equivalent, as for the language of real algebra ( + and •) when one permits all real closed fields as interpretations. Thus, if in our case the class of formulae involved includes formally different definitions like R and 2?+ or Fin and F these pairs will be equivalent in all interpretations Second we pick on several inter­ pretations first and then look for an axiomatic theory which is valid for them but may leave some statements undecided ; and of course even an axiomatically decided statement may have proofs in the special case which are not valid (e P from the axioms for PTOuDS the riffht and left inverses are equal; but alimpler "proof is possible for commutative groups!) This is the common situation in mathematics where different concepts i e IrZlvmZrtZi^tions mav have the same extension in one interp X t o . T u T X e r J e T r x another We are convinced that non-trivial recursion theory is not primarily concerned with construcdvity. But it remains a fact that certain parts of it have a close relation to constructive (intuitionistic) concepts, e.g., the distinction between a finite set given by: its sequence number or by: [x\x<,n & (Ey)(/(y) = x\} where / is recursive cf. Post |"14] This suggests that" correct generalizations may use an axiomatic theory based on the formal rules of intuitionistic logic. Since these formal rules have other interpretations than the intended constructive ones the suggestion is quite consistent with our conviction. We do not pursue this matter here. The

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reader may note that the source of our definitions (described immediately below) is the theory of model-theoretic invariants of [9]. But for an axiomatic theory it is not convenient to have merely these invariants; one wants to develop their logic, i.e., give an appropriate interpretation of logical con­ stants. There is a precise analysis in the literature (for so-called extensionally definite formulae of [8]), and this logic is indeed non-classical. (c) Model-theoretic foundations. The very first appearance of the class of number-theoretic recursive relations and functions was in Godel [5], not in connection with Church's problematic thesis, but under the name: Entscheidungsdefimte Relationen. In the spirit of the times this was formulated proof-theoretically, but the model-theoretic meaning was clear; one has a sequence of terms, the numerals e.g., 0, SO, ..., S»0 in the usual systems of arithmetic with 0 for zero and S for the successor function; granted that in each model sf of $, any given numeral has the same denotation, say the number n, then the formula *{%) is entscheidungsdefinit if the set wr\{x\ \za{x}} is isdependent of s/; we calc it itvariant on w. In Mostowski's exposition of Godel's theory, much of the work is independent of the assumption that the set (of Godel numbers of)
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G. KREISEL AND GERALD E. SACKS

model. But if we consider, as was done in Mostowski [13], second order systems, and all hyperarithmetic subsets of co are invariantly definable, we should regard these definitions of sets as "just as good as the numerals" So, the modification suggested in Kreisel [9], was to use mvaiiants of comodels for recursion theory on hyperarithmetic sets of natural numbers. If, instead of second order arithmetic, one considers a theory of ordinals, one gets precisely the definitions in the introduction for n {-systems by the following Fundamental principle of the generalizaiion: One takes results of ordinary recursion theory, and replaces the word •'recursive" by "invariantly definable on co," some (or all) occurrences of "finite" by "invaraantly definable;" all this for general models. To get the analogue of recursion theory on u>i, one replaces co by coi, and "general models" by "w-models" Other occurrences of ""inite" may be replaced by "bounded" or retained, in accordance with the second approach of (b). I n v a n a n t l y definable

General Models Finite subsets of w (all O)

co-models Subsets of coi which are Ai, where
is nj. Invariantly definable on

ta: recursive subsets of to, where O is r.e.

coi: subsets of coi which are Ai on coii where
To get all subsets of to, resp. coi mentioned in the table, O must of course contain certain minimal axioms. In the case of general models, Robinson's system Q is a well known sufficient set; in the case of co-models one needs the supremum axiom and (certain restricted forms of) the principle of proof and definition by generalized induction. For details, cf. 1.2 (c) of [9], where also a more abstract (and very much restrained) exposition of the general notions above is given in 1.1. 3 A completely parallel treatment can be given based on (transfinite) concatenation theory where an co-model contains the sequence |*i*... of length precisely = co, and instead of the numerals |, [*[, |*|*|, ... one has ex­ plicit definitions of sequences of | of length < a, a < coi. In view of the elegance of Smullyan [19], it may be worth pursuing a recursion theory based on this. Besides generalizations, the model-theoretic foundation leads also to an analysis of familiar concrete results. For, different general notions may have the same extension in a special application; thus given a concrete result one has the problem of analyzing just which general notion is in­ volved. In the case of general models (of arithmetic) or ordinary recursion theory, the collection N"> of finite sets of natural number N has the following properties: its cardinal is Ar, moreover Nm ~ N by functions invarrantty

62 METARECURS1VE SETS

323

definable on TV, N<» respectively; each finite set has cardinal < N, is invariants definable (i.e. included in an invariantly definable set) and conversely. Different proofs of the same theorem involving finiteness, may use different properties of N«, and will generalize to proofs of different theorems in generalizations. Similarly, the distinction between regular and non-regular sets breaks down, a set being regular if its intersection with any invariantly definable set is also invariantly definable. All sets in­ variantly definable on the hard core are automatically regular. In the case of general models every set is regular but not for other classes of models (Intuitionistically! not every r e . set X is regular in the sense that («)(Ea)(a = {xlxtX & x < n\] where a ranges over finite sequences preented explicitV in faction Church's thesis regular r e sets are recursive because a would be a recursive function of n Ifintuitionistic logic should turn out to be exactly correct for the axiomatic theory a little sophisticate wffi be needed to Leroret non^orenex formulas ^M~~(EdVaixxS&x
mejdhg C system^■ i n t u S o S c n e g a ^ n being weak) °rrespond m g " N . B . Our point of view differs from Mostowski [12] who regards the model-theoretic foundation as the approach to recursion theory, while we regard it as a prop for an abstract theory. Consequently, he" formulates explicitly conditions on consequence relations, satisfied by those for general and for ry-models, from which the relevant properties of definability-OK can be derived. The proofs below make the parallel quite clear, but are set out informally; their principal function is to resolve the ambiguities described above in the analysis of familiar results. (d) An equation calculus. For technical convenience we use an equation calculus got from one of Kleene's by taking metafinite sets of premises instead of finite ones. Kripke [10] has developed an equation calculus suitable for recursion theory on arbitrary, initial segments of the ordinals. We are indebted to Kripke [10] for the equation calculus we describe in section 3 below, which is suitable for recursion theory on the recursive ordinals. Now, in the case of ordinary recursion theory we find most expositions based on 3.n equation calculus repulsive because a set of rules is 'put down, without a.ny clear reason for the choice. The justification lies in the theorems

proved e g closure Tinder diagonalization ; but one seldom says in so many

words with"respect to what notion of definability the rules are supposed to be complete For the model theoretic foundation above the situation is much more satisfactory since once one is concerned with general models ^ o r f y « W ™ S Z r i T t t e ^ r f f W t e ^ u S e Here we would argue that at least ttT^^L^o^d^ri^ mmtouo^lv tte restriction^

seems

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G. KREISEL AND GERALD E. SACKS

definable (syntactic) objects are finite. But once one considers w-models, the matter is more delicate, and one has to study just what infinite formulae are appropriate (see [9] § 2.2). (e) Further generalization. The notions of (c) extend to other classes of models, e.g., /J-models; of a theory of ordinals in which < must be wellfounded, i.e., all models considered are abstractly simply segments of the ordinals; or of a set theory in which e must be well-founded. In the latter case we have a recursion theory on the elements of the Shepherdson-Cohen minimal models. But for a systematic treatment, the classes of models considered must not be chosen by ad hoc considerations ("absoluteness" of some historically or intrinsically important notion like finite or wellfounded) but by a more flexible method, e.g., the use of infinite formulae. In fact, an a l t e r n a t e definition of w -modelis this; we consider all general models satisfying the infinite formula (x < co)(x
and <;n on

(It is understnnrl that due respect mu<;t be paid to

the satisfaction relation and not only to purely external (syntactic) properties.) In such transfimte progressions there are usually catching points where new difficulties are met. It is therefore particularly encouraging that a direct development of recursion theory based on arbitrary ordinals is in progress, independently by Kripke [10], based on equation calculi, and by Platek, based on schemata. It is not yet known just how much of what we call non-trivial recursion theory generalizes to (say) uncountable ordinals. But there is reason to hope that these other developments will reveal useful information for abstract recursion theory not provided by (c) above. 3. The equation calculus. Following Kripke [10], we make use of a formalism similar to that of Kleene [6]. For each recursive ordinal, we have a corresponding numeral. (It is convenient to think of the set of numerals as a n ] set of unique notations.) We have a sequence, fo,f1.f2, •••, of function letters, and a sequence, x0, xlx x2, .,., of variables. Additional primitive symbols are: '(successor), E (there exists), and < (less than). The terms are the numerals, the variables, and expressions of the form f where t is a term, or f(h, ..., t„) where / is a function letter and h, ..., tn are terms (n 2: 0), or (Ex
64 METARECURSIVE SETS

325

({Ex < j))(t(x))) = 1 is an immediate consequence of the set of premises {t(y) = 1 1y < p}} The role of R3 is easily understood if one considers a slight extension of Kleene's formalism for recursion theory. Suppose we add propositional connectives, number-quantifiers bounded by terms, and appropriate rules to Kleene's formalism. The resulting formalism can be used to define the recursive functions by finite systems of equations in a manner similar to the use of Kleene's formalism. The two formalisms are equivalent in the sense that both define the same class of functions, the recursive functions; however, their generalizations to metarecursion theory differ in that the extended formalism permits deductions from infinite sets of premises. Rule R3 is nothing more than the familiar rule concerning the use of the bounded quantifier. For each finite set E of equations, we define S E , the set of all equations deducible from E, by transfimte induction on the recursive ordinals. This elegant definition is due to Kripke [10]. Let S E = E, and SE+1 = the set of all conclusions obtained by applying rules R1-R3 to S E ; when we apply R3 to S0, we require that /? < a. If a is a limit ordinal, let S E = U{Sfy3< a}. Finally, let S E = U{S E |a < c^}. Let Q be a n j subset of unique notations for recursive ordinals; it is convenient, but not necessary, to assume Q is a subset of Kleene's 0 linearly ordered by < 0 . Each equation of our formalism can be viewed as a finite sequence of members of Q, and consequently, a member of Q can be assigned effectively to it to serve as its Godel number. For each mtQ, let S E be the set of Godel numbers of members of Sg,,, where \m\ is the recursive ordinal represented by m. Then the predicate nuQ & ntS* is n j ; and there exists a recursive function / such that for each mtQ, S E is a hyperarithmetic set and j(m) is an index for it. The proofs of these last two facts are along the same lines as Kleene's argument [7] that 0 is 11} and Spector's argument [18] that for each btO, 0b is hyperarithmetic. Now let Q serve as the set of unique notations needed in section 1 to define metafinite, metarecursive, and metarecursively enumerable, and think of the Godel numbers as designating recursive ordinals. Then the set of Godel numbers of all equations deducible from E becomes a meta­ recursively enumerable set of recursive ordinals, and for each a, S E becomes a metafinite set. These developments are in accord with the transfer prin­ ciple of section 2; the finite deductions of Kleene's formalism for recursion theory have evolved into the metafinite "deductions" of the formalism for metarecursion theory. A partial function y, from the recursive ordinals into the recursive ordinals is partial metarecursive if and only if there is a finite system of equations E with principal function letter / such that y(a) is defined and

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= 0 if and only if /(a) = J) is a member of S E . A function is metarecursive if it is partial metarecursive and total. (Note that we have required E to be finite rather than metafinite; if E is permitted to be metafinite (or even metarecursive), the class of metarecursive functions is not enlarged. The transfer principle of section 2 allowed for the possibility that certain occurrences of "finite" in recursion theory would not become occurrences of "metafinite" in metarecursion theory. Kripke [10] shows that finite systems of equations suffice even for recursion theory on uncountable initial segments of the ordinals. The definition of maximal set given in section 4 is another example of an occurrence of "finite" in recursion theory which is retained in metarecursion theory.) A non-empty set is metarecursively enumerable if it is the range of a metarecursive function. We show that the above definitions are equivalent to those given in section 1. (Kripke [11] greatly clarifies the connection between recursion theory on arbitrary initial segments of ordinals and the analytic hierarchy.) It is enough to show that a function metarecursive according to section 3 is metarecursive according to section 1, and that a set metarecursively enumerable according to section 1 is metarecursively enumerable according to section 3. Let y> bb e metarecursive eunction defined bb y ainite eystem E. For each recursive ordinal a, let a be the unique member of Q such that \a\ = a. For each equation e, let [e] be its Godel number. Then V(«)

= /?«-. (E«)K

& [/(a) = P] « Sj),

and consequently, Gw = {(a, /?) | y(a) = $} is a n | subset of h{m)eO. The usual inductive definition of 0 can be translated into a finite set of equations in our formalism which define a metarecursive function whose range is 0 By combining these equations with the equations for h we can define a metarecursive function whose range is B. Finally, we can find a metarecursive function whose range is A, sinrp there exists a finite set of equations defining a partial metarecursive

function whose domain is 0 and which takes each member of 0 to the ordimd it representT Note that in the above, we are thinking of the recursive ordinals as an extension of the natural numbers, and that we have shown that each n j set of natural numbers is a metarecursively enumerable set of recursive ordinals. Similarly, each metafinite (or metarecursive) set of natural numbers is hyperarithmetic. Our first theorem is an enumeration theorem for partial metarecursive functions. A proof of it could be given by developing the formalism of

66 327

METARECURSIVE SETS

metarecursion theory along the same lines that Kleene [6] develops the formalism of recursion theory. We regard it, however, as a direct conse­ quence of the enumeration theory for 11} predicates; i.e., each f h predicate P(xi xn) can be put in the form (f)(Zy)T](f(y),e,Xl

xn,y).

THEOREM 1. For each n > 0: there exists a partial metarecursive function
xn) =a
In Theorem l the xt's of course range over the recursive ordinals, and ~ , the symbol for strong equality, has the same meaning it has in Kleene [6]. We need a metarecursive indexing of the metafinite sets. Let j and k be metarecursive functions such that for each metafinite set K there exists a unique a such that (x)[xel< <- (X < ](a) & % , *) = 0)].

For each «, we define Ka = {xx < j(«) & k(a, x) = 0}. For each a, Kx is metafinite, since it is bounded and metarecursive. If K = Ka, we call a the index of K. The existence of / and k follows from the fact that the hyperarithmetic sets are defined by transfinite recursion on 0. The indexing provided by j and k enables us to speak of partial metarecursive functions which take' metafinite sets as arguments. Thus when we say (a) such that q>(K) ~ y ( a ) , where K = K*. Let A and B be arbitrary sets of recursive ordinals, and let a(«) be the characteristic function of A. We say A is weakly metarecursive in B if there exists a partial metarecursive function q> such that (i)«<2(«}[a(a) = * - (EM)(EN) )
N, H) = i&McB

&

NccB)],

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G. KREISEL AND GERALD E. SACKS

where H, M and N range over the metafinite sets. The definition of "metarecursive in" is also analogous to the definition of "recursive in", since in recursion theory, a finite set of questions about A is decided after asking a finite set of questions about B. Clearly, if A is metarecursive in B, then A is weakly metarecursive in B. But the converse is not true! This is a far from abvious result recently obtained by G. Driscoll. In fact, Driscoll shows that the relation "weakly metarecursive" is not transitive, whereas it is an exercise to show that the relation "metarecursive in" is transitive. Let y>(M, N, a) be a partial metarecursive function, where the variables M and AT range over the metafinite sets. By Theorem 1, it has a normal form wle M N a) Let <e M N a. ,y >ennte t t h Godel l u m b e e oo fth ordered 5-tuple,' (e M N , a', y); we assume 0 is not a Godel number. Since is partial metarecursive there exists a metarecursive function tie a) such t h a t : if wle M N a) is defined and equal to v then tie a) = <£ M N a. v> for some a' if tie a) =?t 0 then tie a) = <e M N a. v> for some M V . a n d v and wle M 'N l i s defined and equal to v We deffne a T predicate analogous to that'of Kleene [6] ; let B be an arbitrary set of recursive ordinals T\(B,e,a,a)

~t(e,o)

= <e, M, N, a , y > & M £ J 3 &

N£cB.

Let U(e, a) be a metarecursive function such that if t{e, a) = <e, M, N, <x, y>, then U(e, a) = y. We define two partial functions: {«}*(«) = y - T\(B, e, a, a) & U(e, a) = y; {«}*(«) = y « (Ea)[r}(fi. «, a, a)] & (a)[r|(B,

«, a, a) -► U(e, a) = y].

We have a normal form theorem as a consequence of Theorem 1. THEOREM 2. If A is weakly metarecursive in B, then there exists a finite ordinal e such that {«)[«(■) = {«}*(«)], where a(«) ?'s
68 METARECURSIVE SETS

329

4. A maximal set. One of the deepest results in the theory of recur­ sively enumerable sets is Friedberg's construction of a maximal set [3]. A recursively enumerable set M is said to be maximal if its complement is infinite, and if for each recursively enumerable set R, either R-M or cR-M is finite. The transfer principle of section 2 suggests nine possible definitions of maximal, metarecursively enumerable set of recursive ordinals, since we can replace ''infinite" by "infinite" or "not metafinite" or "un­ bounded", and since replace "finite" by "bounded" or "metafinite" or "finite" Throwing all caution to the winds we choose the strongest of the nine possibil.ties. A metarecursively enumerable set M of recursive ordinals is said to he ■maximal if its romrilement is unbounded and if for

each metarecursivdv enumerable s e t T either R-M or cR-M is finite Note that a maximd set must be unbounded THEOREM 3. There exists a maximal, metarecursively enumerable set of recursive ordinals. PROOF. Let Q be a n} set of unique notations for the recursive ordinals. If e is a notation for a, let \e\ = a; the partial function \e\ is partial metarecursive. We claim Q is not metarecursive. Suppose it is. Then the function h{a), defined by

AM-f'"'£"? v [0 otherwise, is metarecursive. But the range of h restricted to the natural numbers is unbounded. This contradicts (B*). Thus Q is not regular.2 (fi*)) is stated at the beginning of section 6.) Let r(a) be a metarecursive function which enumerates Q without repe­ titions; it follows from {B') that any non-metafinite, metarecursively enumerable set can be enumerated without repetitions by some metarecur­ sive function. Our construction is inspired by that of Friedberg [3]. One difference between our construction and Friedberg's consists of our use of the function r(a) to insure that the complement of M is unbounded. Without r(a) we would have only that cM is not metafinite. We are also indebted to C. E. M. Yates [20] for a modification of Friedberg's con­ struction he devised in order to construct a complete, maximal set. Without Yates' modification, we would have for each metarecursively enumerable set R, either R-M or cR-M is bounded rather than finite. We define a partial recursive function, g(a, e); for every recursive ordinal a and every natural number e: J(a,e) ={|r(r)l|T ^. a & r{r) ^ e}; _ (greatest member of J(a, e) if J(a,e) ^ 0 , g[a, e) — | Q ot jj erw jjse 2 The notion of regularity is defined near the end of section 6.

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For any function /(ex), we say lim„f(a) exists if (Ey)(cr) (a > y-+/(a) = /(y)). It is clear that Ztm0£(cr. e) exists for each e; in fact, limag(a, e) is either 0 or the greatest ordinal which has a notation in Q less t h a n or equal to e. Let g(e) = lim„g{a, e). Then the range of g(e) is unbounded. In addition, if d ^ e, then g(d) ^ g(e). Let 9?i{e, x) be the parttal metarecursive functton of Theorem 1 which enumerates all partial, metarecursive functions of one variable. For each natural number e, let Re be the range of Ix \ tpi{e, x). Thuu R0, Ri, Rz, ,.. is a simultaneous, metarecursive enumeration of all metarecursively enumerable sets. There exist metarecursive functions, u(a) and v(a), such that for each a, u{o)eRvia), and such t h a t for each e, Re = {u(T) | v(r) = e}. For each a and e, let R° = {u{r) | T ^ a & v(r) = «}. In short, R°t consists of all ordinals enumerated in Re at or prior to stage o of our simultaneous enumeration. We will define a metarecursive function M(a) and a partial, meta­ recursive function v{a, e) simultaneously by recursion on a. For each a, M{a) will be a metafinite set of ordinals less than a; M will equal U{M„ | a < wi). We will say r\ is in the z-th estate at stage a if rii\J{M{d) | 6 < a} and

i = 2 {2 f "' 11, e J!J & / £ *}. The e-state of i; at stage a, if it is defined, is a natural number less than 2^'+i. If tj e U{M(<5) | a < a), then we will say rj has no e-state at stage a. The notion of instate is due to Friedberg [3]. Principle (P) is the major reason for the use of the notion. (P) If a c i-th estate at stage a and (S e j-th estate at stage a and i < j , then for each c ^ e, the estate of a at stage a is less than the estate of 0 at stage a. (P) is an immediate consequence of the fact t h a t 1 + 2 + 2 2 + . . . + 2 b < 2 b + 1 We proceed at stage a as follows. It is safe to assume W(a) = U { M ( T ) | T < a) is metafinite, since we are defining a metarecursive function AT I M{r) with the property that for each T, M(T) is metafinite. (This last remark is a typical example of the role of (A*) and (B*), two principles stated at the beginning of section 6, in the definition of functions by metarecursion.) We define via, e) for all finite e by induction on c. Since W(a) is bounded, there must be an e s t a t e at stage a which contains an ?? greater than every member of foio i) I i<e\ U ieio e))' let i(a e) be the highest such e-state We define: v[a, e) = m[ricj(o, e)-th e-state & ??>g(ff, e) & (i)(<e{v(a. j)
70 METARECURSIVE SETS

331

be the largest member of {Umav(a, i) | i<e) U {g(e)}. Let T be so large that

Then for each a>T, /(a, e) is the highest e-state at stage a which contains an r]>y. Let / be the greatest member of {j(a, e) \ o>r). Let a be the least member of [t] | r,>y & (Ea)„ >T ( W --th e-state at stage a)}. Let Ji > T be such that a is in the j-th e-state at stage ax. Then j( ffi, and suppose /(CT, e) = / and »(a, e) = a when a2 > a S e»i. Then a < U{Mf (5) | <5 < az), and consequently, « is in some e-state at stage <72- Since 02 > au the e-state of a at stage a2 cannot be less than the e-state of a at stage a\. Since 02 > T and a > y, the e-state of a at stageCT2cannot be greater than /. It follows 7{a%, «) = 7 and v( g(e) for all e. But then the complement of M is unbounded, since the range of g(e) is unbounded. Following Friedberg [3], we say a is a resident of the i-th e-state if a is in the i-th e-state at stage a for all sufficiently large a. Then aecM if and only if a is a resident of some e-state. Since cM is infinite, it must be that for each e, some e-state has infinitely many residents. We say an estate is well-resided if it has infinitely many residents. LEMMA 2. For each e, there is only one well-resided e-state. PROOF. By reductio ad absurdum. Fix e, and suppose both the z-th and the 7-th e-states are well-resided, where i < 7. It follows that there exist d and c such that d > c ^ e, v(c) is a resident of the i-th e-state, and v{d) is a resident of the f t h e-state. Let a be so large that v(c) is in the i-th e-state at stage a, v(d) is in the j'-th e-state at stage a, and v(a, i) = v(i) for all i g c. Let t* be the c-state of v(c) at stage a, and let 7* be the fi­ st ate of v(d) at stage a. Then i* < j ' by principle (P); and i* = j(a, c), since v(c) = v{a, c). It follows from the definition of j(a, c) that v(d) < v(c), since j(o, c) < 7* and v(d) is in the /*-th c-state at stage a. But v(d) > vie), because d > c. Let i? be a metarecursively enumerable set; then R = Re for some e. If R-M were infinite, then one of the odd-numbered e-states would be wellresided. If cR-M were infinite, then one of the even-numbered e-states would be well-resided. It follows from Lemma 2 that M is maximal.

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5. Applications t o n j s e t s of natural n u m b e r s . In section 3 it was shown that each non-empty, 11J set is the range of a metarecursive function. (One effect of rule R 3 is to make possible the enumeration of II} sets.) The simultaneous, metarecursive enumeration of the metarecursively enumerable sets described in section 4 is easily changed to a simultaneous, metarecursive enumeration of the 11} sets; each Re is replaced by Se, where Se consists of the finite members of Re. By identifying the II} sets with the metarecursively enumerable sets of natural numbers, we strengthen the well-known analogy between the 11} sets and the recursively enumerable sets. In fact, we can now apply the priority method to the II} sets and ODtain new results. We call a n } set M maximal if cM is infinite, and if for each 11} set S, either S-M or cS-M is finite. (C) There exists a maxima,, 11} set. The proof of (C) can be given either as an immediate generalization of Friedberg's construction of a maximal set [3] or as a specialization of the proof of Theorem 3 ; the latter approach requires only t h a t g(a, e) be replaced by e and Re by Se. Friedberg [3] also shows t h a t each non-recursive, recursively enumerable set is the union of two disjoint, non-recursive, recursively enumerable sets. His argument readily generalizes, with the help of metarecursion theory, to a proof of (D). (D) Each non-hyper arithmetic, 11} set is the union of two disjoin,, nonhyperarithmetic, 11} sets. ' (C) and (D) support the conjecture (due to H. Rogers, Jr., cf. [9]) t h a t the ordering of recursively enumerable sets by set inclusion is ismorphic to the oidering of 11} sets by set inclusion, or at worst, isomorphic modulo the finite sets. 6. P o s t ' s p r o b l e m . We now present a solution to Post's problem for the recursive ordinals. Our solution is analogous to Friedberg's solution of Post's problem for the natural numbers. Friedberg [2] makes use of the following combinatorial principle: (A) the union of finitely many finite sets is finite. A collection {K} of metafinite sets is said to be metafinite if {K} = {Ka | aeH} for some metafinite H. The metarecursion-theoretic analogue of (A) is (A*): the union of the members of a metafinite collection of metafinite sets is metafinite. (A*) follows readily from the definitions in section 1 We also need (B*): the range of a metarecursive function restricted to a metafinite set is metafinite. (B*) is the analogue of (B): the range of a function restricted to a finite set is finite. A requirement R = {(Fa, Ha) \atA} is a collection (possibly void) of ordered pairs of disjoint metafinite sets of recursive ordinals. A set T of recursive ordinals meets requirement R if for some atA, FaTT and HarTT = 0 .

72 METARECURSIVE SETS

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Let t = {to, h, g) be a triple of functions defined on the recursive ordinals with recursive ordinals as values. We say t enumerates requirements if for each o ^ O , KMo) and A'(i(„, are disjoint, metafinite sets, and if all values of g are finite. The requirements enumerated by t are R0,RURZ, ...; for each natural number k, Rk = {(F«, H°) | g[a) = k], where F" = KMa) and H« = KMo). With each metarecursive t which enumerates requirements, we associate a set T called the priority set of t. T is metarecursively enumerable, and meets every member of a certain subclass of the class of requirements enumerated by t. T is defined by stages. Stage 0 = 0. Ta = 0. Stage a > 0. TB = \J{TS | 6 < a) if (a) or (b) or (c) is t r u e : (a) there is a f < a such that g(y,) < g(o), y> > 0, F<"£ \J{T6 | 6 < y>}, F*£TV, Hy>n\J{Td \ 6 < a) --= 0 , and H*nF« ^ 0 ; (b) there is a rp < a such that g(v) = g(<j), y> > 0, F*>$ \J{Td | <5 < y}, f » c r , „ and # * n U { r d I <5 < a} = 0 ; (c) # ° n U { F , | < 5 < a } # = 0 . Otherwise, 7 0 = u { F „ | ,5 < a} U F° Let T = U { r « | « < « i } . For each k, we say « „ is met at stage a if a > 0, A = g(a), F°$ U{T6\6 < a} and F«cTc. If Rk is met at stage a, then / / " n l , = 0 , since clause (c) must be false at stage a, and since H°r\F° = 0 for all a. For each k, we say Rk is injured at stage a if there is a y < a such t h a t F * was met at stage y, Hv>nll{T6 \ 6 < a} = 0 and H«>r\Te ^ 0 such that g(o) = A, F° U $ {Td\6 < a}, H°n\J[Td \ 6 < a} = 0, and LnF" = 0 . LEMMA 3. If y
73 334

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t h a t the set S = {a\(Ei)(i
& Ri is met at stage a)}

is finite. We now show that the set {a\Rk is injured at stage a} is a subset of 5. Fix a, and suppose Rk is injured at stage a. Then there is a y < a such t h a t Rk was met at stage y, H*>r\U{Td \ 6 < a} = 0, and HT\Ta =£ 0 ; consequently, 0 < y, g(y) = k, F<*$ U{Td | 6 < y}, and Fv$Tv, If g(y) were less than or equal to g{a), then (a) or (b) would be true at stage a, and Rk would not be injured at stage a. Hence g{a) < k = g(y), and RB(a) is met at stage o. LEMMA 5. 7/ * is metarecursive and enumerates requirements, then T, the priority set of T, is metarecursively enumerable and meets every t-dense requirement. PROOF. It follows from (A*) and (B*) t h a t there exists a metarecursive function / such that T„ = Kf{B) for all a. But then T = U{Km \ a < coi} is metarecursively enumerable. Fix k and let Rk be t-dense. We need a y such that Rk is met at stage y and is not injured at any later stage. By Lemmas 3 and 4, the set K = {6\(Ei)iSk

(Ri is met or injured at stage 6)}

is finite; let n be the largest member of K. Let L = \J{F»\JH« \ 6 S n}l L is metafinite by (B*) and (A*). Since Rk is *-dense, there is a a > 0 such thatg(a) = k, F°£\J{Td\6 < a}, H°n\J{Te\8 < a} = 0 , and LnF" = 0. Clearly, a > n, since F° # 0 and I n i ^ = 0. It follows that Rk is not met at stage a and t h a t F"$Ta. Thus either (a) or (b) or (c) is true at stage a. Clause (c) is false at stage a. If clause (a) were true at stage a, then there would be a y such that g(y) < k, R M was met at stage y, and R was injured at stage a- but a > TJ It follows clause (b) is true at stage c Thus there is a y < a such t h a t g(y) = g(a) = k, Rk was met at stage y, and /?* was not injured at stage 6 for any <5 such that y g <5 < CT. Finally, i?t was not injured at stage d ^ a, since a > i). Let /.(a, 0) be a metarecursive function such that if « x # a2 or £ x # /S2, then />(«i, ^i) # />(a2, /S2). Let qfa) and r(>;) be metarecursive functions such that q(n) is finite for all r\ and such that for each e and a, the set {r)\q(r}) = e & r(ij) = a} is unbounded. THEOREM 4. TAere ex«* too metarecursively enumerable sets such that neither is weakly metarecursive in the other. PROOF. By Lemma 5. We first define metarecursive functions T„, F°, H° and g(a) simultaneously by recursion on a; for each a, T„, F» and H° will be metafinite sets. Stage or = 0. T0 = Fo = H° = tz, and g(0) = 0. StageCT> 0. Let £ = 9 (a), /!„ = {»?|2» ? eU{r i) |<5< C r}},

74 METARECURSIVE SETS

335

and B„ = {V\2r,+

\e\J{Td\d
Case 1. a is of the form r+m, where m is even and finite. Case 1.1. There exist v <-a and y < a such that (1)

T\(A„, e, p{e, v), y) & U(e, y) = 1 & p(e, rf) 4 Ba\JKr(
let y> be the least rj < a such that (1) holds for some y < a. We set: F° = {2p(e, V ) + l } ; fl* = {2^ 4At); g(a) = 2 e + 1 . Case 1.2. Otherwise. We set F« = H° = 0 and g(a) = 0. Case 2. a is of the form r+m, where w is odd and finite. Proceed as in case 1 with A and B interchanged, all members of F° even, all members of H« odd, and g(a) even. Let T„ be defined as in Lemma 5. There exist metarecursive functions, to and h, such that F« = Khia) and H° = Kb{a) for all a. Let t = (t0, h, g) and T = U{Td \ 6 < coi} T is the prionty set of t. Let A = {i)\2i} t T} and B = {r)\2r] + 1} e T. By Lemma 5, T meets every i-dense requirement. We use this fact to show B is not weakly metarecursive in A. A similar argument shows A is not weakly metarecursive in B. Fix e. Suppose (e}^(a) is defined and less than 2 for all a. Let i?2e+ibe /-dense. Thus T meets i?2e+i- It follows that there is a a such that g(cr) = 2e+l, F°sT and fiTiJ = 0. Then case 1.1 must hold at stage a, and there must be a v and a y such that: T\(Aa,e,p(elV),y)&U(e,y)

= 1;

F° = {2p{e,f) + l}; H° = {2r)\r) 4 A„}. Since F« c T, we have ^(e, y>)fB. Since H«n7 = 0 , we have {eY{p{e, y,)) = 1. Now suppose i?2e+i is not i-dense. Note that H«n\J{Te \ d < a] = 0 and F«nU{r,51 a < 0}( = 0 for all a. It follows that there is a metafinite set L such that for all a, either g(a) =£ 2e+l or F* = 0 or F«nL ^ 0 . We use this fact to find a r such that r y F and { ^ ( T ) = 0. Let JJ be such that 2/>(e, 17)+1 < i ; the existence of rj follows from (B*). Let T = />(e, JJ). If T were a member of F , then there would be a a such that g(a) = 2e+ 1, F« = { 2 T + 1 } , and F°nL = 0 . Thus T y Fff for all a. Suppose {e^(r) = 1. Then there is a a > rj and a y < a such that (1) holds and Kr(a) = {a|2a+l eL}. Then g{o) = 2e+\, F° J- 0, and F«r\L = 0. For each finite k, let y(A) be the cardinality of S* = {alF* is met at stage a of the construction of Theorem 4}; by Lemma 4, y(A) is finite. For each infinite a, let y(«) = 0. We claim y is not a metarecursive function.

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Suppose it is. For each finite k, let w{k) be the greatest member of S*; for each infinite a, let w{*) = 0. Then w is metarecursive, since y is. It follows from (B*) that there is a y such that w(k) < y for all finite k. Then T = U{F" I a < y} is metafinite by (A")) and A and B are metarecursive. Kripke [10] introduces the notion of admissible ordinal, and then shows that for each admissible ordinal «, it is possible to develop a recursion theory for the ordinals less than a. He proves that Kleene's an is the first admissible ordinal after wo, and his notion of a. r recursive function is equivalent to our notion of metarecursive function. He obtains appropriate generalizations of Theorems 1 and 2 for all admissible a and he proves Theorem 4 for a large class of admissible a. which includes co\. In [[7] ]i will be shown that the following solution of Post's problem generalizes to all admissible a: there exists recursively enumerable sets A and B such that A is not recursive in B, and B is not recursive. The most striking phenomenon of metarecursion theory is the existence of bounded, metarecursively enumerable sets which are not metarecursive; we have an example of one at the beginning of the proof of Theorem 3. We call a set of recursive ordinals regular if its intersection with every metafinite set is metafinite. A metarecursively enumerable set is regular if every bounded, metarecursively enumerable subset of it is regular. More intuitively, a metarecursively enumerable A is regular if and only if some metarecursive function / enumerates it in such a way that for each a, / eventually stops enumerating new members of A less than a. The automatic regularity of all sets in recursion theory is the main obstruction to lifting theorems of this theory up into metarecursion theory. We say that two sets have the same meladegree ii each ii metarecursive in the other. Dekker [1] shows: if / is a 1-1 recursive function, then the de­ ficiency set of / has the same Turing-degree as /, and if the range of / is not recursive, then the deficiency set of / is simple (in fact, hypersimple). Dekker's result generalizes readily to metarecursion theory if it is assumed that the range of / is regular; without this assumption we are unable to show that the deficiency set of / has the same metadegree as the range of /. In [15] it was shown that each non-recursive, recursively enumerable A is the union of two disjoint, recursively enumerable sets whose Turingdegrees are incomparable If we attempt to generalize this result to a metarecursively enumerable A, we appear to fail unless we assume A is regular. The lack of regularity is not as alarming as it sounds, since it will be shown in [17] that every metadegree which contains a metarecursively enumerable set also contains a regular, metarecursively enumerable set. This last result makes it seem plausible that most, if not all, theorems about Turing-degrees of recursively enumerable sets can be lifted up into metarecursion theory. G. Driscoll plans to discuss this matter further.

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It also will be shown in [17] that there exist infinitely many metadegrees of metarecursively enumerable sets which contain only regular sets. One virtue of the lack of regularity is that it leads to an intricate theory of n} sets of natural numbers. For example, it will be shown in [17] that there exist two 11} sets of natural numbers such that neither is weakly metarecursive in the other. This result can be taken to be the solution of Post's problem for 11} sets. Spector [18] showed that all n} sets lie in two hyperdegrees but in [17] it will be seen that they he in infinitely many metadegrees.

REFERENCES [I] J. C. E. DEKKER, A theorem on hypersimpee sets, Proceedings of the hg American Mathematlcal cal Society, 5 (1954), 791-796. [2] R. M. FRIEDBERG, TWO recursively enumerable sets of incomparable degrees of unsolvability, Proceedings of the Nationll Academy of Science, U.S.A,, 43 (1957), 236-238. [3] R. M. FRIEDBERG, Three theorems on recursive enumeraiion, this JOURNAL, 23 (1958), 309-316. [4] R. O. GANDY, Proof of Mostowski's conjecture, Bulletin de VAcadimieie Polo­ naise des Sciences, 8 (1960, 571-574. [5] K. Go DEL, Vber formal unentscheidbaee Sdtze der Principia Mathematica und verwandter Systeme I, Monatsheftehg fur Mathematik tik und Physik, 38 (1931), 173-178. [6] S. C. KLEENE, Introduction on to Metamathematics, tiks New York, Toronto, Amsterdam and Gronmgen, 1952. [7] S. C. K L E E N E , On the forms of the predicates in the theory of constructive ordinals (second paper), American Journll of Mathematics, ixds 77 (1955), 405-428. [8] G. KREISEL, Set theoretic problems suggested by the notion of potential totality, Infinitistictic Methods, Warsaw 1961, 103-140. [9] G. KREISEL, Model theoretic invariants: applicaiions to recursive and hyperanthmeiic operations, Theory of Modess Symposium, m Berkeley, forthcoming. [10] S. K R I P K E , Transfmite recursions on admissible ordinals, in preparation. [II] S. KRIPKE, Admsssible ordinals and the analytic hierarchy, in preparation. [12] A MOSTOWSKI, Sentences es Undecidableble in Formalized ed Arithmetic, tuic Am­ sterdam, 1952. [13] A. MOSTOWSKI, Represenlability of sets in formal systems, Recursiveive f Function Theory, Proceedings of Symposia ia in Pure Mathematics, cs 5 (1962), 29-48. [14] E L . POST, Recursively enumerable sets of positive integers and their decision problems, Bulletin of the American Mathematicalcal Society, 50 (1944), 284-316. [15] G, E. SACKS, On the degrees less than Ol, Annass of Mathematics, cs 77 (1963), 211-231. [16] G. E. SACKS, The recursively enumerable degrees are dense, Annass of Mathe­ matics, 80 (1964), 300-312. [17] G E. SACKS, Admissible ordinals, Post's problem and regularity, in preparation.

77 338 [18] [19] [20] Duke

METARECURSIVE SETS C. SPECTOR, Recursive well-orderings, this JOURNAL, 20 (1955), 151-163. R. M. SMULLYAN, Theory of Formal Systems, Princeton, 1961. C. E. M. YATES, Three theorems on the degrees of recursively enumerable sets, Mathematical cal Journal, to appear.

UNIVERSITY OF PARIS CORNELL UNIVERSITY

78

POST'S PROBLEM, ADMISSIBLE ORDINALS, AND REGULARITY BY

GERALD E. SACKS(i)

1. Introduction. The basic notions of metarecursion theory were introduced in [7]. Metarecursion theory is an attempt to generalize the ideas and arguments of recursion theory from the natural numbers to the recursive ordinals. Ordinary recursion theory concerns itself with finite sets of natural numbers. Metarecursion theory deals in an analogous fashion with metafinite sets of recursive ordinals. In [7] it was seen that two of the deepest results of ordinary recursion theory, the solution of Post's problem [3] and the maximal set construction [4], gener­ alize to the metarecursive case. Theorem 4 of [7] states that there exist two metarecursively enumerable sets of recursive ordinals such that neither is metarecursive in the other. Kreisel [6] casts doubt on the contention that Theorem 4 of [7] is the correct generalization of Post's problem. He wonders how to correctly formulate the notion of Turing reducibility for recursive ordinals and he dis­ cusses four possible choices. It is not our place to decide the correct notion of Turing reducibility for metarecursion theory; however, in §4 wc show that there exist two metarecursively enumerable sets such that neither is reducible to the other by any of the four methods discussed by Kreisel. In [7] two sets of recursive ordinals are said to have the same metadegree if each is metarecursive in the other. It is not known if the metadegrees of meta­ recursively enumerable sets are order-isomorphic to the Turing degrees of re­ cursively enumerable sets, but all the existing evidence seems to favor an affirm­ ative answer. Driscoll [2] has shown the metadegrees of metarecursively enumerable sets are dense; i.e., between any two comparable ones, there is a third. His proof is not a mere translation of the argument of [12]. There exists a formidable obstacle to the lifting up into metarecursion theory of results about recursively enumerable sets. Call a set of recursive ordinals regular if its inter­ section with every metafinite set is metafinite. In ordinary recursion theory, the intersection of any set of natural numbers with any finite set is finite; this platitude is essential in many arguments about recursively enumerable sets. Unfortunately there exists a multitude of nonregular, metarecursively enumerable sets [7]! Received by the editors September 28, 1965 and, in revised form, February 8, 1966. (') The preparation of this paper was supported by U.S. Army Contract ARO-D-373. The author wishes to thank Professor G. Kreisel for invaluable help in the writing of this paper. 1

79 2

G. E. SACKS

[July

Nonetheless, Driscoll [2] was able to make a long detour around the obstacle of nonregularity with the aid of the principal result of §2: each metarecursively enumerable set has the same metadegree as some regular, metarecursively enumerable set. Spector [14] showed that all nonhyperarithmetic, n } sets have the same hyperdegree. It is possible to interpret his result as the answer to Post's problem for n j sets. In this paper we offer what we hope is a happier alternative. Suppose we identify the natural numbers with the finite, recursive ordinals. Then, as was seen in [7], the IIJ sets of natural numbers coincide with the metare­ cursively enumerable sets of finite, recursive ordinals. In §3 we show that there exist two n j sets of natural numbers such that neither is metarecursive in the other. Kripke [8], [9] has succeeded in generalizing a large portion of recursion theory by introducing the notion of admissible ordinal. An admissible ordinal has the property that the set of all lesser ordinals behaves very much like the natural numbers when one attempts to initiate the pre-Friedbergian arguments of recursion theory. Kripke's definition of admissible ordinal is expressed in terms of closure properties that refer to an equation calculus analogous to the one employed by Kleene [5] for ordinary recursion theory. Kripke's calculus is very convenient for the purposes of metarecursion theory and was utilized in [7]. (It turns out that col, the least nonrecursive ordinal, is the first admissible ordinal occurring after to.) In this paper, our first duty is metarecursion theory, but we cannot resist indicating which of our results generalize to every admissible ordinal. In particular, we give a solution to Post's problem which is uniform with respect to all admissible ordinals; i.e., we give a single argument in which the admissible ordinal a occurs as a parameter. We conclude with a discussion of some open questions. 2. Regularity and metarecursive enumerabiliry. In [7] a set of recursive ordinals was said to be regular if its intersection with every metafinite set was metafinite. Each metarecursive set is regular. Each nonhyperarithmetic, XT* set of natural numbers (regarded as a set of finite, recursive ordinals) is a nonregular, meta­ recursively enumerable set. In metarecursion theory we are forced to deal with bounded, metarecursively enumerable sets which are not metarecursive. This last remark is the reason that the phenomenon of nonregularity makes it difficult, if not impossible, to lift many results of ordinary recursion theory up into meta­ recursion theory. But the situation is not as gloomy as it appears, since we can show by an essentially elementary argument that each metadegree of a meta­ recursively enumerable set contains a regular, metarecursively enumerable set. In §4 we see that there exist metadegrees of metarecursively enumerable sets which contain only regular sets. The overall situation concerning regularity is not clear and is discussed in §7.

80 1966]

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3

THEOREM 1. Each melarecursively enumerable set has the same metadegree as some regular, melarecursively enumerable set.

Proof. The main idea of what follows is inspired by Dekker's notion of deficiency set [1]. We use what might be called a double deficiency set. Let {Kx} be a standard indexing of the metafinite sets; i.e., there exist metarecursive functions j and k such that for each a, Ka = {/i\$ <j(a) and /c(a,/?) = 0}. Let Q be a n{ set of unique notations for the recursive ordinals; let n be a metarecursive function such that for each a, n(a) is the unique natural number in Q which is a notation for a. For convenience we assume that Q is a linearly ordered subset of Kleene's 0. We write A^MB to indicate A is metarecursive in B. Intuitively, A ^MB means there exists a method M for determining the truth or falsity of both of the following two assertions for an arbitrary metafinite K: K c A and K £ cA; the method M must be such that the status of K is decided by the fact that B satisfies some metafinite set of conditions such as those expressed by K^B and K-2 — cB. Thus M can be thought of as a metarecursively enumerable set of 5-tuples of the form <0 K K, K-, /> and <1 K K, K-, i> where K £ A if and only if for some ^ £ f i and K2 £ cB, we have <0,'K,K»K 2 ,0> e M, where K $ A if and only if for some K, c £ and K-,^cB we have (0 K K, K-, 1> eM and where a similar relation holds between the 5-tuples' of the form

k ,

Cy4, and B.ma

We write A SmB (read /I is many-one reducible to JB) to indicate that for some metarecursive function/, we have 0i£A<->/(a)eB. If A ^mB, then A^MB. We write A ^WB to indicate ,4 is weakly metarecursive in B. Let Fa be a standard enumeration of all the finite sets of recursive ordinals. We write A^fB (read A is finitely reducible to B) to indicate A^WB (there is an e such that A = {e}B) and there is a metarecursive g such that for each a, the computation of the truth value of a e A draws only on the information expressed by the membership of Ff(a)sB of and Flia)ccB.ibili If A^fB and Bg„C, then Ag,wC. We need the notion of finite reducibility because Driscoll [2] has shown that the relation ^ w is not transitive.are Suppose A is metarecursively enumerable and we wish to prove A£MB. It is sufficient to find a method M for showing K£cA when and only when K^cA is true. We merely combine the method M with an enumeration of A. If K c A , then eventually we will enumerate all of K in A. If X $ / I , then M will show that for some a e K, we have {a} s c.i4. If K £ c<4, then M will show K^cA. If X $ o 4 , then for some aeX, we will enumerate aeA. We will refer to the above observation as principle P .erable Let A be a metarecursively enumerable set. We aim to find a metarecursively enumerable, regular B in the same metadegree as A. Let A* = {a | ,4 n X„ / 0 } . We check that /I and A* have the same metadegree. Observe that (1) KazcA* +-» | J {KyyyeXa}<=cX ~ *„«, S cX ~ /»(a)eM*,

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where h is metarecursive. (We are using very heavily the fact that a metafinite union of metafinite sets is metafinite.) It follows from principle P and (1) that A* is metarecursive in A. The definition of A* implies that A is metarecursive in A*. L e t / be a one-one, metarecursive function whose range is A*. We introduce the double deficiency set o f / : Dj = {«\(EW(fi)<m&nW))<TiU{*))&P

> «)}■

We show Df is regular by showing Df2C\Ly is metafinite for every y (L, = {a | a < y}). Fix y; we need only find a t such that (2)

D / n Lr = {«| {EP)!<x{fW «)} •

We define a partial metarecursive function d by induction: d(0) £ ^(/J > y & (£«)(a < y &/(/?) < / ( a ) & ij(/(/0) < l(/(«)))); d(n + 1) £ /,/?(/? > d(n)&(m) mgll (/(/?) < / ( * . ) ) V '/(/(«) < MM*")))) & (£«) (a < y &/(/?) < /(a) &IJ(/(/J) < »;(/(«)))).

Clearly, the domain of d is a finite, initial segment of the natural numbers. Let T be the least ordinal greater than y and greater than any element of the range of d. Then T satisfies (2). Note that t , regarded as a function of y, is weakly metarecursive in A*. Then it follows from (3) and principle P that Df is meta­ recursive in A*. (3)

K, E c£>/ « K , n £ > / OL A , } = 0 .

(Recall that X . E Lm and ; is metarecursive.) Now we make an assumption about Df and show A* is metarecursive in Df. Let p be defined by p(a) s /z/?(/(/D > a & >?(/(j?)) > ff(a) &fi$ Df). We assume p is total. Then p is weakly metarecursive in Df, and (1) tells us K. s ^*«-►ft(«)M * ~ /'(«) i { / ( « | P < M*))} ■

Finally, we are faced with the problem that p may not be total. Let y be the least ordinal such that (4)

iP) if(P) £ y V nif(P)) i rfa) ype

Df).

Then there must be a a such that !;(/(/?)) > rj(y) for all /?><x, and (5)

(P),>.V(P)£jVfieDf).

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Let A* = A*C\Ly. We claim A* and A* have the same metadegree. Let A' = A*-{a\a^yV (£*)«„ (a =f(5))}. It suffices to show A' is metarecursive in A*. We make repeated applications of (5): aeA'^(EP)?>a(f(P) PeD/^(E5l)(51

=

a&peD,2),

>P&f(51)

j(a)),

2

51><j&f(5l)>y-+5leD, , 5leDf2^{E52)(52

> <5, &f(S2)
< *K/(A))),

52>a&f(52)>y-+52eDf2,

(E;(«)). But then we have <xeA'~(Ep)(ES)(a

< (1 < 5&tx =/(/?)

(6) &/(^?<«&^(/(<5))

<>?(*))•

Let Fm be the finite set {E | S g y &ri(e) < r](a)}; we can assume t is metarecursive, since our unique notations form a linearly ordered subset of Kleene's 0. Then (6) becomes (7)

xeA'~{EP)(E5)(o


=/(/?) &/(<5)e£, (0) ).

With the help of (1) we obtain (8)

Ka <= cA'~(Ka

- (A* - A')) 5=

cA*~q(.a)ecA',

where q is metarecursive (we assume some repetition in our indexing of the metafinite sets). Principle P, (7) and (8) imply A' is metarecursive in A*. Let B = {>;(a)|ae/ly*}. Since A* is bounded, it must be that B and A* have the same metadegree; in fact, B^mA* and A*^mB. B is useful because it contains only finite ordinals. Let g be a one-one, metarecursive function whose range is B. Let £>, = {a | (Efi)(S(P) < g(a) & / ? > « ) ] . D, is seen to be regular by the same argument we used to see Df2 is regu­ lar. We claim D, and A,* have the same metadegree. De is metarecursive in B, and hence in A*, by the same argument used to show Df2 is metatecursive in A*. It remains only to show A* is metarecursuve in De. We note

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K. £ eA**-*(i)(r eKa&x
=0)

(9) ~

KmmZcA~Hm(a))ecA*,

where m is metarecursive. We can choose h and m so that

(10) (a)(&(m(«)) > y 4 ^ ( m ( i ) ) > n(y)) &(5)d>a(h(m(a)) #/(<5))). Then we have (11) K a <= C / 4 y *^/,( m ( a ))ec/4'. Now B <, w De by an argument similar to the one used to show A* ^ M 0 / . (Actu­ ally, this is the argument used by Dekker [1] to prove the range of a one-one, recursive function is recursive in its deficiency set.) But then Ay*£wDt, since Ay*^mB. It follows from (7) that A'gfA?*; consequently, A'^wDg. This last together with (11) and principle P give A* g MDg. The proof of Theorem 1 suffers from a certain lack of uniformity; this defect is discussed in §7. Let a be an arbitrary admissible ordinal [8], [9]. Kripke calls a set of ordinals a-finite if it is y-recursive and is bounded by some ordinal less than a. Call a set of ordinals (less than a) regular if its intersection with every a-finite set is a-finite. Define the notion of "a-recursive in" by strict analogy with the notion of "meta­ recursive in" as defined in [7]; simply replace "metarecursive" by "a-recursive" and "metafinite" by "a-finite". Let us say two sets have the same a-degree if each is a-recursive in the other. THEOREM 1". Each a-recursively enumerable set has the same a-degree as some regular, a-recursively enumerable set.

Proof. We proceed as in Theorem 1, but some modifications are necessary. Let a* be the projectum of a. (Kripke defines the projectum of a to be the least ordinal /? such that there exists a one-one, a-recursive function whose range is bounded by p.) Note that an a-recursively enumerable set bounded by an ordinal less than a* must be a-finite; this fact is analogous to the truism that every finite set of natural numbers is metafinite. Let f be a one-one, a-recursive function whose values are less than a*. Then f(a) plays the same role that n(a) did in the proof of Theorem 1. The only difficulty lies in showing that A' is a-recursive in A* and that A* is a-recursive in Z>„. Equation (6) becomes (6°)

n E A ■«-»(£/}) (ES) (a
/(/?) &/(<$) ^ y
Let Km

be the a-finite set {e | e g y &/(e) < ffa)}, where s is a-recursive.

(7°)

n 6 A' ~ (E/J) (ES) (o
/(/?) &f(8) e KsM).

The set {t(f(5)) |/(<5) e Ks(n)} is a-finite, since it is a-recursively enumerable and bounded by t(r,) < a'. But then the set {/(<5)| f(5)eKsM) is a-finite. It follows from (7°) and (8) that A' is a-recursive in A* The computation of the truth-

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7

value of ye A' from A* uses only information about the membership of A* in­ volving ordinals s such that t(e) < r(i;). But the computation of the truth-value of tieA' from B uses only information about the membership of B involving ordinals less than r(r\) < a*. Dg is not a-finite, so there must be a r such that #(T) > /(>;) and T^D?. Any member of B less than /(i?) must be enumerated by g before g{x). Thus the computation of the truth-value of r\eA' from Dg is accomplished using only finitely much information about the membership of Dg. In short, A' is weakly a-recursive in D . This last together with (11) and principle P gives A* is sc-recursive in D We can strengthen Theorem 1" as follows. Let a be an admissible ordinal. Let cB, the complement of B, denote {/? |/?< a] - B. Call an a-recursively enumerable set unbounded if it is not bounded by any ordinal less than a. We say an a-re­ cursively enumerable set B is simple if cB is unbounded, and if every a-recursively enumerable subset of cB is a-finite. Then we can prove: each non-a-recursive, a-recursively enumerable set has the same a-degree as some regular, simple, a-recursively enumerable set. This is a consequence of Theorem 1", since the ideas of Dekker [1] work quite nicely on regular sets, and since the deficiency set of a function which enumerates a regular set is regular. 3. MeUdegreesof nJ sets. Before we prove Theorem 2, we recaL some definitions of [7]. The T-prcdicate of [7] is analogous to that of Kleene [5]. its definition is: T,1 (B,e,<JL,o)~t(e,o)

= <e,M,JV,a,y> &M s B&N s cB;

l{e,a) is a special metarecursive function which indexes "deductions". Roughly speaking, T,1 (B,e,a,a) says: B is an arbitrary set of recursive ordinals; e is the Godel number of a finite system of equations £; and t(e,a) is the Godel number of a metafinite "deduction" from E and B whose last line gives the value y of some function at argument a. U(e,a) is a metarecursive function such that if t(e,a) = <e,M,N,a,y>, then U(e,a) = y. Two useful partial functions are: {e}oB(a) = y <-» T, (B, e, a, a) & U(e, a) = y; {e}"(a) = y~(E<j)({e}°(x) = y)&(c)({e}!

is defined -U(e,a)

= y).

Theorem 2 of [7] states that if A is weakly metarecursive in B, then there exists a finite e such that (a)|>(a) = |>} B (a)], where a (a) is the characteristic function of A. The converse is also true. Let M{e,a) and N(e,a) be metarecursive functions such that if ((, then M(e,c)= M and N(e,o) = N. (We are making use of a standard indexing of the metafinite sets which enables us to speak of partial metarecursive functions that take metafinite sets (M,N) as values.) We introduce two partial functions:

85 8 ( P )

G. E. SACKS *[*].'(«) = y~{e}*(x) K[e]»(«) = y~(Ea)

[July

= y&K-(N(e,o)

KJB) is infinite; K

(*[<>>» = ?)&(
The difference between K [e] B and {e}B is the key that unlocks the proof Theorem 2. The "deductions" associated with K [e] B are restricted; each them permits infinitely many members of K not to be in B or in N(e,a) £ The B of Theorem 2 (as well as the A) have the property that K [e] B ~ {e}B for e and certain metafinite K's depending on e.

of of cB. all

THEOREM 2. There exist two U\ sets of natural numbers such that neither is weakly metarecursive in the other(2).

Proof. We intend to metarecursively enumerate two sets, A and B, of na­ tural numbers with an eye to meeting certain requirements. The first set of re­ quirements will insure that A and B are "incomparable". The second set will insure that neither A nor B are metarecursive. The two sets of requirements are redundant, since if A is not weakly metarecursive in B, then A is not metarecursive. Sometimes, however, redundancy (overkill) is useful. We will simultaneously attempt to satisfy both sets of requirements, but we will be unable to prove directly that the first set is successfully met. Instead, a direct argument will be given that the second set is met, and then (and only then) it will be possible to show with the aid of the properties of K [e] B that the first set is met. We handle conflicting requirements by means of a system of priorities similar to that of Friedberg [3]. To be more precise, in the course of the construction, we make only finitely many attempts to satisfy each requirement. Driscoll [2] proves theorems about metarecursively enumerable sets by means of priority arguments in which infinitely many attempts are made to satisfy individual requirements. We define eight partial metarecursive functions, A(a), B{a), K(c,e), Ua.e), p(a, e), q(a, e), s(a, e) and v{a,e) for all a and all finite e by induction on a; p,q,s and v will take finite ordinals as values, while the other four functions will take only metafinite sets as values. Let r be a metarecursive function taking only finite values such that for each c 0. If the partial function XS \ K(5, e) changes value only finitely offen as increases through the ordinals less than cr, let K(o*,e) equal its "last" value; i-e-, (2) It is not hard to show, using Spector [14], that if B is a nonhyperarithmetic, n j set, then every nj set is invariantly definable from B (regarded as a set). This last notion is defined in Kreisel [6].

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POSTS PROBLEM, ORDINALS AND REGULARITY

(E5)s,e)

=

9

K(a*,e)).

If the partial function changes value infinitely often, let K(a*,e) = K(0,e). (We will see that the partial function Xa\K(a,e) changes value only finitely often as a increases through the recursive ordinals.) Define L(o*,e), p(o*,e), q{a*,e), s(o*,e) and v{a*,e) similarly. Let c = r{a) and o = i+j, where t is a limit ordinal and j is finite. Let A(a*) = \J {A{5)\5
e^c;

p(a, k) = /ix[x e L(a, k) - B((r)&(Ey)(y e L{<J, k) - B{a) &x>y>

v(a, k) + v(a*, k) + p(
v(a,k) = ^x[xeL((T,/c)-B((T)&x>p((T,/c-l)]

if

k>c.

Case 1.2. s(( c. If K(a, e) - A{a) is finite for some e, then define q(a, e) = q(a*, e) and s(<x, e) = s{a*,e) for all e. Suppose K{o,e) - A(6) is infinite for all e; then set: q(
/^[xeKter.^-^&^H^eK^fc)-^) &x > j> > s(a,fc) + s(a*,fc) + o((T*,fc))] if fc £ c;

s(q(a ) fc-l)] if

k>c.

Case 2.2. K^*, 0 does not satisfy the hypothesis of Case 2.1. Change nothing and proceed directly to stage a + 1.

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Case 3. j = 2 (4). Similar to case 1 with A and B interchanged; we also interchange s and p, K and L, and p md q. Case 4. ;== 3 (4). Similar to Case 2 with A and B interchanged. If case 4.1 holds and L(a,e) - B{a) is infinite for all e, then sett p(a,e) = p{a*,e) if e ^ c ; v(a,e) = v(o*,e) if e g c + 1; ;>(cr,ft) =

iix[xeL(a,k)-B(o)&(Ey)(yeL(o,k)-B(o) &x>v>

v{a,k) + v(a*,k) + pO*,fc))] if fc> c;

I

p{a, k-1)]

if fc > c + 1.

That completes the construction. Note that we are using the fact that the predicate, a is the index of an infinite, metafinite set, is metarecursive. The system of priorities we are following is buried in the above equations; we make it explicit. Our requirements are: B # {e}A, A # {ef, B # {e}*, and A # {e}*5 where e ^ 0. B ¥= {c}A has higher priority then A =# {e}Bior all e ^ c. (Any conflict between two requirements is resolved in favor of the requirement of higher priority.) A ^ {c}B has higher priority than A / {e}*, B # {e}*, and B # {e}A for all e > c. A ^ {c}* and B # {c}* have higher priority than B * {e}A for all e ^ c. Let /(a) be a function from the recursive ordinals into the recursive ordinals. We s a y / 0 ) changes value only finitely often as a increases through the recursive ordinals if there exists a finite sequence, 0 = <x0 < ox < ■■■ < am = co,, such that for each i < m, f(a) is constant when restricted to
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imply infinitely many changes in Xo\p(o,c) for some c < e by Case 2.1. A similar relationship holds between L and q. LEMMA

2.

For each e and a, K(o,e) - A(a) and L{a,e) - B(o) are infinite.

Proof. Fix e ^ O and <x>0. First we show L{a*,e) - B{a*) is infinite. By Lemma 1, there is a least y < a such that L{o*,e) = L{o,e) = L(y,e) = L{x,e) whenever a > x > y. Fix m\ we locate a member of L(y,e) - B(a*) greater than in. Clearly, f< v(a*,f) < p(a*,f) < v(a*,f+ 1) for all / . Let / be such that e+m 6 > a; then p(x,d) = p(a*,d) = p(S,d) when a > <5 > T and d^f. Then y e L ( T , / ) - B ( T ) , and so y e L(x,e) - B(x), since e• e L(y / ) — B(T) We still must see that y i B(o*) - B(x) Suppose yeB~{5)- B(5*) where a > 5 > x. Then y = p(5*,c) or y = y(<5*,c). But v ^ D(<5* C) because v(8* f— 11 < t»C<5* f ) < v < D(\5* fV similarly v ^ ufo"* cl Finally, y e L(y,e) — B(V), since e < / . Now we show L(
3.

For each c, A * {c}* and B * {c}*.

Proof. Fix c and let s(c) = lim„s(<7,c) (the final value of s(o,c)). Suppose {c}*(s(c)) is defined. Let y be the least 5 such that s(c) = s(<5,c) = s(a,c) for all a > <5. Then s(c) i A{y) by Lemma 2. If there is a a such that {c}*(s(cj) = 1, then s(c)eA by Case 1. If {c}*(s(c)) = 0, then s(c)fA, since s and q have disjoint ranges. A similar argument shows B # {c}*. LEMMA

4. For each c, B*{c}A

and A / {c}B.

Proof. Fix c and let p(c) = \im.p( 5 and e ^ c. Then p(c)p and K(<, * ,r, [c]/ (, ' ,) (p(c)) = 1. It is enough to show N(c,S) r\A = 0, since {c}A(p(c)) is defined. Now K(a,e) C\N(c,5) = 0 for all e^c. But K(a,c)~K(a,c) for all a>a, since a> p; consequently, K(x,e) s K{a,c) for all T ^ «r and e ^ c. Thus Ar(c,<5)r\K(T,e) =0 for all x ^ tr and e ^ c. Then g(i,e) and s(x,e)$N(c,5) for all t S cr and e ^ c. So if 7V(c,(5) n X # 0 , there must be an f
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s(i*J)eN(c,5)r\A(z). Both of these alternatives are ruled out by the fact that o p . Now suppose p(c) i B. We claim {c}\p(c)) # 1; suppose otherwise. Then there is a y and a /? such that 7 < 0 and {c}^"'*)(p(c)) = 1 for all a ^ p; in addition, M(c,y) <= A and JV(c,y) £ c,4. Assume P^p. Then K(cr,c) = K(P,c) for all erS p. Any element added to A after stage jS must be a member of X(/?,c) wiih finitely many exceptions (at most c). Consider K(0,c) - (N(c,y) VA(6)) for any er ^ p; it must be infinite, since otherwise (A - A{P)) - (N(c,y) U U{a)) would be finite and A would be metafinite. But Lemma 4 says A is not metafinite. But then K(a~-ci[cYl"')(p(c)) = 1 for all cr ^ /J, and some instance of case 2.1 would put p(c) in 5 . A similar argument shows /4 # {c} B . Note that it was not sufficient for us to know that various limits such as lim^Kcr.e) and limbe.,e) existed. Our key lemma, Lemma 2, used Lemma 1 very heavily; we had to know that the function Xc\Up,e) changed value only finitely often as a increased through the recursive ordinals. This mode of argument seems to generalize only to those admissible ordinals which project into cardinals. We make a conjecture on this matter in §7. THEOREM 3. Every countable partial ordering is embeddable in the partial ordering of metadegrees of U\ sets.

Proof. Combine the argument of Theorem 2 of §5 of [11] with the proof of Theorem 2 above. The proof of Lemma 2 above becomes more complicated, but no conceptual change is required. 4. Subgeneric sets. The notion of subgeneric set is due to Kreisel. Before we give the definition, we recall some features of Kripke's equation calculus intro­ duced in [8] and sketched in [7]. Kripke's calculus is modeled on Kleene's [5]. The terms include function letters and numerals denoting recursive ordinals. The deduction rules are similar to the substitution rules of Kleene with one exception, rule (R3). By means of (R3) it is possible to deduce (x)x
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In [7] it was observed that a function /(?) = 5 e S f , . We now characterize the relation, "weakly metarecursive in", in a similar fashion. Let B be an arbit­ rary set of recursive ordinals. Let SE'S = £ U {g(a) = 11 a $ 0} V {#(°0 = 01 a e 0}; for each a > 0 , let SE'g equal the set of all immediate consequences of U { S f * | } > < a } . (Do not forget the condition made in the definition of SE concerning the use of (R3).) It is not true that S£'= ( J { S f ' ' | a
a^A~f(a)O0eSE'<-' aMW((a) = leSE—, where SE's-m is the set of all members of SE>< whose deduction-trees are meta­ finite. Clearly, the height of each member of SE-''""is a recursive ordinal. A ^mcB means the membership relation of A is computable from B using the equation calculus and metafinite deductions only. THEOREM 4.

A SWB~A

^mcB.

Proof. If A ^WB, then the computations used to decide A from B are meta­ finite and are metarecursively enumerable; consequently, A^mcB. Suppose A <,mcB by £. Each deduction permitted is a metafinite object; the roots of each deduction of a e A or a £ A consist of a metafinite set of membership conditions satisfied by B. The set of all metafinite deductions is metarecursively enumerable. It is not difficult to define the partial metarecursive functions needed to describe how each decision about the membership of A is determined by a metafinite neighborhood condition satisfied by B. Then A^WB. We say B is subgeneric if for every £ , we have Sf;*= U{Sf-*|a < °>i)» where S%'* includes the diagram of B. Intuitively, B is subgeneric if every equa­ tion deducible from £ and B is deducible from £ and B at some recursive ordinal. It is easily checked that every metarecursive set is subgeneric, and that every bounded, nonmetarecursive, metarecursively enumerable set is not subgeneric (first observed by Kreisel). In a letter to the author, Kreisel asked: do there exist subgeneric, nonmetarecursive, metarecursively enumerable sets? The reason

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[July

for his interest lies in the light shed by the notion of subgenericity on the various notions of reducibility discussed in [6]. It turns out that one can prove the exis­ tence of nonmetarecursive, subgenenc sets by a standard measure-theoretic argument or by a "forcing" argument. (In fact, with probability one, a randomly chosen set of recursive ordinals is subgeneric.) However, a much stronger result, expressed by Theorem 6, can be obtained by the priority method. First we intro­ duce a generally stronger notion. We say B is hyperregular if the range of each function weakly metarecursively in B, when restricted to a metafmite set, is bounded by a recursive ordinal. We will see that a set is both regular and hyper­ regular if and only if all possible deductions from the set are metafinite. THEOREM 5. A regular set of recursive ordinals is hyperregular if and only if it is subgeneric.

Proof. Suppose B is regular and subgeneric. First we show B is completely regular: every bounded set weakly metarecursive in B is metafinite. Let C be bounded and weakly metarecursive in B. Let y be the least upper bound of C. There exist functions m(5) and n(5) weakly metarecursive in B such that the truthvalue of 5 e C is deducible from B (and some £ not depending on 5) using only membership information about B expressed by Km(j) c B and K„(i) c cB. Sup­ pose p = {m(<5) + n(6) I 8 < y) is unbounded. Then Q, a IIJ set of unique notations, is computable from {|t| \tep }by means of the unrestricted equa­ tion calculus. But then Q is computable from B, and consequently, Q is subgeneric. No nonmetarecursive II,1 set is subgeneric, so p must be bounded. It follows from the regularity of B that C is metafinite. Let / be weakly metarecursive in B, and let K be a metafinite set. Then {<|aJ, | / ( a ) j > | a e K } is bounded and weakly metarecursive in B; by the previous paragraph, it must be metafinite. Now suppose B is regular and hyperregular. Let £ be a finite set of equations, and let /? be a recursive ordinal. Suppose each member of {/(y) = 01y < £} is deducible from £ and B by means of a metafinite deduction. Then there exist functions s(y) and r(y) weakly metarecursive in B such that some metafinite de­ duction of /(y) = 0 from £ and B uses only the membership information about B expressed by KHnsB and K,M^cB. But {s(y)|y
Proof. Any subgeneric metarecursively enumerable set is regular. THEOREM 6. There exists a nomnelarcursive, metarecursively regular, hyperregular set of recursive ordinals.

enumerable,

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15

Proof. Let Ra,R,, - , R e , •• • (e < m) be the standard enumeration on the metarecursively enumerable sets; let R° be all the members of Re enumerated prior to stage a; Xoe\R° is a metarecursive function which takes metafiiiite sets as values. Let r{a) be a metarecursive function taking only finite values such that for each c, the set {o|r(o) = c} is unbounded. At stage a we consider putting an element into A in order to insure that ,4 (~\Re^ 0 if Re is unbounded; in this fashion we make A nonrecursive. At the same time we try to preserve all values of {e}A(m) (m < to) as they develop. We do not add an element of Rc to A if ii would change the apparent value of {e}A(m) for any m all members of N(e,8); otherwise, let y{e,a) = 0. We define d(e, a) = px(x ^ y(e, a)&(p)(p < a - x ^ d(e, p))) for each e. Let r(a) = c. If R'enA(a*) = 0 , ana if there is i p sucs that peR"c8t(e)td(e,(j))&p greater than any member of (J {Rri6) n 04(<5) - ,4(<5*)) |5 < a & r(<5) < c} by at least to, add the least such p to A(a*) to obtain yl(cr); otherwise, let A(a) = Let ^ = ( J { X ( c r ) | a < a ) I } . LEMMA

8. for each e, \ima d(e,a) exists.

Proof. Fix e. For each c ^ e, there is at most one a such that at stage c: A is augmented for the sake of Rc. (This is the purpose of the clause R° n 4(ff*) = 0.) Let a be so large that A is not augmented at stage p for the sake of Rc for any p ^ a and c g e. The set {y(e, p) | p S 0 for some <5 ^ or, then y(e,p)^y(e,5) for all p>c5; suppose T^CT is such that y(e,T) is the maximum value. Then d(e,p) = d(e,t) for all p § T. (Note that if a is added to /I for the sake of Rcat stage p ;> tr, then and a > d(e,p).) LEMMA 9.

,4 is nor

metarecursive.

Proof. First we show c4 is unbounded. Fix a; we claim {/?a< fi
93 16

G. E. SACKS

[July

for the sake of Rd, where d> c, then p ^ a + co, since p must be greater than t by at least co. Thus any member of {/? | a < a + co} aaded dt A after rtage e must be added for the sake of Rd, where d limad(e,a)), and p is greater by at least co than any member of |J {R6r{S)n(A(5)-A(5*))\d
Proof. Fix e and suppose {e}A(m) is defined for all m< co. Fix m; we show how to compute {e}A(m) metaeflectively. Let a be so large that A is not aug­ mented at stage p for the sake of Rc for any p ^ a and c g e, and such that d{e,p) = d{e,a) for all p ^ e r . Let t(m) ^ all members of N{e,p). But then d(e,y) > all members of N(e,p) for all y ^ i(w), and so N(e,p) S cA; consequently, {e}^ IIm),)i (m) = {ef(m). It follows from Lemma 10 that A is both regular and hyperregular. If A and B are sets of recursive ordinals, let A ^CB mean that A is computable from B by means of Kripke's equation calculus; of course, deductions whose heights are nonrecursive ordinals are permitted. If B is regular and hyperregular, and A is computable from B, then A is weakly metarecursive in B. Kreisel considers several notions of invariant definability in [6] and in private communications. In a future publication he will present the most appealing of these notions; for now, let us denote it by ^ 7, and let us be content to hear some of Kreisel's observations concerning it; ^ j is transitive; if A and B are sets of recursive ordinals and A ^tB, then A ^CB; if A and B are sets of natural num­ bers, then A-^IB+->A<:CB^A is hyperarithmetic in B. It follows from Theorem 7 that there exist two metarecursively enumerable sets such that neither is invariantly definable from the other. THEOREM 7. There exist two metarecursively enumerable sets such that neither is computable from the other by means of Kripke's equation calculus.

Proof. We lift Theorem 1 of §5 of [11] up into metarecursion theory as fol­ lows: each regular, nonmetarecursively enumerable set B is the disjoint union of two metarecursively enumerable sets of incomparable metadegrees. The regularity of B is all-important; we need to know that for each a we eventually finish enu-

94 1966]

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17

merating all members of B less than a. Except for this point, the lifting up is routine. Let B be regular and hyperregular. Then B = B0nBj, B0^^I =*. and the metadegrees of B0 and B1 are incomparable. It follows that B0 and B^ are regular and hyperregular; but then neither is computable from the other. We conclude the present section by outlining some generalizations to arbitrary admissible ordinals. At the beginning of this section we sketched Kripke's equa­ tion calculus for co,. The only change needed for an arbitrary admissible a is the introduction of numerals forordinals less than a. We define SE and S?'* as before. By the definition of admissibility, we have S a £ = M {Sj| y < a}. Let B be a subset of a. We say B is subgeneric if for every E, S** = U {S*''\ y 8. The following three properties of a regular set B are equivalent to hyperregularity: (a) every deduction from B is a-finite; (b) the range of each function weakly a-recursive in B, when restricted to an a-finite set, is a-finite; (c) the restriction of each function weakly a-recursive in B to an a-finite set is an a-finite partial function. THEOREM

Proof. We indicate only that hyperregularity plus regularity implies property (a). The second half of the proof of Theorem 5 uses no special properties of recursive ordinals, generalizes to an arbitrary admissible a with no change, and gives the desired result. COROLLARY

2. Each regular, hyperregular set is subgeneric.

THEOREM 9. For each admissible a, there exists a non-a-recursive, a-recursively enumerable, regular, hyperregular set.

Proof. First we mimic the proof of Theorem 6; the argument splits into two cases distinguished by the truth-value of the statement: a*, the projectum of a, is less than a. If a* < a, the argument of Theorem 6 is followed closely, and the ordinals less than a* take over the part of the natural numbers. If a* = a, then a slight modification is necessary; instead of preserving values of {e}A(fi) for all j5 < a*, we preserve {r.}A([}) for all /? < c, and make use of the repetition inherent in any standard enumeration of the functions weakly a-recursive in A. Thus we obtain a non-a-recursive, a-recursively enumerable set A with the prop­ erty that any function weakly a-recursive in A, when restricted to an a-finite set, is an a-finite partial function. 5. The nonexistence of maximal sets. Let a be admissible. We say an a- recursively enumerable set M is maximal if cM is not bounded by any ordinal less than a

95 18

G.E. SACKS

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and if for each a-recursivcly enumerable set R, either R - M or cR is bounded by some ordinal less than a. Clearly, a maximal set is unbounded. The maximal set construction of [7] shows that if a* = to, then a maximal, a-recursively enu­ merable set exists. It follows that there exist uncountably many, countable, ad­ missible a such that a maximal, a-recursively enumerable set exists. (In fact, it was shown in [7] that there exists a metarecursively enumerable set M with un­ bounded complement such that for any metarecursively enumerable R, either R - M or cR is finite.) The purpose of the present sectton is to give an example of an important theorem about recursively enumerable sets which does not gen­ eralize to arbitrary admissible ordinals. THEOREM 10. There exists a countable, admissible ordinal a such that no a-recursively enumerable set is maximal.

Proof. First we show that no maximal, aleph-one-recursively enumerable set exists. Assume that V = L. L e t / b e a one-one, aleph-one-recursive function which assigns an aleph-one-finite set of natural numbers to each countable ordinal; the equations defining / merely specify some diagonal procedure. Let <» be the usual one-one function which assigns a member of the closed unit interval to each set of natural numbers. Then h = gof can be viewed as an aleph-one-re­ cursive function, which associates a real number with each countable ordinal. Suppose M is a maximal, aleph-one-recursively enumerable set. Let IU/2,/3,--be the usual enumeration of all open subintervals of the unit interval with rational endpoints. For each n, let J„ = {y | h(y) e I„); then J„ is aleph-one-recursively enumerable if a natural h is chosen; in fact, J„ is aleph-one-recursive uniformly in n. For each n, J„- M or rJn - M is sountable; let K„ bb en - — if J„ - — is countable, and let Kn be cJ„-M otherwise. Let K = \}{Kn\n
96 1966]

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Note that the argument of Theorem 10 shows there does not exist any maximal CA (complementary analytic) set of real numbers. (Call a CA set M maximal if its complement is uncountable and if for each CA set R 2 M, either R - M is countable or cR is countable.) This fact combined with the existence of maximal U\ sets provides strong evidence for the following assertion: the analogy between recursively enumerable sets and IIJ sets is much stronger than the analog between n j sets and CA sets. We will give other evidence in [13]. Of course, one is free to say that the above notion of maximality has no place in the theory of CA sets. The argument of Theorem 10 still works if one adopts the following definition: call a CA set M maximal if its complement is not Borel and if for each CA set J J 2 M , either R - M is Borel or cR is Borell 6. Post's problem for admissible ordinals. Kripke [8] has obtained a solution of Post's problem for a large class of admissible ordinals. We give a solution of Post's problem which holds for every admissible a, but we leave open the struc­ ture of the upper semilattice of degrees of a-recursively enumerable sets. THEOREM 11. For each admissible ordinal a, there exist a-recursively enumerable sets A and B such that A is not a-recursive in B and B is not a-recursive.

Proof. The introductory remarks of §3 readily generalize to an arbitrary admissible ordinal a; the natural numbers are replaced by the ordinals less than a*, the projectum of a. Thus we have a T-predicate whose definition is: T!(B,E,a,o)^t(e,a)

= <£,M,N,a,y>&M

<= B&N £

CB,

where E < a*, B is a set of ordinals less than a, and M and N are a-finite sets; is a special a-recursive function which indexes "deductions." Tx\B,i,<x,o) says: e is the Godel number of a finite system of equations £ and t(e,a) is the Godel number of an a-finite "deduction" from E and B, whose last line gives the value y of some function for argument a. U(E,O) is an a-recursive function such that if r(£,a) = <e,M,JV,a,y>, then U(c,a) = y. Two useful partial functions are: { £ } > ) = y~T1\£,£,«,o)&V(e,o) = y, I(E.O-)

{£}B(a) = y~(£a)({#(«)

= y)&(a)()(}0B(a) is defined - U{z,c) = y).

If A is weakly a-recursively in B, then for some e< a*, we have (y)(a(y) = {s}B(y)), where a(y) is the characteristic function of A. Let {Ry\y
97

20

[July

G. E. SACKS

We simultaneously define four a-recursive functions, A(o), B(a), d{z,o) and K{z,a) by induction on a. Stage o = 0. A(o) = B(o) = 0 , d{e,o) = z, and K(e,o) = Q for all £ < a * . Stage a> 0.Let«5 = r(8),A{o*)=\J{A(p)\ p X(e,ff»))&p greater than double any member of \J{KM n(B(T)-B(x*)|\z d{%, a))

&x> d(e,a*) + least upper bound of A{a*) + 1) if £=

5&B(o)¥=B(o*).

Now we consider {5}^''\d(S,e)) for all y < a. Suppose d(8,o)tA(o*) and there is a y!(&•) U {d(5,a)}. Let X(5,a) = K(8,a*) + least upper bound of N(d,T), and let JKT(E,er) = K(E,
exists

a t

such

that for

all

Proof. Fix 5 < a*. At the beginning of stage a > 0, we consider adding an element to B for the sake of Ry; this operation occurs at most once before the requirement that B(o*)nR* =0. Let S be the set of all y g £ such that at some stage a, a number of R', where y = r(o), is added to B. S is a-recursively enumerable and is bounded by 5 < a*. It follows that S is a-finite. Let T be so large that all members of S have been enumerated prior to stage T . LEMMA 12.

B is not a-recursive.

Proof. First we show cB is unbounded. Fix y < a and suppose {j3| y ^ jS < a} £ B. Let a0 be an infinite ordinal >y such that a 0 e B(o) - B(o*) and da > every member of B{o*). For each n = 0, let <xn+, be the first member of B to be enum­ erated after <x„ with the property that a 0 < <*„+1 < 2a„; our supposition concern­ ing y implies a„ is well-defined for all n < 5(n) and c(n), must be greater than 2a„; consequently, d(n + 1) < 5(n) for all n, a contradiction. Now fix 5 and suppose R„ is unbounded; we show Rs* cB. Let T be as in the

98 1966)

POST'S PROBLEM, ORDINALS AND REGULARITY

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statement of Lemma 11. Let p ^ x be such that \imad{E,o)eA^\\med{w)eA(P*) for all £ ^ 5. The existence of jS follows from the properties of a* and x and the fact that <5 limcK(e>o)), and p is greater than double any mem­ ber of (J {RyrM n(B(y) - B(y*))) r(y) < 5}; the existence of p follows from the nature of a*. Then B must contain some element of R . LEMMA 13. A is not weakly a-recursive in B.

Proof. Fix <5. Suppose {<5}B(y) is denned for all y < a. Let d = lim^c?, <x)e / I . For some a, we must have d = d(S,a)iA(a*) and {5}?(,,,)(<0 = 1 for some x< a. Also, K(<5, a, then r(p) = p£5 and d(5,p) ^ d(5,p*). But p > < and Ay ||f(<5,yyii a nondecreasing function of y, so d(5,p) = d(6, p*). Now suppose d$A. Then {e}V) # 1, since otherwise the second half of stage a would eventually compel us to put dm. A. The proof of Theorem 11 used only one combinatorial fact concerning all admissible ordinals; namely, an a-recursively enumerable set bounded by an ordinal less than a*, the projectum of a, is a-finite. If a* < a, then a* behaves somewhat like co does in metarecursion theory. If a* = a, then all bounded, a-recursively enumerable sets are a-finite, and it is possible to initiate some familiar arguments of ordinary recursion theory. But we are concealing many combin­ atorial difficulties. The proof of Theorem 8 works because at most one attempt is made to satisfy each requirement of the form: if Rt is unbounded, then RrOB#0. We have not found any combinatorial trick for attacking all ad­ missible ordinals with priority arguments that permit more than one attempt to satisfy any requirement. 7. Open questions and conjectures. What follows is not a complete list of all interesting questions concerning the topics of this paper, but is, hopefully, a selection. Ql. If a is admissible, is a*, the projectum of a, admissible? This question was suggested by Kripke, who has shown that a* must be the limit of admissible ordinals(3). Q2. Do there exist a-recursively enumerable subsets of a* of incomparable a-degrees? We think yes. The answer to Ql might be of some help. Q3. Do there exist a-recursively enumerable sets such that neither is com­ putable from the other by means of Kripke's equation calculus. We think yes. Q4. For what a do maximal, a-recursively enumerable sets exist? In par­ ticular, is there a maximal, a-recursively enumerable set for some uncountable a? If A is weakly metarecursive in B, then A = {e}B for some e as described at (3) Kripke has recently shown the answer to Ql is yes.

99

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G. E. SACKS

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the beginning of §3. The reduction procedure used to compute A from B can be identified with the metarecursively enumerable set {t(e,a)\a < » , } . A typical value of t(e, ;if M c B a n d N S cB, then {e}B(a) = y, where Af and N are metafinite sets. We say e is intrinsically consistent, if for all.a, tr, and a2, if r ^ a , ) = <^-Wi.-Ni.a.y1>, f(e,<^) = (e,M2,N2, a,y 2 >, and (Mi u M , ) n ( W , u N 2 ) = 0 , theny, = y 2 . Q5. For each e, does there exist an intrinsically consistent e* such that {e}A~{e*AA for all A? We think not. In ordinary recursion theory the answer is yes, since we can require that M and N define a finite, initial segment of a charac­ teristic function rather than an arbitrary, basic neighborhood. Q6. How does one define the jump operator for sets of recursive ordinals? A satisfactory definition should have the property that if A is metarecursive in B, then the jump of A is metarecursive in the jump of B. An answer to Q5 should help here. Q7. Does there exkt a metarecursive function / such that for all e,RJ(e)is regular and has the same metadegree as /?,? The proof of Theorem 1 does not provide such an / , but we think it exists. An answer to Q8 might help. Q8. If A is a nonregular, metarecursively enumerable set, does there exist a n } set of the same metadegree as Al We conjecture yes(4). In[13] we will give axioms for recursion theory in the sense that one gives axioms for group theory.Any object satisfying our axioms will be called a recursion system and will consist of an initial segment of the ordinaals and some partial functions on that segment and into it. It will be seen that most of the results of this paper hold for arbitrary recursion systems and that the notion of regular, hyperregular set (inspired by Kreisel's notion of subgeneric set) is the key to the clas­ sification of recursion systems. REFERENCES

1. J. C. E. Dekker, A theorem on hypersimple sets, Proc. Amer. Math. Soc. 5 (1954), 791-796. 2. G. Driscoll, Contributions so metarecursion theory, Ph. D. Thesis, Cornell University, Ithaca, N. Y., 1965. 3. R. M. Friedberg, Two recursively ynumerable sets of Incomparable degrees ofunsohability, Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 236-238. 4. , Three theorems on recursive enumeration, J. Symbolic cogic c2 (1958), 309-316. 5. S. C. Kleene, Introduction to metamathematics, Van Nostrand, New York, ,952. 6. G. Kreisel, Model-theoretic invariants, Theory yo Models symposium, Berkeley, Calif., (to appear). 7. G. Kreisel and G. E. Sacks, Metarecursive sets, J. Symbollc Logic 31 (1966)) 1-21. 8. S. Kripke, Transfiniie recursions on admiisible ordinals. .I II (abstracts), J. Symbolic Logic 29 (1964), 161-162. 9. , Admissible ordinals and the analytic hierarchy (abstract), J. Symbolic Logic 29 (1964), 162. (*l Note added in proof. We can nhow the enswer to Q8 is yes.

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10. E. L. Post, Recursively enumerable sets of positive integers and their decision procedures, Bull. Amer. Math. Soc. 50 (1944), 284-316. 11. G. E. Sacks, Degrees of unsohability, AAnals oo Mathematics Studies No. 55, Princeton Univ. Press, Princeton, N. J., 1963. 12. , The recursively enumerable degrees are densen Ann. of Matha 80(1964), 300-312. 13. , Axioms for recursion theory, ((o appear)) 14. C. Spector, Recursive well-orderings, J. Symbolic Logic 20 (1(15), 1515163. CORNELL UNIVERSITY, ITHACA, NEW YORK

101

ON A THEOREM OF LACHLAN AND MARTIN GERALD E. SACKS1

In [3] we raised the following question: does there exist a recur­ sively enumerable degree d such that 0<"> <0 ( " + 1 > for all n ^ O ? This question was answered affirmatively by Lachlan [l] and by Martin [2]. Lachlan's proof combines a familiar priority argument with the fixed point theorem of Kleene. Martin's proof is a new form of priority argument based on Theorem 3 of section 6 of [3]. In this paper we give a vanishingly short proof of the Lachlan-Martin result without any use of priority. Our argument is an exercise in the fixed point theorem. We exploit, possibly for the first time, a uni­ formity concealed in most proofs of the fixed point theorem. Let A be an arbitrary set of natural numbers, and let Wt, Wf, Wt, • • • be a standard simultaneous enumeration of all sets recur­ sively enumerable in A. For each e^O, let

RU{2n\nGAM2n+l\neWA.\. Then 1$, Rf, Ri, • ■ ■ ii a ssandard ssmullaneous enumeration of sets recursively enumerable in A and of degree greater than or equal to that of A. The proof of Theorem 3 of §6 of [3] makes clear that there exists a recursive function / such that (A)(e)[A < 8 « „ < A'

and

(J?«w)' = K?'l,

where A is the degree of A. By the fixed point theorem, there exists a c such that (A)[RAM

= RA}.

The fact that c is independent of A is the key that unlocks the shiny little box containing our proof. It follows that (A)[A

and

(R*)' = R? ].

But then (A)[A < RA

and Received by the editors December 27, 1965. 1 The preparation of this paper was supported by U. S. Army Contract ARO-D373. 140

102 THEOREM OF LACHLAN AND MARTIN

141

(/l)(n)[A<")<(^)
1. A. H. Lachlan, On a problem o/G. E. Sacks, Proc. Amer. Math. Soc. 11 (1965), 972-979. 2. D. A. Martin, On a question o/G. E. Sacks, J. Symbollc Logic 31 (1966), 66-69. 3. G. E. Sacks, Degrees of unsolvabUity, Annals so Mathematics studies No. 55, Princeton Univ. Press, Princeton, N. J., 1963. CORNELL UNIVERSITY

103 FUNDAMENTA MATHEMATICAE L.XI (1967)

A minimal hyperdegree by

R. O. G a n d y and G. E. S a c k s * (Cambridge Mass.) Two sets of natural numbers have the same hyperdegree if each is hyperarithmetic in the other. A non-hyperarithmetic set is said to have minimal hyperdegree if all sets of lower hyperdegree are hyperarithmetic. I n this paper we construct a set t h a t has minimal hyperdegree, and we study a certain class of models of the hyperarithmetic comprehension axiom. We draw upon ideas occurring in Spec-tor's construction of a mini­ mal degree of unsolvability [9] and in Feferman's application of forcing to analysis [2]. Our argument mixes Cohen's forcing method [1] with classical truth considerations; however, the use of forcing is not essential to the construction of a set of minimal hyperdegree. Instead of forcing with finite conditions in the style of Feferman [2], we force with infinite, hyperarithmetic conditions. As one might expect, forcing with infinite conditions is much closer to truth than forcing with finite conditions. A set generic with respect to our notion of forcing must necessarily have minimal hyperdegree. All of our forcing is with respect to a second order language L(S). which is virtually isomorphic to Feferman's language L*(S) ([2], p . 335). L(S) is the language of first order number theory augmented by the constant symbol S, some second order variables, and the membership symbol e. Let Ox be a n\ subset of 0 [4], the set of all notations for re­ cursive ordinals, such t h a t each recursive ordinal has precisely one nota­ tion in 0, [3]; if b is the unique notation in 0a for the recursive ordinal /?, we write \b\ = /?. In addition, the relation \b\ < \c\ is the restriction of some recursively enumerable relation to 01. For each f t O „ L(S) has set variables Xb, Yb, Zb, ...; L(S) also has set variables X, Y,Z,.... number variables x, a. numeral n for each, natural number and symbols for eauality (--=) successor (') addition ( + ) and multiplication (•). For each b e 0lf the variable Xb is said to be ranked; the variable X is said to be vnranked. A formula ft of L(S) is called ranked if every set * The second-named author was partially supported by the Guggenheim Founda­ tion and bv U. S. A. Contract ARO-D.373.

104 216

R. 0. G a n d y and G. E. Sacks

variable occurring in g is ranked. A formula of L(S) is called existential if it is ranked or if it is of the form (EX)g with X the only unranked variable occurring in g. The ordinal rank of a ranked formula g is the least ordinal a such that o > \b| for every free variable X 6 of g and such t h a t a > \b\ for every bound variable X* of g. A formula g is called arithmetical if no bound set variables occur in g. Let X be an arbitrary set of natural numbers. Following Feferman [2], for each b e 0,, we inductively define a structure JK>6(X) = X ft and truth in \J {X a | \a\ < \b\}: (i) A sentence g of ordinal rank < \b\ is true in (J {Jia\ \a\ < \b\) if it is true when S is interpreted as X, the number variables of g are re­ stricted to to, and each second order variable X° of g is restricted to JLa. (ii) For each formula 8(a>) (with only x free) of ordinal rank < \b\, let xQ{x)= \%\ §(n) is true in U {Jia\ \a\ < \b\}\; then Jib consists of all such sets xQ[x). We define *,(X) = U {«*»W) & < 0,}. A sentence g of L(S) is true in JL(X) ( | = X O T « 5 ) ^ it; *s *""» when each unranked variable of g is re­ stricted to X ( X ) and the remaining symbols of Of are interpreted according to (i) above. Let U be a ranked or unranked set variable of L(S), and let t denote a number-theoretic term. We write gr(*9(*)) for the result of replacing each occurrence of t e V in g(Z7) by §(<). If o 0 , Oj, ..., am_x is a finite sequence of natural numbers, then it is effectively represented by the sequence number n{p\+a,\ i < m), where p( is the ith largest prime. Sequence numbers provide a means of referring to finite, initial segments of characteristic functions of sets of natural numbers as if they were natural numbers. We use p, q,r, ... to denote finite, initial segments of characteristic functions. We write p > r if p ? r and p is extended by r. Let P be a non-empty set of finite, initial segments; we say t h a t P defines (or is) a perfect, closed set if (p) p f P(Eg) a < P (Er)„p[g * r&q>

r&r>

q&p>

q&p>r].

We write X e P if infinitely many initial segments of the characteristic function of X belong to P . (If P defines a perfect, closed set, then {X| X e P} is a perfect, closed set in the standard sense.) Let P,Q,R,... denote hyperarithmetic, perfect, closed (h.p.c.) sets; i.e., the set of sequence numbers of the initial segments in P is hyperarithmetic. We write P > Q if (X) (X eQ-+XcP). I t is routine to assign indices to h.p.c. sets so that the following relation in x and T is n\: x is the index of a h.p.c. set P and Y « P . For this reason it makes sense to say t h a t the relation " P is a h.p.c. set and Y t P " is n\. Similarly, it makes sense to say t h a t the set of all formulas of L(S) is n{.

105 A minimal hyperdegree

217

The forcing relation P||-»5, where P is a h.p.c. set and 5 is a sentence of L(S), is defined by means of five closure conditions: (i) P'th% if 5 is ranked and (X)(X C P - * K A U » S ) (ii) P|hA(Ez)g(a:) if %(x) is unranked and P|h*5(«) for some n. (hi) P>:rh(EXb)%(Xb} if g(X6) is unranked and P^h%{xS(x)) for some S(z) of ordinal rank < \b\. (iv) P | K ( E X ) 3 ( X ) if ( E ^ o . t P H E X 6 ) ^ ) ] . (v) P | h h 3 i & 5 2 if 2fi&2f2 is unranked, P | h k g i , and P , h A g 2 . (vi) P J r A ~ g if 5 is unranked and ( < 2 ) P > Q [ ~ < 2 H S ] We defined |UA in terms of closure properties rather than transfinite induction in order to simplify the statements of our proofs (cf. Feferman [2], p. 336). A set S is said to be generic if for each sentence g there is a P such that S e P and [Pjj-/>5 or P\\-h~g]. It is not immediate that generic sets exist, because the definition of \\-h contains some peculiarities, particularly the contrast between conditions (i) and (vi). It is precisely these peculiarities which make possible a quick proof of the all-important Lemma 1. 1. The relation P\\-h$, restricted to ranked g, is n\. Proof. It is readily verified that the relation g is true in jtl(X), restricted to ranked g, is n} in g and X; one need only note that this relation can be defined by Z\ closure conditions (cf. Feferman [2], p. 337), The relation X e P is arithmetical on the n\ set of all hyperarithmetic. perfect, closed P's. (This last point is elaborated in the comments immedia­ tely preceding Lemma 6.) EJreisel's Lemma ([7], p. 307) states: if P{x, y) is n\ and (x)(By)P(x, y), then there exists a hyperarithmetic function / such that (x)P(x,f(x)). Kreisel's argument also establishes a slightly stronger fact needed below: if P(x,y) is n\, then there exists a partial n\ function / such that (x)[(Ey)P(x,y)-+f(x) is defined &P(x,f{x))}. (A partial function is n\ if its graph is n\). The purpose of Lemma 2 (the "sequential" lemma) is to standardize a construction which we use repeatedly, and which is characteristic of forcing with closed, perfect sets. LEMMA 2. Lei {gm} be a hyperarithmetic sequence of existential sentences If (m)(Q)p>Q(ER)Q>R[R\\-h'5m), then (E<2)P>0(m)[<2|u>,gm] . Proof, Suppose (m)(Q)P>Q{BR)Q>Jt[R\'rh'5m]. By Lemma 1 and con­ dition (iv) of the definition of | K , the following relation is n\ in m, Q, and R: P ^ Q > R & Rlrhftm- The argument of Kreisel's Lemma ([7], p. 307) shows that R can be regarded as a partial n\ function / of m and Q: R = f(m, Q)-+P > Q ^ R & RW-nftn. Let us say that Q, and
106 •21S

H. O. (iaudy and G. E. Sack*

disjoint h.p.c. subsets of Q if there exist &,&«<} such t h a t q, * q2, 3 i > ? 2 , 3 2 > ? i , Q i = {p\ pcQ&qi>p), and p). B y iterating the partial n\ function/, we can define a hyperarithmetic, partial function Qf with the following properties: Q°0 = P ; for each m ^ 0 and i < 2m, QT1 and Q?+2l are basic, disjoint, h.p.c. subsets of Qf, Q?+1 - f t g m . and
Then V is a h.p.c. set, P $s (?, and X e § ~ (m) C M W C X * Q f ) Fix m and X*Q. There exists an i such that XtQ?*1 Since g m is existential, ;=^ ( X ) g,». But then $!>-»&».

and 9 f + l : : - A ® M .

LEMMA 3. ( g ) ( P ) ( E < 2 ) P ? 0 [ < ? H g v g ! r A ~ g ] .

P r o o f . Condition (vi) of the definition of f r » makes it safe to assume t h a t g is ranked. Conditions (i) and (ii) of the definition of "-h make it safe to assume t h a t g is in prenex normal form. By the full ordinal rank of g, we mean a function / defined on 0, such that for each b e 0 , , f(b) equals the number of occurrences of b in g. We say f
if

(E6) 6c0l [/(ft) < 5(6) & («)W>|»|(/(<0 = ff(c))| •

By the arithmetical rank of 5 , we mean the number m of occurrences of number-theoretic quantifiers in g. The rank of 5 is (/, m). We say (/> m) < (g, n) if / < g or if / = g and m < » . We prove the lemma for ranked g by induction on the rank of g. Suppose t h a t g is (EX 6 )g(X b ). Let {8,(0-)} be a hyperarithmetic enumeration of all formulas (with just x free) of ordinal rank < \b\. For each i, %{Z9t{x)) has lower full ordinal rank than (EX 6 )g(X 6 ). Our inductive hy­ pothesis is: (i)(P)(BQ)P>Q\Q\-h'S(xQi(x)) or 0 | h * ~ 3 f (&«(*))]. Fix P . If (E;)(E<2)p^[g(*S ( (z))], then all is well; suppose not. Then it follows from the inductive hypothesis t h a t (i)(Q)p>Q(ER)Q>R[R^ll~'5{xQi(x))}. By Lemma 2, there is a Q such t h a t P^Q and (t)|G>-ii~<5(*8i(aO)]; but then e i r » ~ ( E X 6 ) 2 f ( Z » ) . If g is of the form (Ex)%(x), we note that ft (n) has lower arithmetical rank than (Ez)g(tf) and then proceed as above. If g has no quantifiers. then there is a q e P such that the desired Q = {p| ? e P & g :> p}. The existence of generic sets follows from Lemma 3. A standard transfinite induction shows: if 8 is generic, then (g)[ -xwg*— (EP)(S«P&P,;-»5)1. LEMMA 4. 7/ 5 u gcnctric, g ( X ) w arithmetical, and | = ^ ( S ) ( E X ) g ( X ) , tfte# ( E X ) [ g ( X ) & X t* hyperarithmetic & X e X ( £ ) ] .

107 A minimal hyperdegret

219

P r o o f . Since £ is generic, there must be a P such t h a t 8 e P and P - , ( E J 6 ) 5 ( X * ) for some i « 0 , . Since P is a hyperarithmetic, perfect, closed set, there must exist a hyperarithmetic E « P . I t follows from the definition of - „ for ranked sentences that (EI)[X 35), we can find a ranked formula8 (a?) such that J T = in] != « fr ,S(n)} holds for every X. LEMMA 5. .For each x\ set A, there exists an existential formula P{x) such that for all generic S, (n)[n c A~ = MS) P(n)]. P r o o f . By Gandy [3] there exists an arithmetical predicate B(x, Y) such that for all »,

n e A ->(EY)[B(n.

Y) & Y is hyperarithmetical] .

At the end of the proof of Lemma 4, it was noted that every hyper­ arithmetic set belongs to MX) for every X. I t follows from Lemma 4 that for all n.

»HA~=^S)(EY)B(u,

Y)

For the sake of Lemma 6, it is necessary to sharpen our previous observations concerning the assignment of indices to hyperarithmetic, closed sets. Kreisel ([7], p. 307) mentions an arithmetical formula A (x, Y) with the following property: if x e 0, then (E,Y)A {x,Y) and (Y)[A{x, Y)-* ^ Y is a hyperarithmetic set of the same Turing degree as Ex]. Since each hyperarithmetic set is recursive in Ex for some x, it is possible to assign indices to hyperarithmetic sets and to obtain an arithmetical formula B(x, Y) such t h a t the set of indices I is n} and such t h a t : if x e I, then (EjY)JB(ar, Y) and (Y)[B(x, Y)-+(Y)0 = the hyperarithmetic set whose index is x], where ( Y ) 0 = {m| 2m t Y). I and B can be modified to obtain a n\ set 1, of indices for hyper­ arithmetic, perfect, closed sets and an arithmetical formula B^x, Y) with the following properties: if x e I,, then (E1Y)B1(x,T) and (Y)[BL(xt Y)->(Y)0 = the h.p.c. set whose index is x]. Let D(Y,Z) be an arithmetical formula which says: (Y)0 contains infinitely many sequence numbers which represent initial segments of the characteristic function of Z. Then x 17, &(Y[[B 1 (x, Y)^D(Y. Z)] says: x is the index of a h.p.c. P and Z t P . LEMMA 6. Let A be a n\ set (of indices) of hyperarithmetic, perfect. closed sets, and let S be generic. If (P)(EQ)P>0(Q e A), then (EQ)(Q c A & Proof. By Lemma r. there is a formula P(.r) such that for all ge-

108 220

R. 0. Gandy and G. E. Sacks

neric S, (n)[« e J.«-M=^ ( S ) P(n)]. Let g denote the following formula of L(S):

(Ex)\P(x)&(T)(B1(x,

Y)^D(Y,S))].

Then for aU generic S, (EQ)[Q eA&Se Q] H = J I G ( S ) 5- Now fix the generic set S. To prove the lemma, it is enough to find a P such t h a t P\-h g and £ € P. Suppose (for the sake of a contradiction) t h a t no such P exists. Then there must be a P such t h a t P h „ ~ g and 8 e P. There must also be a Q,, such t h a t P > Q1 and Q1 e A. By Lemma 3, there exists a generic set S' e Qt. Since P|-A~5 a n d P > & , it follows & !j- f t ~ g . Thus |= jt(so~5> and consequently, ~(E#)[# eA&S'eQ]. But & e .4 and S' e Qt. LEMMA 7. ie« 5(a;, Y) 6e a formula of L(S) wfcose only free variables are x and Y and whose only unranked variable is Y. If P'(t-h(x)(EY)%(x, Y), then(EQ)P>Q(Bb)b(0l[Q\'rh(x)(EYbm(x,Yb)l P r o o f . Let P|L-A(a:)(EY)3r(a;, Y). Then Lemma 3 and the definition of |f-» imply (n)(Q)p> 0 (EP) 0 > f i [P|H(EY)5(7i, Y ) ] . I t follows from Lemma 2 t h a t there is a Q such t h a t P > 0 and ( n ) [ < 2 K ( E Y ) g ( « , Y)]. This means t h a t (»){E6)6€0l[)]. By Spector [8], there is a btO^ such that (n)(\f{n)\ < |6|). But then C H - h ^ K E Y * ) ^ * , Y*). LEMMA 8. Let S(x) be a ranked formula with only x free. For each P, there exists a Q such that P^Q and either (i) or (ii) holds: (i) (X)(XeQ-+X

is hyperarithmetic

in {n|| = J I C ( X ) g(«)});

(ii) ( X ) ( X e Q - > { n | |=jt (X) S(7i)} ts hyperarithmetic}). P r o o f . We proceed in the spirit of the proof of Lemma 2. C a s e 1. (Q)p>QCEQ1)Q>Qs{BQ2)Q>Q,CEn)[Qii[-hQ{n)&Q2frh~Q(n)]. I t follows from Lemma 1 and the argument of EjeiseFs Lemma ([7], p . 307) t h a t we can regard Q,,Q2, and n as partial n\ functions of Q. By iterating these partial n\ functions, we can define a hyperarithmetic, partial function Q? with the following properties: Q°0 = P; for each TO > 0 and i < 2m, Q?+1 and Q?+2l are basic, disjoint h.p.c. subsets of Q?, and (En)[<2r+1l^e(«)&^hA~S(n)]. We define Q by: p e G « - ( m ) (E x »)«,«(? € # • ) • Then $ is a h.p.c. set, P ^ Q, and X e # <- (m)(E 1 i), <2 m(X t QT) ■ We say t h a t t puts X in # if (w)(X e
109 A minimal hyperdegree

221

condition (i) of the lemma. Fix XeQ. Let S = {n\ \=ji(X)^(n)}. To see t h a t X is hyperarithmetic in 8, it is enough to see t h a t the'unique t which puts X in Q is hyperarithmetic in S. Consider the following definition of t: t(0) = 0 &(*(»+l)

= t{m)vt(m

+ l) = t{m)+2n)

;

t (m + 1 ) = t (m) ♦-. (En) [Q^1 [-» 8 (n) & Qjffi+v. I - * ~ 8 (n) & n e S] . The above equations define t, because S(«) is a ranked formula, and consequently, [X eQ?+1 & Q?+l''r-h^{n)]-> n c S. But then M s hyper­ arithmetic in 8, since the relation P | | - * g is hyperarithemtic when P is restricted to be a member of {Qf\ i<2m&m^0} and g is restricted to be a member of {8(1}, ~ S ( n ) | n > 0}. C a s e 2. (ER)P>R(Ql)R>Ql(Q2)„>Ql(n)~[Q1\-hQ(n) &Qr-h~Q{n)]. By Lemma 3, (»)CBQ) J & Q [«|!-»9(n)vg;r-*~S(«)]. For each n, let Pfl

- 1 ~B(n)

i-

( B ^ ^ o W i h * - S(SJ] .

The defining property of R guarantees t h a t P„ is well-defined; in ad­ dition {F„} is a hyperarithmetic sequence of ranked formulas. The nature of R also guarantees that

I t follows from Lemma 2 that (EQ) K>c >(w)[<2l- h F„]. If XeQ, then '=JUx)8(»)*-*
{Q\

Y). Let

(X)(XeQ-+XtP)},

and let i Let A be Lemma 1 must first of Lemma

!

= W | P ^ &

(Eb)b(0l[Q -h(x)(EYh)%(x,

Y»)]} .

the set of indices of all members of At^At. I t follows from that A is n\. I t is our intention to apply Lemma 6 to A. We show (Q)(ER)Q>Ii(R e A). Fix Q. The last sentence of the proof 4 implies there exists a ranked sentence g such that (X) [X t P <-

110 222

R. 0. G a n d y aDd G. E. Sacks

*-»f=Jt.(X)S]. By Lemma 3 , there exists a Q, such t h a t Q>Q„ and Q^ri.% or Q, . n A ~ g . If Q, - A 5 , then P > (?,. and by Lemma 7, (BR)Ql>R{R e A2). K & : - f t ~ & then Q1^A1. By Lemma 6, there is a Q2 * A such t h a t S*Q2. Since <S « P , it must be t h a t Q2eA2. But then ( ^ ^ H E Y 6 ) ^ * , Yb) for some ft £ 0U and consequently, h ^ z H E Y 6 ) ^ * , Y 6 ). LEMMA 10. If S is generic, then S has minimal hyperdegree. P r o o f . Let S be generic. First we show that S is not hyperarithmetic. Suppose t h a t S equals the hyperarithmetic set E. Following Kleene ([4], p. 35), we can find a ranked formula xQ(x) such t h a t E = {n\ |=X(X)S (»)} for all X. Since S is generic, there exists a P with the property that P '-h S = a£(z). But then (X)(X eP-+X = E). This last is absurd, since P is uncountable. Now suppose t h a t K is hyperarithmetic in S. I t follows from Lemma 9 and the argument of Kreisel ([6], p. 114) t h a t K e Ji{S). Let Q{x) be a ranked formula such that H f I « | = A ( S ) 8 ( » ) . Let ^ =

\Q\ (X)(XeQ^X

is hyperarithmetic in {n\ :=JC ( 2 : ) S(«)})},

and let -A,= {$i (X)(Xe<2->{w|

Krfux) 8 (")} is hyperarithmetic)}.

Let A = At ^A2. Then the following observations imply t h a t A is n\: the relation "X is hyperarithmetic in Y" is n]; the relation " k * , x , S ( r c ) " is hyperarithmetic (in X and «). By Lemma 8, (P){'EQ)p>Q(Q € A). By Lemma 6, (~EQ)(Q cA&ScQ).1iQc J.,, then S is hyperarithmetic in K. If QcA2, then .£ is hyperarithmetic. THEOREM. There exists a set of minimal hyperdegree less than the hyperdegree of 0. P r o o f . The relation P . - , , g, where P is a h.p.c. set and g is an arbi­ trary sentence of L(S), is hyperarthmetic in 0 (cf. [2], p. 337). Let {5ml m ^ 0} be an enumeration hyperarithmetic in 0 of all sentences of L(S). By Lemma 3, we have* (m)(P)(EQ) P > 0 [P - „ 5 m \ ' P ! - f t ~ g m ] . The argument of Kreisel's Lemma ([7], p. 307), relativized to O, makes it possible to regard Q as a partial function of m and P whose graph is n] in O: {m)(P)[P^f{m, P) & (/(m, P) - A g m v / ( ? » , P ) : - > , ~ g m ) ] . By iterat­ ing f, we can define a function P,„ hvncrarithmetic in O with the following AV properties: P w > P m + 1 &[P m -ASm\'P,„|K~<5mJc define £ by the formula: »ifS-(Em)[Pm-An«S]. Then £ is generic and hyperarithmetic in O. {S is, in fact, the uinque X such that (m){XePm).) By Lemma 10, S has minimal hyperdegree. In [2], p . 340, it was observed t h a t O is not of minimal hyperdegree.

Ill A minimal

hyperdegeer

223

We conclude with a word on the elimination of forcing from the construction of a set of minimal hyperdegree. The key lemmas underlying the proof of the theorem are Lemmas 7 and 8. Lemma 7 provides the means of insuring that mf, the least ordinal not recursive in S, equals coj, the least non-recursive ordinal. Lemma 8 provides the means of insuring that S has minimal hyperdegree, given that cof = a*. Both of the key lemmas are consequences of the nature of the forcing relation restricted to ranked sentences, where it coincides with the classical truth relation. These considerations, if pressed hard, lead to a construction of a set of minimal hyperdegree recursive in 0 without any recourse to the idea of forcing. References [1] P . J . C o h e n , The independence of the continuum hypothesis, Proc. Nat. Acad. of Sci. U . S. A. 50 (1963), p p 1143-1148 (Part I) and 51 (1964), p p . 105-110 (Part II). [2] S. P e f e r m a n , Some applicaiions of the notion of forcing and generic sets, F u n d . Math. 56 (1965), p p . 325-345. [3] R. G a n d y , Proof of Mostowski's conjecture, Bull. Acad. Polon. Sci., Ser. M a t h . 8 11960), p p . 571-575. [4] S. C. K l e e n e , Hierarchies of number-theoretic predicates, Bull. Araer. Math. S o c , 61 (1955), pp. 193-213. [5] — Quantification of number(tlieoretic funciion., Compos. Math. 15 (1959), pp. 23-40. [6] G. K r e i s e l , Set theoretic problems suggested by the notion of potential totality, Proc. Symp. Infinitistic Methods, Warsaw 1961, p p . 103-140. [7] — The axiom of choice and the class of hyperarithmctic funciion,, Indag. Math. 24 1962), p p . 307-319. [8] C. S p e c t o r , Recursive weU-ordering,, J . Symb. Log., 20 (1955), p p . 151-163. [9] — On degrees of recursive unsolvability, Ann. of Math., 64 (1956), p p . 581-592. UNIVERSITY OF MANCHESTER .ind MASSACHUSETTS INSTITUTE OF TECHNOLOGY

H
U 17. .'' 1907

112

MEASURE-THEORETIC UNIFORMITY IN RECURSION THEORY AND SET THEORY BY

GERALD E. SACKS(')

1. Introduction. The notion of measure-theoretic uniformity was presented in [23], [24]. In this paper the notion is first utilized in an arithmetic setting and then extended to hyperarithmetic theory and set theory. The recursion-theoretic concepts touched on are: the arithmetic, hyperarithmetic and analytic hierarchies of Kleene [12]; ^-models of the hyperarithmetic comprehension axiom of Kreisel [16]; recursive ordinals [11]; and hyperdegrees [13]. One of the recursion-theoretic results we prove is: if P(X) is a U\ predicate with free set variable X and the set XP(X) has positive Lebesgue measure, then P(A) holds for some hyperarithmetic set A. A corollary of this result is: the set of all A" such that the ordinals recursive in X coincide with the recursive ordinals has Lebesgue measure 1 (see footnote (9), below). In the area of set theory we are largely concerned with showing how relative consistency results follow in a natural manner from the notion of measure-theoretic uniformity. We develop Cohen's independence results [2] as well as a result of Solovay [29], [30] on the extendability of Lebesgue measure to all sets of reals. Solovay makes use of Cohen's forcing method, but he ingeniously replaces Cohen's finite forcing conditions by closed sets of positive measure. He shows: if ZF (Zermelo-Fraenkel set theory) is consistent, then ZF + "there exists a translationinvariant, countably additive extension of Lebesque measure defined on all sets of reals" + "the dependent axiom of choice" is consistent^). We reprove Solovay's theorem with emphasis on the notion of measure-theoretic uniformity and with the help of some elementary properties of uniformly distributed, independent random variables; however, the fine details of our argument are not substantially different from those originated by Solovay. §4 of this paper, which deals solely with set theory, can be read independently of §§2 and 3, but it is intended to be read as a natural continuation of the earlier sections. Received by the editors March 13, 1968 and, in revised form, August 1, 1968. (') The preparation of this paper was partially supported by U.S. Army Contract ARO-D373 and by NSF grant GP-6982. The author wishes to thank Professor A. Nerode for many helpful conversations on uniformity and definability, and Professor D. Lacombe for enabling the author to present the paper in a series of lectures [24] at the University of Paris in November, 1966. (2) R. B. Jensen has shown that the dependent axiom of choice is stronger than the count­ able axiom of choice. Solovay has also shown: if Z F + "there exists an inaccessible cardinal" is consistent, then so is Z F + "the dependent axiom of choice" + "all sets of reals are Lebesque measurable".

381

113 382

G. E. SACKS

[August

Our interest in applying measure-theoretic ideas to questions of mathematical logic began with the reading of Spector's measure-theoretic construction of two incomparable hyperdegrees [32]. Spector showed, using Fubini's Theorem and the measurability of analytic sets, that the set of all pairs (X, Y) such that X and Y are hyperarithmetically incomparable has measure 1. Shoenfield extended Spector's argument to show there exists an uncountable set of pairwise incomparable hyperdegrees. Feferman [3] obtained two incomparable hyperdegrees by means of a forcing argument, but Spector's proof, although less constructive than Feferman's, is much shorter. In [22] we showed: if X is not recursive, then the set of all Y such that the Turing-degree of Y \s incomparable with that of A'has measure 1. In §3 we show: if X is not hyperarithmetic, then the set of all Y such that the hyperdegree of Y is incomparable with that of X has measure 1. There is a significant conceptual difference between these last two results. The proof of the former of the two relies on the countable additivity of Lebesgue measure and on the following fact: if X is Turing-reducible to (i.e., recursive in) Y, then X is reducible to Y by means of one of countably many reduction procedures whose nature does not depend on Y (i.e., X={e}r). In contrast, if X is hyperarithmetic in Y, then X is reducible to Y by means of a hyperarithmetic reduction procedure whose nature depends on the ordinals recursive in Y. Since the ordinals recursive in Y can be arbitrarily large countable ordinals, the proof of the latter of the above two results has to take into account uncountably many hyperarithmetic reduction procedures. This difficulty is overcome in §3 by showing that if one ignores a set of F s of measure 0, then one can assume the ordinals recursive in Y are the recursive ordinals. It seems impractical at this point to give a general and formal account of the notion of measure-theoretic uniformity. Instead, we give a variety of examples and applications in the belief that the notion is simple enough to abstract from the examples. Let N be the natural numbers. We put the usual probability measure on 2" as follows. Let m be the unbiased measure for 2 = {0, 1}: w({0, 1})=1, w({0}) = w({l}) = i, and m(0) = O, where 0 denotes the empty set. Let /x be the product measure induced on 2" by N. Of course h is merely the product-space version of Lebesgue measure on the unit interval [0, 1] of the reals [10, p. 157]. What we need of measure theory can be found in Halmos [10]. Let R(T, x, y) be a recursive predicate [12] of the set variable T and the natural number variables x and y. Thus the truth-value of R(T, x, y) for any actual choice of Te2N and x,yeN is determined by finitely much information about the membership of T and the values of x and y. A familiar uniformity can be put as follows: (T)[(x)(Ey)R(T, x, y) -> (£/)(/eecursive in T & (x)R(T,

x,f(x)))}.

(In fact, f(x)^wR(T,x, .>>)•) Another way of expressing this uniformity is: for each T, if the predicate (x)(Ey)R(T, x, y) has a Skolem function, then it has a Skolem function recursive in T. Thus in our study of the predicate (x)(Ey)R(T, x, y),

114 1969]

383

MEASURE-THEORETIC UNIFORMITY

we can reduce our attention from arbitrary Skolem functions to Skolem functions recursive in T. It is now quite natural to ask: can we reduce our attention still further to recursive Skolem functions? The answer is yes, if we ignore a set of Ts of measure 0, and if we replace Skolem functions by bounding functions: f is a bounding function for (x)(Ey)R(T, x, y) if (x)(Ey\ s ,<X>R(T, x, y). The existence of a bounding function is equivalent to the existence of a Skolem function. PROPOSITION 1.1. Let R(T,x,y) be recursive. If the set of all T such that (x)(Ey)R(T, x, y) has measure 1, then the set of all T such that (Ef)[f recursive & {x){Ey)vtknx)R{T, x, y)] also has measure 1.

Proof. It is enough to show: for each rational 8>0, there exists a recursive/ such that {T\(x)(Ey\inx)R(T,x,y}} has measure at least 1-8. A nonempty, basic open subset of 2" consists of all T satisfying a consistent conjunction of finitely many atomic conditions of the form me TOT their negations. The measure of a nonempty, basic open set is 2"n, where n is the number of distinct atomic conditions that define the basic open set. Thus ^({T \ m eT &m+ 1 $ T})= 1/4. Since R is recursive, there exists a recursive function b?-" such that for each m and n, b?-n is a basic open subset of 2", and such that for all T, (m)(n)[R(T, m, n) <-+ (Ei)(Te b?-")]. For each w, let f{m) = MtWJ {#"" | i £ * Sin £ *}) £ 1 -S/2 m + 1 ]. For each m,f(m) is defined because K U W , n | i>0 & «>0})= 1. (Read >**' as "the least t such that".) A stronger and more attractive proposition than 1.1, and one closer to the idea of measure-theoretic uniformity would be: the set of all Tsuch that (x)(Ey)R(T, x, y) -*■ (£/)[/recursive & {x){Ey\ilix)R{T,

x, y)]

has measure 1. Unfortunately, D. A. Martin has shown that for some recursive R this stronger proposition is false. We have to go to the language of arithmetic to realize the full flavor of measure-theoretic uniformity. Theorem 2.2 of §2 states: let B(T, x,y) be arithmetic [12]; then the set of all Tsuch that (x)(Ey)B(T, x, y) -> (£/)[/is arithmetic & (x)(Ey)^nx)B(T,

x, y)]

has measure 1. In short, if we restrict T to a set of measure 1, then we can restrict ourselves, in our study of arithmetic predicates of T, to arithmetic bounding functions. The reduction from bounding functions arithmetic in T to arithmetic bounding functions is the essence of measure-theoretic uniformity in the arithmetic case. 2. The arithmetic hierarchy. Let 5(T) be an arithmetic predicate of T. The set fB(T) is a Borel subset of 2" of finite rank and consequently is measurable. Let r

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be a variable ranging over the rational numbers in the unit interval [0, 1]. By making use of some standard Godel numbering, it is possible to regard the predi­ cate ^{fB{T))£r as a number-theoretic predicate whose free variables are B (ranging over the arithmetic predicates) and r (ranging over the rationals in [0, 1]). For each «> 0, an arithmetic predicate is said to belong to -° (11°)ifit is expressible by means of a recursive matrix and a quantifier prefix beginning with an existential (universal) quantifier and passing through n-\ alternations of quantifiers. Re­ cursive predicates are said to belong to both Eg and n°. LEMMA 2.1. (i) For each n^O: the predicate fi(fB(T)^Sr, restricted to Bel,°, is n°. (If) For each n ^ 0: the predicate n(fB(T)) g r, restricted to B ell°, is U°+1. (iii) The predicate n{fB(T)) g r, where B is arithmetic and otherwise unrestricted, is hyperarithmetic.

Proof, (i) and (ii) are proved simultaneously by induction on n. First let « = 0. Then B(T) is a recursive predicate, and there exists a finite union U {bt |i0 and B{T) e 2°. Then B(T) is of the form (Ex)B{T, x), where B(T, x) e n°.lt and KtBiT))

^ r~(m)Mf((Ex)x±mB(T,x)))

^ r].

The inductive hypothesis for (ii) implies that the right side of the above formula is n°. Suppose instead that B(T) e n°. Then B(T) is of the form (x)B(T, x), where B{T,x)el,°_1, and H(TB{T)) S r^(8)(Em)[f,(T((x)xSmB(T,x)))

Z r+8),

where 8 is a variable ranging over the positive rationals. The inductive hypothesis for (i) implies that the right side of this last formula is n°+1. To see (iii) observe that the "effective" nature of the inductive step of the proof of (i) and (ii) implies the existence of a recursive function / with the following property: for each n>0, f(n) is the Godel number of a n° predicate Pnn){b,r) such that PfUb, r)<-» b is the Godel number of B(T) & B(T) eH°n& p{fB(T)) g r. The predicate PHn)(b, r), with n, b and r as free variables, is hyperarithmetic, since the truth set for arithmetic is hyperarithmetic. In Theorem 2.2 we attempt to capture what we mean by measure-theoretic uniformity in the arithmetic case. A standard uniformity for arithmetic predicates of the form B(T, x, y) can be put as follows: if (x)(Ey)B(T, x, y), then (Ef)[f is

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385

arithmetic in T & (x){Ey\ s Hx)B(T, x, y)]• The corresponding measure-theoretic uniformity is obtained by demanding that the bounding function/be arithmetic rather than arithmetic in T. The cost of such a demand is a set of T's of measure 0. THEOREM 2.2. Let B{T, x, y) be arithmetic. Then the set of all T's satisfying the following condition has measure 1: if (x)(Ey)B(T, x, y), then (Ef)[f is arithmetic & (x)(Ey)vSnx>B(T,x,y)].

Proof. Let 8 be a positive rational. We define an arithmetic/such that Kf(x)(Ey)„snx)B(T,x,y))+&

§

^f{x){Ey)B{T,x,y)).

Let b(x,r) be the predicate: li.{f{Ey)B{T, x, y))>r. By Lemma 2.1, b(x, r) is arithmetic. Then/(x) is the least j such that for all r, Ux,r) -* tfftEy)a

s

,B(T, x, >•)) > r - 8/2* + 1 .

COROLLARY 2.3. IfB(T) is arithmetic and the set fB(T) has positive measure, then B{A) holds for some arithmetic A(3).

Proof. First we consider a special case, where B(T) is U°2, and then we show how to reduce the general case to the special case. Let B(T) be (x)(Ey)R(T, x, y), where R(T, x,y) is recursive. By Theorem 2.2, there is an arithmetic function/such that n(f(x)(Ey)vsnx)R(T,x,y))>0. There exists an arithmetic function cx such that for each x, cx is a finite union of basic open subsets of 2" and (Ey)ytnx)R(T,x,y)^Tecx. For each x, let dx = C\ {c, | i^x}. Note that for each x, the closed set dx is non­ empty, because n(fB(T))>0. If A is an arithmetic set such that (x)(A e dx), then B(A) holds.'We define such an A by an induction on n: neA^

Kf((x)(T e dx) & (/),<„(/ e T+-* z e A) & n 6 T)) > 0.

A is arithmetical, since it is arithmetic in the function dx. Now we show how to reduce the general case to the special case by consider­ ing an illustrative example. Let B{T) be (x)(Ey)(u)(Ev)R(T, x, y, u, v), where R(T, x, y, u, v) is recursive. By Theorem 2.2, there is an arithmetic/such that f(x)(Ey)ySHx)(u)(Ev)R(T,

x, y, u, v)

has positive measure. By using a trick of Kleene [12], we can pass a bounded existential quantifier through a universal quantifier: (E)'Xinxlu)(Ev)R(T, x, y, u, v) <-> (u)(Ey),&nx)(Ev)R(T, x, y, («)„, v). (3) Corollary 2.3 was independently obtained by H. Tanaka [34].

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By Theorem 2.2, there is an arithmetic g such that the set f(x)(«)(Ev)vSII(xM(Ey)sSMR{T,

x, y, («)y, v)

has positive measure. But now the argument of the special case can be applied. Let us say that B(T) holds for almost all T if v{tB{T))= 1. Then Corollary 2.3 is equivalent to the statement: if B{T) is arithmetic and holds for all arithmetic 7", then it holds for almost all T. It follows immediately that the predicate "T is arithmetic" is not arithmetic. This last result was first proved by Addison [1] by means of a forcing argument. Corollary 2.3 is an example of what might be called a measure-theoretic basis result. Gandy's basis theorem [7] for arithmetic predicates states: if B(T) is arith­ metic and (ET)B(T), then B(A) holds for some A recursive in Kleene's O, the set of all notations for recursive ordinals. (Kleene [13] gives an example of an arithmetic predicate B(T) such that (ET)B(T) but B(T) holds for no hyperarithmetic T.) Call a set T fundamental (with respect to arithmetic) if B(T) holds for every arithmetic B such that n{f£(T))=l. It is possible to construct hyperarithmetic, fundamental (with respect to arithmetic) sets with the help of Lemma 2.1, but we prefer to deduce the existence of such sets as a corollary to Theorem 3.9. They behave very much like the generic (with respect to arithmetic) sets of Feferman [3]. In particular, they are not implicitly arithmetically definable; that is, they are not unique solutions of arithmetic predicates. 3. The hyperarithmetic hierarchy. An analytic predicate P(T,x) (Kleene [12]) is said to be U\ if it is expressible in the form (Y)A(Y, T, x), where A(Y, T, x) is arithmetic. LtlP(T, x, y) be nj. A familiar uniformity due to Kreisel [16] can be put as follows: if (x)(Ey)P(T, x, y), then {Ef)[f is hyperarithmetic in T & (x)(Ey)ySIMP(T,

x, y)].

The corresponding measure-theoretic uniformity is obtained by first discarding a set of F s of measure 0 and then requiring that the bounding function/be hyper­ arithmetic rather than hyperarithmetic in T. We develop this result (Corollary 3.12) and several related ones by studying a relativization of Kleene's ramified analytic hierarchy [14]. Very briefly, the ramified analytic hierarchy is obtained by starting with the empty set and iterating the process of analytic defin­ ability through the recursive ordinals(4). Kleene [14] showed that the ramified analytic hierachy is identical with the set of hyperarithmetic sets. Following Feferman [3], we relativize by starting with an arbitrary set T rather than the empty set(5). (*) Many persons now use the term "ramified analytic hierarchy" to denote the result of iterating the process of analytic definability past the recursive ordinals to its endpoint ft,. (5) It might seem more natural to relativize by replacing not only the empty set by T but also the recursive ordinals by the ordinals recursive in T, however, it follows from Lemma 3.5 that this relativization is identical with the one we adopted for almost all T.

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The relativization is described by a ramified, second-order language -S?(JT), which differs little from Feferman's language &*(Sf) ([3, p. 335]). S?(f) is the language of first-order number theory augmented by the constant symbol &~ denoting an arbitrary set of natural numbers, some second-order set variables, and the membership symbol (e). Let 01 be a T\\ subset of O such that Oy is linearly ordered by < 0 and has order-type «i (the least nonrecursive ordinal). (The existence of Ox is proved in [4] and in [6].) Thus each recursive ordinal has just one notation in O^; if b is the unique notation in Ot for the recursive ordinal /3, we write \b\=0. For each b e 0lt &{3~) has ranked set variables X", Y\Z\...; &{3T) also has unranked set variables X, Y,Z,..., a numeral n for each natural number n, and symbols for equality ( = ), successor ('), addition ( + ), and multi­ plication (■). A formula J5" of JS%T) is said to be ranked if every set variable occurring in F is ranked. The ordinal rank of a ranked sentence & is the least a such that a > \b\ for every variable X* occurring in 3F. Let T be an arbitrary set of natural numbers. Following Feferman [3], for each b e O b we inductively define a structure J(b(T) and truth in the structure 1J {J?0(T) | \a\ < \b\}\ (i) A sentence J^ of ordinal rank ^ |Z>| is true in U {JtJJ) \ \a\ < \b\} if ii ts true when 3T is interpreted as T, the number variables of SF are restricted to w, and each set variable X" of P is restricted to Jftt(T). (ii) For each formula ^(x) (with only x free) of ordinal rank < \b], let xS?(x) denote the set {« | <${n) is true in | J {J<JJ) \ \a\ < |fr|}} ^ ( T ) consists of all lets of natural numbers defined in this manner. We define Jf{T)*=(J {J(£T) \ b e 01], Let ^ = ^ ) = the set of all hyperarithmetic sets. A sentence 3F of -^(JT) is true (in symbols, JZ(T) N &) if it is true when each unranked variable is restricted to Jt{T) and the remain­ ing symbols of F are interpreted according to (i) and (ii). It is a routine matter to choose a Godel numbering for the formulas of &{$~) with the following proper­ ties: the predicate "e is the Godel number of a ranked formula of <e(3T)" is U\; for each b e 0 1 ( the predicate "e is the Godel number of a ranked formula of £e{$~) of ordinal rank < \b\" is recursively enumerable (uniformly in b). The assign­ ment of Godel numbers makes it possible to classify various relations involving the formulas of &(&~). We will occasionally blur the distinction between formulas and Godel numbers of formulas. The full ordinal rank of a ranked formula 3? is a functton/: 01 -* w such chat for each b £ Oi,f(b) is the number of occurrences of (Xb), (EY>).... in ^ If / and g are full ordinal ranks, t h e n / < r g if (Ec)[c G 01 &f(c) < g(c) & (d)(c <0ld-+f{d)

= g(d))].

The relation < r is a well-ordering of the set of full ordinal ranks. If 9(x) is a ranked formula (with only x free), then P(X9(x)) denotes the result of replacing each occurrence of t e Y in J^( Y) by ^(t)- (t is a number-theoretic term.)

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3.1. Let &(V>) be a ranked formula in prenex normal form whose only free variable is Y". Let {«?,(*) \|^n) be a finite sequence of formulas with only x free of ordinal rank g \b\. Then the disjunction Vis* ^"(iS?,M) is equivalent to a prenex normal sentence of full ordinal rank less than that of {EYb)^{Yb). PROPOSITION

Proof. By illustrative example. Let J^Y") be (EXcXYiXEZb)R(Xc,

Y*,Z", Y%

where R contains no quantifiers, and let « = 2. The following formulas are equiva­ lent:

V (ExcXYi,)(Ezb)R(xc, Yd,z», **&)); (£*<)[ V (Y*)(EZ>)R(X<, Y",Z\ *9t(x))\; (EXc)(Y$)(Yt)\ V (EZb)R(Xe, Yf, Z\ **,(*))]; (EX<){ YSK Yt){EZ")\ V R{XC, Yf, Z\

j&fflfl.

The block of quantifiers (Yg)(Yf) can be replaced by a single quantifier (Yd) by means of a standard trick: each occurrence of /e Yg(t e Yf) is replaced by It e Yd(2t+\ e Y"). The formula V*(*c,

Yf,Z\x$t(x))

iS1

is equivalent to a prenex normal formula Q(XC, Yd, Zb) such that all of its quanti­ fiers have superscripts which are notations for ordinals < \b\. Clearly, (EYt)(EX%Y*)(EZ''yR(Xc,

Yd,Zb, Y")

has greater full ordinal rank than {EX% y)(£Z») Q(XC, Y", Zb). For each sentence & of <£(T), we define p{&), the probability that F ii stue in Jt{T), to be the Lebesgue measure of {T | J({T) V 3?}. (It is easily seen by transfinite induction that {T \ JK(T) 1= &} ii a Borel lubset of 2".) LEMMA

3.2. JTie predicate p(S?)Zr, restricted to ranked & and rational r, is

Proof. Spector observed that the intersection of all sets satisfying a £ | condition is n | ; i.e., if A(X) is Si, then fj {* | ^(J)} is OJ. We give an inductive definition of p(S^)>r, restricted to ranked, prenex normal 3F and rational r, and then we note that the cases of our inductive definition can be expressed by means of I\ closure conditions. The number-quantifier rank of SF is the number of occurrences of (A), (£>•),,.. in JT The rank of & is the ordered pair (/, m), where / is the full ordinal rank of & and m is the number-quantifier rank of &. We say (/, m)<(g, n)

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389

i f / < , g or iff=g and m < n. The inductive definition ofp(F) ^ r proceeds according to the rank of &. Case 0. f has no quantifiers. The predicate p{3?) S r, restricted to & without quantifiers and rational r, is recursive; i.e., p(fn e3T) = \ etc. Case 1. ^ is of the form (EX^^iX"). Let {^(x) | ;^0} be the uniform (in b) recursive enumeration of formulas with only x free of ordinal rank g \b\. Then p(jF)Sr is defined to be WEm)[p(Vm*[(**«(*)))

£ r-«].

where 8 is restricted to the positive rationals. By 3.1, Vis» ^C*^i(*)) is equivalent to a prenex normal sentence of full ordinal rank less than that of (EX^^X"). Case 2. & is of the form ( J H - W ) . Then p(2T)^r is defined to be l

/

\lSm

J

By the dual of 3.1, A i s - ^ ( » i W ) is equivalent to a prenex normal sentence of full ordinal rank less than that of (X")^(Xb). Case 3. & is of the form (Ex)^{x). Then p{3F) Z r is defined to be

W&n)\piy L \lSm

m))

* r~8\, /

J

where S is restricted to the positive rationals. By the same argument used to establish 3.1, V.sm^O") is equivalent to a prenex normal sentence of lower rank than that of (Ex).?\(x); in fact, the full ordinal rank will be unchanged but the number-quantifier rank will be less. Case 4. & is of the form (x>^(x). Dual to Case 3. Case 2 can be rephrased as a Y\ closure condition as follows: If J^ is a ranked, prenex normal sentence of the form (Xb)3?W>) and (m)[p(\/l£n ^ ( x ^ , ( x ) ) ) S r ] , where {9,(x) \ i^O} is the uniform (in b) recursive enumeration of formulas with only x free of rank < \b\, then/>(J*") > r. The hypothesis of Case 2 is n j in variables 3F and r. LEMMA 3.3. Let #"(*, Y) be a formula of
p{{x){EY)^{x,

Y)) g r^p((x)(EYc)3?(x,

Yc)) > r(e).

Proof. Fix rand suppose p{{x)(EY)^{x, Y))^r. Define (Y)& to be {/1 pl+< e Y], wherep n is the nth smallest prime (^ = 2,^ = 3,...). Then M[/7((£y)(Am^(".(n.))) £ ' ] • (6) It is not possible in genera! to improve 3.3 by finding a c that is independent of r.

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It follows that

where S is restricted to the positive rationals. It follows from 3.2 that the predicate Tce Ox &.pKEYt)t A *(** (Y%)\\

5 r-81

is n j in c, m and 8. By Kreisel's Lemma [16, p. 307], there exists a hyperarithmetic function c(m, 8) such that ( m ) ( S ) W ( £ r * « ) ( A^'(«.(>'" mi< ')n))) £ r - 8 J . By Spector [31] there is a c e 0 2 such that (m)(S)(|c(m, 8)| g |c|). But then («)(8)[/»((£y0(A m ^(«,(n»)) S r - 8 ] , and consequently, /?((*)(£ yc)-^(*> l * ) ) * * By an instance of the hyperarithmetic comprehension axiom, Kreisel [16] means the universal closure of any formula of the form (x)[{EY)A(x, Y) <-» (Z)B(x, Z)] ->• (EX)(x)[x e X <-y (EY)A(x,

Y)),

where A(x, Y) and B(x, Z) are arithmetic predicates that may contain free set variables other than Y and Z. LEMMA

3.4. lf& is an instance of the hyperarithmetic comprehension axiom, then

p(Sn=l. Proof. Let A(x, Y) and B(x, Y) be formulas of Sf{f) whose only free variables are x and Y and whose only unranked variable is Y. Let K be the set of all T such that (x)[(EY)A(x, Y)<->(Z)B(x,Z)] is true in Jt(T). Then (x)(EY)[A(x, Y) v ~5(x, K)] is true in Jf(T) for all Te K. Fix 8>0. By 3.3 there i s a c s O , such that (x)(EYc)[A(x, Yc)v~B(x, Yc)} is true in Jl{T) for all TeJzK, where H(K-J)<8. But then (x)[(£yc)^U, yc) *-*(£/)/* (*, 7)] holds in ^ ( J ) for all TeJ. Since (EYc)A(x, Yc) is a ranked formula with only x free, it follows that {EX)(x)[x E I H (EY)A(x, Y)] holds in ^ ( 7 ) for all TeJ. Lemma 3.4 suggests: if a set of T s of measure 0 is avoided, then Jt{T) is very similar to Jt. The similarity is pursued further in Lemma 3.5, and is exploited to prove 3.7. The underlying cause of the similarity is expressed by Lemma 3.2, which says that the probability that a ranked sentence SF is true in Jt(T) can be "com­ puted" in Jt. The use of the term "computed" is appropriate for two closely related reasons: the predicate p(^)^r, restricted to ranked & and rational r, is SJ in the sense of Jt (i.e., the existential set quantifier is restricted to the sets of Jt); and the predicate is metarecursive [17] as well.

122 1969] LEMMA

MEASURE-THEORETIC UNIFORMITY

391

3.5. For almost all T, Ji{T) is the set of all sets hyperarithmetic in 7( 7 ).

Proof. The argument of Kleene [14, p. 35], relativized to T, shows that the sets of J?(T) are just those sets each of which is recursive in HJ for some be 0^0T{B). According to Kreisel [15], if T belongs to an w-model of the hyperarithmetic comprehension axiom, then every set hyperarithmetic in T also belongs to that co-model. Now apply 3.4. PROPOSITION 3.6. Let {Jft be a hyperarithmetic sequence of ranked sentences of <£(9~). Then there is a ranked sentence F such that for all T, J?(T) 1= F if and only if (i)[JK(T)¥ &ft. Proof. Let H be a hyperarithmetic set such that n e H <-» n is the Godel number of a member of {^}. By Spector [31] there is a c e Os such that the ordinal rank of J^ is 5 |c| for all I Let VJ be the set of Godel numbers of all formulas of ordinal rank ^ \c\ which are true in JT(T). By adapting the argument of Kleene [14, p. 36], one can show that VJ is recursive in HJ for some c in 01 £ 0T (see footnote (8)). Clearly, ( / ) ( / e H ^ i e VJ) is equivalent to {i)[J?{T) 1=ffift.Again by adapting the arguments of Kleene [14, p. 35], one can show the existence of ranked formulas @i(x), 9t(x), and &2(x) of S?(f~) such that for all T: H = {n\ JK(T) 1= ®i{n)}', HJ = {n | Jf(T) ¥ &2(n)}; VJ = {n | JfQT) 1= »«(»)}• The desired !F is ( x ) ( ^ W -+ %{x)). The Godel number of 2? can be found effectively from the Godel number of {^}, since one can pass effectively from the Godel number of {Jft to c, c', and the Godel numbers of Sfx and 9a. LEMMA 3.7. For each ranked sentence & and rational S > 0, there exists a ranked sentence <S such that p{3? & ~ SF) < 8 and such that {T \ J((T) t <#} isa closed subset of{T\Jt{T)tST}.

Proof. Let P(ft S, S?) denote the following number-theoretic predicate: 3? is a ranked sentence, S is a positive rational, S? is a ranked sentence, />(#" & ~S?)< 8, and {71 ^ ( 7 ) N S?} is a closed subset of {T | ^(7") 1= J5"}. With the help of 3.2 and the fact that Jt(T) ¥ F, restricted to ranked &, is n}, it is not hard to check that ?{&, S, W) is n i . We prove by induction on the rank of P that {h)(E&)P{&, S, S?). The cases are the same as those of 3.2. (We again assume that P is in prenex normal form.) The most (7) Feferman [3] showed by means of a forcing argument with finite conditions that for all r outside a first category subset of 2", -#(D is the set of all sets hyperarithmetic in T. (8) There is no harm in thinking of O, as a subset of OT, since there exists a recursive func­ tion/with the following properties: for all T, a<0ib^f(a)<0rf(b), and aeO^\a\ =

\M\o*.

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interesting case is when & is of the form ( A ^ - W O - Let {»,(*) | />0} be the uniform (in b) recursive enumeration of formulas (with only x free) of ordinal rank £ |d|. For each i, jri(x&,{x)) has lower full ordinal rank than {Xb)^{Xb). So by the induction hypothesis, (i)(8)(£jn/»(^(iSr,(x)), 8/2'♦», X). Since P is n\, it follows from Kreisel's Lemma [16, p. 307] that there exists a hyperarithmetic sequence {X, | | £ 0 } of fanked formulas such that

By 3.6, there is a ranked formula Jf equivalent to the "infinite conjunction" of the •*?s. Then P{&, 8, Jf). 3.8. If & is a ranked sentence of &(ST) and p(Sr)>0, then Jt{T) t= & for some hyperarithmetic T. LEMMA

Proof. By 3.7 it is safe to assume that {T \ Jt{T) ¥ JF} ii a closed subset of 2". We inductively define a sequence {$?„ | nSO} of atomic sentences: <Sn = nz3r = n$P

i{p(f

&(i)l
kneST)

> 0

otherwise.

By 3.2 the set / / = {« | S?„ N » e ST) ii hyperarithmetic. Clearly, J?(H) ) J^ THEOREM 3.9. ///>(*) is FIJ <WK* rte Lebesgue measure of XP(X) is positive, then P(H) holds for some hyperarithmetic H{9).

Proof. By Gandy [6] or Spector [33], there is an arithmetic predicate s/(X, such that

Y)

(X)[P(X) <-> (EY)( Y is hyperarithmetic in X & st(X, Y)}. It follows from 3.5 that for almost all T, [Jt(T) V (EY)s/(tr,

Y)] «-> P(T);

consequently, p((EY)s^(^,

Y)) = ix(XP(X)) > 0.

By 3.8, Jt(H%) ¥ (EY)st(3r, Y) for rome hyperariihmetic //,. Since eveey member of J({Hd is hyperarithmetic in Hu there must be a hyperarithmetic H e Ji{Hx) such that .*/(#!, H), and hence /»(//), hold. Theorem 3.9 is another example of what might be called a measure-theoretic basis theorem. Shoenfield's basis result [27] for 11} predicates states: if P(X) is U\ and (EX)P(X), then P(C) holds for some C constructible in the sense of Godel [9]. (9) Theorem 3.9 first appeared in Sacks [24]. Subsequently, H. Tanaka [34] independentty developed another proof of 3.9 in which Corollary 5 of Saaks s233 ]Corollary y.10 0o thh eresent paper) is quoted but which avoids the formalism of the ramified analytic hierarchy.

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Theorem 3.9 can be relativized in a reasonable manner. Let P(A, X) be nj with A and X as free set variables. For each A, if the Lebesgue measure of XP(A, X) is positive, then P(A, H) holds for some H hyperarithmetic in A. It follows that if a complementary analytic set of reals [28] has positive Lebesgue measure, then it contains a real number hyperarithmetic in the determining system [28, p. 34] of its complement. Another way of expressing Theorem 3.9 is: if P(X) is n j and holds for every hyperarithmetic X, then P(X) holds for almost every X. We have suppressed a considerable amount of uniformity inherent in 3.9. It is not difficult to check: there is a partial recursive function/such that if e is the Godel number of a III predicate P(X) and the Lebesgue measure of XP(X) is positive, then/(e) is defined and is the Godel number of a hyperarithmetic set H such that P{H) holds. The least ordinal not recursive in X is denoted by wf, the least nonrecursive ordinal by wx. COROLLARY

3.10. The set {X | wf = Wl } has Lebesgue measure 1.

Proof. First observe that wf >a>j is U\. By Gandy [6] or Feferman-Spector [4] there is a recursive linear ordering < R of the natural numbers whose maximal well-ordered initial segment has order-type wx. Then wf >wy ii squivalent to the following U\ predicate: (Ee)[e is the Godel number of a recursive-in-A-welllordering of the natural numbers (<„)] & (/)[/is not a 1-1 order-preserving map of <e onto an initial segment of
Proof. Suppose M({* | A£kX})>0. By 3.5, ^{{T\Ae then exist a ranked formula V(x) such that l*({J I A = {« | uT(D

^(F)})>0. There must

1= 3?(B)}}) > 0.

-

Let A"c{r| /4 = {n | ^(T ) 1= «?(«)}} be such that n(K) = r>0, where r es rational". Let J be a finite union of basic open subsets of 2" such that ^{KU)
(T)[TeJ++Jt{T)Y/l Then n e A
^ 2r/3,

and consequently, /I is hyperarithmetic by 3.2. (10) S. Kripke has greatly strengthened Corollary 3.11 in the following direction: let A^2" be a set of measure 0 such that 4 t K ana such chat a t a set of tht same hyherdegree as some member of K is also a member of K\ then the set of all X such that some Y e K is hyper­ arithmetic in X has measure 0.

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COROLLARY 3.11' (CARL JOCKUSCH). LetP(X)be T\\. Jf(Y)(Ye then the Lebesgue measure of{X \ (EY)(P(Y) & Y
HYP-+

~P(Y))

Proof. By 3.9 it is enough to show (EY)[P(Y) & Y^hX] is U\. This last predi­ cate is equivalent to (Eb)[b e Ox & Y is recursive in Hf & P(Y)). A set A is said to be implicitly arithmetically definable if it is the unique solution of some arithmetic predicate; i.e., there is an arithmetic B{X) such that (E1X)B(X) and B(A). It is well known that each hyperarithmetic set is recursive in some implicitly arithmetically definable, hyperarithmetic set. Feferman [3] showed by means of a forcing argument with finite conditions that there exists a hyper­ arithmetic set which is not implicitly arithmetically definable. The notion of funda­ mental (with respect to arithmetic) was defined at the end of §2. COROLLARY 3.12. There exists a hyperarithmetic set which is fundamental {with respect to arithmetic). Each such set is not implicitly arithmetically definable.

Proof. A set T is fundamental (with respect to arithmetic) if for every arithmetic predicate B(X),

K*W)=1-*5(D. By 2.2 (iii) and 3.9, there exists a hyperarithmetic T which is fundamental with respect to arithmetic. If B(X) is arithmetic and B(T) holds, then XB(X) must have positive Lebesgue measure. THEOREM 3.13. Let P{T, x,y) be Yi\. Then the set of all T satisfying the condition below has Lebesgue measure 1:

(x)(Ey)P(T, x, y) -> (Ef)HYFW(£y)„

s Hx)P(T,

x, y).

Proof. By Gandy [6] or Spector [33], there is an arithmetic formula tf{T, x, y, Y) such that for all T, m, and n: P(T,m,n) <-+(EY)[Y <;hT & rf(T, m, n, Y)]. By 3.5, we have for almost all T and all m, n: P(T, m,ri)<-» J((T) V {EY)tf{2T, m, n, Y). Then for almost all T: (x)(Ey)P(T, x, y) «-> Jt(T) t= (x)(Ey)(EY)js/(^, x, y, Y). Let 6 be a positive rational. We define a hyperarithmetic function/such thai the set of all T satisfying the following condition has measure at least 1-8S (x)(Ey)P(T, x, y) ~> (x)(£y)„ s ,ix)P{T, x, y). Let b(n, r) be the predicate p((Ey)(EY)rf(y,n,y,

Y)) > r,

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where r is rational. By 3.2, b{n, r) is hyperarithmetic. Then/(«) is the least j such that for all r, b{n,r)-*p{{EyX&l{EY)tf{F,n,y,

Y)) > r - 8 / 2 n + 1.

Theorem 3.13 is the instance of measure-theoretic uniformity we associate with hyperarithmetic theory; by suppressing a set of T's of measure 0, we are able to restrict our attention from bounding functions hyperarithmetic in T (provided by Kreisel's Lemma [16]) to bounding functions that are actually hyperarithmetic. A sharpening of Theorem 3.13 will be useful. COROLLARY 3.14. Let P(T,x,y) and Q(x, y) be U\ formulas such that (x)(z) ■{Ey\SvQ(x, y). Then the set of all T satisfying the following condition has Lebesgue measure 1:

(x)(Ey)[P(T, x, y) & Q{x, y)] -> {Ef)HYP(x)(Ey)„iHx}[P(T, x, y) & Q(x, f(x))]. Proof. By 3.13, for almost all T there exists a hyperarithmetic g such that (x)(Ey)[P(T, x, y) & Q(x, y)] -» (x)(Ey)ySsix)[P(T, x, y) & Q(x, y)]. Since {x){z){Ey)zivQ(x, y), it must be that (x\Ey)[g{x) g y & Q{x, y)]. By Kreisel's Lemma [16], there exists a hyperarithmetic/such that (*)[*(*) S f{x) & Q(x,f(x))]. But then for almost all T there exists a hyperarithmetic/such that (x){Ey)[P(T, x, y) & Q(x, y)]->(x)(Ey)„ * Hx)[P(T, x, y) & Q(x,f(x))]. Let T0, Tu T2,... be an arbitrary seqsence of sets of natural numbers. We turt our attention to the structure Jf(T0, Tu T2,..)) introduced by Feferman [3]. The language <£{STa, fu f2,..2) differs from <£(ST) (described at the heginning of §3) only in that the set constant F has been replaced by the set constants $~o, V\, - ^ 2 • .. the structure JK(T0, Tu T2,...) ii deffned in the same manner as Jf(T). Each set constant ^ >s of course interpreted as T,. Each member of J?„(T0, Tu T2,..)) is defined by a formula &(x) of ordinal rank ^ \b\, where be Oa. Note that each such formula &(x) can explicitly mention only finitely many ^j's; thus it is possible that Jt{Ja, Tlt T2,..)) will not, in general, have a member in which all the T,'$ are recursive. We think of (Jo, T,, T2,.. .) as being a member of (2")w. Lebesgue measure M for 2" was defined in §1 as the product measure induced by the unbiased measure m on each factor 2 = {0, 1}. Lebesgue measure p for (2")N is defined similarly. If ^ is a sentence of JSP(.r0) STU 2T2... .), then the set {(T0,Tu ...) \ J?(T0, Tu .. .) t 3T)

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is a Borel subset of (2")" and so is Lebesgue-measurable; lttp(&~), the probability that & is true in JK(T0, Tu ...), be the measure of this set. Let v and w be recursive functions such that the map h: N -> NxN, defined by h(n) = (v(n), w{ri)), is one-to-one and onto. Let x e (T\ be defined by (En)[n eT& v(n) = x & w(n) = y]. Then the map h: 2" -> (2")N defined by A(r)=((D<» CDi. (r)a,...) is one-to-one, onto, and measure-preserving; i.e., if A^V, then ^ is a Borel subset of 2" of measure m if and only if h[A] is a Borel subset of (2")w of measure m (see foot­ note (12)). LEMMA 3.15. The predicate p(3tr)^r, where P is restricted to the ranked sentences ofSe(Ta, f[, S-2,.,,) and r is rational, is U\. Proof. Same as 3.2. By an instance of the 2} axiom of choice, Kreisel [16] means the universal closure of any formula of the form: (x)(EY)A(x,

Y)-+(EY)(x)A(x,(Y)x),

where A(x, Y) is arithmetic and may contain free variables other than x and Y. Feferman [3] showed that the 2} axiom of choice holds in Jt(T0, Tu...) for all (T0, r „ , . . ) outside a first category subset of (2")" by means of a Cohen-type forcing argument. THEOREM 3.16. If P is an instance of the £1 axiom of choice, then F is true in •*(T0, Tu T2,...) with probability 1.

Proof. Let s/(x, Y) be a formula of SC(^0, flt...) whose only free variables are x and Y, and whose only set variable is Y. Let T be such that J?((T)0,(T)1,...)Hx)(EY)s/(x,Y). We intend to show that almost every such T has the property that Jf((T)0, (T)x. ...)V (EY)(x)sf(x,

(Y)x).

The lemma then follows from the measure-preserving property of the map h(T) = ((T)0, (T)u ...). Let {&b(x) | b e O J be a U\ enumeration of all ranked formulas with only x free of -S%5T0, S\, S~%,...). Then (n)(Eb)[b e O, & J(((T)0, (T)x,...)

1= tf(h, Wb(x))\.

If T lies outside a certain set of measure 0, then it follows from 3.14 that there exists a hyperarithmetic function/such that (n)(Eb)bsfM[f(n)

6 Ox & J(((T)o, ( T ) l t . . . ) ¥ j/(fi, iST6(jr))].

By Spector [31], there i s a c e O , such that f(n)<0lc (n)(Eb)[b SDic&

J?((T)Q, (T)lt...)

for all ». Thus 1= s/(n, x9>(x))].

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There exists a ranked formula Jf (x) of i?(.r 0) ^ , ^" 2 ,)..) such that for all T, n, andb^0lc: Jf((T)„, (T)lf ...)♦= »»(«) «-» ^((7") 0 , , H X , . . . . 1= = £ {x34?(x))b. The existence of Jf follows by the same argument used in 3.6. But then •*«?%, (T)u ■ •.) «■ W ( £ > K ( x , C*^(*)U It easily follows that -*((D<>, (^)i- •••)•■ (£y)(Jc)J^(x, {Y)x). The proof of Theorem 3.16 was little more than an application of Corollary 3.14, which was only a refinement of Theorem 3.13, the statement of measure--heoretic uniformity for the hyperarithmetic case. In the same manner, Lemma 3.4 and Theorem 3.9 can be viewed as consequences of Theorem 3.13. In short, if one keeps in mind the hyperarithmetic version of measure-theoretic uniformity, as expressed by Theorem 3.13, one can readily recover all the results of §3. 4. The constructible hierarchy. Let Jt be a countable initial segment of L, the class of constructible sets, such that Jl is a model of ZF, the axioms of ZermeloFraenkel set theory, and hence of V=L, the axiom of constructibility. (Jl = F"a for some countable ordinal a, where F is defined in Godel [9].) We utilize measuretheoretic ideas in the style of §3 to study the effect of adding an arbitrary set T of natural numbers to Jl. The resulting structure is denoted by Jl(T). Lemma 4.2 is the instance of measure-theoretic uniformity we associate with the constructible hierachy of Jl, and it readily implies that the replacement axiom holds in Jl(T) with probability 1. In order to give a complete proof of Lemma 4.1, it is necessary to give some details concerning the structure of Jl(T). For this purpose the symbolism of Tharp [36] is convenient. The symbols of the language -270 are: e (membership); unranked set variables x,y,z,...; ranked set variables x", yu, z\... for each ordinal a o f ^ ; propositional connectives, and existential and universal quantifiers for both ranked and unranked set variables. The atoms of 210 are of the form t1 e t2, where tt and t2 are variables. The formulas of &° are constructed from the atoms and the logical symbols in the standard manner. We define a class ^ of constants (intended to name the members of Jl(T)) by induction on the ordinals of Jl. V{0) = {n\n}. «■(<*+1): ST is a member of V(*+ 1). .et 4>(xu, y, yn) be ) formula of ^° whose only free variables are xB,yl7.. .,yn (n&O), and whose only quantified variables are of the form xB for /S less than or equal to a. Then x°(x", clt..., c„), where each c, (1 £i | P<x) f o r e a c h l i m i t ordinal AA

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Let Sf=(J {<£{<£) | a € Jf). The symbols of JS%T) ara those of i?° together witw the members of <S. The atomic formulas of -S?(J") are of the form r, e / 2 , where /, and f2 are variables or members of V. A sentence & of JS?(^) is said to be ranked if all its variables are ranked. The ordinal rank of a ranked sentence 3?, denoted by o(&\ is the least upper bound of {jS | (A-*) or (£>•«) occurs in ^ } u {jj + 1 1 xe occurs in 3?}. If / e <£, ,hen the ordinal rank of t, denoted by o(f), is the ordinal rank of the sentence 0 e /. For each set T of natural numbers, we simultaneously define several concepts by induction on the ordinals of J(: c e <£(a) and b e J?a(T) and c denotes 6; o(&)gaand^ is true in 1. a=0. JS0(T) = w and d denotes «. JF is true in ^ 0 ( r ) if it it true ehen J7" is interpreted as T, n is interpreted as n, and the variables x°, y°, z ° , . . . are restricted to a.. 2. a = y+ 1. Let c 6 ^(a) be of the form xv(x>)) Then c denotes i, where fc = {o | a e Jtr(T) & (£(d))}. Jta(T) is the set of all such b'%. Let o(&) < a. Then & is true in ^ a ( J ) if it is true when each quantified variable x> of 3? is restricted to J?t(T), and each constant c of Sf(j9) occurring in JF is interpreted as the b e JfB(T) it denotes. 3. a = A, where A w a fimft ordinal. Jfh(T)=\J {J?e(T) \ | < A}. Let ^ ( D = !J {^(r) | a e Jf). Let i^ be an arbitrary sentence of <e(T). 3? is true in Jf{T) {Jt{T} N JF) if ii ii srue when each unranked, quantified varrable is restricted to Jl(T) and the remaining symbols are interpreted as above. If 3F is ranked, then & can be put in prenex normal form without any increase in ordinal rank. Assume & is in prenex normal form. The quantifier alternation rank of J ; denoted by a{3T), is the number of alternations of quantifiers in the prefix of &. The leading quantifier rank of &, denoted by 1(3?), is the number of quantified variables occurring in the prefix of 3? on the left of the left-most alternation of quantifiers of 3? if a(3^>0. If a(^) = 0, then 1(3*) is the number of quantified variables occurring in the prefix of P. Thus if & is (xa)(xe)(ya)(Ey')&, where <$ has no quantifiers, then l{3*) = 3. For each sentence &, the countability of J( implies that {T | ^ ( 7 ) N JF} is a Borel subset of 2". For each &, the probability that F is true in Jt(T), denoted by />(JF), is defined to be the Lebesgue measure of the Borel set corresponding to JF We have just given what might be called the external definition of the function p(3?). In Lemma 4.1 we give the internal definition, i.e., a definition inside Jf. If <S is an instance of the replacement axiom, then the fact that the replacement axiom is true in Jl together with the fact the probability function p(&) is ^-definable will suffice to show/>(«?)= 1. Hopefully, we will see this matter as a typical instance of the phenomenon of measure-theoretic uniformity. Of course the ^/-definability of p(3sr) plays the same role here as the ^-definabliity of the forcing relation in Cohen [2]. The difference, if there is one, is that/>(^) has a trivial external definition.

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Let A be an arbitrary set. We say A is ^-definable if there exists a formula &{x) of ZF with constants denoting elements of J( such that A equals {a | a e J( & a satisfies &(x) in uT}. 4.1. For eacA n S 0: The function p{F), restricted to sentences S? of Se(3T) having at most n unranked quantifiers, is ^-definable. LEMMA

Proof. We consider only n=0, since Cases 1 and 2 below make clear how to handle unranked quantifiers. We assume & is in prenex normal form and define p(&) by induction on the rank of J2" = r(^ r ), where r{&) is the triple (o(^), a(F\ l{Sf)). (The triples are ordered lexicographically.) Case 1. P is ( ^ M . Then

X^) = gib j ^ / . W ) | n < « , c, 6 »(«)}, where gib denotes "greatest lower bound". If c,(JSn) e ^(o), then /\tsn ^(ci) can be put in prenex normal form g? such that o(«)So(J). We claim r(<S) 1, then a(S?) = a(J*") and 1{&)<1{&). I f / ( J r ) = l , then a(&) 0, then & is equivalent to a sentence of lower ordinal rank, as is indicated in Cases 3b, 3c and 3d. Owe 3a. p{men)=\ ifm(>" e«) = 0 ifmS«. Case 3b. o(/1){ty). 0(1, er 2 )=)5+l and £feo(#(*j)). Case 3c. ofo)><>(**). Then /, e r2 is equivalent to (Ey°)[(x°)(x° e >>" «-» ^(x8)) &y"e t2], where /, is of the form xa{xa). o{i% e t2) = <x+1, but the ordinal rank of the above equivalent formula is ^a. Case 3d. 0
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Both 2.2 and 3.13 have the form: for almost every T there exists a n / e t c , but 4.1 has the form: there exists an / s u c h that for almost all T etc. The difference is a consequence of the strong "closure" properties of the constructible hierarchy. LEMMA 4.2. Let P(x, y) be a formula of&{&) whose only free variables are x and y. Then there exists an J(-definable function f such that for allaeJf with probability 1: (x°)(Ey)P(x°, y) <-► (x°)(Eyna))P(x", /<«>).

Proof. Let g(a, £) =p((xa)(Eyt)P(xtt, / ) ) ; g is ^-definable by 3.1. For each a, Af | g(a, £) is a nondecreasing ^-definable function from the ordinals into the unit interval. Since the replacement axiom holds in Jf, there must exist ^-definable functions h{a) and/(a) such that h(a) is the least upper bound of {g(a, ft) | p e Jf) andg(a,/(a)) = h(«). LEMMA

4.3. Let & be an instance of the replacement axiom. Then p(3F)=\l

Proof. The replacement axiom for J((T) can be split as follows: (i) the range of a function restricted to a set is contained in a set; (ii) let P(x) be a formula of &{$-) whose only free variable is x; then (y){Ez)(x)[x ey&P{x)

HIEZ],

By 4.2, (i) holds in J/(T) for almost all T. But (ii) follows from (i) in J((T) by standard arguments implicit in Godel [8, 9]. Let y e J?(T). Then it follows from (i) by means of a Skolem-Lowenheim argument that there exists a v e Jf(T) such that >'er,>'<=t,, a n d i n ^ C r ) , (x){xey&P(x)

<^-xey&P°(x)),

where P\x) is the result of restricting the quantifiers of P to v. Then z^x(xey&. P\x)) is a first-order definable subset of v; and ZEJ<(T), since every first-order definable subset of a member of J((T) is a member of JOJ). PROPOSITION 4.4. Let {^\new}be a sequence of J( of sentences of<£(T) of countable (in the sense of Jf) ordinal rank. Then there exists a sentence & of count­ able ordinal rank such that for all T,

J((T) ¥ JF <-> (En\J((T) In addition, 3F regarded as a function of{^n\ne

V Pn). w], is Jf-definable.

Proof. Let a be a countable (in the sense of Jt) ordinal such that o(SQ«x all n. Then J{(T) N &n <<» J?a(T) \= .^„

for

for all n and all T. By standard arguments there exist a countable ordinal £ > a and a formula P(x°) of SP(f) of ordinal rank jS such that for all n and all T, Jfa{T) r- 3Fn <~ JfB(T) V N(n).

132 1969]

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401

The existence of P(x°) follows from two facts: (a) truth in J?JT) has a first-order definition (uniform in T) with quantifiers restricted to Jttt+JJ); (b) every firstorder definable subset of a member of Ji(T) is a member of JK{T). The desired SF is (Ex°)P(x0). LEMMA 4.5. For each ngO, fA*«? exwfc an Jt-definable function \3F\3F*, defined for all sentences 3? of ¥{3~) having at most n unranked quantifiers, such that p(& <_> jr*) = i and j r * has countable (in the sense of J() ordinal rank.

Proof. Similar to 4.1. We consider n = 0 only, assume J5" is in prenex normal form, and define 3?* by induction on the rank of 3*. If JT has countable ordinal rank, then ^ * is 3?. Let 3? have uncountable ordinal rank. Suppose 3* is (Ex^^x"). Then p(SF) = lub {/>( V * i f o ) ) I *(' ^ ») £ C M } where lub denotes "least upper bound". As in 4.1, V<s* 3[fe) is equivalent to a prenex normal sentence of lower rank than (Exa)^1(xa). Then

/>(V^fo)+-*(V^te))*) = i. where (V.sn-^Cd))* has countable ordinal rank. It follows from the ^-defina­ bility of \3F \p(3?) (4.1) that there is an ^-definable sequence {c, | i<m}QC(a) such that # )

= lub { P ( V ^ ( C ) ) I / < - } •

Let .F* be equivalent to V,<» (Vi«. ^fa))*. &* exists by 4.4. If & is ( ^ J ^ ) , we proceed in a manner dual to the above. If & has no quantifier prefix, then 3* is equivalent, as we saw in 4.1, to a sentence of lower ordinal rank. LEMMA

4.6. Let a be a cardinal ofJt.

Then p{a is a cardinal of

Jt(T))=\.

Proof. We sharpen the argument of 4.2. Let a be an uncountable cardinal of M, and let p e «. We show p(fi can be mapped 1-1 onto a) = 0. Let / b e a constant < 0}. By 4.1, Ay | £(y) is ^-definable. Since / denotes a function and p is countably additive in the sense of J?, it follows that k(y) is a countable set of ^ for all y e ft But then (J {/t(y) | y e f t is a subset of a of cardinality (in the sense of Jf) less than that of a, and consequently, ^ ( { / ( y ) | y e f t C U { A - ( y ) | y E f t # « ) = 1. LEMMA

4.7. Witf probability 1: //ie ;wver 5
Proof. Our argument is related to the one given by Godel [9, p. 46] to show that

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the power set axiom holds in L. Let d be a constant of $f. It will suffice to find an ordinal y of J( such that for all c e ^",

p(c£ d-*ceJty(T))

= 1.

Let de #(a+1 )-«•(«*). Then the members of the set denoted by d can be denoted by constants of #?«). By 4.5 there is an ^-definable function Xbc | (b e c)* *uch that

p(4ec«(|,e

c)*) = 1

for all b,ce<€, and such that (b e c)* has countable ordinal rank. For each let &(c) = kb | (6 e c)*, where b is restricted to S<(«). If A:(cc) = it(c2)) ,hen

csV,

p(d £ d & c2 £ (b e cf) = 1, and (b e c)* has countable rank. As in 4.7, if k(c1) = k(c2), then p(Cj £ a & C2 £ a -► c, = C2) = 1. As in 4.7, {*(c) | c e
such that for all

Let d* denote the set of sets of JOJ) denoted by the constants of d. Then for all ce<€, p(c £ a--ce„ it follows that the cardinality of d (in JT) is at most a>f=2B=a+. But the cardinality of d* (in uT(r)) is at most that of d with probability 1. THEOREM 4.9 (R. SOLOVAV). With probability \:J((T) + AC +"cardinals are absolute" + V^L.

is a model oj ZF+GCH

Proof. Apply 4.3, 4.6, 4.7, and 4.8. For all T, J((T) Y F e L if and only if TeJt. Clearly p(& e L) = 0, since . * is countable.

134 1969]

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We now develop Cohen's results concerning the independence of the continuum hypothesis by making purely notational changes in the proof of Theorem 4.9. Let y be an uncountable cardinal of Jt. We study the structure J?(T% where V is an arbitrary subset of y. The language &{?->) is similar to the language <£(ST) (=&{&<*)) except that f is replaced by y and f(0)={« | n e w} is replaced by {/3 | /S)) then the set

is a Borel subset of 2"; its Lebesgue measure, denoted by p(&), is the probability that P is true in J/(T"). THEOREM 4.10 (R. SOLOVAY). Let y be an uncountable cardinal of Jt'. With probability 1: Jl(V) is a model of ZF+ AC +"cardinals are absolute" + V±L. If y is not cofinal with w (in the sense of Jf), then 2<" = y holds with probability 1 in Ji(T").

Proof. Ranks are assigned to the sentences of &{&-*) in precisely the same way they were assigned to sentences of <£(T). There is no difficulty encountered when y is replaced by JT' in 4.1, 4.2, 4.3 and 4.4; 4.5, 4.6, and 4.7 are unaltered by the substitution of $-> for S~, since they depend only on the countable additivity of the measure. Let g:yxo>->y be the canonical 1-1 correspondence between y x a , and y. For each 8>y, let a6 be a constant of V which denotes {n \ g(8, n) e T>). If 8^82, then p(n sa6l^fie ai2) = \ for each n, and consequently p(atx = ati) = 0. a It follows that p(2 Sy)=l. Now assume y is not cofinal with w. The argument of 4.8 serves to show p(2a = y) = 1. For each c e <€, let k(c) = \n\(ne c)*, where ne u> and (n e c)* has countable ordinal rank. \S k{c1) = k(c2), then p(d £ w & c2 c u, ->- Cl = c8) = 1. The cardinality of the set of sentences of J2%r') of countable ordinal rank is y; so k{c) can be regarded as function from u> to y. It follows {&(c) | c e W) is a set (in ^ ) of cardinality at most yu = y, since (in Jf) y is not cofinal with co and the generalized continuum hypothesis holds. But the cardinality of 2" (in J/(T')) is at most that of {k(c) \ceV} with probability 1. Let (T0, Tu T2,,..) be a sequence of arbitrary subsets of 2". We now study the effect of adjoining (T0, Tlt T2,...) to Jf in the same measure-theoretic fashion we studied the effect of adding an arbitrary T to Jf. The language i%7"0, ST,, 3T2,...) is obtained by altering 2(T) as follows. Replace F by yo, 5 ^ , ^ , . . . ; add primi­ tive symbols + (addition) and • (multiplication) with the proviso that they are to operate only on number-theoretic terms. ^(0) is again the set of finite numerals. «f(a+ l) is defined as at the beginning of §4 save that $~ is replaced by r0, SI, 3T2,.. ., STU . .. (i< cu) and the formula <j> can have occurrences of + and . Thus

135 404

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the constants of V{\) now denote all the subsets of w which are arithmetical in some finite subsequence of T0, Tu T2,...; truth in the structure Jt{T<>, Tx, T2, . . .) and the relation " c e tf & b e J((J0, Tlt T2,...) & c denotes b" are defined as before. Regard (7~0, Tu T3,...) as a point in the space (2")", and give (2")w the product measure induced by the measure m on each factor 2. (m({0})=w({l})=4.) Then for each sentence & of X{r0, S~x, F%,...), the set {(ro, Tx, T2,...)\

JT(TD, Tu

T*,...)**)

N N

is a Borel subset of (2 ) ; the measure of this set, denoted by p{&), is the proba­ bility that 9F is true in Jt(T0, T» T2,.. .)• The argument of 4.1 now shows: for each n^O, the function p(3T) restricted to sentences . F of &(ra, S~x, F2....), is ^-definable. The arguments of 4.2, 4.3, 4.4, 4.5, 4.6, and 4.7 still work, and together they show: with probability 1, J?(TD, Tu T2,...) is a model of ZF + "cardinals are preserved". Actually, it is necessary to restate and reprove 4.4, 4.5, and 4.7 as 4.17.1,4.17.2, and 4.17.3. That task is postponed until 4.17 is established. It is instructive to consider why 4.8 now fails. The last line of the argument of 4.8 assumes the existence of an ^(7>definable function which assigns to each set of JK(T) a constant of V that denotes it; the function is obtained by assigning y to Tand then exploiting an ^-definable well-ordering of the formulas and con­ stants of &($~). In the present case, the language if(^"0, Xt, JT 2 ,...) has an ^-definable well-ordering, but with probability 1 (as we shall see below) there is no Ji(T0, TUT2,.. .)-definable function which assigns ^ to Tt for all i<w. Jt(T0, Tu ...) has certain symmetries studied by Cohen [1], Feferman [2], and Solovay [29], [30] which we shall study in terms of measure-preserving transforma­ tions of (2")". Let r be a function whose domain is {^ | i e «■} and whose range is a subset of fJ) = c(K^o), • • •, /"(.TJ); in the same manner, r can be extended to a function from the sentences of i%r 0 , Tx,...) into itself. If r is a map from «"(!) into ^(1), then r induces a map r* from (2")" into (2")" as follows: n e r*(Tt) «-+ J?(T0, Tu...)VSe

r(^);

r*(T0, Tu ...) = (r*(r0), r*(7' I ),,..). We say r is a map from ^(1) onto #(1) if r is a map from <€{\) into #(1) and if for each iew, there is a M, e< r{&) <-» J?(r*(TB), r*(Tx), ...)(= J^/br a// (r 0 , 7 \ , , . . ) .

(") Let c,dzV. We say r = d\t c and . . .) for all (7-0, r „ T2l. ..). J. Rosenthal [21] has shown that this notion of equality is uf-definable. His result is not needed in the present paper.

136

1969]

MEASURE-THEORETIC UNIFORMITY

405

Proof. By induction on the rank of 3F. (We use the same notion of rank employed in 4.1.) It is sufficient to consider two cases. Case \. & is n e STt. Then J?(T0, Tt>...) 1= r(n e &[) <-> Jt{T<>, Tu...)¥n 6 r{$~{) HBEr*{T % ) *-* J({r{T0), r(F,),,..) ¥ n e J~t. Case 2. 3F is (£x-")S?(x\ b), where b is the only member of <€ occurring in &. It is easily checked that r maps V(a) onto V(a); i.e., for each deV(a), there is a c 6 ^(o) such that r(c) = d. The following formulas are equivalent: J?(r*(Te% r*(7-0, ■■■)*= ( £ 0 * ( * * . *); (£c)[r £ V(a) & ur(r*(7i), r*(7V), ■-.)*= 9f(c, 6)]; (£c)[e e
Proof, r* is 1-1 and maps (2")w onto a set whose complement has measure 0. Now apply 4.11. Feferman's transformation lemma [3, 4.8] is a forcing-theoretic version of our Lemma 4.12. Let
It is clear that V has a constant which denotes JT(a; T0,..., Tk. J ; let us adopt the convention of denoting JT(a; 7"0,..,, 7 ; . , ) by uT(o; ^"0, ...,3Tk^\ Similarly, let ^ " o , • • -, -*"*-i) = U W « ; *"o, • • -, **k-j) I « 6 - # } ,

and let ^ ( J ^ , . . . , 3Tk^) denote 1J {^(o; Ta,.. .,Tk_l)\

ae Jt).

PROPOSITION 4.13 (R.SOLOVAY). Foreachk^O: Jf(X0l. ..,Srk_J with probability 1.

is well-ordered

Proof. Let -T = ^/(T0, Tu Ta,...) be a model of ZF with the allowable exception of the power set axiom. Fix ArfeO. In yT we can define truth in Jr for all ranked sentences whose .^-symbols are included in the list Fa,.. .,3Tk_x. It follows by standard arguments that there exists a 1-1 ^T-definfible map hk of Jf{T0, ...,Tk-x) into tf(JT0, • • •, -^c-i), ie., each set of J({T0, ...,Tk.1) is mapped to a constant of V(3~0,. .., Fk _,) which denotes that set. But there exists an .//-definable (hence ^-definable) well-ordering of <£. The definition of hk has T0,...,Tk., as param­ eters.

137 406

G. E. SACKS

[August

LEMMA 4.14 (R. SOLOVAY). Let &(x°) be a formula of £C{^0, &l,...) whose only free variable is x", and whose ^-symbols are included among P0t..., ^ _ a . Then with probability 1,

(£x°)W)

♦- (Ex°W(x°)

&x"eJ?(«;r0,...,

rk)].

Similarly, with probability 1, {ExW{x)«-

(£x)[Jf(x) &x e Jf{X0

rj\.

Proof. Let k = 1. To prove the first half of the lemma, it is enough to prove: for each sequence c0, c% cn e f(«), there exists a sequence r(cQ), r(cx),..., r(cn) e
P(r(c())\.

cn e #(«; ^" 0 ,..., ^"J. Consider the following

r(f0) = JT0; r ( ^ + m + 1 ) = ^ + 2 rW+1)

= x0(mx0+jetri)

(i S 0);

{] < m).

It is not difficult to check that r maps #(1) onto #(1) and that r* is measurepreserving(12). Clearly, r(Cj) e «f(o; 5 ^ ^1) and r(J^(c,)) is ^(rfo)) for each ign. Now apply 4.12. We specify some maps of #(1) into Sf(l) needed for the proof of Lemma 4.15. For each « > 0 and /SO, let (•^"m)i = i 0 K2 f + l Ar°+2«-l)6^;]j (•^m)n =

X0[(2"A:0 + 2 ' , - 1 ) £ ^ ; ] .

For each m . ^ 0 , we define a map r* of #(1) into #(1): rJCJJ) = 5J iWm+d

= (-^J.

tfC^.+ . + k) = fm+k

(/ < m); (' < «); (k 5 1).

It is not hard to see that (r»)*: (2")" -► (2")" is measure-preserving (see footnote (12)). We claim that r? maps <'° e ^ m + n & ^° = 2 " / + 2"-1) v (Ei)esi2)Cf. Halmos[10,p. 159(7)].

= 2, + V + 2 ' - l ) .

138 1969]

MEASURE-THEORETIC UNIFORMITY

407

Clearly, r?(rf?)=^,. Thus r™ is a map of ^(1) into S?(l) which meets the hypotheses of Lemma 4.12. LEMMA 4.15 (R. SOLOVAY). With probability 1, the dependent axiom of choice holds in Jt{T0, Tu T2,.,.).

Proof. Let &(x, y) be a formula of <£(2Ta, &"%,...) whose only free variables are x and y and all of whose 57-symbols are included among JT0, ^r1,...,^m.1 (m^O). We intend to find a n / e ^ ^ , , . . . , $~m) such that/denotes a function on ^ and such that with probability 1, (x)(Ey)^(x, y) -> (x°)JRy(;c0),/(;c° + 7)). For the sake of notational simplicity, let m = 1. By 4.14, for each n g 0, the following two sentences are equivalent with probability 1: (x){Ey)[x e Jt^,...,

J») -► ^ ( x , j ) ] ;

(x)(£y)[* e ^ ( y 0 , . . . , jTn) -+ J^(x, j ) & j e ^ ( y o , . . . , f n + x)]. With the aid of 4.12, we apply r* + 1 to the above pair of sentences to obtain another pair equivalent with probability 1: (x)(Ey)[x e JH^, (^)0,..., (x)(Ey)[x e Jt^,

(Jr1)B_1) -* F{x, y)];

(*i) 0 > ..., ( ^ ) „ . x) -> ^ ( x , ^) & j> e ^ ( ^ 0 , ( ^ ) 0 , , . . . (^i)„)).

(A^e: ^(^" 0 ,(^"i)o,...,(^"i)n) denotes the collection of all sets of J?(T0,Tly T2,...) denoted by constants of the form ri+1(c) for some c e <€{2Ta, &u ..., fn+1).) It follows from 4.13 that the sequence of sets Jt{f0, (^)0 (^)n) (ngO) can be well-ordered uniformly in n with probability 1. But then for some/e y), then /(0) e ^(^" 0 ), /(n+T) e ^(^- 0 , ( . ^ V . . . , (iTl)n)

(„ > 0),

(x°)^(/(x°),/(x° + I)). We can insist t h a t / e tf^, ^ ) , since the constants that denote the well-orderings needed in the definition of/are members of V(3TQ, $\). The argument of Lemma 4.15 can be greatly simplified, if one merely wishes to establish the countable axiom of choice (see footnote (2)). In fact, Proposition 4.13 and Lemma 4.14 readily imply: with probability 1, the product of a family of non­ empty sets, indexed by a well-ordered set, is nonempty. A very simple notion of conditional probability of truth in J?{T0, Tu T2,.. .) will make it possible to strengthen Lemma 4.12. Let {^[\ i e K) be a finite set of .^-symbols; let A^w for each ie K. The conditional probability that 3? is true in J((T0, Tu T2, ...), given that T{ = AKfor all i e K, is denoted by p^F I 3T = A„ i e K),

139 408

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[August

and is defined as follows. Let (2")2 be the set of all sequences (7 0 , Tlt Tit...) such that Tt=Ai for all i e K. Think of (2NfK as the product of countably many copies of 2* and finitely many one-point spaces, one one-point space {At} for each i e K. Give each copy of 2" the usual Lebesgue measure, and give each one-point space measure 1. Let (2*)g have the product measure induced by the measures assigned to its factors. Then p(& | 5 > A , ie K) is the measure of the set of all sequences (T0, T1,Ta,...)of (2")£ such that & is true in Jt(T0, Tu T2,...). Let r be a map of <€(\) into < (2")" is measure-preserving. Suppose that fixing {S^\ie K] has the effect of fixing

Then for each sentence & of X(F0,

{r(^}

| i e Kr}.

&l,

9~%,...),

p(r(SF) | ^, = A„ i e K) = p(& \ f, = Bu i e KT), where for each ie K„ B, is the set denoted by r(^t) when T{ = AJor

all i e K.

Proof. We extend the argument of 4.12. The map r*: (2N)" ~* (2*)" is 1-1. Let r£ be the restriction of r* to (2")%. Then r? is a 1-1 map of (2")£ into (2 w )g r . The lemma will now follow from 4.11, after we show r% is measure-preserving. Let fa | i e m) be a sequence of constants of <S which denote subsets of a,. We say {c, | i£n) is a sequence oiindependent, uniformly distributedw-terms if /?(»?! £ c„ & w 2 6 ca &• • • & mk e clk) = 1/2" whenever 0 < i I S f 2 ^ ••• g i i k g n and m, = mp->ij?ip for \<,jk. We say {c, | / 6 cu} is a sequence of independent, uniformly distributed o,-terms if {c,| i^n} is such a sequence for all n > 0 . It follows from 4.12 that {r(^7) | r e w} is a sequence of independent, uniformly distributed w-terms. This last fact implies r% is measurepreserving (cf. Halmos [10, p. 191-192]). COROLLARY 4.17. Let {f,\ isK) include all the ^-symbols occurring in the sentence & of SC(3Ta,
140 1969]

MEASURE-THEORETIC UNIFORMITY

409

Proof. Let b denote an arbitrary prepositional combination of sentences of the form m e Sx, R $ S„ where i,j i K. Then p{b) can be thought of as the measure of an arbitrary finite union of basic open subsets of (2")». ((2*)Jf is defined just before 4.16.) It follows from 4.16 that p{S &b\St

= AuieK)

= p(b)p(S \ St = Au ie K).

For example, if K={0}, b is in e Su and r is the map defined by r(S1) = w-S1 r(^) = St(i*\), then

and

p(S & m e St | S0 = /4„) = i>(-^ & * * ^ I SB = ^ 0 ) by 4.16, and consequently, />(^ &meS1\S0

= A0) = \p(S

\ Sa = A0).

For each 8>0, there exists a * such that p(S^b\St

= At,ieK)

& 1-8.

But then/>(jr | St=A„ i e K) must be a solution of AT = X 2 . Suppose {y; | ; e K} includes all the ^-symbols occurring in S, where S is a sentence of JSP(^"0, •*!» -^i, • • •)• Then Corollary 4.17 tells us that the truth-value of S in the structure Jt(T0s Tu T2,,..) is determined by the choice of {Tt \ieK} for almost all sequences (T0, Tu T2,...). Let (2")K be the product of #-many copies of 2", one copy of 2" for each i e K. Give (2")K the usual product measure. Identify the sequences {At \ieK} with the points of (2")*. Then by Fubini's theoremp(S) is the measure of the set of all sequences {At\ieK} such thatp(S \S=At,ieK) = 1. PROPOSITION 4.17.1 (CF. 4.4). Let {Sn\new}be a sequence of M of sentences ofSf(S0, Sx,...) of countable (in the sense ofJT) ordinal rank such that for some k, no Scsymbol with i^k occurs in any Sn. Then there exists a sentence S of countable ordinal rank containing no S-symbols not occurring in at least one Sn and such that for almost all (T0, Tu ...),

J?(T0,

TX,...)¥S

(En)(J?(T0, Tx,,..)

In addition, S regarded as a function of{Sn \new)

1= Sn).

is Jf-definable.

Proof. The argument of 4.4 establishes the first part of the proposition, namely, that the desired S exists. The ^-definability of S as a function of {S„ | n e w} follows from the ^-definability of the probability function p restricted to sentences of Se(S0, Su...) of countable ordinal rank and from the following equations which characterize S up to a set of measure 0:

«[p(V ■ * - * ) " >]; (h)(Ei)\plS-+

V Sn\ ^ 1-81;

8 is a variable ranging over the positive rationals.

141 410

G. E. SACKS

[August

LEMMA 4.17.2 (CF. 4.5). For each /iSO: there exists an ^-definable function XS? | JF*, defined for all sentences & of ^{3T0, &l,...) having at most n unranked quantifiers, such that p(& <-*&*)= land such that F* has countable (in the sense of JT> ordinal rank and contains no ^-symbols not occurring in 3F. Proof. We proceed by an induction on rank after the fashion of 4.5. Let F be a prenex normal sentence of uncountable ordinal rank of the form (Exa)3P1(xa) such that f0 is the only J^-symbol occurring in ST. By 4.14,

p(&) = lub fpl V *i(c,)) | c, £ <$(a; f0,

fl)\,

where lub denotes "least upper bound". As in 4.1, \fISn J^(Ct) is equivalent to a prenex normal sentence of lower rank than (Ex")&i(x"). Then

where (V.sn-^fo))* has countable ordinal rank and contains no ^-symbols other than > 0 or rx. It follows from the ^-definability of p that there exists an ^-definable sequence {ct | /««}£< then the desired F* is S?. Suppose 9 contains both JT0 and 3TX. We know that p(f <-> SF) = 1 and that the only ^-symbol occurring in ? is ^"0. It follows from 4.17 that/>(S? | &o = T0) equals 0 or 1 for almost all T0. The function Ar0 | p(& \ Ta = Ta) is Jt(T0, Tu .. .)-definable for almost all (T0, 7 \ , . . . ) ; its definition contains no ^-symbol other than ^"0) and has countable ordinal rank because IS does. Thus there is a formula 3t(3Ta, x°) of countable ordinal rank such that for almost all (T0, Tu...): Jf{J0, Tu...)¥

Jt(S~0, 0) <-» p\<3 | ;T0 = T0) = 0;

uT(r0) 7",,...) 1= Jt(y0,

I) <->i>(2? I ^"o = T0) = 1.

Then the desired .F* is J f ( ^ T). Note that the above argument merely establishes the existence of 3F* and not its ^-definability. The ^-definability of J r * is a consequence of the ^-definability of the probability function p (restricted to sentences of if(5"0, &[,. ..) having at most n unranked quantifiers) and the following characterization of 3?* up to a set of measure 0: p{& *-* 3F*) = 1; J^* has countable ordinal rank and contains no &\symbol not occurring in $Z LEMMA 4.17.3 J((T0,Tx,...).

(CF.

4.7). With probability 1: the power set axiom holds in

142 1969]

MEASURE-THEORETIC UNIFORMITY

411

Proof. Same as 4.7 save that 4.5 is replaced by 4.17.2. Feferman [3] showed that if T0, Tu T2,... is a sequence generic in the sense of Cohen (i.e., generic in the sense of forcing with finite conditions), then in Jf(T0, Tu T2,..,), the Boolean algebra of all subsets of w has no nonprincipal, maximal ideal. The proof of 4.18 is a measure-theoretic analogue of his argument. THEOREM 4.18. With probability 1: The Boolean algebra of all subsets ofm has no nonprincipal, maximal ideal in the structure Jt{T0, Tu T2,...).

Proof. Let c e # be a constant which denotes a set of subsets of m. Suppose p(c is a nonprincipal, maximal ideal of 2°>)>0. Let S^ ..,, 3Tn^ include all the .^-symbols occurring in c. By 4.17 and Fubini's theorem, there must exist sets A0,...,An^ such that p(c is a nonprincipal, maximal ideal of 2°> | ^ = Au i < n) = 1. Let a" denote w-d.V/e

intind to show

p{3Tn

6

c& &: e c | SI = Au i < n) = l1

Either K - ^ e c | $~i=Al,i
= At,i

< n,STn = A) = 11

By 4.17 and Fubini's theorem, p{J~ne c\Srt = Aui
= At,i
/(A)).

But p(Sn ec ^f{3Tn) ec\3Tx = A„ i < n) = 11 since c denotes a nonprincipal maximal ideal with probability 1 when S[ is fixed at At for all i
= Au i < «) = 1.

In order to develop Solovay's relative consistency result on the extendability Lebesgue measure, we shall replace
of Su be as

143 412

G. E. SACKS

[August

possible from a measure-theoretic point of view. This device was originated by R. Solovay. Let c0 = c0{T0, • •., fn) be an co-term of <€, that is to say, a constant of V which denotes a subset of w. (We can assume that the ^-symbols occurring in c0 are 3r0, ^ T% for all sufficiently large n, since we can always add ^-symbols to c0 in a spurious fashion.) Recall the notion of "sequence of independent, uniformly distributed co-terms" introduced at the end of the proof of 4.16. We say c0 = ca{&~0,..., S^ is a projective, uniformly distributed co-term if there exist co-terms ct(y0,.,., Fn) (I Si£n) and kt{^0,..., JQ (i|n) of # such that {c, | /<«} and {A:, | i ...,kn) for all igw. (k,(c0,..., c„) is the result of substituting c4 for ^7 infc,.)Note that if c0 is a projective, uniformly distributed co-term by virtue of the existence of c, (1 g i g n ) andfc,(i < n), then c, (1 g ig/i) and A:, (/ ^ n) also are projective, uniformly distributed co-terms. It is tedious but routine to check that (fm)„ (^m)n and df, which were defined immediately preceding the proof of Lemma 4.15, are projective, uniformly distributed co-terms. The collection of all such terms is ^-definable, since P{&), restricted to the ranked sentences of &{r0, flt 3T2,...), is ^-definable. Let 0) and ^*(A) (A a limit ordinal) are defined in the same manner as V(y+1) and T(A). The constants of tf*(y+l) (y>0) denote the first-order definable subsets of J??+1(T0, 7\, T2,...). The atomic formulas of ^*(^~o. ^ i , ^"a, ■ • •) are of the form t, e t2, where h and i2 are variables or con­ stants of V*. Let Jt%(T0, Tx, T2,.. .) = co. Let t e «*(1); suppose {^ | / e * } is the set of all .^-symbols occurring in t. By Lemma 4.17,

p{net\ft

= Tt,ieK) = 0 or 1.

We say that t denotes {n\p{n et | ^ = r „ j e K)= 1}. -//f(r 0 , 7^, T2,...) is the set of all sets denoted by some r e «"*(1). J?% .(To, Tx, T%,...) (y>0) is defined in the same inductive fashion as ^% + 1 (r 0 , 7\, F 2 , . . . ) . Thus the only conceptual differ­ ence between Jf(T0, Tu T2,...) and *#*(T0, Tx, T2,..,), or between JSf(5"0) ^ 5 ^ , . ..) and &*(f0, ^ i , ^ a , . . . ) , is the difference between %\\) and $f*(l). A routine transfinite induction establishes the existence of an ^/-definable f-
144 1969]

MEASURE-THEORETIC UNIFORMITY

413

is an m-term; c and f(c) have the same occurrences of ^-symbols; the set denoted by c in Jf*(T0, Tu ...) equals the set denoted by f(c) in JT(T0, Tu ...) for all (T6, Tu ■ • •)■

For each sentence SF of Sf(f0, f[, f a , . . . ) , the probability that SF is true in Ji*(T^ Tu T2,...), denoted by z?*^), is the measure of the Borel set {(7"0, Tlt T2,...)

| ^ * ( r 0 , rx, r a > . . . ) * * } .

The argument of 4.1 shows: for each n>0, the function p*(.F), restricted to sentences of
{r(^) | i e AT,}. rAcw/or eac/i 5«i/«»ce ^ of^*(T0,

*l, 2T2,.. .),

p(r(i^) | rt = A„ i e Jf) = /»(#" | ^ = St, i 6 £ ) , where for each i e K„ B, is the set denoted by r(f,) in Jt*(Ta, Tu T2,. ..) when Tt = AjorallieK. Proof. Same as 4.16. The arguments of 4.11 and 4.12 are correct for J?*(T0, Tu T2,...). The differences between V(\) and <
LEMMA

Tu

Proof. Same as 4.15. But it is necessary to check that the maps of «"(1) onto tf(l) employed in the proofs of 4.14 and 4.15 are also maps of 'if*(1) onto fc*(1). From now on we will identify elements of 2" with real numbers in the closed unit interval [0. 1] by means of the usual dyadic expansions. Thus if A, Be 2s, then A + B (mod 1) will correspond to the arithmetic sum (mod 1) of the real numbers corresponding to A and B. A similar remark applies to u,-terms oftf*.

145 414

G. E. SACKS

[August

LEMMA 4.22. (a) Let c(Ta, ...,^n)e tf*(l); then there exists a map r ofV*{\) onto «*(1) such that r{3T0) = c(3Ta,.. .,f„), r ( ^ ) = ^ " B + 1 , and the induced map r*: (2N)N ->■ (2N)N is measure-preserving. (b) If t^,..., ^»_ 1 ) is an w-term oft*, then there exists ac„e «*(1) such that c, = ^ , + r(^"0, • •, -^n -1) (mod 1) with probability 1. In addition, there exists a map r oft*{\) onto «"*(!) 5«cA f/ifl/ r(^) = ^ (/<«), r(^,) = Cn, and the induced map r*: (2")N -► (2")" w measure-preserving.

Proof. Since c0=c(^"0» ■ •., K) 6 ^ * 0 ) , there must exist c,(^" 0 ,..., fn) and t f t , . . , ^ (i<:n) such that {c, | i
(Jfc > 1),

( 1 S / S n),

maps «f»(l) onto V*(l). The induced map r*: (2Nf -+ (2"f is measure-preserving, since {r(^) \ i e w) is a sequence of independent, uniformly distributed co-terms. (Cf. end of proof of 4.16.) By the remarks immediately preceding 4.19, there exists an co-term t ^ , . . . , ^r„_1) e <€ such that 3Tn + t{^0, ...,J'n.1) denotes the same set in J(*(Ta, Tx,. ..) that ^ - t - ^ O T o , . . . , ^"n_,) denotes in ^(7" 0 , T , , . . . ) . We must show cn = ^"n + /i is a projective, uniformly distributed co-term. For this purpose, it is convenient to think of the co-terms of V as random variables [5] defined on the sample space (2N)». An co-term is uniformly distributed in our sense if and only if it is uniformly distributed in the standard sense when it is interpreted as a random variable. Thus if c0, • • •, cn is a sequence of uniformly distributed co-terms such that c, and c, (0
is a sequence of independent, uniformly distributed co-terms. The claim is a con­ sequence of the following type of standard argument. Let X0, Xu Y be random variables on (2T such that X0, X1 are independent and uniformly distributed, and such that X0, Y are independent. Then X0+ Y(mod 1), Xx are independent and uniformly distributed, since />(*<,+ Y=a&

X, = c) = ^p(Y=a-b

&X0 = b& X1 = c)

b

= p(X0 = a)p(X1 = c)Jtp{Y=a-b

| X0 = b & X1 = c)

b

= p(X0 = a)p(Xi = c). To check that cn = ^ , + / 1 (^ r o,,.., ^ , - 0 is projective and uniformly distributed,

146 1969]

MEASURE-THEORETIC UNIFORMITY

415

let c( = kt=&l (i< n) and let k^K-tx(fa,..,, Tn_x) (mod 1). The desired map r is defined by r(^) = ^7 (rV«) and r(^"„) = c„. Let A:" be a restricted variable of Sf*(f0, Fx,...), restricted to the subsets of w in J?*(T0, 7\,.. .)• Thus we can denote each subset of 2" in J(*(Ta, Tu...) by a constant of the form x^x"), where JF(xn) is a formula of ^*(^" 0 , -^i, • • •) n whose only free variable is x . For each (T) = (T0,T,,T2,.. .), we define the afoo/wre (in T) measure of x l ^ x " ) , denoted by ^ ( x ^ x " ) ) , as follows: ^ ( x ^ x " ) ) = / > * ( . ^ ) | ^ = T« i e K), where; $ K and {57 | ie K] is the set of all ^-symbols occurring in & (By 4.19, the value of j is irrelevant so long as jf K.) For each (r) = (7"0, Tu T2,...), we say ^ * ( r 0 , 7\, T2,...) is measure-complete if for every ^(x") e ^*(^" 0 , -^i. • • •), we have

/£(*V(x°)) > o -* ^*(r0, r„...) N (^^^(x"). The concept of absolute (in F) measure has been defined for definitions of subsets of 2N in J(*(T0, 7 \ , . . . ) rather than subsets of 2"; for each (T) = (T0, Tt,...), let us say £ is well-defined on all subsets oflN if all definitions of any given subset of 2" in J(*(T0, 7 \ , . . . ) have the same absolute (in T) measure. 4.23. With probability 1: (a) Jt*(T0, Tlt T2,...) is measure-complete; (b) ifJe*(T0, Tu T2,...) ir measure-complete, then £ is well-defined on all subsets of 2»inJIP(T0,T1, T2,...). LEMMA

Proof. First we do (a). Let T0 be the only 57-symbol occurring in &(&, .T0). We must show: for almost all (T) = (T0, Tu T2,...), if /xJ(x".F(xn, 5ro))>0, then J(*(T0, Tu T2,...) V{Exn)&{xn, JT0). Suppose not. Then by 4.20 and Fubini's theorem, there exists a set K^2" of positive measure such that for all T0 e K, p*(~(Exn)^(xn,

3Ta) | ;T0 = T0) = 1,

P*{P(rlt sra) | JT0 = T0) > 0. Now we do (b). Let J ^ x " , 3Ta) and J^(x n , S\) be formulas such that 2TX is the only ^-symbol occurring in Jftx", J^) (i<2). By 4.20 and Fubini's theorem, m0 = /xJ(xnJ?o(xn, 5"0)) = p*(5F0(3T2, 3T0) \ Ta = T0, S"t = Tx) for almost all (T) = (T0, Tlt T2,.. .). Similarly, mi

= ^ ( i " ^ ( A n , * i » = p * ( ^ ( ^ a , * y | 5"o = r 0 , ^

for almost all (7). Let J^ A ^ almost all (T-), if m4

= r,)

denote (.F0 & ~ ^ ) v ( ^ & ~ ^ 0 ) . Then for

= M ^ ' W * " , f0)

A ^(*n,

*i»)

= 0,

147 416

G. E. SACKS

[August

then m0 = w1. Finally, for all (T) = (T0,TUT2,..,), measure-complete and J(*{T0, Tu Ti,..))

V (xn)(^0(xn,

if J?*(T0, Tlt J a , . . ) ) is

STa) = ^(xn,

Si)\

then w A = 0 . Lemma 4.23 strongly suggests that MJ is a good candidate for a countably ad­ ditive, translation-invariant extension of Lebesgue measure to all subsets of 2" in J(*(T0, Tlt T2,..)) for almost all (T) = (TB, Tx, T2,...). We ehall lo fourse e^arry out this suggestion, but there is a delicate problem, easily overlooked, that we must face immediately. The definition of yl was given by means of terms denoting sub­ sets of 2" in ^•(Tg,Tl,T2,...). With probability 11 there is no map in J?*(T0, Tx, T2,..)) which associates with each subset of 2N a term denoting that subset; this follows from the argument of Theorem 4.18. Nonetheless, we are able to define p* in Jt{T0, Tlt..)) for almost all {T) = (T0, Tu ...) by exploiting the measure-theoretic symmetries of V*(l). In fact, Lemma 4.24 is the sole reason we insisted that every co-term of ^"*(1) be projective and uniformly distributed. LEMMA

4.24. With probability 1: & is Jf(T0, Tu .. .)-definabk.

Proof. Let 5 be an arbitrary subset of 2" in JK*(Tn, Tu T2,...). For almost all {T) = (T0, Tu T2>...), there exist Jt*{T0, 7\, T2,.. .)-definable m a p s / a n d b such \hz.\.f{s) = R(xn, y) is a ranked formula of &*{f0, S\,..)) in which no ^-symbols occur, i ( j ) = / i i s a subset of to in Jt*{Ta, Tu T2,...), and xnR(xn, y) denotes s in Jt*{T0, Tu Tu,..)) when y is interpreted as b. The set * is merely an encoding of the finitely many 77s that are denoted by .^-symbols in some term denoting s; it is easily checked that all such encodings can be accomplished by projective, uniformly distributed co-terms (cf. d% defined immediately preceding 4.15). Thus s ii sennted by some term c{&~Q, S~x, J~2) e ^"(r), a,d M!(J) = P*(R($~* c{sr0, jru ^

\^

=

Ti,i

^ 2).

Unfortunately there is (with probability 1) no Jt*(T0, Tu .. .)-definable map h such that h{r) = c(3~a, Px, ST2). But, thanks to the symmetry yf
Proof. Let &(xn, f0, x°) be e aormula of if*(^"0, 3TU ...) whose only ^-symbol is y0- Let yf!=co be such that for all w # « , p*((x") ~ W(xn, 3T0, m) & 3F(xn, Sr0, n)]I :T0 = A) = ]I

148 1969]

MEASURE-THEORETIC UNIFORMITY

Let (T)={A,TUT2,...).

417

Then

n

n

= ^*((£A"°)Jir(^"1> 5"o, x°) \f0 T

n

a

= fi a(x {Ex°)^(x ,

= A)

0

3Ta, x )).

The countable additivity of & for almost all {T) = (T0, Tlt T2,..)) now follows from 4.20 and Fubini's theorem. We establish the translation-invariance of £ for almost all (T) by a variation of the argument of 4.24. Let S^(xn, Sro) be a formula of Jf*(^"0, Plt...) whose only .^-symbol is .T0, and let c(JT0, ■*!) be an a-constant whose only ^-symbols are ^"0 and ^ . Then Mc

= ^ ( ^ J F ^ + c ^ , *i\ f0)) = p*{3F{3r2 +c{y0, sr^, 3r0)\yi = Th i S 1)

for all ( D = (r-0,7-,,..); and ix = H.Ta{xn^{xn, 3T0)) = p*(Sr(F2, 3Ta)\srt = T„ i < 1) for almost all (T) by 4.20 and Fubini's theorem. By 4.22 (b) and 4.19,

p*(SF(sr2+c{sr0, s]), r0)\r,

= TU i g 2)

= p*(P{3r2, r0)\rt

= Th * < 1, T% = r 2 + c(70, 7\))

for almost all (F). But then by Fubini's theorem ^ = M for almost all T. For almost all (T), £ agrees with Lebesgue measure on all Lebesgue-measurable sets because nl is countably additive and agrees with Lebesgue measure on all basic open subsets of 2". (Of course we are using the fact that every countably additive measure on 2" is regular (Halmos [10]); this fact requires the countable axiom of choice in its proof; fortunately, 4.21 provides us with the countable axiom of choice for almost all (T).) 4.26 (SOLOVAY [29], [30]). If ZFis consistent, then ZF+"there exists a countably additive, translation-invariant extension of Lebesgue measure to all sets of reals"+ "dependent axiom of choice" is consistent. THEOREM

Proof. By 4.21, 4.24 and 4.25, J(*(T0, Tt, T2,...) is the desired model with probability one. Scott and Solovay [25], [26] have devised a very valuable way of viewing forcing arguments in terms of homomorphisms of Boolean algebras. Consider what happens from their point of view when a Cohen-generic 7 c N is added to Jt'. Cohen [ 1 ] defined a aorcing relation, , ¥ P, where q is a finite tes of conditions on T and & is a sentence of &(?). Suppose q is interpreted as a basic open subset of 2" Then ${$?) = \J{q\q¥

~~ST}

149 418

G. E. SACKS

[August

defines a map from the sentences of &{T) into RO, the Boolean algebra of regular open subsets of 2". RO is a complete, Boolean algebra, and 4> is a homomorphism in the sense of complete Boolean algebras. By that we mean, (S^ v ^ ) ^ ^ ) V^(J^. #(~J^) i n t e r i o r of 2"-^), and )) = \J {(c) \ ce0, the map 4>, restricted to sentences having at most n unranked quantifiers, is ^-definable. The ^-definabliity of <j>, together with the fact that is i hohomorphism in the hense of complete Boolean algebras, as ill that is used to show (^)=\ for each inssance & of the replacement axiom. Our probability function p{&) is a map from the sentences of <£(?) into the usual linear ordering of the reals in [0, 1]. Now p is not a homomorphism, since p(£\ v P2) need not equal maxW^),^). But p is a "near-homomorphism"; it preserves negations and monotonic unions: p(-~3F)=\-p(&\ and p{{Exa)^{xa)) = least upper bound of {/>0F(co)v • • • V^(c n )) | I, e ef(a)}. The eact thatt? ii sot t aomomorphiim ii sxactly tth eeason that prenex normal form was exploited so heavily in the proof (Lemma 4.1) of the ^-definabllity of p. The ^-definabllity of 4>, together with tht fact that p is a "near-homomorphism", was enough to show (Lemmas 4.2 and 4.3) that p{SF) = 1 for each instance 5? of the replacement axiom. Thus the measure-theoretic uni­ formity approach seems to suggest that the Scott-Solovay Boolean-valued model point of view can be generalized still further, possibly by dropping homomorphisms in favor of "near-homomorphisms". Solovay [29] significantly extended Cohen's forcing method by using closed subsets of 2" of positive measure as forcing conditions. His forcing relation, q Ih 3?, is identicaa with Cohen's save etat t is s aet to faturaa lnmbers so J( which encodes a closed subset of 2" of positive measure. The proof of Solovay's Theorem (4.26) is based on this notion of forcing. Call T^N Solovay-generic if it is generic in his sense of forcing. Call T fundamental if for all sentences 3? of ZF, p(&) = 1^Jt(T)Y3r. (By yorollary y.47, p(F)=\ oo r 0f & has sn ^-symbols.) Solovay has observed that T is fundamental if and only if T has the same degree of nonconstructibility as some Solovay-generic T. (In this case, T and T have the same degree if each is constructible from the other via the ordinals of**.) Thus the transition described in the previous paragraph from homomorphisms to "near-homomorphisms" might correspond to a transition from sets to degrees of sets. REFERENCES 1. J. W. Addison, The undefinability of the definable. Abstract 622-71, Notices Amer. Math. Soc. 12 (1965), 347-348. 2. P. J. Cohen, The independence of the continuum hypothesis. 1, II, Proc. Nat. Acad. Sci. U.S.A. 50(1963), 1143-1148; 51 (1964), 105-110.

150 1969]

MEASURE-THEORETIC UNIFORMITY

419

3. S. Feferman, Some applications of the notions of forcing and generic sets, Fund. Math. 56 (1965), 325-345. 4. S. Feferman and C. Spector, Incompleteness along paths in progressions of theories, J. Symbolic Logic 27 (1962), 383-390. 5. W. Feller, An introduction to probability theory and its applications, Wiley, New York, 1957. 6. R. O. Gandy, Proof of MostowskC's conjecture, Bull. Acad. Polon. Sci. Ser. Math. 8 (1960), 571-575. 7. , On a problem ofKleene, Bull. Amer. Math. Soc. 66 (1960), 501-502. 8. K. Godel, Consistency proof of the generalized continuum hypothesis, Proc. Nat. Acad. Sci. U.S.A. 25 (1939), 220-224. 9. , The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory, 4th printing, Princeton Univ. Press, Princeton, N. J., 1958. 10. P. R. Halmos, Measure theory, Van Nostrand, Princeton, N. J., 1950. 11. S. C. Kleene, On the forms of the predicates in the theory of constructive ordinals (second paper), Amer. J. Math. 77 (1955), 405-428. 12. . Arithmeiical predicates and function quantifiers, Trans. Amer. Math. Soc. 79 (1955), 312-340. 13. , Hierarchies of number-theoretic predicates, Bull. Amer. Math. Soc. 61 (1955), 193-213. 14. , Quantification of number-theoretic functions, Compositio Math. 14 (1959), 23-40. 15. G. Kreisel, Set theoretic problems suggested by the notion of potential totality, Proc. Sympos. Infinitistic Methods, Warsaw, 1961, pp. 103-140. 16. , The axiom of choice and the class of hyperarithmeiic functions, Indag. Math. 24 (1962), 307-319. 17. G. Kreisel and G. E. Sacks, Metarecursive sets, J. Symbolic Logic 30 (1965), 318-338. 18. S. Kripke, Transfinite recursions on admissible ordinals. I and II (abstracts), J. Symbolic Logic 29 (1964), 161-162. 19. R. A. Platek, Foundations of recursion theory, Ph.D. Thesis, Stanford Univ., Stanford, Calif., 1965. 20. H. Rogers, Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967. 21. J. Rosenthal, Addition of relations in model theory; Truth in all of certain well-founded countable models arising in set theory, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1968. 22. G. E. Sacks, Degrees of unsolvability, Princeton Univ. Press, Princeton, N. J., 1963. 23. , Measure-theoretic uniformity. Bull. Amer. Math. Soc. 73 (1967), 169-174. 24. , Higher recursion theory, Lecture Notes, Massachusetts Institute of Technology, Cambridge, Mass., 1966. 25. D. Scott, Lectures on Boolean-valued models for set theory, Amer. Math. Soc. Summer Institute on Axiomatic Set Theory, UCLA, 1967. Lecture Notes.( M ) 26. D. Scott and R. Solovay, "Boolean-valued models for set theory" in Axiomatic set theory, Proc. Sympos. Pure Math., vol. 13, Amer. Math. Soc., Providence, R. I. (to appear). 27. J. R. Shoenfield, The problem ofpredicativity, Essays on the Foundations of Mathematics, Magnes Press, Hebrew Univ., Jerusalem, 1961, pp. 132-139. 28. W. Sierpiriski, Les ensembles projectifs et analytiques, Memor. Sci. Math., no. 112, Gauthier-Villars, Paris, 1950. (") This reference is a preliminary version of [26].

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29. R. Solovay, The measure problem. Abstract 65T-62, Notices Amer. Math. Soc. 12 (1965), 217. 30. , The measure problem, (to appear). 31. C. Spector, Recursive well-orderings, J. Symbolic Logic 20 (1955), 151-163. 32. — , Measure-theoretic construction of incomparable hyperdegrees, J. Symbolic Logic 23 (1958), 280-288. 33. — , Hyperarithmetical quantifiers, Fund. Math. 48 (1959), 312-320. 34. H. Tanaka, Some results in the effective descriptive set theory, Publ. RIMS, Kyoto Univ. Ser A 3(1967), 11-52. 35. , A basis result for Tl\-sets of positive measure, Hosei University, 1967. 36. L. Tharp, Set theory. Lecture Notes, Massachusetts Institute of Technology, Cambridge, Mass., 1965. MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE, MASSACHUSETTS

152 Aaes, Congres intern, malh., 1970. Tome 1, p. 251 a 254.

RECURSION IN OBJECTS OF FINITE TYPE by

GERALD

E. SACKS

My hope here in Nice is to draw attention to the work of S. C. Kleene [7] on recur­ sion in objects of finite type. In pursuit of that hope I will touch lightly on some related developments in generalized recursion theory. My Nicene creed is: Kleene's notion of recursive object of finite type and Godel's notion of constructible set are of similar, but not of the same, substance. An Athanasian might see them as the same after reading Shoenfield [20] on hierarchies, but the Arian view is more balanced in the light of Moschovakis [11, 12] on hyperprojective sets. I owe much to R. Gandy, T. Grilliot, and P. Hinman, who patiently explained to me the concept of recursion in objects of finite type, and to G. Kreisel [8], who taught me that such things as " concepts " exist in the context of recursion theory. An object of type 0 is a natural number. An object of type n > 0 is a total function whose arguments and values are of type < n. U,V,,.. denote objects of finite type. Kleene [7] introduced a transitive relation U «J V {to be read U is recursive in V). If U and V are objects of type 1, then < coincides with Turing reducibility. For each finite type, 0 ambiguously denotes the function of that type which is everywhere equal to 0. If U < 0, then U is said to be recursive. If U < V and V < V, then U s V (to be read U and V have the same degree). X, Yt... denote members of 2" called reals, and F,G, H,... denote total functions from the reals into the reals. For each n > 0, *_E is the characteristic function of equality for objects of type < n. Thus 2E(X, Y) = 0 if X = Y, and = 1 otherwise. 2E has the same degree as the Turing jump operator. A result of great internal beauty obtained by Kleene [7] is: the objects of type < 2 recursive in 2E are just the hyperarithmetic ones. S^U, the k-section of U, is the set of all objects of type k recursive in U. Kleene [7] asked: do there exist Fs such that (1) S,F consists of the arithmetic reals?, (2) SjF consists of the Aj reals? Recently Grilliot [5] answered (1) negatively by showing: if S,F is closed under the Turing jump, then 2E < F. (2) is answered affirmatively below. Platek [13] calls a transitive set A admissible if A is closed under finitary set opera­ tions and all instances of the D, reflection and A0 comprehension axiom schemas are true in A. A function / from A into A is called ^-recursive if the graph of f is a £ j subset of A. For every F it is possible to construe SjF as a countable transitive set AS,F by exploiting the standard encoding of hereditarily countable sets by reals. An immediate consequence of Shoenfield [20], Hinman [6], and Grilliot [5] is: AS,F is admissible if and only if 2£ < F. It follows from Gandy's work [3] on selection operators that if 2E < F, then AS,F satisfies the £ , dependent choice axiom schema. THEOREM 1 [17]. — (i) and (ii) are equivalent.

153 252

G. E. SACKS

(i) A is a countable admissible set that satisfies the £ , dependent choice axiom schema, and every member of A is countable in A. (ii) There exists an F of type 2 such that 2E < F and A = AStF. For each ordinal a let La be the set of all constructive sets of constructible order
His proof is an annlication of GodeT
nwenheim

nrinrinle

for L ■

Z subsystems of L are isomorphic to initial segments of L Code's principle (with L replaced by L J is central to current work on admissible ordinals- it plaJs an unexpected part in the solution of Post's problem [19] I say unexpectecI " b e ™ he u * o m o L T e o r e t i c i d e a s i n recursion theorv was at one time a surprise o me

O^SeThe^dKrd^

from llZ BarwTse^nTwJIbvmeZ of a a ^ ^ ^ I Z ^ ^ A h T ^ u S ^ S i ' sl^e set then T l m J t e ^ ^ f o S m S E o n s from A ]Z AYTIeauTva en tntheA r e c u r v e ^ n n c t i Z (The e m . W l e n c e M " f a 1 , » uncountable A>« wnLlTke.rrecommen^rL . « I JiJZ, ™ f Z Z n„ P S , ? E , i t t h e g r e T v W f v of' i d « , ! ™ m generalized[^recursion theory

as COROLLARY 2. — If a is a countable admissible ordinal, then there exists an F of type 2 such that L,n2* = S^, and such that for every G of type 2 and of lower degree than F, L , n 2 V S,G. COROLLARY 3. — If n > 0, then there exists an F of type 2 such that the reals recur­ sive in F are just the Aj reals. THEOREM 4. — If U is of type n and "£ < I/, then SjU = SXF for some F of type 2. The above four results are proved with the aid of Godel's Skolem-Lowenheim principle for L, Cohen's forcing method, and Grilliot's hierarchies based on objects of finite type [4]. The next theorem combines forcing with the Friedberg-Muchnik priority method. Platek [13] calls X F-recursive in Y if A" # F, Y Two reals have the same F-degree if each is F-recursive in the other. Hinman calls a real F-recursi­ vely enumerable if it is the range of a partial function of type 1 recursive in F. A wellknown result of Spector [21] can be extended to show: if 2 £ < F, then all non-F-recursive, F-recursively enumerable reals have the same F-degree. I say X is Zj in Y over AS,F if A" is a I , subset of ASSF(Y) where AS^iY) is the result of adjoining Y to /IS^Flmd closing under A0 comprehension. THEOREM 5. — If 2 £ < F, then there exist two F-recursively enumerable reals such that neither is A, in the other over AS,F.

154 RECURSION IN OBJECTS OF FINITE TYPE

253

Kleene [7] showed that the 2£-recursively enumerable reals were just the n } reals. Theorem 5 for F = 2E was proved in [15]. The superjump is a fundamental object of type 3 introduced by Gandy [3]] it llfts F to F 1 . Let { e }r(X) denote the value (possibly undefined) of the e-th partial func­ tion of type 2 recursive in F for real argument X. The value of F\e, X) is 0 if { e }r(X) is defined and 1 otherwise. 2£' is the hyperjump and has the same degree as £ , , an object of type 2 associated with the Souslin operation and introduced by Tugue [23]. Gandy [3] showed: if F < G, then F1 < G1 Hinman has asked: is there a condition on G that implies the existence of an F such that F1 = G? Hinman's question was inspired by Friedberg's classic result [2V if JO < X then there exists a Y such that JY= X, where 0 is the empty set and J is the Turing jump. THEOREM 6 [18]. — Assume the continuum hypothesis. such that (G\EF)[H <, G - F 1 = G].

Then there exists an H

The Fs of Theorems 1 through 5 are constructed in countably many steps, but the F of Theorem 6 is constructed in. uncountably many steps. If the continuum hypo­ thesis is dropped, then Theorem 6 can be approximated in the sense of Theorem 7. The continuum hypothesis is needed to make the approximations cohere with one another. THEOREM 7. — If SjG is closed under hyperjump, then there exists an F such that S,Fl = S,G. The next theorem is intended to suggest that the Tugue hierarchy for S j ^ , is similar to the Shoenfield hierarchy for StF* whenever 2E < F; it was proved in [16] for the case of F = 2E. THEOREM 8. — If 2E 5, F, then the F-degrees of S,F' have a minimal, but no least, upper bound. Most of the results of this paper have the following form: a structure B associated with some generalization of recursion theory is given; then an object U of type n is constructed such that the members of B coincide with the objects of type < n that are recursive in U. Since Kleene's definition of relative recursiveness is inductive, it follows that B can be defined by an induction based on U. If enough results of the above form can be found, it may be possible (as Kreisel has suggested) to prove theorems about structures occurring in generalizations of recursion theory by thinking of them as having been built up by inductive definitions based on objects of finite (or higher) type. Among the means to that end would be various sharpenings of Theo­ rem 4 The sur>erjump 3S is an object of tvDe 3 of lower degree than 3 £ but an arralication of Corollary 2 above to Platek [14] provides an F of type 2 such that S, 3S =7^. So it seems likely that the hyDOthesis " "£ < U " of Theorem 4 can be renlaced hv some­ thing of wider scope. Theorem 4 can be extended from 1-sections To fc-sections as follows. For each n there is a V of type n such that for all U of type rr if V < U and It < n then 9 11 = S W for «nme W of tvnp k 4- 1

It is nnssible that "£ ITUV miffirc

for / b u t al: the moment ^ e e d a V whcTdecree arrears tc be hicher tha7»£ saw when*=■ 1 (Addrf ™proof: if G f i d e l ' s ^ o m ^ S ^ W I i t y 3 ^ ™K1"5

155 254

G. E. SACKS

BIBLIOGRAPHY [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

J. BARWISE, R. GANDY and Y. MOSCHOVAKIS. — The next admissible set. Jour. Symb. Log., to appear. R. FRIEDBERG. — A cirterion for completeness of degrees of unsolvability, Jour. Symb. Log., 22 (1957), pp. 159-160. R. GANDY. — General recursive funciionas of finite type and hierarchies of functions, University of Clermont-Ferrand (1962). T. GRJLUOT. — Hierarchies based on objects of finite type. Jour. Symb. Log., 34 (1969), pp. 177-182. —. — On effectively discontinuous type 2 objects, Jour. Symb. Log., to appear. P. HINMAN. — Ad astra per aspera: hierarchy schemata in recursive function theory, Ph. D. Thesis, Berkeley (1966). S. C. KLEENE. — Recursive functionals and quantifiers of finite type, Trans. Amer. Math. Soc, 91 (1959), pp. 1-52 ; 108 (1963), pp. 106-142. G. KRBSEL. — Model theoretic invariants: applications to recursive and hyperarithmetic operations, The Theory of Models, North Holland (1965), pp. 190-205. — and G. E. SACKS. — Metarecursive sets, Jour. Symb. Log., 30 (1965), pp. 318-338. S. KRIPLE. — Transfinite recursions on admissible ordinals, Jour. Symb. Log. (Abssracts), 29 (1964), pp. 161-162. Y. MOSCHOVAKIS. — Hyperanalytic predicates, Trans. Amer. Math. Soc, 129 (1967), pp. 249-282. —. — Abstract first order computability, Trans. Amer. Math. Soc, 138 (1969), pp. 427-504. R. PLATEK. — Foundations of recursion theory, Ph. D. Thesis, Stanford (1966). —. — A countable hierarchy for the superjump, Proceedings of the 1969 Manchester conference, to appear. G. E. SACKS. — On the reducibility of n J sets, Advances in Math,, to appear. —. — Countable admissible ordinals and hyperdegrees, to appear. —. — The 1-section of ■ type n object, to appear. —. — Inverting the superjump, to appear. —. — and S. SIMPSON. — The a-finite injury method, to appear. J SHOENFIELD. — A hierarchy based on a type 2 object, Trans. Amer. Math. Soc, 134 (1968), pp. 103-108. C. SPECTOR. — Recursive well-orderings, Jour. Symb. Log., 20 (1955), pp. 151-163. G. TAKEUTI. — On the recursive functions of ordinals, Jour. Math. Soc Japan, 12 (1960), pp. 119-128. T. TUGUE. — Predicates recursive in a type 2 object and Kleene hierarchies. Comment. Math. Univ. St. Paul (Tokyo), 8 (1960), pp. 97-117

P-P69 Department of Mathematics, MIT Cambridge, Mass 02139 (U.S.A.)

156

FORCING WITH PERFECT CLOSED SETS GERALD E. SACKS1 0. Introduction. Forcing with perfect closed sets has its origin in Cohen's invention of forcing with finite conditions [1] and in Spector's construction of a minimal Turing degree [28]. Cohen taught that truth can be approximated more easily and more completely in the intermediate stages of a construction than any recursion theorist would have believed possible, and Spector revealed that infinite approximation by means of perfect closed conditions is radically different from finite approximation. The difference, it seems to us, is manifested in the sequential lemma, Lemma 1.4 of § 1 (see Footnote 2, p. 335). In § 1 we study perfect closed forcing in the context of set theory. We show: if ZF (Zermelo-Fraenkel axioms for set theory) is consistent, then so is ZF plus there exist precisely two degrees of nonconstructibility. Two sets of natural numbers have the same degree of nonconstructibility if each is constructible from the other. All the constructible subsets of a> constitute one degree of nonconstructibility denoted by 0. A degree c is said to be less than or equal to a degree d if some set of degree c is constructible from some set of degree d. A nonzero degree is called minimal if the only degree less than it is 0. We lift Spector's construction of a minimal Turing degree from recursion theory to set theory in order to force all the nonconstructible subsets of a to occupy the same degree. We also show by means of a transformation lemma that sets which are generic with respect to perfect closed forcing have the following curious property: if T is generic, then every set which has the same degree of nonconstructibility as T is generic. In § 2 we examine perfect closed forcing in the hyperanthmetic case. The 1 The preparation of this paper was supported in part by U.S. Army Contract DAHCO-4-67C-0052. The author wishes to thank Professor J Barkley Rosser for introducing him to Spector's minimal degree argument.

331

157 332

G. E. SACKS

principal result of [5] is reviewed in order to bring out certain technical difficulties which crop up in the hyperarithmetic case and which have no counterpart in the set-theoretic case. It has to be shown in the hyperarithmetic case that the forcing relation, restricted to sentences of recursive ordinal rank, is IIJ, and this is seen to be the case only after the forcing relation is denned with certain precautions; if it is defined by routine analogy with the set-theoretic case, then its classification becomes uncertain. The corresponding problem of definability of the forcing relation in the set-theoretic case presents no problem at all. In § 3 we touch on the arithmetic case and introduce the notion of pointed, perfect closed set. Using that notion, we obtain an order-theoretic characterization of the Turing degree of the truth-set for arithmetic in terms of the degrees of the arithmetic sets and the jump operator. We also briefly describe Spector's minimal Turing degree construction in terms of perfect closed forcing as well as several results based on forcing with pointed perfect closed sets whose detailed proofs will appear elsewhere [23]. Let 2 = {0, 1} have the discrete topology and let 2" have the product topology [8]. Let P,Q,R,. .. be perfect closed subsets of 2". We conceive of a typical P as a tree with binary branching; each path of the tree is an infinite sequence of O's and l's and is viewed as the characteristic function of a subset of w. Every path of P branches infinitely often; this is equivalent to saying that P is perfect. Suppose we adopt some standard recursive one-to-one correspondence between all finite sequences of O's and l's and w. Then P can be encoded as a set of sequence numbers corresponding to finite, initial segments of paths of P. We will use P, Q, R, . . .ambiguously to denote perfect closed subsets of 2m and their encodings as subsets of to. Whenever we classify P, we refer to the encoding of P. Thus if we say P is arithmetic, we mean that the encoding of Pas a subset of w is arithmetic. Let T be an arbitrary subset of w. If we say TeP, then clearly we are thinking of P as a subset of 2". If we say P £ Q, then we mean {T)(Te P-+TeQ). Let p,q,r,... ambiguously denote sequence numbers and the finite, initial segments of characteristic functions they encode. We say p > r if p = r or if p is extended by r. Let P be a nonempty set of sequence numbers. Then P is an encoding of a perfect closed subset of 2" if and only if

O'WPfWO'WI/' We sayTep of T.

>q&p>r&q^r&r^q).

(or Tsatisfies/?) if/; is an initial segment of the characteristic function

1. The set-theoretic case. Let ./// be a countable, transitive set which is a model of ZF plus V = L (Godel's axiom of constructibility [6]). Let T £ w. ,//(D is the set of all sets constructible from 7" via the ordinals of Jl. In order to make the definition of Jt(T) more precise, we adopt the symbolism of Tharp [30]. The symbols of ^ ° are: e (membership); unranked set variables x,y, z, . . . ; ranked set variables V, y,z*, . . . for each ordinal j f i ' ; propositional con­ nectives; and quantifiers for both ranked and unranked variables. The atoms of <£» are of the form /] 6 /2, where 7, and t2 are variables. The formulas of SC° are

158 FORCING WITH PERFECT CLOSED SETS

333

constructed from the atoms and the logical symbols in the usual fashion. A class

«"(« + l):f6?(« + 1); let M#,yu . . . . v j be a formula of £"> whose only free variables are JC,^ > ' „ ( « > 0) and whose quantified variables are all of the form x" for /?'s < ot; then #^(x«, eu .. . . c„), where each c, (1 < i < n) is either^7" or a member of U W ) [ £ < <*}, is a typical member of If (a + 1). »(A) = U W ) I /S < A} for each limit ordinal A. Let » = U{«X«) | a e UT}. The symbols of ^ ( ^ ) are those of if 0 together with the members of ) or (3xf) occurs in J^} U {/? + 1 | £> occurs in J^}. If t e <€, then o(r) = o(0 e r). For each T £ w it is a routine matter to simultaneously inductively define the following concepts: c e #(a) and beJta(T) and c denotes b; o(&) < a and & is true in Jt (T). Of c o u r s e d is interpreted at T, x* ranees over Jt (T), and x ranges over Jt(T) = U K . C O I « e - * } • We write ^ ( T ) |= JF to mean .F is true in ^ ( P ) . Let 0 denote the empty set, and let Jtt be Jta(0). Then

j( = \}{Jtt\

« e 4

Now we define the forcing relation P If J% where P e ^T and .F is a sentence of i f (#"). Of course P £ Jt means P i s a perfect closed subset of 2°> whose standard encoding as a subset of m is a member of Jt. The definition is orthodox save for the interpretation of P. (1) P (r ( 3 x ) ^ W if P If- *-(c) for some c e V. (2) P If (3*«).F(x«) if P l h F ( c ) for some c e V(«). (3) P H - J F & S T i f P H - . F ' a n d P l f 2?. (4) /> If- — ^ if (Q)[P => g -

~ ( G H" J^)].

(5) P If tfeJF if ( r ) [ r e P - ^ ( F ) M e « n (6) Pfrmenifm

< n.

(7) P H- f, € h if O (r0 < o(t2), h is f *(**) and P If #*,). (8) P If ^ e /, if o(h) > o{t2), tx is Jc-^(x-) and P If (3/)[(x-)(jf e / «-> *(*«) &y*et2\. (9) P If /x e / s if o(r,) = o(/2) > 0. Similar to (8). We say T is generic if for each sentence JF of i f (7) there is a perfect closed Pej? such that P e P and either PH-JF or P H - ~ i F . Standard arguments [7] suffice to show: (i) (f)(P) ~ [P \Y 3? and P l h ~ J ^ ] ; (ii) (P)(&)®Q)[P => g & g H- JF or g H- <*-jrj; (iii) (P)(jF)(g)[P 2 g & P H - . F — g H - J H ; (iv) generic P's exist. A delicate but not troublesome point arises in the course of the proof of (v): if T is generic, then Jt(T) \ .F if and only if ( 3 P ) [ P E P & P If JF]- The difficulty can be put as follows. Suppose T is generic, TeP n g , P If .F, and

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0 H- y ; then there should be an R £ P n Q such that T e R and A H- & & y . In the case of Cohen-forcing with finite conditions, it was safe to let R be P O Q, but now a different argument is required, because the intersection of two over­ lapping, perfect closed sets need not be perfect. It is not difficult to show by a Cantor-Bendixson analysis that if T$J( and TeP r\Q, then TeRc p n Q, where R is the perfect kernel of P O 0. In short, if a perfect closed set is encoded by a subset of w belonging to Jt', then so is its perfect kernel. But Kreisel [13] has shown that this type of argument will not work in the hyperarithmetic case of § 2. His result is: there is a hyperarithmetically encodable, closed set whose perfect kernel is not hyperarithmetically encodable. The proof of proposition 1.1 (ii) does work in the hyperarithmetic case. 1.1. (i) / / T is generic, then T$Jt. n Q, then TeRforsome R £ p n Q.

PROPOSITION

TeP

(ii) / / T is generic and

PROOF. Let c e V denote a subset of a>, and let c(x°) denote the sequence number which represents the characteristic function of c restricted to arguments <x°. To prove (i) let T be generic and suppose T=deJ(. Then there is a P such that TeP and (a) P H- ( 3 x » ) ( J M # 0: the relation PH-jF, restricted to sentences & having at most n unranked quantifiers, is ^-definable.

LEMMA

of&{f)

The proof of 1.2 is standard [7]. It follows from 1.2 that if T is generic, then the replacement axiom holds in M. It suffices to prove: let &{x,y) be a formula of Se(&~) whose only free variables are x and y: if P If {x*)($yW(x',y), then P K- (jc")(3/)i r (x°, / ) for some /S. P has the property that (c)«¥<«)(0)p2o(3*)O2ij(3^rfetf,a)[*

H-.F(r, 0 ] .

By 1.2, d can be regarded as an ^-definable function of c and 0. Since <
160 FORCING WITH PERFECT CLOSED SETS

335

LEMMA 1.3. For each n > 0: There exists an Jt-definable function AjF | 3?*. defined for alt sentences ? of,j2'(Jr) having at most n unranked quantifiers, such that

(P)[P^SF^PVr^*) and the ordinal rank ofF* is less than w, (in the sense of Jt). PROOF.

Standard. Let K = {P | PVr &}.

By 1.2, K is a subset of 2" in

Jt.

By Godel [6], KeJtt for some a < co2, where w, is the w2 of Jt. Lee J27" be (1P)(P £K&LST eP). Clearly the ordinal rank o f V * < OJ2. Suppose P H- JT*. Then for some Q, P H- g 6 /C & 3T e Q. It follows £eK, since 2 and AT :re members of Jt and no falsehood about Jt can ever be forced. P E 0 , because f l f J " e 2 implies (/>)(/> eP^peQ). But then /> H- J*". If one forces with finite conditions in the manner of Cohen, then the « 2 in Lemma 1.3 is replaced by w,. On the other hand it can be shown that if one is forcing with perfect closed conditions, then the co, of Lemma 1.3 cannot be reduced to W!. We turn now to the problem of showing for generic T that the cardinals of Jt(T) are the same as those of Jt. In the case of Cohen-forcing with finite conditions [1] or Solovay-forcing with closed sets of positive measure [25], [26], the preservation of cardinals in generic extensions is a consequence of the countable chain condition: if J is a set of pairwise incomparable conditions (i.e., P, QeJ &P*Q^P$Q8cQ$P), then J is countable. It is not difficult to see that the countable chain condition fails for perfect dosed conditions. Nonetheless, cardinals are preserved, thanks to Lemma 1.4, the sequential lemma.2 LEMMA 1.4. Let {J% \ i e co} be a countable (in the sense ofJt) set of sentences of Se(3T). Let P have the property that

(0(0F2o(3*)02B[Ktt" &iiThen there exists an R c P such that for each i,

R=\J£%

and K ' h ^ ,

for all) < 2 \ In addition, R) n R'k = 0 if j * k. PROOF. Suppose (/)(0)i>=oO*)O2R[*lr-.FI]. By 1.2, there is an Jtdefinable function/(i, 0) such that if P 2 0, then 0 2 f(i, 0) and / ( / , Q) h^t. Let us say Qt and 02 are basic, disjoint perfect closed subsets of Q if there exist sequence numbers qx and q., e Q such that ql £ q2, <ja%qt, Ql = {p\peQ & qi>p}, and Q, = {p j p e P & q2 > p). By iterating f(i, Q)) ,t ii not difffcult t t 2 Lemma 1.4 is called the sequential lemma in order to be consistent with [5], where a hyperarithmetic version (Lemma 2.2 of the present paper) of 1.4 occurs. Mathias [IS] formulates 1.4 more abstractly and calls the result the fusion lemma. Shoenfield [24] invokes the term "splitting" to describe the proof of 1.4.

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develop an ^-definable function ).ij\ Q) with the following properties: Q% £ P and ^ H - ^ o ; for each i a n d ; < 2 \ Q)+l and Q'+\, are basic, disjoint perfect closed subsets of Q); Q}+l^^i+u and QJ+J, H-^ {+1 . Let * = (-),- Uy weakly forces F (P H-* F ) if P H- ~ ~ F . An immediate corollary of Lemma 1.4 is

Another way of expressing the message of 1.4 is: for all P and all sequences {JF, I i e to) (in the sense of Jt) of sentences of &{T), &R)p2n(i)[R

B " * ^ V R lr* ~ j F , ] .

Thus perfect closed forcing gives us the power to "decide" every one of a countable set of sentences by means of a single forcing condition. If we concentrate on sentences of countable ordinal rank, then we can do even more; this additional power is important in the hyperarithmetic case (§2), where the sentences of recursive ordinal rank behave very much like the sentences of countable ordinal rank in the set-theoretic case. LEMMA 1.5. Let & be a sentence of &(?) of countable (in the sense of JT) ordinal rank. Then for each P there is a Q £ P such that either (1) or (2) holds:

(i) (r)(ree-ur(r) r Jf); (2)

{T)(TeQ^Jt{T)\~3r).

PROOF. By transfinite induction on the rank o f & . We give only the principal inductive step. Let & be (3X«)^"(JC«), where a is countable in the sense of Jt. Let {c( \i€w} be an enumeration of »{s); tf(oc) is countable (in the sense of Jt) because a is. For each /, J*"(cj) has lower rank than (3xa)&(x'). It follows from the inductive hypothesis that for each / and each P, there is a Q £ P such that either (1) or (2) holds: (1) {T)(TeQ^Jt(T)r-F(cO);

(2) (T)(Te Q -* Jt(T) \- ~JF(c,)). Fix P. If there is a Q £ /> such that (1) holds, then the argument is complete; suppose there is no such Q. Then it follows that (>KQ)i-=>Q(3*)(;=!R(T)[Te R -+ J?(T)

\~^(Cl)].

To obtain a Q c /> such that (/?)[J6 Q-+Jt{T) (=(*") ~ F ( . v < ) ] , one repeats the construction used to prove Lemma 1.4 with one small change. The Jtdefinable function / ( / , Q) now has the property that if P 2 Q, then Q 2 / ( / , 0 and (T)(Tef(i, Q)-+Jt(T) h-M#"(C,.)).

162 337

FORCING WITH PERFECT CLOSED SETS

LEMMA 1.6. Let {&,) be a countable (in the sense of Jt) set of sentences of £f(3T) of countable ordinal rank. Let P have the property that O)(0P2QO*)O2K[*^.]-

Then there exists aQ£

PROOF.

P such that

We claim that P has the property that

Fix i and Q. Then there is an R s Q such that RVr^,. an tf, £ R such that either (1) (T)(TeRl^^f(T) M^or

By Lemma 1.5, there is

(2) (T)(TeR1-*Jir(T)Y~&d. Since there is a generic Te * x £ K and/? H-J5",-, ii ti clear rhat t(1 holdss But the existence of the desired Q follows from our claim and a repetition of the argument occurring at the end of the proof of Lemma 1.5. LEMMA 1.7. If T is generic, then Ji and Jt(T) have the same cardinals. PROOF. Let a be an uncountable cardinal in the sense of Jty and let /J < a. L e t / e # denote a function which maps /9 into a. For each y < /S, let k(y)

= {r | r < a & (3P)(/> H-/(y) = r ) } .

By Lemma 1.2, k(y) eJ(; and the cardinality of k{y) is at most wX) then K = U W y ) | y < 0} has cardinality less than a. If T is generic, then in J((T) the range of/is a subset of /T. Suppose a = coj and £ = a>0. Now we need the sequential lemma, 1.4. Suppose for the sake of a contradiction PVrf is a 1-1 map of w0 onto w:. It suffices to find an R £ P and a y < wx such that # H-* range o f / is bounded above by y. P has the property that ( 0 ( G ) ) 2 C ( 3 i O o 2 H [ J ! H- ( 3 T ) ) / ( / 0 = r <

Wl )]]

By Lemma 1.4 there is an P £ P such that for each i, R = U Al

and R) H-/(i) = T" < W l

;<2'

for all ; < 2', in addition, R) n J?< = 0 if; # it. Let y = lub {TJ | f e co &/ < 2"'}. Then y < w, since Ay | rj is ^/-eefinable; and P lh* range of/ is bounded above by y. The next lemma lifts the heart of Spectous minimal degree argument [28] from recursion theory to set theory. Let c e W denote a subset of « . For each T, generic or not generic, let c(T)= {n\J!(T) [nee).

163 338

C. E. SACKS

LEMMA 1.8. Let c e % denote a subset of to. For each P there exists aQs P such that either (1) or (2) holds: (1) (T)[Te Q &Tis generic — TeJ?(c(T))] (2) (3b)(T)[T e Q &Tis generic -* c(T) = be Jt\. If in addition c has countable (in the sense of Jt) ordinal rank, then either (V) or (2') holds: (V) (T)[Te Q TeJtai(c(T))); (2') (lb){T)[T e Q -* c(T) = beJ(]. PROOF. We continue to exploit the construction occurring in the proof of the sequential lemma, 1.4. Case 1. (Q)p^Q(3Ql)Q^Q(^Q2)Q2Q2(3n)[Q^nec & Q^n^c]. By Lemma 1.2, Qu Q2 and n can be regarded as ^-definable functions of Q. Iter­ ation of these functions in the style of the proof of Lemma 1.4 leads to an Jtdefinable function lim \ Qf with the following properties: Q\ £ />; for each m > 0 and f < 2m, Q?+1 and Q-+& are basic, disjoint perfect closed subsets of

0f.and miQr'H-riec&Qr^H-fitc]. We define Q by TeQ^(mX3iOt
&QZ)

ff " e c -* n G c(7).

The assumption of genericity can be dropped if c has countable ordinal rank. Because then Lemma 1.5 makes it possible to choose Q™+1 and 0™+l so that there exists an n such that for all T, TeQ?+*-+J?(T)\>8ec, T 6 e,m++2"' -> Jt{T) b

n$c.

Note that if X, Ye Q and * jt Y, then c(JT) 5* c(Y).

164 FORCING WITH PERFECT CLOSED SETS

Case 2.

339

Case 1 fails; then there is an R £ P such that (0I)R2

©,) ~ [ & H- « £ c & 02 H- « € c].

For each « e m, let f*"

if(3e)«=0[GH-»6f]

[n$c

if(B0B2eIGH-ii^c].

The defining property of i? guarantees that ^

is well defined and that

By Lemma 1.4 there is a (? £ tf £ /> such that («)[0 | £ . F ] . If Te 0 and J is generic, then So 0 satisfies condition (2) of the lemma, since 0 |£ n e c is ^-eefinable. Suppose c has countable ordinal rank. By Lemma 1.6 there is a 0 £ * such that

(«)(D[ree->^(io^j. It follows that (n)(T)[T<=Q -+ (n e c(T) «-»& n is n £ <•)].

So 0 satisfies condition (2') of the lemma, since An | i%, is .//-definable. THEOREM 1.9. / / T is generic, then Ji(J) is a model of ZF which satisfies the following sentences: every constructive cardinal is a cardinal; there exist exactly two degrees of nonconstructibility. PROOF. We noted in the remarks subsequent to Lemma 1.2 that the replace­ ment axiom holds in Jt{T). Clearly, the following also holds in Jt{T): there is a set T £ a> such that every set is constructive from T. By Lemma 1.7 the cardinals of Jt(T) are the same as the cardinals of Jt. But then the argument of the last chapter of Godel [6] shows that the generalized continuum hypothesis holds in JOJ). So the power set axiom must hold in J((T). (Of course one can show the power set axiom holds in J({T) without using 1.7 or the sequential lemma.) By Proposition 1.1 T is not constructible in the sense of Jt{T). Suppose AT £ to and AT £ Jt(J). Let r e ? deffne K: neK^ Jt{T) \-nec. Consider the sentenced:

ST eJt(c($~))

V c{S~) £ Jt.

By Lemma 1.8 there is no P that forces the negation of &. Since T is generic, it follows that

T£Jt(c(T))

or

c(T)eJt.

In other words, T e J/(K) or K £ Jf. THEOREM 1.10. If constructible to, is countable, then there exists a minimal degree of nonconstructibility.

165 340

O. E. SACKS

PROOF. By Lemma 1.3 r is generic if and only if T is generic wiih respect to all sentences of ordinal rank less than the w2 of Jt'. So if constructive w2 is countable, then generic T's actually exist and are of minimal degree by Theorem 1.9. It would be interesting to find some axiom of infinity which, when added to ZF, decides every elementary question about the partial ordering of degrees of nonconstructibility. R. Solovay suggested that a measurable cardinal might suffice for this purpose. If 3"is Cohen-generic (i.e., generic in the sense of forcing with finite conditions), then it is easy to see that Tdoes not have minimal degree in Jt(T) by looking at the even and odd parts of T. R. Solovay has shown: if 7"is Cohen-generic and T'eJ({T), then there exists a Cohen-generic T" such that J({T) =Jf(T"). It follows that minimal degrees of nonconstructibility do not occur in Cohen-generic extensions of M. Thus far we have been a little vague about how to encode a perfect closed ? £ 2 " a s a set of sequence numbers. Now we choose a particular method of encoding for the sake of the proof of Lemma 1.11, a transformation lemma associated with perfect closed forcing. As in § 0 , p , q , r, ... ambiguously denote sequence numbers and the finite, initial segments of characteristic functions represented by those sequence numbers. We encode P in the form of a function A/7 | pii with the following properties: />° is the null initial segment; for eachy > 0 and i < 2 \ p{ ^ p?1, p\ > p\++\„ p?1 £ p%\, and pt+\> £ pf\ Then

(T)[TP^^O)(310(i
For each TeP, let hP Q(T) be the unique member of OVP Q(PO | ' < 2' & Tep>}. Then hP Q is a 1-1 map of P onto Q. We will call hP Q a canonical homeomorphism of P onto Q. If R £ P, then hP,Q(R) = {hP 0(T) \ T e R} is a perfect closed subset of Q. The map hP Q induces a map hP Q:&(5?~) — i f (5") aa follows. Let ST ep denote a finite conjunction of sentences of the form n ' 5 " a n d m ^ J with the defining property that

{T){Tep~J({T)\2Tzp). For each n, let {r} be the finite set {pn+1 | i < 2" + 1 & (r)(r

e ^+1 - „ e 7)},

and similarly, let {s»} be {P:+11 / < 2" +1 & (T)(T 6 p:+1 - n $ T)}. Then for every

TeP: nGT^

(3ii)T G rf) & (i)(T $ s,").

166 FORCING WITH PERFECT CLOSED SETS

Let c be a number-theoretic term of &{f).

341

Then A , , g ( c € J ) is denned to be

(iOt^" e hPiQ(ret)] & (f)[S~ £ fcP>g(j9]; if . F is a formula of ^ ( i ^ ) , then h,, 0{3?) is the result of replacing each occurrence of c e ^ in j r by A, 0 ( i e V ) ; h(d) where rfe flf, is denned similarly. If Ae ^ e #(1), then the following transformation lemma (cf. Feferman [2]) can be estab­ lished by induction on the rank of J5": for all TeP, Jt(J) \& if and only if Jt{hP.Q(T))\>hPQ{&). LEMMA

each TeP,,

1.11. Let h be a canonical homeomorphism of P, onto Qx, Then for T is generic if and only ifh(T) is generic.

PROOF.

Let K be an ^-definable set of P's. We say K is dense if (R)(3P)[R 2

P&PeK].

It is standard to observe that 7 is generic if and only if Te [J{P\P e K}tor every ^-definable, dense A". Clearly, if A" is ^-definable and dense, then there cannot be an R that forces

^i\J{P\PeK}; and for each sentence & of &{T),

the set

{P\P*F

or

Plr-JF}

is ^-definable and dense. Fix Pv We say K, is P 2 P & P e JCJ. The above standard observation can be relativized to P,: TeP, is generic if and only if Te [J{P \PeKx} for every ^-definable K, dense in P,. Suppose that TeP, is generic, and that AT, is dense in Pv There cannot be an R such that P e P and RVr^eP,

& ^"^U(P|PeA'1}.

If there were such an P , then by Proposition 1.1 there would be an P, s P O P,. But then the density of A", in P, would provide a P s P, such that

P^^e\J{P\PeK,}. Since h is a canonical homeomorphism of P, onto Qu U(P | P 6 A",} is dense in Pt if and only if U W ) | P e iC,} is dense in Q,\ furthermore, if K2 is dense in Qu then A*2 = U W ) | ^ e #x) f«r some A-! dense in Pj. THEOREM

1.12.

If T, is generic and T2 e Jt{T,) - Jt, then T2 is generic.

167 342

G. E. SACKS

PROOF. Since Jt and JHJd have the same cardinals, T2 = ciTJ = {n | Jt{Jx) Yn e c} f°r some c e fffw,). By Lemma 1.8 there is a Q such that 7\ e Q and

(r)xre^re^ i W r)). Recall the end of Case 1 of the proof of Lemma 1.8. Q has the property that (X)( Y)[X, Y G Q & X * Y-* c(X) * c( Y)]. Let R = {c(T) [TeQ]; then c is a 1-1 map of Q onto /?. We claim a small addition to Case 1 of Lemma 118 guarantees that c can be construed as a canonical homeomorphism of Q onto R. The addition is: for all m and / < 2m+\ require that (i) or (ii) hold:

Then Lemma 1.11 implies that T2 = c(J,) is generic. For more information concerning the properties of JUT) for generic T see Prikry [17] and Mathias [15]. The minimality of the degree of a generic T was established in the proof of Theorem 1.9 by clauses (1) and (2) of Lemma 1.8. If one wishes to build a more complicated model of ZF in which the degrees of subsets of w are controlled by perfect closed forcing, one cannot in general prove Lemma 1.8 (1) or Lemma 1.8 (2). In that emergency one falls back on clauses (1') and (2') of Lemma 1.8, which suffice if cardinals are preserved. Since the relation A is constructive from B is analytical (in fact, 2») in the sense of Kleene [9], it can be used to study the independence of various questions concerning the analytical hierarchy. A. Levy has shown by means of his cardinalcollapsing method that if ZF is consistent, then ZF plus the negation of the axiom of choice for some n^ predicate is consistent. The axiom of choice for a predicate P{n,X) is: (n)(3X)P(n, X)-+(3X)(n)P(n, (*)„), where (X)n = {m \ 2m • 3" 6 X). II Levy's model, ,onstructible wn „i souutable for all finite /;. His result limiting the provable versions of the axiom of choice can be obtained by extending the arguments of this section to build a model JV in which the degrees of nonconstructibility are isomorphic to w and in which every constructible cardinal is a cardinal. Thus it is true in JT that: for each n > 0, there is an X such that (*)„, {X\, . . . . (X)n have distinct degrees, but there is no X such that for all n > 0, (X)0, (X), , (X)„ have distinct degrees. S. Kripke and D. A. Martin carried on in the above vein and built a model of ZF in which a AJ subset of w failed to be constructible.3 Mathias [15] gives an intuitive account of how this is done. The general idea is as follows. Choose an appropriate sequence (^B) of logically independent sentences from the theory of partial order. Than a set A £ w can be encoded as an analytical set in a model JV s The question: is every analytical subset of w constructible'' was first suggested to us by G. Kreisel.

168 FORCING WITH PERFECT CLOSED SETS

343

by taking steps so that n e A ifand only if Sr*n is true in the partial orderingof degrees of nonconstructibility of JT. Jensen succeeded in using these ideas to obtain the best possible result: he lowered A} to A' and arranged for the axiom of choice to be true in his model. Jensen's result was first obtained by Solovay. who managed it without any use of degrees. Silver has found another way of building a model in which minimal degrees of nonconstructibility occur. He makes use of a special kind of perfect closed condition he calls coinfinite. A coinfinite condition P = (A,B)is given by a pair A, B £ w such that A n B = 0 and to - (A U M) is infinite; we say TeP if A £ T and T n B = 0. The sequential lemma (1.4), mildly altered, holds for coinfinite conditions. (For more information about coinfinite conditions, see Mathias [15].) Coinfinite conditions were discussed but not used in Spector's paper on minimal Turing degrees [28]. They were used to prove Theorem 2 of [21]. 2. The hyperarithmetic case. In the present section our purpose is to expound forcing with perfect closed sets in a context where the definability of the forcing relation is a more delicate matter than it is in the set-theoretic context of § 1. With this end in mind we review the minimal hyperdegree construction of [5] and add some intuitive remarks that were unfortunately left out of [5]. The language &(S~) is little different from Feferman's language &*(Sf) [2, p. 335]. &(?) is first order number theory augmented by a constant set symbol 3T, certain set variables, and the membership symbol e. Let O, be a U\ set of unique notations [3] for recursive ordinals: if b e 0 , is the unique notation for 0, then we write \b\ = /?. In addition we suppose there exists a recursively enumerable relation R(c, b) such that for all b e Ou {c | R(c, b)} = {c \ \c\ < \b\}. For each b e 0 „ ^(T) has ranked set variables X\ X", Z\ . . . ; ^(ST) also has unranked set variables X, Y,Z,... , number variables x,y, z, . . . , a numeral n for each natural number n, and symbols for equality, successor, addition and multiplication. A formula 3? of &(?~) is said to be ranked if every set variable occurring in & is ranked. A formula of i f (JT) is said to be existential (or SJ) if it is ranked or it is of the form (3XW with Jf the only unranked variable occurring in &. The ordinal rank of a ranked formula & is least ordinal a such that a > \b\ for every free variable Xh of & and such that a > \b\ for every bound variable X" of &. Let T be an arbitrary subset of w. Following Feferman [2], fot each i e O . w e inductively define a structure Jtb(T) and truth in \J{J?a{T) \ \a\ <\b\). (i) A sentence & of ordinal rank <\b\ is true in \J{Jta(T) \ \a\ < \b\} if it is true when 3~ is interpreted as T, the number variables of ^ are restricted to w, and each ranked variable X" of & is restricted to Jfa(T). (ii) For each formula 9(x) (with only x free) of ordinal rank <|6|, let x9(x) be {n | 9{n) ii ttue in \J{J?a(J)\ l«| < l*|}}; then Jtt(T) consists of aal such sets x9(x). Let Jf(T) be \J{Jfb(T) | b e O,}. Then & is true in Jf{T) (written u f ( r ) (•#") if it is true when each unranked variable of & is restricted to J((J) and the

169 344

G. E. SACKS

remaining symbols of & are interpreted as in (i) above. If T is hyperarithmetic, then Jt{T) is the set of all hyperarithmetic sets, which we denote by Jt. 3F{x<$(x)) denotes the res"ult of replacing each occurrence of / e ^ in .F(<8r) by &(t), where / is a number-theoretic term and W ii s aet tariable. As in § 0 we use P,Q,R,... to ambiguously denote perfect closed subsets of 2<" and their encodings as subsets of to. But now we insist that P, Q, R, . . . , be hyperarithmetic. Thus we can write, as we did in §1,?eJt. It is routine to assign indices to hyperarithmetic, perfect closed sets so that the following relation in x and y is n j : x is the index of a hyperarithmetic, perfect closed set P and y e P. For this reason it makes sense to regard the set of all hyperarithmetic, perfect closed (h.p.c.) sets as a n} subset of w. Similarly, one regards the set of all formulas of & (ST) as a U\ subset of w. It is instructive to see what happens when the analogy between the constructible hierarchy of § 1 and the hyperarithmetic hierarchy of the present sectton is pursued. In both cases a perfect closed set is acceptable as a forcing condition only if it can be encoded by a subset of w belonging to Jt. The analogy suggests that the forcing relation PfrP, where P is h.p.c. set and & is a sentence of ^(JT), should be defined by the following inductive clauses: (1) P H- (1X)^(X) if (3b\,0i[P H- {lXb)&{Xb)] (2a) P H- (3A"')Jr(A"') if P H- JF(.Y3?(A-)) for some 9(x) oo frdinal lank k \b\. (2b) P H- (3x)#(.v) if P H- 3?(n) for some n 6 to. (3) />H-jFv <$\f PVrF or \
170 FORCING WITH PERFECT CLOSED SETS

So we drop the analogy and define H-„ in the manner of [5]. (i) P If-, &X)&(X) if {m„0l[P »■* (3Z»>F(J^)]. (iia) F(f ; ,(3A" , )J E "(^) if &{X") is unranked and Pfrk&(*9{x) ^(x) of ordinal rank < | 6 | .

345

for some

(iib) P Vrh ($x)3F{x) \l3F(x) is unranked and P rf-„ SF{n) for some n e w. (iii) P H-„ J5" V 9 if J*" V <$ is unranked and P H-„ JF or /> lr„ ^ . (iv) P (fft ~ i F if 3F is unranked and (Q)PSQ (v) P»-h & if JF is ranked and (T)[re/»-

~ [2 H-» . F ] . uT(7) ^ .

LEMMA 2.1 [5], 7Vi
is U\.

The proof of Lemma 2.1 is nothing more than inspection of clauses (v) and (i) of the definition of H-„. But we must pay a price, fortunately within our means, for the use of clause (v). It is no longer obvious as it was in § 1 that because clause (iv) does not apply to ranked sentences. But this difficulty can be overcome by a sequential lemma (2.2) argument. LEMMA 2.2 [5]. Let {&,} be a hyperarithmetic sequence of 2* sentences of &{5T). Let P have the property that

Then there exists aQc

p such that

Note that Lemma 2.2 is closer in flavor to Lemma 1.6 than to Lemma 1.4. But the proof of 2.2 is essentially the same as that of 1.4. The only real difference is that the ^-definable function/(/, Q) of 1.4 becomes a partial n\ function defined for all i and all hyperarithmetic Q £ P. (A function is called partial U\ if its graph is IIJ.) The existence of/follows from Lemma 2.1 and an important lemma of Kreisel [14] which says: if P(x, y) is II}, then there exists a partial U\ function g such that (x)[(3y)P(x,y)->g(x)

is defined & P(x, g(x))]*

The desired Q £ p is built, as it was in Lemma 1.4, by iterating/. To see that Q is hyperarithmetic one has to use the fact that a partial FI} function, restricted to a hyperarithmetic subset of its domain, is hyperarithmetic. LEMMA 2.3 [5].

(f)(P)QQ)es
K ^

VQ

Vrh~&l

The proof of Lemma 2.3 is very much like that of Lemma 1.5 and proceeds by transfinite induction on the rank, appropriately defined, of &. The only interesting 1 Kreisel's lemma generalizes a familiar lemma of recursion theory which says the same thing with II; replaced by recursively enumerable.

171 346

G. E. SACKS

inductive step occurs when J5" is a ranked sentence of the form (IX^iX"). If there is a 9{x) of ordinal rank < \b\ and a Q £ P such that Q H-„ &{x9(x)), then Q «-„ OX")^{Xb). Suppose there is no such #(x) and g. Let {9t(x)} be a hyperarithmetic enumeration of all 8?(jt)'s of ordinal rank <|6|. Then the induction hypothesis implies Now the sequential lemma (2.2) can be applied, as it was in Lemma 1.5, to obtain a 0 £ P such that 2 K" (AT4) <- ^{Xb). T is said to be generic if for every sentence & of JSf (^") there is a P such that Je P and either P tf-h J^ or P H-A ~ J r . The existence eo feneric P's sollows srom Lemma 2.3. The minimal degree argument of Lemma 1.8 is easily modified to prove the next lemma. LEMMA 2.4 [5]. Let <${x) be a ranked formula of&($~) with only x free. For each P there exists aQ£ p such that either (I) or (2) holds: (1) (T)[Te Q^Tis hyperarithmetic in {n \ Jt(T) \ 9f(«)}]; (2) (3H)(T)[Te Q—{n\ JK{T) \^{n)) = H is hyperarithmetic]. LEMMA

2.5 [5]. If T is generic, then every set hyperarithmetic in T belongs to

J((T). Before we discuss the proof of Lemma 2.5, we make some related observations without proof. For every T it is the case that every member of Jt{J) is hyper­ arithmetic in T. If T is generic, then wf = w„ where o f is the least ordinal not recursive in T. According to Kreisel [13] the conclusion of Lemma 2.5 is equivalent to the statement that the hyperarithmetic comprehension axiom (h.c.a.) holds in J((J). A typical instance of the h.c.a. is the universal closure of a formula of the form (x)[(3 Y)s/(x,

Y) ^ (Z)@(x, Z)] - (3r)(x)[x e X <-» (3 Y)sf(x,

Y)],

where sf(x, Y) and 8{x, Z) are arithmetic formulas which may contain free set variables other than Y or Z. By an argument of Feferman [2, p. 339] which holds for all T, the hyperarithmetic comprehension axiom is true in Jt{T) if the following Sj-bounding principle is true in JK{T): let 3F{x, Y) be a formula of -2%r) whose only free variables are x and Y and whose only unranked variable is Y, if b b (JC)(3 Y)3F(x, Y), then (x)(3 Y )5?{x, Y ) for some b e Ov The principle holds in Ji(T) for all generic T by Lemmas 2.6 and 2.7. LEMMA 2.6. Let jF(x, Y) be a formula of &(?) whose only free variables are x and Y and whose only unranked variable is Y. If P H-„ (A)(3 Y)&{X, Y), then Q H-„ (*)(3 Yb)3^{x, Y")for some Q £ P and b e 0,.

The proof of Lemma 2.6 is another application of the sequential lemma (2.2). Suppose P \rh (x)(3 Y)^(x, Y), where &(x, Y) ii sa si Lemma a.66 Then ("X0i)i.3o(3*)o=*[*M33Wn,

Y)l

172 347

FORCING WITH PERFECT CLOSED SETS

By Lemma 2.2 there i s a g g f such that (»)(3A)[* 6 01 & 0 H-, (3 7*)JF(fl, r>]. By Lemma 2.1 and Kreisel's lemma (as stated just before 2.2), there is a hyperanthmetic function/such that (n)[f(n) eO.&Q

If,, (3 F ( »>)jF(„,

Y'M)).

By Spector [27] there is a b e Ot such that (n)(\f(n)\ < \b\). So Q H-, (x)(3 Y»)F{x.

Y%

Lemma 2.6 almost establishes that the ^-bounding principle holds in Jt{T) for all generic T. What is lacking are some technical details concerning definability that are immediate in the set-theoretic context of § 1 but that require proof in the hyperanthmetic case. Let A be a set of indices of hyperanthmetic perfect closed sets. We write P e A to mean that the index of the perfect closed set P, encoded as a hyperarithmetic set of sequence numbers, belongs to A. A is said to be dense if

(P)(3Q)[QeP&QzPj. A set A is said to be .-//-definable if there is a formula JF(X) of &(f) in which 3~ does not occur such that A = {n \ Jt\ ^(n)}. For each T: a set A is said to be .//(7>definable if there is a formula p(x) of if(.T) such that ^ = {/,|^Cr)r-.F(*)}. /4 is called genetically persistent5 if there is a formula J^f.v) of i f ( ^ ) such that for all generic T, A = {« | Y/(7) >(/7)J. LEMMA 2.7. T is generic if and only if T e U [P \ P e A} for every dense A arithmetic in Kleenes O. LEMMA 2.8. A is arithmetic in Kleenes O if and only if A is ^-definable if and only if A is genetically persistent.

Let T be generic; we use Lemmas 2.6 and 2.7 to show the SJ-bounding principle holds in . # ( n . Let P(x, Y) be as in the hypothesis of 2.6. Let A be the set of all QS such that for some b e 0 , . Q K-, (,v)(3 Y)P(x.

Y) — (.v)(3 Y*)&(x, Yl).

A is arithmetic in Kleene's O, since both 01 and the indexing of all Fs is U\. A is dense by 2.6. Then 2.7 provides a Q e A such that TeQ. To complete the proof of Lemma 2.5 we need only prove Lemmas 2.7 and 2.8. 5 Persistence is a term used by H. Putnam [18] to describe a well-known property of the hyperanthmetic sets. Kreisel [13] formulates the persistence property as follows: a set is hyperarithmetic h a n d only if it belongs to every cu-model of the hyperarithmetic comprehension axiom. Another way of putting it is: a set A is hyperarithmetic if and only if there is a ranked formula S{x) oi^{jT) such that for all T, A = in \ Jf'T) V .*m\. This formulation of persistence is a consequence of a slight modification of Kleene [10, p. 35).

173 348

G. E. SACKS

Suppose A is dense and arithmetic in Kleene's O. By 2.8 A is generically persistent. Let s/(.x) be a formula such that for all generic T, A = {n j Jt{T) f=j/(n)}. We claim there is an 3F such that for all generic T, J({T)\

JF «-♦ T e \J{Q\ Q e ,4}.

The desired J^ is (3x)K(.v) & (Y){38{x, Y)^3>(Y,$~))], where # ( * , K) and 9{Y,Z) are arithmetic formulas with the following properties. Let / be the U\ set of indices of hyperarithmetic perfect closed sets. If r, e /, let P„ be the hyperarithmetic perfect closed set whose index is n. For each n e /, (3i Y)SS{n, Y) and (Y)[@(n, Y) — (Y)B = P„].e 3>(P, T) says 7"£i>. To see that every generic Te \J{Q | Qe^}, it suffices to see there is no P such that PVrh ~ J T If there were such a />, then by the density of A there would be a 2 — /> such that g e ^ and 2H- ~ ~ . ^ Suppose 7 is arbitrary but that Te (J{2 | 2 e >4} for every dense -4 arithmetic in Kleene's 0 . T is generic if r 6 \J{Q | 2 H-A ^ or 0 H-- ~ J r } = AT ^or every sentence J*" of &{&), But it is easy to check that K? is dense and arithmetic in Kleene's O. We turn to Lemma 2.8. It is enough to show: (i) O is ^-definable; (ii) O is generically persistent; (iii) if A is generically persistent, then A is arithmetic in O. We dispose immediately of (iii). Suppose A = {n \ Jt(T) \^{n)} for all generic T. Then n e A <-> (3P)(P H-„ s#{n)). A is arishmetic en O sincs P H-„ J r , restricted to J ^ s of the form ssf{n), is arithmetic in O. Mostowski's conjecture [4] provides an arithmetic formula C(X,n) such that for all n, n e O <-^ {3X)[X e HYP & C{X n)] Thus (i) O is ^-definable by a 2 l formula O will then be generically persistent if every arithmetic formula and every generic T are such that if Jt(T\V(3X\C(X) then (3X)lXe HYP & C(XW SuDDOsePH- (3X)C(X) Then pi Cimx)) for some ranked
2.9 [5]. / / T is generic, then T has minimal hyperdegree.

If T is generic, then T is not hyperarithmetic by the same mode of argument employed in Proposition 1.1 to show that a set generic in the sense of § 1 is not a constructible set of Jt.. The fact that T has minimal hyperdegree follows from Lemmas 2.4, 2.5 and 2.7. THEOREM

2.10.

If T is generic, then every set of the same hyperdegree as T is

generic. Let 5 have the same hyperdegree as some generic T. Then where &(x) is a ranked formula. By Lemmas 2.4 and 2.7 there is a Q such that TeQ and Q satisfies clause (1) of 2.4. Then the argument of Theorem 1.12, ' The existence of £ follows ffom the uniform implicit arithmettc deffnabiliiy of hyperarith­ metic sets of arbitrarily high Turing degree, cf. Kreisel [7, p. 307].

174 FORCING WITH PERFECT CLOSED SETS

349

routinely modified, establishes that S is generic. The modification includes Lemmas 2.7 and 2.8. 3. The arithmetic case. In accord with §0,P,Q,R,... ambiguously denote arithmetic perfect closed subsets of 2" and their encodings as arithmetic subsets of w. One of the principal objectives of the present section is to introduce pointed, perfect closed sets and to apply them to a problem concerning the Turing degrees of certain hyperarithmetic sets and in particular the truth set for arithmetic. The arithmetic language i f (J~) is the language of first order number theory together with a constant set symbol T and a predicate letter St denoting a certain recursive predicate. &(£T) has symbols for membership (e), addition ( + ), multiplication (•), and successor ('), and a numeral ft for each natural number n. St{3~) ii not eesentially different ffom Feferman's llnguage L*(S) [[2 p. 328]] The predicate letter M is associated with the jump operator [19] for Turing degrees. By the enumeration theorem there is a recursive predicate 8t(T, xu x2, y) such that for each recursive predicate &>{T, xt,y) there is an e with the property that (T)(x2)[(3y)<%2(T, e, x2,y) *-> (3jOW,

xt,y)].

The jump of T, denoted by T. can be defined by 2« • 3" e f «

t3y)8,{T,

m, n, y)

with the understanding that every member of T is of the form 2m ■ 33 " T "cn be thought of as the disjoint recursive union of all sets recursively enumerable in T. T{n). the «-jump of T, is defined by: r°» = T; T(H+1) = {TM)'. 0 is the empty set. A set is arithmetic if and only if it is recursive in 0"" for some n. T{"], the (o-jump of T, is the disjoint recursive union of all /7-jumps of T: We call 0<w) the truth set for arithmetic. It is essential to note the existence of a recursive predicate 3?{T, n, x,y) such that for all 7 and n: neTm *-+{3x)(y)3t(T,n,x,y). The language SPif) includes * ; thus the 2-jump of T is definable by a E2 formula of &(&-). It is convenient to assume that the truth or falsity of @{ T,n,x, y) for each choice of T, n, x and;- is determined by an initial segment of the character­ istic function of T of length at most y. Let S? be a sentence of &{&"), and let T be an arbitrary subset of w. We say T makes & true (or more briefly, F is true) when & is true subject to the inter­ pretation of F as Tand to the standard interpretation of the arithmetic symbols and sM. The forcing relation. P H- 5?, where & ii s aentence of &{JT) and P ii sn arithmetic perfect closed set, is defined by: P (r 5? «-> >T)[TeP — JF], Since we require every T e P to make & true, it is perhaps not appropriate to say P forces J r . On the other hand, if forcing is thought of as an approximation of truth, it seems reasonable to expect perfect closed forcing to provide a perfect approxi­ mation of truth.

175 350

G. E. SACKS

We say P is generic if for each & there is a P such that T e P and either P H- 3? or P H- -&" The existence of generic P's is an immediate consequence of Lemma 3.1. We could prove 3.1 by the sequential lemma approach of § 1 and 2, but it is more instructive to invoke what might be termed the local forcing approach. Let p,q,r,... be finite, initial segments of characteristic functions of subsets of w. {p>q and Tep are defined in §0.) The definition of p H- &, where & is a sentence of &{T)% is given by four inductive clauses: (i) p H- y if 9? is a true equation of ariihmetic, or if & is of the form ft e .T or J>(^\ m, n, c) and every Pe/> makes J5" true. (ii) p %■ (3x)^(.v) if (3/7) [^ H- .T(n)]. (Hi)/>K-~ar i f ( ? ) B & , ~ [?»-*]. (iv) p\rPxk.Px if/> H- ^ , and /? H- -F 2 . In short, pfr& follows the rules of Cohen's forcing with finite conditions as formulated by Feferman [2] in the arithmetic case. Let P be an arithmetic perfect closed set. The forcing relation/? Vrp JF, where/? £ P and JF is a sentence of &{$~), is defined exactly as p H- 3F waa deffned above except that tcause (iii) becomees (iii)^/?H-p~^

if

(?),>,[? 6 J P - » ~ ? l H p g F ] .

We call /? H-p & the localization of p B- #" to P. The idea of local forcing was used by Spector [28] in his construction of a minimal Turing degree. Spector's funda­ mental lemma [28, p. 588] was in essence the following: let & be a £ 2 sentence of ^{ST), and let P be a recursive perfect closed set; then there exists a recursive perfect closed Q S P such that either ( r ) ( P e <2 — J*") or (P)(Pe 2 — ~ ^ 0 The details of Spector's proof coincide with the details of the local forcing argument given in Lemma 3.1 for the case & 6 £ 2 . Spector uses his lemma to construct a sequence P0 2 Pj 2 P2 2 • • • of recursive perfect closed sets with the following properties: for all P £ P„ if {e}T is total, then {e}T is recursive or 7 is recursive in {e}T ({e)T is the eth function partial recursive in T according to the standard enumeration); for all TePe, Pis not the eth partial recursive function. Thus if T £ fl [P, | e > 0}, it is clear that T has minimal Turing degree. The properties of Vr1' are virtually the same as shose of H-; ;ntuitively, ,his si she case because P and 2" are homeomorphic by means of an arithmetic map. T is generic for P if for each sentence & of &($~) there is a p e P such that Tep and either pVr1'^ or /> H-''~.F. If Pis generic for P, then j £ P . and P makes ^ true if and only if there is a /? £ P such that P £ /? and /> K-;' ^ . LEMMA 3.1. Le.' P 6e an arithmetic perfect closed set. and let JF be a sentence of the arithmetic language &(5"). Then there exists an arithmetic perfect closed Q £ P suc/i r/w/ e/7/ier (a) or (b) /io/A: (a) (T)(TeQ-+&);

(b) (70(Tefi

—~^").

Furthermore, if 3F is S 2 , //ie/; 2 fa recursice in P

176 35

FORCING WITH PERFECT CLOSED SETS

1

There musl be a p e P such that either (a) p H-p & or (b) p Vr1' ~$rSuppose (a). We develop a Q c P satisfying clause (a) of the lemma. Let {^m | m ^ 0} be an enumeration of all sentences of &{&) having no more quantifiers than J^ does. The relation q H-^ &m, viewed as a relation on q and m, is arithmetic, because a finite bound has been imposed on the number of quantifiers that can occur in Wn. It follows there must exist a partial arithmetic function f(q, m) such that if a eP, then/(a, m) eP,q > f(q m) and either fid m) Vrp &' or f(q, m) If' ~ J F m By iteration of/, it is"possible to define a partial arithmetic function Xim.\ qm with the following properties- a" = v for each w > 0 and PROOF.

?f+1

„.r _ j r m 5

a n d ?f+ti

^ ^

m

o r 9 r + + 2 j„ frP

_ ^ .

Then

g =

{q?

|w >

0

&

/ < 2"} is an arithmetic perfect closed set. If 7 6 Q, then T is generic for P with respect to all sentences in {.FJ, and consequently, Tmakes & true since Tep = riand,lH-jr. Now suppose ^ i s S Let 9 be ( 3 X ) ( J ^ J ; ) where ^ x , j ) is a quantifierfree formula of <£{J ) . Suppose again that (a) pl-T. This means /J H-J [y)P{il,y) for some «, and so (™)(«)rep{3r)rep[p > 9 -

9

> r & r (fp &(*, m)}.

The relation r H-p y<« w") viewed as a relation on r B and m is recursive in P because ^ has no quantifiers. It follows there exists a function partial recursive in P such that if q e P and p >q then /Y/w o l e ? a > f(m a) and f(rn, q) VrF Sf{n, m). Let &m be ^ ( « , «l)7and define ten, | ?« by iterating/Cm, ? ) as in the previous paragraph. Then Q = {q™ \ m > 0 & i <2m} is recursive inP, and every TeQ makes (3x)(y)^(x,y) true. Finally, suppose (b) p Vtp (x)(3y) ~y(x,y). Then (n)(q)oePC3m)(3r)r€P[p > q-+q

> r & r \\-p ~2>{n, m)].

As above, we can make m and r into functions of n and 9 partial recursive in P, functions which are defined whenever q e P and p >q. Then as above r is iterated to obtain a 2 recursive in P such that every TeQ makes (x)(3j) ~ Sf(x, y) true. One of the advantages of using local forcing instead of an appropriate sequential lemma to prove Lemma 3.1 is that local forcing makes it easier to see Q is arithmetic. We say P is a pointed, arithmetic perfect closed set if (T)(T e P — P is recursive in T). Every perfect closed P has T% such that T is recursive in P; let Tr be the canonical one obtained by traveling up P viewed as a tree and turning left at every branch point. If P is pointed, then P and 7> have the same Turing degree, and that degree is less than or equal to the degree of every TeP. Pointed, perfect closed sets are useful for forcing the degree of a generic 7" to be an upper bound for some given countable set of degrees. The notion of pointedness is used in [23] to show amon" other things: (1) every countable admissible [12], [16] ordinal > oj \$ of

177 352

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the form w, then wf = a for some F with the minimal property that for every J ' of lower hyperdegree than 7\ wf <<x. Recently Friedman and Jensen found a very interesting proof of (1) which replaces forcing by a compactness theorem of Barwise for infinitary languages. Their compactness approach does not seem to yield the minimal property of (2). A typical consequence of (2) is: the set of hyperdegrees of the A' sets has a minimal upper bound. The idea of pointedness goes back to the Kleene-Post proof [11] that there exists a set of lower degree than that of 0("" but of higher degree than that of 0 (m) for all m, i.e., the degree of the truth set for arithmetic is not a minimal upper bound for the degrees of the arithmetic sets. We say T is pointedly generic if for every sentence & of <£{$~) there is a pointed, arithmetic perfect closed P such that Te P and either f l f ^ or P H- ~ J f . The existence of a pointedly generic T follows from 3.1 and 3.2 and the mode of argument occurring in Lemma 3.3. We state without proof the fact that the Turing degree of any pointedly generic T is a minimal upper bound for the Turing degrees of the arithmetic sets, since it was shown in [20] that every count­ able set of Turing degrees has a minimal upper bound. PROPOSITION 3.2. If P is a pointed, arithmetic perfect closed set recursive in the arithmetic set A, then there exists a pointed, arithmetic perfect closed set Q £ P such that A and Q have the same Turing degree. PROOF. Although P may be encoded in a rather arbitrary way as an arithmetic set of sequence numbers, we can always re-encode P as {p™ | m > 0 & i < 2"1}, where kirn | p™ is recursive in the given encoding of P and is such that for all m and i < 2m: pm > p'"~\ p? > p"\tk, p?+l £ pZtl and pZt~ > £m+1. Since e is pointed, the re-encoding of P has the same Turing degree as the given encoding of P; i.e., the given encoding is recursive in the canonical path TP, and TP is recursive

in ).im | p™

We define Q inductively: p°0eQ; if m + 1 1s odd and p™ e Q, then p^\ pZ+l €Q\ if m + 1 1s even,/;™ e Q and (m + \)!2eA, then/),"1*1 e Q; if m + 1 is even, p™ e Q and (m + 1)12$A, then p"£l e Q. Clearly, Q £ />, and Q is recursive in P, A. But P is recursive in A, so Q is recursive in A. Consider the canonical path TQ £ Q. TQ is recursive in Q, and T0eP; so P by virtue of its pointedness is recursive in TQ and hence in Q. But then A is recursive in Q, since A is recursive, in P, Q as a consequence of (m +

l)/2eA~(li)(i<2"'&pr1ee).

To see that Q is pointed, fix Te Q. A is recursive in T, P as a consequence of (m + l ) / 2 6 / l ^ ( 3 / ) ( i < 2 " ' & T 6 p r 1 ) Then Q is recursive in T, P, since Q is recursive in A. But P is recursive in T because TeP; so 2 's recursive in T.

178 FORCING WITH PERFECT CLOSED SETS

353

3.3. There exists a T such that Tm has the same Turing degree as 0"" , and such that 0("> is recursive in Tfor all n. THEOREM

1

PROOF. Let {JFJ be an enumeration of all S 2 sentences of &{$~). We intend to define two functions, km \ Pm and hn | i(m), each recursive in 0""> and with the following properties: P, = 2"; Pm 2 Pm^; Pm has the same Turing degree as 0"">; Pm+l If &m or Pm+1 H- ~ ^ m ; r(m) = 0 «-> P m + 1 H- JF m . Assume that Pm has been defined and that Pm is a pointed, perfect closed set which has the same Turing degree as 0 (m) . By Proposition 3.2, there is a pointed Rm £ F m whose Turing degree is 0(m+1> By Lemma 3.1, there is a />ra+1 £ Km such that />m+1 is recursive in Rm and either P m + 1 H- J ^ or Pm+l If ~ J r t n . Since /?„, is pointed, it follows that every P £ # m and recursive in Rm is pointed and recursive in Rn. (This fact was demonstrated in 3.2.) So Pm+i is pointed and has the same Turing degree as 0<mJ-1' If />„_, H- J r m , put r(m) = 0; otherwise, put t(m) = 1. We obtain Aw | Pm and Am | t(m) recursive in 0«»> by appealing to various uniformities. Examination of the proof of Proposition 3.2 reveals that Rm is recursive uniformly in P'm; i.e., Rm is the result of applying a fixed recursive reduction procedure to P'm, a procedure independent of the value of Pm. To develop Pm+1 and t(m) from Lemma 3.1, we must first decide if there is a p e Rm such that p IH8- W„ Since . F m is 2 * the answer as a function of m is uniformly recursive in R%>. Thus Pm+1 is uniformly recursive in P<*\ and Aw | t(m) is re­ cursive in Aw | *»». Since F0 is recursive and Aw | 0 (ml is recursive in 0««», it follows that Aw | Pm and Aw | t(m) are too. Since {.Fm} includes all sentences of the form tfe-T, it must be that

D{P m | w > 0} has a unique member T recursive in <)«"•>. And for each w, 0""' is recursive in T since 7"e Pm and P m is pointed and of the same degree as 0,m». Since the recursive predicate # is part of ^f(#"), w e J r ( 2 ) is a S 2 formula of J ? ( ^ ) . Assume JF2rn is w £ ^" (2) for all w. Then w e T2'«-»r(2w) = 0, and consequently, Tw is recursive in 0("». Putnam [18] has shown: for all 7", if (w)(0<"" is recursive in T), then 0("" is recursive in Ti2). For each set A, let A be the Turing degree of A. After Kleene and Post [11] showed that 0 0}, Spector [28], proved that {0'"" | w > 0} has no least upper bound. Putnam [18] calls A the n-least upper bound for {B, | i > 0) if A is the least member of {C"" | (i)[B, < C]}. Thus Spector ruled out a 0-least upper bound for {0<m) | w > 0], It is not difficult to show in the style of Theorem 3.3 that {0!w) | w > 0} lacks a 1-least upper bound. And of course 3.3 says that 0 U ) is the 2-least upper bound of {0(m) | w > 0}. By Spector's uniqueness theorem [27], it is acceptable to define 0M for each recursive ordinal a to be the Turing degree of the hyperarithmetic set H„, where b is any notation for a. Then the argument of Theorem 3.3 is readily extended to give the following result.

179 354

C. E. SACKS

3.4. / / a is a recursive limit ordinal, then 0(o) is the 2-least upper {0««|j8
THEOREM

bound of

It can be shown that the set of all Turing degrees of hyperarithmetic sets has no 2-least upper bound.7 This fact together with Theorem 3.4 leads to a tidy character­ ization of the hyperarithmetic hierarchy in terms of the jump operator and the notion of 2-least'upper bound. H. Putnam [18] has extended Theorem 3.4 in a nontrivial fashion to the entire constructible hierarchy. A set T is said to be implicitly arithmetically definable if there exists an arith­ metic predicate J*{X) such that J*(T) and (^X)s/(X). Every implicitly arith­ metically definable (i.a.d.) set is hyperarithmetic, and every hyperarithmetic set is recursive in some i.a.d. set. Feferman [2] found a hyperarithmetic set (in fact recursive in 0(lul) that was not i.a.d. We say two sets have the same arithmetic degree if each is arithmetic in the other. THEOREM 3.5. There exists a nonarithmetic, hyperarithmetic set T such that every nonarithmetic set arithmetic in T fails to be implicitly arithmetically definable. PROOF. Let {J%J be an enumeration of the sentences of the arithmetic language 2(T). We claim there is a function In | Pn recursive in 0<") such that for each n, Pn 2 Pn+1, P„ is an arithmetic perfect closed set, and either P„ H- J F „ or Pn H- ~JF„. The claim follows from some uniformities implicit in the proof of Theorem 3.1. Assume & is in prenex normal form and let T(JF) = 1 plus the number of alternations of quantifiers in the prenex of JF. Then the set {p |p H-* &} is recursive (uniformly) in the set PWiJC)) Jt follows that the first conclusion of Lemma 3.1 can be made to read: there exists a Q £ p such that either P H- & or P H- ~ J F and such that Q is recursive (uniformly) in p<^^». But then the sequence P0 2 P1 2 P2 2 • • • can be constructed recursively from m'"'. Let T be the unique member of f){P„ | n > 0)} Tis genericc ,nd the argument of Theorem 1.12, routinely modified, establishes that every set of the same arithmetic degree is generic. (T is of course of minimal arithmetic degree.) So it remains only to see that 7" is not implicitly arithmetically definable. This last is clear since if .&(T) holds, then there is an arithmetic, perfect closed P such that T e P and

REFERENCES 1. P. J. Cohen, The independence of the continuum hypothesis, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 1143-1148 and 51 (1964), 105-110. 2. S. Feferman, Some applications of the notion of forcing and generic sets, Fund. Math. 56 (1965), 325-345. 3. S. Feferman and C. Spector, Incompleteness along paths in progressions of theories, J. Symbolic Logic 27 (1962), 383-390. 4. R. Gandy, Proof of MostowskiS conjecture. Bull. Acad. Polon. Sci Ser. Math. 8 (1960), 571-575. 5. R. Gandy and G. E. Sacks, A minimal hvperdegrec, Fund. Math. 61 (1967), 215-223. In fact it has no /Meast upper bound for any recursive /?.

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6. K. Godel, The consistency of the axiom of choice and of the generalized continuum hypothesis, Princeton Univ. Press, Princeton, N J „ 1966. 7. R. B. Jensen, "Concrete models of set theory" in Sets, models and recursion theory, Amsterdam, 1967,44-74. 8. J. L. Kelley, General topology. New York, 1955. 9. S. C. Kleene, Hierarchies of number-theoretic predicates, Bull. Amer. Math. Soc. 61 (1955), 193-213. 10. . Quantification of number-theoretic predicates, Compositio Math. 15 (1959), 23^10. 11. S. C. Kleene and E. L. Post, The upper semi-lattice of degrees of recursive unsolvability, Ann. of Math. 59 (1954), 379-407. 12. S. Kripke, Transfinile recursions on admissible ordinals, I and II (abstracts), J. Symbolic Logic 29 (1964), 161-162. 13. G. Kreisel, Set theoretic problems suggested by the notion of potential totality, Proc. Sympos. Infinitistic Methods, Warsaw, 1961, 103-140. 14. , The axiom of choice and the class of hyperarithmetic funciion,, Indag. Math. 24 (1962), 307-319. 15. A. R. D. Mathias, A survey of recent results in set theory, these Proceedings, part II. 16. R. A. Platek, Ph.D. Thesis, Stanford University, 1965. 17. K. Prikry, Models constructed using perfect sets, Lectures Notes Axiomatic Set Theory (University of California, Los Angeles, 1967) pp. IV-K-1—1V-K-3. 18. H. Putnam, Collected papers on hierarchy theory, to appear. 19. H. Rogers, Jr., Theory of recursive funciions and effective computability, McGraw-Hill, New York, 1967. 20. G. E. Sacks, Degrees of unsolvability, 2nd ed., Princeton Univ. Press, Princeton, N.J., 1966. 21. — , Posf s problem, admissible ordinals, and regularity, Trans. Amer. Math. Soc. 124 (1966), 1-23. 22. , Measure-theoreiic uniformity in recursion theory and set theory, Trans. Amer. Math. S o c , 142(1969), 381-420. 23. , Countable admissible ordinals and hyperdegrees, to appear. 24. J. R. Shoenfield, 1967 UCLA Recursion Theory Seminar Notes, Appendix B. 25. R. Solovay, The measure problem, abstract 65T-62, Notices Amer. Math. Soc. 12 (1965), 217. 26. , The measure problem, to appear. 27. C. Spector, Recursive well-orderings, J. Symbolic Logic, 20 (1955), 151-163. 28. , On degrees of recursive unsolvability, Ann. of Math. 64 (1956), 581-592. 29. , Hyperarithmelical quantifiers, Fund. Math., 48 (1959), 313-320. 30. L. Tharp, Set theory lecture notes, Massachusetts Institute of Technology, 1965. MASSACHUSETTS INSTITUTE OF TECHNOLOGY

181 Reprinted from Annals of Mathematical Logic 4 (1972) 343-367 with kind pemiission of Elsevier Science - NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.

THE Q-FINITE INJURY METHOD G.E. SACKS and S.G. SIMPSON * Massachusetts Institute of Technology. Cambridge, Mass. 02139, U.S.A.

Received 10 April 1972

§0. Introduction

Recursive functions of ordinals were invented by Takeuti [ 1 ] ]nd subsequently specialized by Kripke [2] and Platek [3] to Z, admissible initial segments of the ordinals, and by Kreisel and Sacks [4] to the re­ cursive ordinals. Takeuti's principal result was: if/is recursive and 0 is an infinite ordinal, then cardinality/0 < cardinality 0 . The above inequality follows from a downward Skolem-Lowenheim argument similar to the one originated by Godel [5] in order to show the axiom of constructibihty implies the generalized continum hypoth­ esis. Takeuti's argument succeeds because the graph of a recursive func­ tion is Z ] ; his inequality fails dramatically if/is replaced by the Z 2 function that enumerates the constructible cardinals in increasing order. Thus right from the start, the generalization of recursion theory from the integers to the ordinals was concomitant with the model theoretic properties of the constructible sets. ' In this paper the Friedberg-Muchnik solution of Post's problem is

* The preparation of this paper was partially supported by NSF GP29079. 1 Browse in (rather than read) the monumental survey of generalized recursion theory by Kreisel [8] in order to grasp the ontological questions ignored here.

182 344

G.E. Sacks and S.G. Simpson, The a-finite injury method

lifted from to to a, where a is any £j admissible ordinal, with the help of some downward Skolem-Lowenheim arguments. Let La be the set of all sets constructive via ordinals less than a. a is Sj admissible if L„ satisfies the Sj replacement axiom schema of ZF. A partial function / : a - a is a-recursive if its graph has a 2 j definition over La (with parameters in L a ). A set A is a-recursively enumerable if it is the range of a partial a-recursive function. Theorem 4.1 furnishes two a-recursively enumerable sets such that neither is a-recursive in the other. It is hoped that the proof of 4.1 reifies the a-finite injury method. (An abstract account of the method would be premature, hence abortive, at this time.) A set H is a-finite if H c a and H G L a . a-finite sets inter­ act with a-recursive functions, when a is Zj admissible, in much the same way finite sets interact with recursive functions, but not for the same reasons. The a-finite injury method specializes, when a = CJ, to the finite injury (or priority) method of Friedberg and Muchnik. They reasoned along the following lines. A recursively enumerable set A is sought which will satisfy a recursi­ vely enumerable sequence {RmIm< CJ} of requirements. Each Rm is a recursively enumerable sequence of neighborhood conditions, i.e. pairs <E Rn and no member of K* is yet in A. In that event it is tempting to add all of H* to A at stage / so as to satisfy Rn. But suppose H* D K is nonempty. Then the addition of H* to A will injure what was done at stage s to satisfy Rm . The situation is summarized by saying Rm and Rn conflict at stage /. Friedberg and Muchnik resolved the conflict in favor of Rm if m < n, and in favor of Rn otherwise. In other words they assigned higher prior­ ity to the requirement with lesser subscript (or index): if m < n, then H* is not added to A at stage f;ifm > n, then//* is added and Rm is injured in order to satisfy /?„. The principal Friedberg-Muchnik lemma states: the number of stages at which Rmis injured is less than 2'" 2 2

l o r a more detailed account of the finite injury priority method see Lachlan [9| or Sacks [ 10, p. 431.

183 §0. Introduction

345

It is proved by induction on m. It implies that Rm can be satisfied if there are infinitely many stages at which fresh opportunities to satisfy Rm present themselves. No difficulty arises when lifting the above argument to a if LQ satis­ fies the Z 2 replacement axiom schema of ZF. Recursive and finite are lifted to a-recursive and a-finite, and the proof of the Friedberg-Muchnik lemma remains valid, largely because the function fb = first stage after which Rb is not injured is X 2 . 3 If LQ does not satisfy Z 2 replacement, then the FriedbergMuchnik reasoning can fail totally. Stillwell [11] conjectured that some modification of the finite injury argument would show that the ordertype of the set of stages at which RB is injured is less than 26 , where 26 denotes ordinal exponentiation. But R. Shore found a coun­ terexample that created a vacuum into which the present paper rushed. The details of the a-finite injury method for a given a depend on the £1 substructures of La. L„ is a Zj substructure of L a if 0 < a and each S, sentence (in the language of ZF with parameters from L^) is true in L.,3 if and only if it is true in LQ ; in symbols, Lp < j L a . Let Sa = {0\p
LJ .

The ordertype of Sa determines the outcome of the method when ap­ plied to a. Section 1 reviews a-recursion theory and draws attention to those parameters that occur in the Zj definitions of certain key a-recursive functions. Such parameters are important ingredients of Z, substructure arguments. Section 2 develops the model theoretic properties of LQ needed in Sections 4 and 5. Section 3 touches on some intuitive aspects of the a-finite injury method, and Sections 4 and 5 apply the method to Post's problem. Section 6 adds a notion related to forcing to obtain the strongest possible incomparability. And Section 7 discusses further re­ sults and open questions. 3

Similarly no difficulty arises when lifting the infinite injury method ([10, p. 86]) to a if la satisfies the E j replacement axiom schema of ZF.

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G.E. Sacks and S. G. Simpson, The a-finite injury method

§ La-recursion

The details missing from what follows can be extracted from [4, 12, 13]. The set theoretic definition of a-recursive given in section 0 must be rephrased in terms of computations before familiar recursion theoretic ideas can be applied. The Kleene-Kripke equation calculus for a inclu­ des numerals that name the ordinals less than a, substitution rules, and an infinitary rule needed to evaluate supremums as in the definition of addition of ordinals by transfinite induction. Sf is the set of all equa­ tions computable from E (a finite initial set of equations) in less than a steps. An important convention requires that every ordinal mentioned in S„ be less than a The natural enumeration of Sf is 2j over L It follows that a is Zj admissible (as defined in section 0) if and only if SE •, = SE for rll E. II nadition a aartiaa lunction/C tt a ex ai Cc -recursive if and only if there is an E such that

fl = 5 * - £i = 5 e 5f for all 7, 5 < a. From now on a is 2 j admissible. A subset of a is a-recursive if its representing function is a-recursive. A set is a-finite if it is a-recursive and bounded by some ordinal less than a. If/is partial a-recursive and defined on an a-finite K, then f[K) is a-finite. There exists an a-recursive function k(y, 77) such that (i) if k(y,77) = 0, t h e n 7 < T ? ;

(ii) if A" is a-finite, then K = {{\k(7, rj) = 0} }or russ one e7 7 <; (iii) the E that defines k mentions no infinite ordinals. (Thus the same £ works for every a.) If K = {7\k{y, 77) = 0}, then T? is said to be the canonical index of K, in symbols K = Kn . A property of a-finite sets is a-recursive if the equi­ valent property of canonical indices is a-recursive. If an equation can be computed from £ in less than a steps, then it owes its existence to some a-finite, wellfounded tree called a computa­ tion. The set of all computations is a-recursively enumerable, so there exists a simultaneous enumeration of all a-recursively enumerable sets. To be more precise, there is an a-recursive function r(a, e) such that

185 347

§l.a-recursion

(v) if B is recursively enumerable, then B = U {Kr(oe)\o < a} for some e < a; (vi) the E that defines r mentions no infinite ordinals. Let R° = Kr(oe) and Rf = U {R°t I a < a). Then {Rt I e < a} is a stan­ dard enumeration of the a-recursively enumerable sets. Similarly there is a standard enumeration {{e}l e < a} of the partial a-recursive func­ tions. There is no harm in blurring the difference between ordinals and ordered n-tuples <5 a , ..., 5„> of ordinals. With that in mind let C c a anddefine[e]£(7) = 5 b y ( E P ) ( E T ? ) [ < 7 , 5, P,T?> e R°t & Kp C C & Kv

C

a-C].

A useful convention is: the computation of [e)^(y) - 6 mentions no ordinal > a. A partial function/C a X a is weakly a-recursive in C (/^wfl, O if there is an e < a such that for all 7, 5 < a, /(7) = 8 ~ (Ea)[[e] 0 c (7) = 6] . It is possible for [e}£(7) and [e] f (7) to differ in value; if no such difference occurs and [e] £(7) is defined for some a, then [e] c ( 7 ) is defined. T h u s / < W Q Cif and only i f / = [ e ] c for some e. B <wa Cif the representing function of B is weakly a-recursive in C. Driscoll [14] showed <wa is not transitive on the a-recursively enumerable sets for many a's. The notion of "a-recursive in" ( < a ) is transitive. A such that: / CA *-> (EH)(EK)[(J, H,K) = 0&HCB&KC JCa-A<-+

(EH)(EK)[
1 &H C B &K C

a-B ] a-B],

for all a-finite J; H and K are a-finite set variables. A and B have the same a-degree if each is a-recursive in the other.

186 34 8

G.E. Sacks and S. G. Simpson, The a-finite injury meehod

§2. a-stability

Again a is Ej admissible. The projectum of a, denoted by a*, is the least 0 such that 0 is the supremum of the range of some one-one (into) a-recursive function. Clearly a*
is

187 §2.a-stebmty

349

a-cardU{/ p X

{p)\p
On the other hand / is one-one, so a-card U {Jp I p < v} = fi. Let d e be the set of all equations in the equation calculus for |3 as sketched in section 1. Assume 7 > 0 > w. |3 is said to be 7-stable if Sf = 5 f n d for all finite £ c d , . (Neither 7 nor r 3re essumed dt ob e£ admissible.) It is readily verified that 0 is 7-stable if and only if Lfi<1 Ly (cf. Section 0) and (3 < 7. Jensen and Kripke have refined Godel's downward Skolem argument in L to show: every a-cardinal > GO is a-stable; a = a* > u if and dnly if a is the limit of a-stable cardinals. The next lemma is the only instance of the above two facts needed in this paper. Lemma 2.4. If a* = a > u> and there is a largest a-cardinal, then a is the limit of a-stable ordinals. Proof. Let N be the largest a-cardinal, and assume a > 7 > N with the intent of locating some a-stable 0 > 7. Define H to be the set of ordinals of the form/(7j, ...,7„), where: n< CJ;J{, ...,yn < 7; a n d / i s a partial a-recursive function whose defining set E of equations mentions no in­ finite ordinals. (H is the Zj Skolem hull of 7 in L tt .) Fix o e H to see that H is an initial segment of a. Each member of H has a Z, definition with parameters < 7. Let p be the least canonical index 5 (cf. Section 1) such that K5 is an a-finite function from N onto a. Then peH.It fol­ lows that the partial a-recursive function K can be defined without mentioning any ordinals > 7, and so K [ N ] C // Thus H equals some /3 < a. 0 < a because a* = a and H can be map­ ped by a one-one partial a-recursive function into the a-finite set of all finite sequences of ordinals less than 7. Fix £ c C p and 2 e Sf n C0 to see that (3 ii a-stable. z ii she last tlne of some a-finite computation whose roots are in E. Let g be an a-recur­ sive function that enumerates (the Godel numbers of) all a-finite com­ putations of z from E. Choose g, as in clause (vi) of Section 1, so that the only ordinals occurring in the definition of g are those in z. Every such ordinal is less than 0, and sog can be defined without mentioning any ordinals > 7. Hence #0 < 0 and z € Sff

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§3. Towards Post's problem

Assume a is Z, admissible. Two a-recursively enumerable sets ,4 and fl, and witness functions/andg, are sought such that for each e < a (i)/ee/4H-(Eo)([e]£(/e)=1); (ii)gee/?~ (Ea)([e]2(ge)=1). If A and B meet the eth instance of requirement (i), denoted by fe, then fe is a witness to the fact that A is not weakly a-recursive in B via Godel number e, as defined in section 1. Conflicts are inevitable because requirement fe (respectively ge) is met by adding fe to A (respectively ge to B) and excluding some a-finite set from B (respectively A) in order to preserve the value of [e]f (respectively [e]„ ). They are resolved by appealing to priorities: it is permitted to add/e to A — even ii fi meens injuring requirement g5 which at some earlier stage excluded fe from A - when/e has higher priority thangS. The obvious wellordering of priorities/D < g0 < f\ < g\ < (fO has highest priority, g0 next highest etc.) fails if a* < a, but the remedy is simple enough. Let t : a -*■ a* be a one-one a-recursive function Then say fe has higher priority than e5 if te < tB. Thus when a* < a the wellordering of priorities has order t v n p IPSS t h a n rv All POP<; well as will he seen in s e c t i o n 4 laruply hf>

cause of Lemmas 2.2 and 2 3.,(If a* = I berg suffice )

the combinatorics of Fried-

If a* = a and the set {0I j3 < a & / is s-stable) }as srder rypp ea ,hen the obvious wellordering of priorities succeeds - thanks again to 2.2 and 2.3. The most interesting case is when a* = a and the set of a-stable ordi­ nals has order type less than a. In that event the predicate *♦/e has higher priority thang6" is not a-recursive, and it becomes necessary to guess at what the priorities are. Thus in addition to the expected convergence problems associated with the finite injury method, namely the limits (if any) of [e]f and [e]^ as a increases towards a, there are convergence problems associated with the transfinite sequences of guesses of priorities.

189 §4.

When a" < a

351

§4.

When a* < a

The following theorem is now proved under the assumption a* < a. Theorem 4.1. There exist a-recursively enumerable sets A and B such that neither is weakly a-recursive in the other. Let Cc a and define {e}£ for each e < a* as follows. Fix t : a - a* so that t is one-one and a-recursive. If r ' e is undefined via computa­ tions whose Godel numbers are less than a, then so is {e)ca . If r le is so defined, then {e)c0 is [r" l e]£ (as defined in section 1). Define {e}c similarly. a-recursive functions A", Ba, f(o, e) andg(o, e) (o < a, e < a*) are defined by recursion on a < a. {A° I a < a} and {B° I a < a) will be nondecreasing sequences of a-finite sets. A will be U (A0 I a < a} and 5 will be U {B° I a < a}. For each e < a*;f(a, e) and g(a, e) will be nondecreasing functions of a. It will turn out that/e = lim /(a, e) and ge = lim g(o, e) exist and a a are less than a, and that:

fe£A~(Eo)({e}*

(/e)=l);

g e e 5 - ( E o ) ( { 6 ^ (ge)= 1). 0 [

« I

mlr «f f ft l\ I „> WI ^

^ S 1

ne witness iunctions/^a, must tne following > i every member o t y ( T , e ) | Te) and e) gl< a, e)satisiy similarly. proviso (Wrr) (e)(e')(a)(7)[e # e' -* /(a, e) * / ( r , e')]

and similarly for*. So let {Zf I e < a*} be an a-recursive sequence of disjoint, unbounded a-recursive subsets of a, and insist /(a, e), g(o, e ) e Z ( .

190 352

G.E. Sacks and S. G. Simpson, The a-finite injury method

CaseO. a = (a*'n)+ 2e for some /i < a a n d e < a * . Set 5° = B<0, and/(a, e') = / « a, e') for all e' < a*. If/(a, e) £ ,4 < 0 and

{e}f (/(a, e)) = 1 , then set 4"

=/4
u { / ( a , e)}

g(a, e')=g(< o,e')

for

e' < e ,

and chooseg(o, e') (e < e' < a*) so thatf(a, e') > a + g « a, e') and the witness function proviso (WFP) forg is met. Computations are preserved by increasing the witness function. Thus if e < e' and T>O, then g(r, e') > a and the addition of g(r, e') to B cannot injure the computa­ tion of {e}f (/(a. e)) = 1, since that computation depends only on B° n o. Otherwise set A° = A
(g(ose))=

1 ,

then set B° = B
for

e < e ,

and choose/(o, e') (e < e' < a*) so that/(a, e') > a + / ( < a, e') and the witness function proviso for/is met. Otherwise set B° = B<0 and /(a, e') = / ( < a, e') for all e' < a*. That concludes stage a. Define / 2f = {olo = (a*-/j)+ 2e&yl° * A
19) §4. When a*
353

The next lemma corresponds to Friedberg's result that card /„ < 2" when a = co. Lemma 4.2. lfv<$and?> is an infinite regular a-cardinal. then Iu is afinite and its a-cardinality is less than j3. Proof. By induction on v. Fix v and assume for each p < v that /p is afinite and its a-cardinality is less than p\ By 2.3, U {/p I p < v) is a-finite and its a-cardinality is less than 0. Consequently its ordertype is less than p. The construction of f(o, e) and g(o, e) guarantees that any two distinct members of /p are separated by some member of U {/„I v < p] ; e.g. if p = 2e', r < a and T, a e /p , then / ( r , e') < f(a, e') and there is an e < e' and a 7 such that r < 7 < a, /(r,e'K/(7.e')
a, e) = /(a, e) whenever a $ U {/„ I v< 2e). So by 4.3

/e = l i m / ( a , e ) < a . The same holds forge = lim g(o, e). o

Lemma 4.4. For each e < a*: fee A —(Ea)({e}f (fe) = 1); geeA

— (Ea)({e}Aa

(ge)=1).

192 354

C.E. Sacks and S. C. Simpson. The vfinite injury method

Proof. Suppose fe eA. Let a be the first stage such that fe e A°. Then a e I2( and fe = / ( T , e) for all T > a. Also {e}f (/e) = 1 . Suppose for a reducio ad absurdum it is not the case that {e}f (fe) = 1. Then for some T > o, it is not the case that {e}f T (/e) = 1; let r' be the least such r. Then r' e / 2 f . + 1 f°r some e' > e, since/(a, e) = /e. At stage T', g(r', e') is added to B a. But g(r', e') > a by definition of g, since e < e' and r' > o. Now suppose (Eo)({e}f (/e) = 1). Choose T large enough so that {e}f (/e) = l and /(a, c) = fe for all O>T. Then for some a>r, case 0 applies a n d / e e / 4 ° .

either fe e A<0 or

Theorem 4.1 is an immediate consequence of Lemma 4.4.

193 §5. When a = a-

355

§5. When a = a* The proof of Theorem 4.1 is now completed under the assumption o* = a. First suppose there is no largest a-cardinal. Then a is the limit of re­ gular a-cardinals. We now proceed as we did in Section 4 when a* was the limit of regular a-cardinals. Let t : a ■* a be an a-recursive function that enumerates every ordinal less than a unboundedly often. Stage a. If to = 2e (respectively 2e + 1), proceed as in Section 4 when a = a* • u + 2e (respectively a* ■ •x + 2e + 1). The proofs of 4.2, 4.3 and 4.4 remain valid: 4.2 and 4.4 needed no assumption on a*; 4.3 did not require that a* be less than a, only that a* be a regular a-cardinal or the limit of regular a-cardinals. Now suppose for the remainder of this section that there is a largest a-cardinal; call it S. Let S = {/33K
for all £ < X and 7 < N. Thus A :i s aunction ffom N • X onto a. Note that k cannot be a-recursive if X < a. k wiil be the source of a strange wellordering of priorities of length N • X, ,a predicced in Section 33 .I will be seen that the set of stages at which requirement e (e < N • • £nd £ < X) is injured is bounded by 0 { . From now on let e be a variable that ranges over the ordinals less than « • X. It will be shown in a moment that k is a S 2 function, i.e. its graph has a Z 2 definition over (with parameters in) L a . The strength of k is its tameness (as the Good Book predicts), a term suggested by M. Lerman [ 15]. Not every Z 2 function is tame. Let/ be any function from N • X onto o. It is not hard dt see that / /' SS 2f and dnly if there is an a-recursive/(o,e) such that

194 G.E. Sacks and S. G. Simpson, The a-finite injury method

356

( e ) ( E T ) ( o ) 0 > f [/(o, e ) = / e ]

/ is said to be tame 2 2 if there is an a-recursive /(a, e) such that for each p < K • X, (ET)(e) e

r [ / ( o , e ) = / e ] . Lemma 5.L k is tame I2. Proof. Let 5° be {0I K < 0 < a & 0 is a-stable}. Arrange 5° as 0g < ft < ... < ft < ... (f < X°), where X° is the ordertype of S°. Let ft = o when X° < £ < X. Let Af" be the a-finite function from N onto ft of least possible canonical index. Define k(o,K-$ + y) = K°(y) for all £< X a n d 7 < K. Assume a > ft to see that ft = ft for all 5 < £. Clearly 06 is a-stable, since j36 < o < a and j36 is a-stable. It follows that ft < ft . But then ft is a-stable, because ft is ft -stable (or is ft ) and j36 is a-stable. If a > ft, then K.% = Jf. since ft = ft and « < ft. Note that the astability of p\ +1 puts the canonical index of Kt below ft+1. (This last point is a consequence of the fact that the a-recursive assignment of canonical indices is defined without any infinite parameters.) Thus k(a, N • £ + 7) = k(K • £ + 7) for all a > ft. Now a-recursive functions A", Ba,f(a. e) andg(o, e) are defined as in Section 4. Let t: a -* H • A be an a-recursive function that enumerates every e < « • X unboundedly often. Case 0. to = 2e. Let e* be of the form H ■ £* + 7* (£* < X and 7* < K) Hft. = 0<,° =sup{ft.\T< a},set/(CT,e*)=/«a,c*).If/J°. * Pf", choose/(o, e*) > 0 + /"(< 0, e*) and in accord with the witness func­ tion proviso (WFP) of Section 4. Set B° = B<0. If/(a, e)tA
(/(o.6))=l,

195 §S.Whena

= a*

357

then set A°=A<°

u{f(o,e)}

g(o,e')=g(
for e' < e ,

and choose g(o, e') (e < e' < K • X) so that g(a, e')> o+ g« a, e') and the witness function proviso (WFP) for g is met. Case 1. to = 2e + 1. Let e* be of the form K • £* + 7* (£* < X and 7 * < N). If/J£ =/3<»°,setg(o. e*) = s ( < o , e*). If 0f, * j3f.°, choose g(a, e*) > a + f(< a, e*) and in accord with the witness function pro­ viso. Set A" =A<0. If g(o,e)tB
(g(o,e))=

1 ,

then set B°= fl<0 u{f(a,

e)}

/ ( a , e ' ) = / « a , e') for e' < e , and choose / ( o , e') (e < e < K ■ X) so that / ( a , e') > a and the witness function proviso f o r / i s met. Now define the injury sets: I2e = {o\to = 2e&A°

*A<°}

7 2e+1 = {ol/a= 2e+ 1 & 5° * 5<°} Lemma 5.2. / / £ < X ffcen U{/K.^lT
.

Proof. By induction on £. Fix £ and assume U {/„l i>< N • $} C 0 r Clearly s is the largest 0 t + 1 -cardinal, since/L +] isa-stable. Fix 7 < «

196 358

G.E. Sacks and S.G. Simpson, The a-finile injury method

and assume for each 5 < 7, / M . e+s \/3j is a 0 | + 1 -finite set whose jS(+1cardinality is less than K. As in the proof of 4.2, any two consecutive members of /« t ^ \ ( 8 t + 1) are separated by s o m e a 6 U { / I K H-f + 7). By the induction hypothesis on *, a £ U {/„ I „ < K • *}. So a e U {/M.e+8\|5t I 5 < 7}. The existence of a is termed an interlacement. By induction on7,{/ K . t + s \(S t I 5 < 7} is a sequence of simultaneously |3£+1-recursively enumerable sets. Assume « is regular. Then 2.3 implies u ^K.f+«^t I S < 7} is 0 t + 1 -finite and its 0 t + 1 -cardinality is less than K. It follows from the interlacement described above that ^ . ^ ^ is a |3 {+1 -finite set whose 0 t + , -cardinality is less than K. If H is not regular then it is the limit of regular a-cardinals. Let K be a regular a-cardina) below K. Fix 7 < K and assume for each 5 < 7, 4 . t + ^ & is a P*+\ "finite set whose & . . -cardinality is less than K. Then apply 2.3 as above. It follows from 5.1, 5.2 and the definition of/(a, e) that f(a, e) = /"(< a, e) whenever a > j3 {+1 , where e = K • £ + 7. Define /e = lim/(a, e)< a a

Define ge similarly. Lemma 5.3. For each e < a*: / e e / l ~ ( E o ) ( [ * e l £ ( / e ) = 1);

geeB^(Eo)([ke]A0

(fe) = 1).

Proof. Suppose fee A. Let a be the first stage such that / e 6 / l 0 The witness function proviso (WFP) implies \k(e,a))f

(fe)=\

Since f(r, e) = fe for all r > er, it follows that 0° = (3. and /c(e, o) = fee. Suppose for a reductio ad absurdum it is not the case that [ke] f (fe) = 1. Then, as in 4.4, there is a r' > a such thatf(r\ e') is added to B<7' and

197 §5. Whena = a'

359

g(r', e') < a. Now e' > e since / ( a , e)=fe. But then g{f, e') > a by definition off, since e' > e. Now suppose (Eo)([*e]£ (/e) = 1). Choose T large enough so that [*(0,e)]f(/(a,e))=l, /(a, e) =/e, and *(o, e) = A:e for all O>T. Then for some a >T, either / e £ / l < 0 or case 0 applies and fe e A °. Theorem 4.1 is an immediate consequence of Lemma 5.3.

198 360

G.E. Sacks and S. G. Simpson, The ctfmite injury method

§6. Hyperregularity

G. Kreisel, a hard man to satisfy, pointed out that Theorem 4.1 is not a totally satisfying generalization of the Friedberg-Muchnik solu­ tion of Post's problem because <wa (weakly a-recursive in) is only one of several possible generalizations of Turing reducibility. So in this sec­ tion we develop what appears to be the strongest possible incomparability result (Theorem 6.2) for a-recursively enumerable sets. If Bc a, then the diagram of 5 , denoted b y A g , i s {g_(7) = 01 7 6 B) U {£(7) = 1) 7 $ B) . If £ is a finite set of equations whose parameters are ordinals less than a (i.e. EC Ca), then SE'B is the set of all equations in £a deducible from E u Ag in any number of steps. A partial function fc a X a is a-calculable (
/(7) = ^^f())

= ^eS£'u

for all 7, 5 < a. A
B if and only iff is a-finitely calculable from B.

Proof. Same as that of [12]] Kreisel has proposed a model theoretic notion of reducibility based on that of invariant implicit definability (cf. Kunen [16] for the defini­ tion); by Barwise completeness the notion coincides with
199 §6.Hyperregularity

361

Theorem 6.2. Suppose a is S j admissible. Then there exist a-recursively enumerable sets A and B such that A %ca B and B %ca A. A relatively simple case of 6.2, a = cof*, was proved in [ 17]. There as here the key notion is hyperregularity. B C a i s hyperregular iff[7] is bounded whenever 7 c domain ff<wa B and 7 < a. B is regular if B n 7 is a-finite for every 7 < a. Lemma 6.3 ([12], [17]). B is regular and hyperregular if and only if all computations from B are a-finite (i.e. SE'B = REB for all finite EC £ J. Lemma 6.4 ([12], [17]). If B is a-recursively enumerable and hyper­ regular, then B is regular. It follows from 6.1, 6.3 and 6.4 that one way of proving 6.2 is to find hyperregular, a-recursively enumerable A and B such that A £Wa B and B £wa A. The next lemma implies there is no other way. Lemma 6.5. If B is a-recursively enumerable but not hyperregular, then every a-recursively enumerable set is a-calculable from B. Proof. Similar to that of Spector's classical result that every n} subset of CJ is hyperarithmetic in every nonhyperarithmetic n{ subset of w. Since B is not hyperregular, there is mf
—*(Ep)p <7 pE/3) p

The last formula implies each membership fact about A can be calculated from /, hence B, by a computation of height at most a + 1. In order to prove 6.2, something a shade stronger will be proved.

200 362

G.E. Sacks and S. G. Simpson, The ^finite injury method

Theorem 6.6. If a is 2j admissible, then there exist a-recursively enu­ merable sets A and B such that A£wa B, B %wa and C={2v\uGA)u

{Iv+UveB}

is hyperregular. The proof of 6.6 resembles that of 4.1 save for some additional pre­ servation requirements designed to make C hyperregular. Since the pre­ servation requiremants of 4.1 were met by increasing the values of the witness functions/(a, e) and g(o, e) at appropriate o\ the additional requirements will be met by increasing them still further. An auxiliary function p(a, e) will embody the additional requirements. / ( < a, e) and g(< a, e) must now satisfy: /(p(o,e) + lim / ( T , e)iflim /(r, e) $ Ap(a, e) + lim g(r, e) if lim g(r, e) $. B<0 . r
T
Essentially p(a, e) is the supremum of all ordinals mentioned in certain computations worthy of preservation at stage a. The definition of p(o, e) splits into five cases, the first two being the most instructive. Case 1. a* < a and a* is a regular cardinal, e ranges over the ordinals less than o. Define C<0 = {2v\veA<°)

u {2v+

y(a, 7, e) = HTT<0 ({e}^"

\\veB<0)

(7) is defined)

m(a, e) = M77
201 §6.Hyperregularity

363

D e f i n e / « o, e) andg(< a, e) in accord with the inequalities preceding case 1 if they apply, otherwise as in Section 4. Then define A", B°, /(a, e) and g(a, e) as in Section 4. Do not forget to honor the witness function proviso (WFP). I2e and / 2 e + 1 have the same meaning they did in Section 4. So the proofs of 4.2 and 4.3 remain valid. Thus U {/p I p < v) is a-finite for each v < a*. But now a further argument is needed to show lim /(a, e) a

exists cind is less thnn ct. Lemma 6.7. Suppose U {/„ I v < 2e} C a and y < m(a, e). Then for all r>a: y(a, j , e) = y(r, j , e) and y < m(r, e). Proof. Fix r>O. The only way y(r, y, e) can fail to be y(o, y, e) is if some ordinal less than v(a, 7, e) is added to C at stage r. But any such ordinal is of the form 2/(T, e*) or 2g(r, e*) + 1 for some e* > e, hence >p(o,e*)^y(a,y,e). Lemma 6.8. For each

e
{y(a,y,e)\o
& e' < e &

y<m(a,e')}

is a-finite. Proof. Fix e < e*. By 6.7 there is a a 0 such that m(a, e') is (for each e' < e) a nondecreasing function of a for all a > o 0 . Let me' = lim m(o, e'). Clearly me' < a*. By 2.1 {e' < el me' = a*} ii sa finite, and so {e' < el me' < a*} is a-finite. Now apply 2.3 and remem­ ber that a* is (by assumption) a regular a-cardinal to see that the partial function {(me',e')le'<e) is a-finite. Then 6.8 follows from 6.7. The existence of lim/(o, e) (=/e < a) is a consequence of 6.8 and the a-finiteness of U {/ \°v < 2e). Simiiarly lim g(o, e) (= £e < a) exists. o

Then as in 4.4, A and B are a-recursively enumerable sets such that

202 364

G.E. Sacks and S.G. Simpson, The a-finite injury method

neither is weakly a-recursive in the other. It remains only to see that C is hyperregular. Lemma 6.9. C is hyperregular Proof. Suppose {e} c (7) is defined for all 7 < a*. By 6.7 and 6.8 {e}o°(y) = {e} c (7) for all 7 < a* and all sufficiently large a. Consequen­ tly {{e}c(7)l 7 < «*} is bounded. Case 2. a* < a and a* is not a regular a-cardinal. Same as case 1 save that the definition of m(o, e) now reads: m(ff,f) = ( c r ) T < ( W a , r , e ) = o ) . Thus the value of {e} c (7) is preserved for 7 < e instead of 7 < Q *. The proof of 6.7 remains valid. The proof of 6.8 can no longer apply 2.3, but all is well by 2.1 since me' < e'. The proof of hyperregularity of C undergoes a slight change. Suppose {e}c(7) is defined for all 7 < a* and {{e}c(7)l 7 < a*} is unbounded. Let p be a regular a-cardinal less than a*. There is an e p > p such that (e}C(T) = {ep} C (7) for all 7 < p. The construction of case 2 tends to preserve {e } c (7) for all 7 < e , hence the argument of 6.9 shows {<7, {e}C(7P)>l 7 < p} is an a-finite partial function equal to «7,{e}f(7)>l7O,

{e)C7\l)-{e)C(l) for all 7 < p6. Clearly /; <wa C. The argument of 6.9 shows the range of h restricted to 5 0 is bounded. It follows that the range of {e}c restricted to a* is bounded. Case 3. a* = a, there is a largest a-cardinal K, and K is regular, e ran­ ges over the ordinals less than K • X, where X is the ordertype of the set of a-stable ordinals beyond K. Define k(o, e) as in Section 5. Let

203 §6.Hyperregularity

365

y(o, 7, e) = H7r
= o).

The proof of 5.2 remains valid. The statement of 6.7 is altered some­ what since y(a, 7, e) may change when 0 increases because of a change in k(a, e). In the statement of 6.8, "a-finite" is replaced by "j3 £+] -finite" where e = K • $ + 5 for some 8 < K. In short the argument of case 3 pa­ rallels that of case 1 with 0 t + 1 corresponding to a, and H to a*, the pa­ rallel being sound by virtue of the a-stability of j3 f+1 . Case 4. a* = a, there is a largest a-cardinal N, and N is not regular. Define .1(0, 7, e) as in case 3, but set m(a, e) = ixyy<6 (y(a, 7, e) = a) , where e = N-£ + 6, 5 < K , £ < A a n d X i s defined as in case 3. Proceed as in Section 5 and in case 2 to show all the functions of interest associated with e converge below j3 £+1 . Then proceed as in case 2 to show C is hyperregular. Case 5. There is no largest a-cardinal. Define y(a, 7, e) and m(a, e) as in case 2. (e now ranges over all the ordinals less than a.) Proceed as in Section 5 when there was no largest a-cardinal. The arguments of case 2 still work since a is the limit of regular a-cardinals.

204 366

C.E. Sacks and S. S. Simpson, The a-fmite injury method

§7. Open questions

1. a denotes a £ j -admissible ordinal. Do there exist (asks Kreisel) semi-invariantly implicitly definable subsets of a such that neither is invariantls implicitly definable from the other? (See [16] for the necessary definitions.) The answer is yes when a is countable, thanks to Theorem 6.2. 2. Suppose A is a non-a-recursive, regular, a-recursively enumerable set. Do there exist disjoint a-recursively enumerable sets B and C such that 4 =Bv C,Btwa C and C^fl? The answer is yes when a* = OJ. 3. Do there exist finite Godel numbers m and n such that for all a, Rm and Rn are a-recursively enumerable sets whose a-degrees are incomparable? The arguments of Sections 4 and 5 fail to produce such an m and «, because of the nonconstructive split into cases therein. An affirmative answer is likely. 4. Does there exist a generalization of the infinite injury method ([ 10, p. 86]) to every 2 j admissible a powerful enough to show the a-recursively enumerable degrees are dense? It is likely that a powerful generalization exists for every Z 2 admissible a. 5. It is obvious that Sections 4 and 5 can be adapted to show: for each n and each 2„ admissible a, there exist two Z„ subsets of a such that neither is \ in the other (over L J ?

205 References

367

References (1) G. Takeuti, On the recursive functions of ordinals, J. Math. Soc. Japan 12 (1960) 119-128. [21 S. Kripke, Transfinite recursions on admissible ordinals, J. Symbolic Logic (Abstracts) 29 (1964) 161-162. [ 3 | R. Platek, Foundations of recursion theory, Ph. D. Thesis, Stanford (1966). [4| G. Kreisel and G.E. Sacks, Metarecursive sets, J. Symbolic Logic 30 (1965) 3 1 8 - 3 3 8 . [5] K. Godel, The consistency of the continuum hypothesis, Ann. Math. Study 3 (Princeton, 1940). [6 ] R. Friedberg, Two recursively enumerable sets of incomparable degrees of unsolvability, Proc. Natl. Acad. Sci. U.S.A. 43 (1957) 2 3 6 - 2 3 8 . [71 A.A. Muchnik, Negative answer to the problem of reducibility of the theory of algorithms, Dokl. Akad. Nauk SSSR 108 (1956) 194-197. [8] G. Kreisel, Some reasons for generalizing recursion theory, in: Logic Colloq. '69, eds. R. Gaudy and C.E.M. Yates (North-Holland, Amsterdam, 1971) 139-198. [9] A.H. Lachlan, The priority method I, Z. Math. Logic 13 (1967) 1-10. [10] G.E. Sacks, Degrees of unsolvability, Ann. Math. Study 55 (Princeton, 1966). [11] J. Stillwell, Ph. D. Thesis, M.l.T. (1970). [12J G.E. Sacks, Metarecursion theory, in: Sets, models and recursion theory, ed. J.N. Crossley, (North-HoUand, Amsterdam, 1967). [13J G.E. Sacks, Higher recursion theory (Springer, Berlin, to appear). [141 G. Driscoll, Metarecursively enumerable sets and their metadegrees, J. Symbolic Logic 33 (1968)389-411. 115] M. Lerman and G.E. Sacks, Some minimal pairs of o-recursively enumerable degrees, Ann. Math. Logic 4 (1972) 4 1 5 - 4 4 2 (this issue). 116] K. Kunen, Implicit definability and infinitary languages, J. Symbolic Logic 33 (1968) 446-451. [17J G.E. Sacks, Post's problem, admissible ordinals and regularity, Trans. Amer. Math. Soc. 124(1966) 1-24.

206 JE.Fenstad. P.G.Hinman feds.), Generalized Recursion © North-Holland Publ. Comp., 1974

Theory

THE 1-SECTION OF A TYPE n OBJECT Gerald E. SACKS ' Harvard University. Massachusetts Institute of Technology

1. Introduction This paper is a puffing up of the proof of the plus-one theorem for the case k - 1. Let "E be the representing function of the equality predicate for all objects X, Y of type less than n: nE{X, Y) = 0 if X= Y, and = lifX^Y. Plus-One Theorem. Let U be of type n. Suppose "E is recursive in Uandk< n. Then there exists a V of type fc+1 such that the objects of type k recursive in Vare the same as those recursive in U. Furthermore k+lE is recursive in V. Recursion in objects of finite type was discovered by Kleene [2]. An equivalent formulation, needed for the forcing argument of section 4, is given in section 2. The proof of the plus-one theorem for the case k > 1 will be given in [3]. It is largely a consequence of a stability lemma described in section 5. kxU is the set of all objects of type k recursive in U, and is called the ksection oft/. The plus-one theorem states that all A:-sections generated by finite type objects (in which the appropriate equality predicates are recursive) are generated by type k + \ objects. The first result on A:-sections was Kleene's [2]: j sc 2 £ "s the set of hyperarithmetic reals. He asked if the c\l2 rea's consti­ tuted the 1-section of some type 2 object. They do by virtue of the complete characterization of 1-sections of type 2 objects (in which 2E is recursive) developed in section 4. 1

The preparation of this paper was partially supported by NSF contract CP-29079. Its principal results were announced in 11 J. The author is grateful to T. Grilliot for steer­ ing him towards i-section problems. 81

207 82

G.E. SACKS

Section 2 redoes some of the elements of recursion in objects of finite type. Section 3 introduces the notion of abstract 1-section and shows that many familiar collections of reals, among then the 1-section of nE(n>2) and the set of lightface AJ reals (min (/,/) > 1), are countable abstract 1 -seetions. Section 4 proves every countable abstract 1-section is the 1-section of some type 2 object in which 2£ is recursive by means of a forcing argument of the sort associated with generic classes rather than sets. Section 5 describes some further results based on the technique of section 4, and speculates on the nature of abstractfc-sectionswhen k > 1. 2. Recursion in higher types The objects of type 0 are the nonnegative integers. An object of type n > 0 is a total function whose arguments range over all objects of type < n and whose values are objects of type < n. Any object of type i can be inflated to an equivalent object of type/ > i by adding dummy arguments. Any object of type n > 0 is equivalent to one of type n whose values are either 0 or 1; e.g. the function F(x) is equivalent to the representing function of the predicate F(x) = y. Any finite sequence of objects is equivalent to a single object; e.g. the pairFoOc),^*) is equivalent to H(n,x), whereH(n,x) = Fn(x) for n<2, and = 0 otherwise. The previous three sentences should make the ambiguities in what follows tolerable. Fix n > 0, and let F, G,... be objects of type n+2 called functional*. The defimtion of G 0 and are nonnegative integers when n = 0. Fix Fand f. The hierarchy {S0} of sets, defined by induction on o, is designed to define "G is recursive in F, n+ 2E, g via e" by induction on n. Stage 0. SQ contains < l,a> for every a, and < (,a,m) whenever ga = m.(\,a) is an index for S() for every a.

208 THE 1-SECT10N OF A TYPE n OBJECT

83

Stageo+l.<2e,a)isanindexfor S0+l if(i) (2e,a) <$S0, (ii) (m,a)isan index for Sn for some integer m, and (iii) there is a function/recursive in S„, "+'£,™. Clause (iii) makes sense by induction on n. If n = 0 the clause states/is Turing reducible to 5 via e. The/of clause (iii) is denoted by Xb{e}s°"*lE-a(b). S0+1 isS 0 augmented by: (1) all indices for Sa+l ;(2) all quadruples <3 e ,<2,6,w)if(2 e ,fl>isanindexfor5 0+] and

T

= m;

and (3) all triples (5e,a,m) i((2e,a) is an index for S0 and F(Xb{e}S°^°(b)) \<e,a)\ = a means (e,a) is an index foiSa. Stage X, where X is a limit. (le a) is an index for 5, if (2e.a) is an index for some 56+1 < X and Xb {ef° '""^(h) is the characteristic function of a set T of indices such that X= sup

{\b\\beT}.

Sx is U{5'6|5<X} augmented by all indices for5 x . / i s said to be recursive in F. n+2E,g,avia e, written / = {e}F»+iE,g,a if (2e,a) is an index for some V i and/isXb {e}S""+lE'\b). recursive in F, n+2E, g via e, written

G is said to be

G= {e} F "+2£'.*

if G(h) is {ey^.s>H0) n+2

for Mh.

R is recursive in F, E if its characteristic function is. *sc(F, n+2E) is the set of all objects of type k recursive in F,'1+2E. If n+2E is recursive in Fin the sense of Kleene [2] then ksc(F,"+2E) = kscF. R is recursively enumerable in F, "+2Evia e if/? = {a\(2c,a) is an index}. This notion of recursive enumerability coincides with Kleene's [2]. Thus/? is

209 84

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recursive in F, n+2E if and only if both R and its complement are recursively enumerable in F,"*2E. The principal fact needed in the next section is the existence of a selection operator discovered by R. Gandy [7], who had to wrestle with Kleene's defi­ nition. His result follows more readily from the hierarchial definition of this section.2 Gandy's Selection Operator. There exists a recursive function \e\e* such that if T is a nonempty subset of co recursively enumerable in F, "+2E via e, then {e*}F"*2E(0) is defined and belongs to T. 3. Abstract 1-sections Let A be a nonempty transitive set. A is said to be an abstract 1-section if it is closed under the operations of pairing and union, and satisfies axiom (1) and schemas (2)-(3), where a G A and g(y) and 9(x,y) are A0 formulas of ZF (i.e. formulas whose quantifiers are restricted, cf. Levy [10]) with parameters in A. (1) Local countability: (x) (x is countable). (2) A0 separation:

(Ex)(y)[yeX^yea

& g(y)).

(3) AQ dependent choice: (x)(Ey)g(x,y)->(EhKn)g{hnMn + D), where/?: w - , 4 . If A is an abstract 1-section, then A is an admissible set as defined by Platek [8], and every member of A is hereditarily countable. Each hereditarily countable set x can be encoded by a real number Y. Let m be the function that takes a code Y to the set mY encoded by Y. The rela­ tion Y is a code & mY = x

2 Not surprising, since his argument is based on a shadowy hierarchy of computations arising from the Kleene schemas of recursion. The existence of Gandy s selection operator was proved by him when n = 0, by Moschovakis 116) when n = 1, and by Platek [ 8 ] when n >).

210 THE 1-SECTION OF A TYPE n OBJECT

85

is defined by induction on the rank of x: Yis a code for {m(Yn)\n 1 and U is a type n object in which "E is recursive. Then the \-section of U is a countable abstract l-section. Proof. It is immediate that 9/f(j set/) is closed under the operations of pairing and union. To check A0 separation let 9(>") be a AQ formula and d S jsct/ be a code. It suffices to find a code c G j scU such that (y)[y £ mc+-+y e md & $(y)] holds in 9ff(j self). The predicate P(e), defined by (En)[e = dn) & 9(me), is recursive in U by Proposition 3.1. The desired c is such that {cn\n
211 86

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holds in9ft(jsc£/). Suppose {p}u is a code. Let Q be the set of all n such that {n}u

is a code & 9 (m{p}u,

m{n}u) .

Qp is recursively enumerable in {/(uniformly inp). Candy's selection operator yields a partial function t recursive in 3 U such that tp S Qp whenever {p}u is a code. Define g recursively in U by: gO = eQ ({eQ}U is a code for 0), g(n + l)=t(gn) g(n+l) = t(gn) Then (n)S(m{gn}u,

m{g(n + l)}u).

For each ordinal a let Z,Q be the set of sets constructive in the sense of Godel [11] via ordinals less than a. a is said to be Ej admissible if La satisfies the Ej replacement axiom schema of ZF. Proposition 3.3. If a isaL^ admissible ordinal, then La Pi 2^ is an abstract 1-sect ion. Proof. Godel [11] showsL 0 2 " = ! ^ ( 1 2 u , where Wj is the least ordinal not countable in L. His argument restricted to La shows

where w? is the least ordinal not countable in Ln. It follows L i

a

local countability. Godel's wellordering of L restricted to L

a

a

satisfies

wa

is Ej overZ,

a.

Let A.- be the set of all lightface A ■ reals. Proposition 3.4. If mm (/,/) > 1, then A ■ ("s a countable abstract \-section. Proof. If / = / = 1, then the proposition follows from Spector's boundedness theorem [9] forE, subsets of Kleene's 0 and Kreisel's selection operator [12] for II] predicates of numbers. I.e. the graph of t is recursively enumerable in U.

212 THE 1-SECTION OF A TYPE n OBJECT

87

Assume max (»"/) > 1. Let HC be the set of hereditarily countable sets. The Kondo-Addison uniformization of n j predicates of reals implies HC satisfies A0 dependent choice. Consequently HC is an abstract 1-section. It suffices to show9ff(AJ) is a L^ substructure of HC. Let 9(x) be a AQ for­ mula with parameters in9tf (A') such that

HCh(Ex)2«. There exists an arithmetic predicate A(Y) such that [7 is a code & A(Y)] *-* HC

|=$(mY).

The set parameters occurnng in 9(x) correspond to \'f •odes occurrrng in A(Y). The Kondo-Addison uniformization supplies a code Z such that 2 satisfies A(Y) and is \\ in theA' codes occurring in A{Y). Since max(/,/)> 1, ZSAJ.Hence9K(AJ)MEx)9(x). 4. Generic type 2 objects Let K be a countable abstract 1-section. Suppose Fmaps to^ into co and is 0 off A". If Fis generic in the sense of the following forcing relation, then the 1-seetion of(F.2E) isK. Let p be a partial function from w w into w. p generates a hierarchy {ro} of reals as defined below. If p is total, then the 7 y S are equivalent to the 5 0 's of section 2 when n = 0 and F = p. If p is not total, then there may be a o such that r o + i has an index but is not total. Stage 0. T0 = {1}. 1 is an index for T0. T0 is totall Stage o+l. 2e is an index for T„+1 if T0 has an index, T0 is total, 2e $T„, r 7 0 is the unique partial function from h . and W ° ( m ) is defined for all m.({e} " *" co into co recursive in T0 via Godel numbers.) Ta+l is total if it has an index and p(Aw{e}r<>(m)) is defined whenever 2e is an index for Ta+X. T0+l isTn augmented by: all indices for Ta+l; all triples (3e,m,n) such that {e}To(m) = n and 2f is an index for T0+1; and all pairs <5e,n) such that p(Xm{e}T»(m)) = n and 2 f fs an index for Ta+X1 |wI= o means m is an index for Tn .

213 88

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Stage X (limiis). 7 e is an index for 7\ if 2e is an index for Tb++ for some 5 < X and \m{e}THm) is the characteristic function of a set R of indices such that X=suu{|w||mG/?}. 7\ is total if it has an index. Tx is U{Tb\6 < X} augmented by all indices for p is said to generate T0 if r„ has an index and is total. Fact // is easily proved by induction on a. FactH. If a u nWK; a and T0 for each ordinal a eWK; and 5 for each S e 2" n9KK. The vari­ ables of JC(A') are: x,y,... (numbers); fi, v,... (ordinals); and 7*M, Tv,... (sets). The atomic formulas of £(K) are of the form: |x| = JI, I A | = M, M < f, and S< ' M . The sentences of JC(A') are built up from the atomic formulas by sub­ stitution of appropriate individual constants for variables and by application of propositional connectives (& and ~) and existential number and ordinal quantifiers. 9 is a ranked sentence of rank o if 9 contains no ordinal quantifiers and a is the least ordinal greater than every ordinal occurring in 9 • Such an 9 >s true m{Tb\5
214 THE 1-SECTION OF A TYPE n OBJECT

89

m is an index for Ts, and S -* u belongs to K. Fix p; since F is generic it suffices to find a q C p such that

q\\-s
215 90

G.E. SACKS

p(\m{e}Ty(m)) is undefined. Let An \e„ be a recursive function such that

a for all X C w and n € w. Clearly {en}x is total if and only if {e}x is. It follows 2e" is an index for T^x because 2 e is. In addition p(\m{en)Ty{m)) is undefined, because the domain of p is an initial segment of Turing degrees. Choose q C p so that the domain of q consists of all functions Turing re­ ducible to Ty, and so that

a for all n € w. Thusq generates T^ but not T^2- since q(T^) And S is Turing reducible to Ty+l since

is undefined.

a

for all n and r.

Lemma 4.3. If F is generic, then j sc(F, 2£) C /T. Proof. Suppose S € j sc(F, 2f) - AT for the sake of a reductio ad absurdum. Then S is Turing reducible to some T0 generated by F but not inlKK, o £9ff A" since F is generic. Let a be the least ordinal not in "WAT. Then F generates some Ta with index le Thus 2e is an index for some Ts+l generated by FmWK, and {e}Ti> is the characteristic function of a set/? such that a=sup{|n||«etf}. Let felUK that:

enumerated. Since Fis generic there is a p satisfied by Fsuch

216 THE 1-SECTION OF A TYPE n OBJECT

91

(a)p||-(*)(E/0[IAI»rt; (b)p||-0i)(E*)(E^
p m | r - | / m | = ow

and(m)(E«)[5 < o m 2 and U is a type n object in which nE is recursive. Then there exists a type 2 object V such that

217 G.E. SACKS

92 1ScU=iScV

and 2E is recursive in V. Corollary 4.6. Suppose a is a countable E, admissible ordinal. Then there exists a type 2 object V such that Lan""

= lxV

and 2E is recursive in V. Corollary 4.7.7/min (/,/) > 1, then the set of all lightface A' reals iishe 1section of some type 2 object in which 2£ is recursive. 5. Further results The method of section 4 is applicable to the study of Gandy's superjump [7]. Theorems 5.1 and 5.2 are typical results of [14] and were inspired by some questions raised by P. Hinman at the 1969 Manchester Logic Colloquium. Let F and G be objects of type 2, G' the superjump of G, and £ , the superjump of 2E. j scG is said to be closed under hyperjump if tSc^t.JSQCiScG

for every

X&jscG.

Theorem 5.1. Suppose j scG is closed under hyperjump. Then there exists an F such that jScG=]sc(F').

Theorem 5.2. (Assume 2W = w,.) There exists an H such that {G)(EF)[H 1. A stability result is needed to overcome prob-

218 THE 1-SECTION OF A TYPE n OBJECT

93

lems caused by gaps in the hierarchy of section 2, gaps that fall between ob­ jects recursive inF," +2 E when n > 0. Call R subrecursive in F,"+2EifR is recursive in some S0 (as defined in section 2) with an index of the form <2V>, where r is a snbindividual. The stability result in question says: each nonempty recursively enumerable (in F,"+2E) collection of subrecursive (in F,"+2E) sets must have a recursive (in F,n+2E) set among its members. At this writing it is not known if there exists a decent notion of abstract fc-section when k > 1. Decency requires that Theorem 4.4 remain true when "1-section" is replaced by "it-section" and " 2 " by "k + V\ References (1 ] G.E. Sacks, Recursion in objects of finite type, Proceedings of the International Congress of Mathematicians 1 (1970)251-254. [2] S.C. Kleene, Recursive functionals and quantifiers of finite type, Trans. Amer. Math. Soc. 91 (1959) 1-52, 108 (1963) 106-142. [3] G.E. Sacks, The it-section of a type n object, to appear. [4] J. Shoenfield, A hierarchy based on a type 2 object, Trans. Amer. Math. Soc. 134 (1968)103-108. |5] T. Grilliot, Hierarchies based on objects of finite type, Jour. Symb. Log. 34 (1969) 177-182. [6] G.E. Sacks, Higher Recursion Theory, Springer Verlag, to appear. [7) R. Gandy, General recursive functionals of finite type and hierarchies of functions, University of Clermont-Ferrand (1962). [8] R. Platek, Foundations of Recursion Theory, Ph.D. Thesis, Stanford (1966). [9] C. Spector, Recursive well-orderings, Jour. Symb. Log. 20 (1955) 151 - 1 6 3 . [10J A. Levy, A hierarchy of formulas in set theory, Memoirs of the American Mathe­ matical Society, Number 57 (1965). [11] K. Godel, The Consistency of the Axiom of Choice and of the Generalized Con­ tinuum Hypothesis (Princeton University Press, Princeton, 1966). [12] G. Kreisel, Set theoretic problems suggested by the notion of potential totality, in: Infinitistic Methods (Pergamon Press, Oxford, and PWN, Warsaw, 1961) pp. 1 0 3 - ' 140. [13] J Shoenfield The problem of predicativitv Essays on the Foundations of Mathe­ matics '(Magnes Press, Jerusalem, 1961 and North-Holland, Amsterdam 1962) [14] G K Sacks Inverting the superjumt) to annear [15 1 S.Feferman, Some applications of the notion of forcing and generic sets Fund Math. 56 (1965)325-345. [16] Y. Moschovakis, Hyperanalytic predicates, Trans. Amer. Math. Soc. 129 (1967) 249-282.

219 HISTORIA

HATHEMATICA 2 (1975)

.

523-528

REMARKS AGAINST FOUNDATIONAL ACTIVITY BY GERALD SACKS, HARVARD UNIVERSITY & M.I.T. I want to discourage people -- mathematicians and others -from trying to build foundations for all of mathematics. There is no need for such an enterprise. Moreover, any such attempt is bound to assume a negative puritanical flavor: "don't do this"; "you can't do that"; etc. Puritanism is not fruitful for mathematics. However, foundational activity is of interest in some limited spheres. For example, when there were difficulties in set theory, a few small changes were needed to straighten things out. There were difficulties, but not paradoxes; there was no need to write Princiaia Mathematica to straighten them out. It was necessary to clarify ideas -- and that is when foundational activity is of interest. Another example is category theory in logic. (Reference is being made to the new theory of "topoi", developed by Lawvere and others. The role of category theory in logic is not to be confused with its role in mathematics, as described for example in S. MacLane, Categories for the Working Mathematician, Springer, 1971.) There are a number of paradoxical constructions which can probably be straightened out by a minimal effort. There is no need to write down axioms for category theory. Foundational activities should be very narrow, very restrained. They should be geared to solve the difficulties at hand. Since there have not yet been any substantial paradoxes, there has not yet been any need for a wider sphere of foundational activity. For example, there were confusions, not paradoxes, in late nineteenth and early twentieth century mathematics, despite all claims to the contrary. There are no substantial paradoxes in the history of mathematics. Sometimes results are called paradoxes, but not for long. For example, the Banach-Tarski paradox is actually just a theorem. [Or consider eighteenth century geometry and Kramer's paradox -interjected by DieudonnS.] When one called these paradoxes, one simply meant that the conclusion was startling, e.g. Peano's paradox of the space-filling curve. After a while such results become accepted, and don't seem nearly so strange. Most mathematical logicians today are not interested in foundations. They are more interested in seeing what ideas from mathematical logic can be applied to other areas of mathematics. These days mathematical logic is a peripheral field of mathematics, and its utility is in its applicability to the more central fields. However, that may change. A good example of this is the work of Abraham Robinson. He was aware of the foundational problems and discussed them with Reprinted with permission from Academic Press.

220 524

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various people. Yet if one peruses his publications, one will find few investigations of such problems. Instead one finds an interest in applications of logic to certain problems in algebra and analysis. And one of his final interests was applying his notion of nonstandard analysis to algebraic number theory. You can think of Robinson's work in nonstandard analysis as foundational if you like, namely, as an alternative foundation of analysis. However I am convinced that this was not his motivation at all. His interest in nonstandard analysis lay in its applications to classical analysis. His idea was to get new theorems for classical analysis by looking at classical problems in a slightly different way. His approach has been fairly successful For example for quite a few years the most useful way of considering the invariant subspace problem was through nonstandard analysis even though this approach has been recently superseded Also there are applications of nonstandard analysis to mathematical economics, unexpected as they may be. There are many other examples of the sorts of services that logic can provide to other fields of mathematics. One of the most paradigmatic is due to a result of GOdel. GOdel defined a certain class of statements which, if provable using the Axiom of Choice, would be provable without the Axiom of Choice. Logic is needed to make precise just what this class of statements is. One consequence is that if you could prove the Riemann hypothesis using the Axiom of Choice (there is no reason to believe that you can), you could prove it without the Axiom of Choice. This method due to GOdel was actually used in a paper by Ax and Kochen [a], although a direct method was eventually found to reprove their result without the Axiom of Choice and without GOdel's method. Also exemplary of the use of logic in other fields of mathematics is its use in making sense of Lefschetz's Principle and the idea of a universal domain. Essentially, Lefschetz's Principle [b] says that any algebraically closed field of characteristic zero and of infinite transcendence rank is as good as any other as far as algebraic geometry is concerned, i.e. if you can prove a statement over one of these fields, it is true over all others. Attempts have been made by Feferman and, more recently, by Barwise to make sense of this notion, by defining a broad class of statements (including all those relevant to algebraic geometry) which are based on Lefschetz's Princinle. Although there haven't been any substantial paradoxes'in mathematics, there have been errors. It is important that histories of mathematics be written which discuss real, substantial errors made by mathematicians! A classic example is related to a mistake made by Lebesgue when he tried to show that the projection of any Borel set in the plane on the line is Borel. The motivation behind this was to show that the projection of a Borel set is measurable -- which, of course, it is, but by a more intricate argument. The error was in print for 10 or 12 years before it

TTA HM 2

Remarks Against Foundational Activity

525

was discovered by Souslin [c]. This led to the definition of analytic set and to the development of descriptive set theory. Histories of mathematical logic should be wary of over-stressing foundational aspects, although the various attempts to erect foundational systems should be included. The axiomatic aspects of mathematical logic has encouraged clarity and precision to a dangerous degree. The attempt to provide maximum precision has a killing effect upon mathematics. Clarity should come at the end, not at the beginning of a mathematical endeavor. Consider Euler's discovery of all the integral solutions to the equation x3 = y 2 + 2. This was accomplished with an argument which must have been incomprehensible, if not meaningless, to most people of his time. As DieudonnS points out, Euler himself did not know what he was doing when he broke expressions into complex factors. Students today are discouraged from proceeding as Euler did. Too often, they are unwilling to move from one point to the next until they have painfully analyzed the minutiae of each argument. The effect is altogether too often, a long and uninspired thesis. NOTES

a. Annals of Math. 83 (1966), 437-56. b. As formulated in A. Weil, Foundatisns of Algebraic Geometr,, Am. Math. S o c , 1946. c. H. Lebesgue, M. Souslin, C.R. Paris 164 (1917), 88-; N. Lusin, Fund. Math. 10 (1927), 1-95. DISCUSSION

Dreben: Sacks is right in saying that mathematical logic today is no longer concerned with foundations in the traditional sense, and has not been since World War II. Sacks might have misled some people, however, into thinking that the reason Russell and Whitehead wrote Princiaia Mathematica was primarily because of the paradoxes. That is not true: The reason that Russell this type of enterprise was not in trying to do mathematics per and Whitehead, or Frege, or their immediate followers engaged in

a These are called paradoxes -- and not contradictions -because most people see what Sacks was saying, i.e. they are

222 526

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genuine mathematical results in set theory. What is somewhat misleading is the view -- a view currently held by many logicians that there never was any real problem in Cantorian set theory, because set theorists are talking about the iterative notion of a set (the notion that you start at the bottom and build up) first made explicit by Von Neumann. This iterative notion underlies the contemporary view of set theory. The truth is, as GBdel has pointed out, that there are really two basic notions of set. There is the logical notion of a set; not in the narrow sense of mathematics but in the broad sense that Frege and Russell intended it to be used. This is a continuation of the medieval idea of the extension of a predicate. Then there is the iterative notion of set, implicit in Cantor and used in mathematical work. This notion never had any troubles in it. GBdel thinks we still don't fully understand the logical notion of set. Moore: (to Sacks) You had said there have been no real, substantial paradoxes in the history of mathematics. What would constitute a real paradox? Dreben: That's a contradiction in terms. Sacks: I think it would have to have the nature of an experience that every mathematician has at least once a year, that of proving a theorem one day and refuting it the next day. Like this -- except that nobody can find any mistakes in either the proof or the refutation. It might mean we would have to retreat in our mathematics to a level where the paradox does not appear. Kline: I believe that historically you are incorrect. The word "paradox" does mean a seeming contradiction which has been resolved. But paradoxes like Russell's class of all classes or the Richard paradox were genuine contradictions as far as the mathematicians were then concerned. They used the "paradox" to ease their troubled souls because the word "paradox" implied that they were going to resolve the problems. Dreben: We should make this clear. The Russell paradox, the Richard paradox, the Burali-Forti paradox -- these are all logical contradictions. Up until the turn of the century, up until the statement of the paradoxes, people talked of wanting to have a general logic. Even mathematicians of the first rank like Zermelo, those building axiom systems, were under the illusion that their work was connected with general logical investigations. Zermelo and even Von Neumann in his 1925 paper say explicitly that the only justification for his setting of the axioms is that they make the paradoxes appear not to exist. Von Neumann, and probably Zermelo, knew that you could have a working set theory based on the iterative notion of set (cumulative type theory) in which there is no contradiction. This is what underlies Von Neumann's proof that the axiom of foundation is consistent with the other axioms of set theory. This is such a powerful notion that all

223 HM 2

Remarks Against Foundational Activity

527

young people in mathematical logic today take it for granted that this is what set theory is about. These two notions of set come together in working set theory although they should be kept separate. Yet, as Gtfdel pointed out in his 1947 paper [C10], there never have been any contradictions in the mathematical notion of set. Dieudonne: What caused people to think of the set of all sets? Dxeben: About 1899, Cantor probably had two notions of set theory: absolutist set theory, which is what we have in mind when we think of sets as extensions of predicates, and cumulative set theory which could be done just for mathematical purposes. It was the absolutist notion that would lead to the paradoxes. Von Neumann (1925) says that some omission is essential to avoid the paradoxes; what that omission is, is arbitrary. He was the first to formulate the cumulative notion of set explicitly [C31, vol. 1, 35-57; Cll, pp. 393-413]. ...Mr. Bishop is engaged in foundational work, a reconstruction of mathematics with some picture in mind. Proof theory is not this. Kline: What was the purpose of Hilbert's work if not to prove consistency? Reconstruction would have been useless to Hilbert if he had not had his Beweistheorie (proof theory). Dreben: The original motivation for proof theory was to prove the consistency of a formalization of mathematics, but after GBdel that kind of motivation was pre-empted. But this does not stop proof theory from being interesting. Kline: Thus, we must distinguish between proof theory before and after GOdel. Dreben: Mathematical logic grew out of philosophical purposes, but it is absolutely essential to realize that even though people use the same words, e.g. "foundations", they may mean entirely different things by them today and have entirely different purposes. Putnam: There are still conceptual problems with the mathematical, cumulative notion of a set. The problem of a working set theorist is this: he wants to have things both ways. At that point, you are right back in the Russell paradox. There was a good discussion of this at the May meeting of the American Philosophical Society, to be published in Nous. There is still a vestige of the paradox hanging over the mathematical notion of a set, although it does not threaten contradiction. Dieudonne: For example, when one is talking about an arbitrary group, he tends to think of the set of all groups. But it is just an illusion that you must talk of this "large" set (of all groups). Birkhoff: Weierstrass provides a good example of the good influence of precision coming at the end of a mathematical development. Here precision seemed to crystallize analysis. Also, constructive proof is better than non-constructive proof, other considerations aside. It would be of service, and not much effort, to indicate whether proofs are constructive or need be non-constructive. I still feel uneasy about the Banach-

224 528

G. Sacks

HM 2

Tarski decomposition; the "components" defined (non-constructively) just do not seem very meaningful. In response to Birkhoffs comments, Sacks said that in closely following GOdel he represents the majority vi^w. Birkhoff agreed. Dieudonne: Since Cohen's work, we have as many different mathematics as we want, for we have as many choices of alephs for the continuum as we want... [Ed.: His implication seems to be that this freedom of choice makes the resulting theories not very meaningful.] Kahane: Hausdorff mixed his set theory with measure theory and topology. For his predecessors, set theory was the study of linear sets. When did the change come so that the study of abstract sets became part of mathematics? Sacks: Cantor speaks of quite arbitrary sets (absolute, general sets). Dreben and Browder added that, in a general working sense, it was in Frechet's Thesis (1906) that abstract spaces were first studied, and that he had applications to functional analysis in mind. [See Part E of the Proceedings.] May.- To clarify the discussion, we should distinguish the question as to when certain individuals thought of sets and spaces abstractly from the question as to when the idea of considering abstract topological spaces and measure spaces became generally diffused. Widespread diffusion of the abstract approach may not have come until the 1920's. The discussion concluded with an exchange of views between Profs. May and Dieudonne concerning a point made by Sacks: that axiomatization often comes late in the historical development of a subject. The following is a statement of consensus written by the editor. Sacks' assertion was notably true of Euclidean geometry and group theory. Likewise, Boolean algebra was decades old before it became axiomatized, first incompletely by C.S. Peirce and then almost too completely by E. SchrBder. ( In a fruitful compromise, Dedkind used variants of the axioms of Boolean algebra to define lattices, modular lattices, and distributive lattices.) But by itself, axiomatization usually contributes more to the precise formulation of a theory than to its development. Carried to excess, it may even tend to make a subject seem moribund, though it certainly did not do this to the theory of finite groups'. What seems to be true is that, for an axiom system to have continuing interest, it must have many variants, each with important realizations. (Bourbaki sometimes refers to the mathematical structures which, dually, satisfy several axiom systems, as "mother structures".) Axiomatization also makes it possible to synthesize whole theories of mathematics from a few basic ingredients Once such a synthesis has taken place, however, its fundamental concepts tend to become static.

225

R e p r i n t e d from ADVANCES IN MATHEMATICS

Vol. 20, N o . 2, May 1976

Countable Admissible Ordinals and Hyperdegrees GERALD E. SACKS* Department of Mathematics, Harvadd University, and Massachusetts Institute of Technology, Cambridg,, Massachusetts 02139

1. INTRODUCTION

T h i s paper began sometime ago with the question: Does every countable set of hyperdegrees have a minimal upper b o u n d ? T h a t question is still open, however the following partial answer is provided by Section 3. COROLLARY 3.14. If A is a countable admissible set that satisfies Zx dependent choice, then the set of all hyperdegrees contained in A has a minimal upper bound. T h e search for minimal upper bounds uncovered the answer to a question raised by R. Platek: Is every countable admissible ordinal beyond w of the form « / for some X C w ? (tu/ is the least ordinal not recursive in X. Platek's question was prompted by his observation that wxx is admissible for every X)) Theorem 4.26 was proved by techniques originally developed to handle some special cases of Corol­ lary 3.14. THEOREM 4.26. If a is a countable admissible ordinal greater than w, then there exists an X C to such that (i)

w / = a, and

(ii)

w/ <<x for every Y oofower rhper degreeetan XX

Subsequently a model theoretic proof of 4.26(i) based on Barwise's compactness theorem was found by Friedman and Jensen [1]. T h e i r argument was simplified by Grilliot [2] and reduced to its essentials by Keisler [3]. T h e model theoretic approach, primarily that of omitting * The preparation of this paper was partially supported by NSF Grant P 29079.

213 Copyright 6 1976 by Academic Press, Inc,

226 214

GERALD E. SACKS

a type, does not yield the minimality property expressed by 4.26(H) 1 . That sort of minimality is a natural side effect of forcing with pointed perfect conditions, the principal technique of this paper. Let P be a perfect set of reals. P is said to be hyperarithmetically pointed if the standard encoding of P by a real is hyperarithmetic in every member of P By 2.2 the hyperdegrees of the members of P are the same as those greater than or equal to that of P. Thus P can be thought of as a condition that forces the hyperdegree of some generic real to lie in the cone of hyperdegrees greater than or equal to that of P without ruling out any member of the cone The key technical problem associated with hyperarithmetically pointed P's arises in the proofs of splitting lemmas a«s indicated in 3 8 T h e prnhlem is rn find precautions

a to take in the course of defining a contracting seouence of hyperarith-

make th^intersection nerfert neve ca\

for much effort but to make

Recursively pointed P's were applied in [4, p. 353] to show certain sequences of Turing degrees have 2-least upper bounds. 2 They were also used with very little understanding of their nature in [5] to prove every countable set of Turing degrees has a minimal upper bound. Recently, Friedman [6] made use of A^ pointed P's to show that every countable set of A^ degrees has a minimal upper bound. One of the reasons that the minimal upper bound question has been answered completely for Turing degrees and AJ degrees but not for hyper­ degrees is that the problem of showing certain intersections of hyper­ arithmetically pointed P's are hyperarithmetically pointed has no counterpart in any of the existing applications of recursively pointed or A J pointed P's Sequences of hyperdegrees are surprisingly less tractable than sequences of Turing or J2> degrees. Section 2 is largely groundwork for Sections 3 and 4; it covers: perfect sets, a brief review of the definitions of admissible set and hyperdegree and of the expressibility of a 77/ relation on reals restricted to an admis­ sible set A as a 27, relation on A, the use of local Cohen forcing to con-

1 Simpson [22] needs 4.26(H) to show: If V is a generic extension of L, then the upper semilattice of hyperdegrees of V does not include a cone of minimal covers. 2 Deeper applications of recursive pointedness to show certain sequences of Turing degrees have definable upper bounds have been announced by Jockusch and Simpson [23].

227 ORDINALS AND HYPERDEGREES

215

struct uniformly hyperarithmetically pointed perfect sets, and an applica­ tion of unbounded Levy forcing. Section 3 is mainly concerned with the existence of minimal upper bounds via hyperarithmetically pointed forcing; it deals with: The forcing relation H-"4 for an admissible set A that satisfies El dependent choice but is otherwise arbitrary, the nature of WA when A is an L(<x), the forcing relation H+', and the location of least upper bounds. The principal technical lemma of Section 3 is 3.11, which establishes the equivalence of genericity in the sense of WA with that of Hf*. Section 4 is singlemindedly devoted to the construction of a minimal solution of a = *♦>,*; it defines: perfect conditions on K (where K is a certain type of collapsing map), JC-uniform pointedness, and forcing with complex conditions of the form (H, &), where H is a perfect condition on K and 9 is a term forced by H to denote a hyperarith­ metically pointed perfect set. The principal technical lemmas of Section 4 are 4.13 and 4.21, each of which surmounts some of the obstacles encountered in iterated forcing with pointed perfect sets. Section 5 hints at some further results and mentions some open questions; it touches on In admissibility, the use of constructibly pointed conditions to collapse uncountable regular cardinals to a>x , and the relation between forcing and compactness arguments.

2.

MACHINERY

2.1. Pointed Perfect Sets Let p, q, r,... be sequence numbers that encode finite initial segments of characteristic functions of subsets of w. p, q, r,... are called finite conditions. A set T satisfies p (in symbols T e p) if p encodes a finite initial segment of the characteristic function of T. p is extended by q (in symbols p D q) if every T that satisfies q also satisfies p. p and q are incompatible (in symbols p | q) if neither extends the other. Suppose Ay ; py is a function such that pf is a finite condition for all i < w and ;' < 2 \ Ay | pf encodes a perfect subset of 2" if pi? and p$ls are incompatible extensions of pf for all i < w and ; < 2'. The perfect set encoded by Xij | />/ is {T\{i)(Ej)(j

< 2'& 7e/>/)!.

Each perfect set P has a standard encoding readily defined by thinking

228 216

GERALD E. SACKS

of P as a tree with binary branching: The branches of the tree correspond to the members of P; the branch points correspond to the finite conditions in the range of the standard encoding of P. Note that the standard encoding of P is recursive (uniformly) in every encoding of P. Let P, Q, R,... ambiguously denote perfect sets and their standard encodings. Thus "TeP" means T belongs to the perfect set P, and "P < f t T" means the standard encoding of P is hyperarithmetic in T. =sCr is a reducibility relation on the reals if it is reflexive, transitive and satisfies (2.1.1) and (2.1.2): (2.1.1)

X recursive in F -* X < r F ,

(2.1.2) X ^rZ& Y
< r T];

i.e., the standard encoding of P is /--reducible to every member of P The notion of pointedness was inspired by the Kleene-Post construc­ tion [7] of a Turing degree above those of the arithmetic sets and below that of the truth set for arithmetic. Their construction is a forerunner of forcing with recursively pointed P's whose standard encodings are arithmetic. It will follow from Propositions 2.2 and 2.3 that forcing with r-pointed perfect conditions imposes no upper bound on the r-degrees of generic reals. PROPOSITION 2.2. IfP is r-pointed, then the r-degrees the same as those of {X \ P ^ r X).

of{T\TeP}are

Proof. Let Ay | pt* be the standard encoding of P. Fix an X such that P < r X. Define h by

m = 0, h(i+ 1) = 2{hi) = 1 + 2(hi)

if isX, if i $ X.

Let T be such that Tep*u for all i. Clearly, T < r P ® h. Hence, T < r X, since P © h < r X. On the other hand, X ^ r T, since

229 ORDINALS AND HYPERDEGREES

217

X < r P © T and P < r T. Thus X =r T. In short, the natural homeomorphism of 2" onto P preserves all /--degrees greater than or equal to the r-degree of P. PROPOSITION 2.3. / / P is r-pointed and P ^ r Y, then there exists an r-pointed Q C P such that Q=rY.

Proof. There exist functions / and g recursive in P such that for each peP: fp and gp are incompatible extensions of p in P. (p eP means p encodes an initial segment of some T £ P.) q> (* < « & / < 2{) is defined by induction on ». ?o°e-P,

*L+1 =//
if if

ieY, i^y,

?!&=/#?/ = ««,<

if if

«'ey, .-#y.

Let g be the perfect set encoded by Ay | q>. Clearly, QC? and 0 ^ r Y. Fix T e ^ ) to see O is r-pointed. Let hi be the u n i q u e ; such that T s ? / . A simple recursion on i defines A» | q)u from 7\ / , g. Conse­ quently y is recursive in T, f, g because

ieY^q^eiffqLjgqti}. P < r T since 7 e P. Hence y < r T and so 0 ^ r T. F
3F(x) & x e a],

where a e A and &(x) is a A0 formula with parameters in A. A typical instance of A0 bounding is (xUa(Ey) P{x, y) - (Eb)(x)„a{Ey)wb &{x, y), where ae A and W{x, y) is a J 0 formula with parameters in A.

230 218

GERALD E. SACKS

The economy of the above definition sometimes obscures the fact that admissible sets satisfy Ax separation and Zx bounding. Consequently many familiar constructions can be carried out within an admissible set A. For example each wellordering in A is order isomorphic to an ordinal in A. For the remainder of this paper every admissible A satisfies the axiom of infinity, i.e., u>eA.It follows that if X and Y are reals such that X is hyperarithmetic in Y and Y e A, then X e A. Thus it makes sense to speak of hyperdegrees contained in A. A typical instance of Sx dependent choice is (x){Ey) &{x, y) - (Ef){n) ^(fn,f(n

+ 1)),

where / denotes a function with domain w and S?(x, y) is a J 0 formula of ZF with parameters. Friedman has shown there exists an admissible A that does not satisfy Sl dependent choice. PROPOSITION 2.5. Let A be an admissible set and let D be a 11^ set of reals. Then A n D is Zx over A.

Proof. Since D{X) is JIXX there must be an e that satisfies (2.5.1) and (2.5.2) for all X: (2.5.1) {e}x (the cth binary relation partial recursive in X) is a linear ordering of w. (2.5.2)

D(X) <-> {ey is a wellordering.

Consequently D(X) &XeA (2.5.3) (E8)(Ef)[f ordinal 5] holds in A.

holds if and only if

is an order preserving map of {e}x onto the

2.6. Admissible Ordinals For each ordinal a let L(«) be the set of all sets constructible in the sense of Godel via ordinals less than a. a is admissible (Kripke [10], Platek [8]) if 1(a) is an admissible set, or equivalently, ifL(a) satisfies the Zx replacement axiom of ZF (with parameters in L(«)). From now on a always denotes an admissible ordinal, b is a-finite means b eL(a). 2.7. L(a, T) and £>(a, jT) As in 1 1 , T C to. L(«, T) is the set of all sets constructible from T via ordinals less than a. The language &(a, &") is the syntactical cconterparr

231 ORDINALS AND HYPERDEGREES

219

of the structure L(a, T). The primitive symbols of J?(a, 3T) are: e; unranked set variables x,y,z,...; ranked set variables xB, yB, zs,..,, for each fi < «; existential quantifiers for all variables; and the propositional connectives & (conjunction) and ~ (negation). A set «"(a) of constants intended to name the members of L(«, T) is defined by recursion on

P < a.

<<(0) is [n | n < ui}. 3- belongs to ^(jS + 1). Every other member of W(p + 1) is of the form i»F(*»), where J^(xB) is a formula built up from the primitive symbols F, and the constants in (J {<«%) | y < £}, and such that all quantified variables in F{xB) are of rank at most p. jf J J^ is defined as is the ordinary Cohen forcing relation overL(a) except that all finite conditions must belong to P. For example: pH-,'»eJ

iff peP and {T)(Tep-~ns

PW:~F

iff

^Pand(

?

T).

uf^[^:.^].

It follows that any T generic in the sense of H-f must belong to P.

232 220

GERALD E. SACKS

2.9. Uniform Hyperarithmetic Pointedness P is hyperarithmetically pointed if (T)[TeP-+P

< „ 7],

where T), where w/ is the least ordinal not recursive in T. For each 8 < «,* let {8}T be the Sth set in Godel's standard wellordering of L(w/, T). His wellordering of L ^ , T) is uniform in T, since {§} can be thought of as an instruction given by a ranked formula of J2?(
&(*)},

where &(x) is a formula of J§f<w/, JT) of rank $ < m// P is uniformly hyperarithmetically pointed if (E8)(T)[T e P-+P = {8}Tk8 < «/].

(2.9.1)

In other words, P is hyperarithmetically recoverable from every T via a procedure independent of T. Note that 8 must be less than a./, since 7V (the leftmost branch of P) is recursive in P; it follows that (2.9.1) expresses a T^/ property of P. The next lemmas shows that the notion of uniform pointedness is not as restrictive as it first appears to be. LEMMA 2.10. Suppose P is hyperarithmetically pointed. Then there exists a uniformly hyperarithmetically pointed OCP such that Q =h P.

Proof.

Let K(b, T) be TsP-*beOT&iP

is recursive ei HJ.

Clearly (T){Eb)K(b, T) since P is hyperarithmetically pointed. K(b, T) is n^ with parameter P, hence by Kreisel [12] there is a function bT, hyperarithmetic in P, such that {T)[bTeOTkK(bT,T)].

(2.10.1)

Since O' is not 2 ^ in P, it follows from (2.10.1) that there is a y0 < such that j bT | < y0 for all T. In other words, (r)[7-eP_^PeZ.(yo,r)].

«/

(2.10.2)

233 ORDINALS AND HYPERDEGREES

221

Let &(u>/, f , 0°) be the language that results from adding a new set constant & to the language i f ( w / , ST). The constants of if(o»/, 3T, &) name the sets constructive from T © P via ordinals less than « / . Let a = « / . The local Cohen forcing relation of 2.8, H-f , is easily extended to all sentences of .S?(a, $~, 3») by inteipreting 0> as P. Thus, for all qeP, qWlneP

iff

aeP,

where "n e P" means n belongs to the standard encoding of P by a real. Let T0 be generic in the sense of H-f with respect to all sentences of &(*,&;&). Since r 0 £ P , it follows from (2.10.2) that P = {8}r« for some 8 < y 0 . Since T0 is generic, there is a p0 satisfied by T0 such that A H-f^ = {»}*♦.

(2.10.3)

The definition of Mj | §♦/, an encoding of the desired £ , is by induction on i.

qlf and j g j , are incompatible extensions of tf, and for each m < 2, either q\?+m Wfa ^ or q\f+m ^ ~ ^ , where {$ri\i< o>} is an enumeration of all sentences of ^ ( c u / , ST, &) of rank < 8 + 1. Since 8 < « / , the restriction of Wl to { ^ 11 < co} is hyperarithmetic in P. It follows that 0 can be defined hyperarithmetically from P. Every o,' belongs to P, so O C P. Every 7 e Q satisfies p0 and is generic with respect to all sentences of rank < 8 + 1, hence (2.10.3) implies P = {8}' for all T e O. Consequently 0 is uniformly hyperarithmetically pointed. P is hyperarithmetic in the leftmost branch of O, and hence in Q. 2.11. If P is hyperarithmetically pointed and P < A Y, then there exists a uniformly hyperarithmetically pointed QCP such that Q =h Y. LEMMA

Proof. First 2.3 then 2.10. PROPOSITION 2.12. For each XeL(a) n 2» there is a YeL(oc) n 2racA that X is recursive in Y and Y e L ( w / ) .

Proo/. Suppose X e L(8 + 1) - L(S). According to Boolos and Putnam [13] there is a ZeL(8 + 1) - L(8) such that 8 < wxz. Let

Y = X®

Z.

234 222

GERALD E. SACKS

2.13. L[cc, K] and &[*, j f ] From now to 4.1, K is always a function from (a - m) x m into a such that for each jS 6 (a - w), Am | AT(/S, w) is a one-one map of to onto jS. L[<x, K] is defined by recursion on p ^ a L[0, A'] = a,,

JC<» = # n ((j8 - to) x m x (S).

LTjB + 1, K] is the set of all subsets of /,[£, AT] first order definable overL[/S, K) with parameters inL[jS, AT] U {KB}. L[X, K] = \J {L[)S, K]\p < A},

for every limit A A <x.

L[oc, K] will be useful in Sections 3 and 4 solely because every member of L[«, K] is countable in L[a, AT]. If a ii admissible and AT ii generic in some reasonable sense, then L[<x, K] ii sa ndmissible eet. One reason­ able sense is defined in the next subsection and another, perhaps less reasonable, in Section 4. The language &[a, Jf] is developed by adding constants to the language of ZF (as in Subsection 2.7) so that all the members of L[«, K] have names; in particular KB is named by JTB. i f [a, yf] also iicludee a two-place function symbol X. 2.14. Unbounded Levy Forcing Let s, t,u... denote finite conditions on K. s is a finite condition with domain r, and of length / if: (2.14.1)

7) is a finite subset of a - co.

(2.14.2)

/ is a finite initial segment of w.

(2.14.3)

s mapsr, x /into <x.

(2.14.4)

For each |9 e,?, Am | iflS, m) ii s aoe-one map po f /nto j8.

* is extended by t (in symbols s ^ t) if the graph of i is contained in the graph of t. K satisfies s (in symbols K e s) if the graph of s is contained in the graph of A". The Levyesque forcing relation s (+«, J5", where J is a finite condition on AT and ^ is a sentence of ^[oc, AT], is defined in the standard manner. For example: 5 H-„ JT(/S, fn) = S 5+a~sf

iff i(jS, m) = 8,

iff (t)s>( ~ [/ H-- arj.

235 ORDINALS AND HYPERDEGREES

223

K is generic if for each & there is an s such that K e s and either s H-„ & or s H-. ~ ^ Suppose s has domain 77 and length /. Then s<s denotes the restriction of s to 7j n ]8. Clearly *<« > 5. The notion of ordinal rank for sentences of JS?[«, J f ] is essentially the same as that for 3?(a, F) in Subsection 2.7. Let it suffice to say that a sentence of ordinal rank at most j8 is interpretable in L[ft K], and that its interpretation requires values of K(y, m) only for y < ft PROPOSITION

2.15. 5«^>are iAe ordtna/ rawA o/ JF w a/ mo*/ ft If

s hK «F, then s
Proo/. By induction on the ordinal rank and logical complexity of 3F. The most interesting case occurs when 3f is of the form ~ 5 ? . It is enough to check there is no t < s
2.16.

The forcing relation s ffa . F , restricted to Zx jF's,

is Ex over La. Proof. It suffices to consider ranked JF's only. Then, the definition of H-« proceeds by transfinite recursion on the ordinal rank and logical complexity of P. The recursion establishes that the forcing relation s Yr.&t restricted to J^'s of rank at most ft is Ax over Jg?(«) uniformly in ft Proposition 2.15 is needed to reduce unbounded universal quantifi­ cation over finite conditions to bounded quantification. For example, suppose & is ~ ^ and is of ordinal rank ft Then, by 2.15, J H-8 J5" iff

a LEMMA 2.17. SU#KW* ^ " ( ^ y ) w a /ormii/a o/ Sf[a,3T\ with no unranked quantifiers. If s B-„ (*)(#y>F(je, j ) , then s H-a {x°){Ey°) jF(x», y") for some p < a.

Proof. By 2.16, there exist functions h and y JE, over L(a) such that for each constant c e «♦(«) and * < s: h(c, t)^tk

h{c, t) H-. (£r (c,1 >) ^ ( c f " ' 1 1 ) .

236 224

GERALD E. SACKS

Since a is admissible there must be a p < * such that: * = *<», if c e ^(p) and t<° = t < 5, then

(2.16.1)

A(c, r) = h{c, t)<> and y(c,,z < p. (2.16.2)

Fix t e tffo) and ( < s to see that there is a K> ^ r such that w H-. ( £ y > F ( f , y ) . By (2.16.1), *<» < *, hence by (2.16.2)

h{c,t<")wAEy)^{c,y). Take w to be *(«, r<°) U (r - «<"). LEMMA 2.18. Su/tfXw* AT a ^«wnc m iAe * w o/ H-„ (2.14). Then, £,[a, K] is an admissible set that satisfies 27, dependent choice, and a is the least ordinal not countable in L[<x, K]. Proof. Suppose L[°c, K] satisfies (x)(Ey)^(x, y), where f(x, y) is a formula of jSf[«, J f ] with no unranked quantifierss .I tollows ffom 2.17 that L[p, K] satisfies (x°){Ey>)P(x°, y) for some p < a. Now L[p, K] is a member of L[a, K], hence countable in L[at, K]. Thus there is a function f:w^L[p, K] such that / e L [ a , JT] and L[p, K] satisfies

(n)^(fn,f(n +1)).

3. MINIMAL UPPER BOUNDS

3.1 Hyperarithmetically Pointed Forcing Throughout Section 3, A is a countable admissible set that satisfies the 2?i dependent choice schema of 2.4. The ostensible purpose of Section 3 is to show that the set of all hyperdegrees in A has a minimal upper bound. Its real purpose, perhaps confessed too soon, is to demon­ strate that certain perfect set constructions, readily performed when A = L(a) with the aid of a natural forcing relation fH, can still be performed, but less gracefully, when A ^ L(a) by means of an unnatural forcing relation brH defined in Subsection 3.9. Let P, 0, R,... be uniformly hyperarithmetically pointed perfect sets as defined in 2.9. Throughout Section 3, all perfect sets belong to A; i.e., their standard encodings are reals in A. Let a be the least ordinal not countable in A. Clearly ac is admissible. The forcing relation

237 ORDINALS AND HYPERDEGREES

225

P ff1 f , where & is a sentence of the language &(a, 3~) defined in 2.7, is denned by recursion on the logical complexity of SF. (3.1.1)
JF is ranked. P H-* & iff the ordinal rank of & is less than (T)[TeP-vL(a,

(3.1.2)

7) (= J*"]-

JF& 2? is not ranked. P H-A S? k & iff P HA & and P H-" @.

(3.1.3) (Ex^i*») for some c e »(/5).

is not ranked. P \j-A (Ex6)^^)

iff P hH ^ " ( 0

(3.1.4)

P H-" (£x)JF(x) iff P H-' (Ex*)JF(x°) for some /S < a.

(3.1.5)

SF is not ranked. P H-" ~ ^ iff (£?) O 0 ~ K? H-* &]■

PROPOSITION

3.2.

The forcing relation P WA &, restricted to Sx J^'s,

is Sx over A. Proof. Clause (3.1.4) implies that only ranked JF's need be con­ sidered. Let & be any ranked sentence, and let By be the set of all uniformly hyperarithmetically pointed Z's (not just those in A) such that the ordinal rank of & is less than wf and (T)[TeZ

—

L(<x,T)\=^].

Dyr is J7,1 uniformly in any Y such that a . / exceeds the ordinal rank of P. Now apply 2.5 relativized to a suitable YeA. Then D? is 27x over A uniformly in JF and Y. 3.3. For each F and P, there is aQCP or O H-" ~SP. LEMMA

such that Q \YA &

Proof. Clause (3.1.5) makes it safe to assume JF is ranked; suppose its rank is 8. Since 8 < a, and a is the least ordinal not countable in A, there i s a n l e i n 2 " such that both P and 8 are recursive in X. By 2.11, there is an R C P such that R s A X The virtue of R resides in the fact that 8 < w , » . It follows that the local forcing relation of 2.8, p ff* 9, restricted to STs of rank at most 8, is hyperarithmetic in R.

238 226

GERALD E. SACKS

Define Ay | qt\ an encoding of the desired Q, so that g 0 »H-.*^.

or

?0"H-f~^-

qlf and $ & are incompatible extensions of qf. For each m < 2, either ? | £ r o H-* ST, or $ * , , [+.* ~9t, where {Sf4 | i < w} is an enumeration of all sentences of «S?(«, JT) of rank at most §. Since 8 < oV, the enumeration {3FJ can be taken to be hyperarithmetic in P , and consequently Q can be defined hyperarithmetically from R.QCR since every q/ belongs to P . It follows that Q is uniformly hyperarithmetically pointed because R is. Every TeO satisfies ?0° and is generic in the sense of H-* with respect to all sentences of rank at most 8. Consequently Q hf4 & or Q hM ~&. Let {8}r be defined as in Subsection 2.9. LEMMA 3.4. Suppose 8 < «. i w eacA P rAere is a Q C P satisfying (3.4.1) or (3.4.2):

(jEJTJjK^rWfSjr

= *].

(T)T,O[T
(3 A 1) (3.4.2)

Proo/. The idea, as in Spector [11], is to find a Q such that the restriction of {8} to Q is continuous and either constant or one-one. As in the proof of 3.3, P can be contracted to an R such that 8 < « x * and the local forcing relation p f-ff @, restricted to STs of rank at most 8, is hyperarithmetic in R. Case 1.

Assume there h zp E R such that ( < ? W ' W " ) ~ fa H-f « 6 {S}^ and r H-? H * {«}' ]♦

Let q» = /> and define Ay | q>, an encoding of Q, as in the proof of 3.3. Let X be

{n\(Eq)[qeQkqH-!!nem. Then {8}T = X for all TeO, since otherwise, there would be a an r in 0 , and an n, such that pDq,pDr, qti-twe&r As in 3.3, OCR,

and

?

and

rrr-f^W.

0 <,, P , and O is uniformly hyperarithmetically

239 227

ORDINALS AND HYPERDEGREES

pointed. R < „ 0 because the leftmost branch of Q is recursive in Q and belongs to R, and R is hyperarithmetically pointed. It follows that

a

Case 2. There is no p that satisfies the assumption of Case 1. Thus, for each p e R, there exist q, r and n such that pDq,

pDr,

qH-^neW,

and r |+f »*{8}^.

Define Ay | tf, an encoding of O, and Ay | « / hyperarithmetically from R so that: For each m < 2, aft* C ? / , and either ott* hf * ST, or ? W H-.* ~ » , , where {ST,} is an enumeration of all sentences of rank at most 8; also

ff^H-.'-i'ePK

and

« & hf-*,* * A T

It follows that {8} is one-one on O since the results of applying {8} to any two distinct members of Q will differ on some « / . Fix TeQ to see that the one-one-ness of {8} on O implies T ^h {8}T, Q. There is a unique r: co - * w such that rO = 0 and: t(i+})

= 2(ti)

f(i -h 1) = 2(f0 + 1

if

»| i e{8} r ,

if

n\.i{h}T.

Clearly T e 9/ iff / = ti, hence P ^ „ t, O. Now * < * Ay | « / , {8} r ; and Ay I nf* s$„ R. Thus t < f t O, {S}7-, since R ^h Q as was shown in Case 1. 3.5.

H~A Genericity

A set D is said to be ^-definable if there is a formula parameters in A) such that

JF(JC)

of ZF (with

D = {b ! A e A and .4 (= Jf(J)}, where i is a constant that denotes b. Suppose D is a set of P's; Z) is said to be dense if (0(£P)[P6Z)&ODP]. 7 is H-" generic if T e (J {P 1 P e D} for every dense ^-definable Z). PROPOSITION 3.6.

7/ T is WA generic, then:

240 228

GERALD E. SACKS

(3.6.1)

X < A T for every X e A n 2";

(3.6.2) for each 8 < a, there exists an X e A <~\ 2» such that either {S}T = XorT < * {8}T, X. Proof. Fix Xe An 2". The set of all Q such that X < „ Q is dense by 2.11 and ^-definable by 2.5. Hence, there is a 0 such that T e 0 and X < A O. Then 0 < f t P since O is hyperarithmetically pointed. Let D be the set of all Q such that Q satisfies either (3.4.1) or (3.4.2). D is dense by 3.4 and ^-definable by 2.5. Hence there is a Q such that either {8}T e A n 2" or 7 < 7l {3}r, 0 . 3.7. J 0 Bounding Assume T is ff-< generic. Theorem 3.13, the principal result of Section 3, will follow from 3.6 after it is seen that
(3.7.1)

where S^{n, x) is a formula of &(a, ST) with no unranked quantified variables. (BP) is an immediate consequence of its generic counterpart (GBP): If P H-" (n)(Ex)S?(n, x), then there is a Q C P such that for some S < a. The proof of GBP is based on some rather obscure manipulations in Subsection 3.11, which become considerably simpler when A = L(«). The problem of proving GBP reduces to the problem mentioned in Section 1 of what steps to take in the construction of a contracting sequence of pointed perfect sets so that its intersection will be pointed. 3.8 When A = L(ac). Assume P h f (n)(£*)jF(n, x) to find a O that satisfies GBP. Thus, (n)(0)PDO(ER)ODK(E8)6
and

PnDPn+1,

Pn+,WA{Exs")^(n.xs").

(3.8.1) (3.8.2)

241 ORDINALS AND HYPERDEGREES

229

Let O = D {P„ i « < «} and S = sup{8" \ n < w}. Clearly Q is a closed subset of P. If O were perfect and uniformly hyperarithmetically pointed, t hen it would be the case Now it is routine to arrange that O be perfect by means of the splitting trick of [4, 16]. For each n, Pn = \){Pn> \j < 2"}, and P*J+1 and P ^ 1 are disjoint perfect subsets of P„>. The splitting trick raises no dust since any perfect set can be split effectively into two disjoint perfect sets by a finite condition. Thus the only substantial problem that arises in the construction of Q is how to ensure that O be hyperarithmetically pointed. (2.10 eliminates any concern about whether or not Q is uniformly pointed.) The solution consists of requiring that Q be absolutely hyperarithmetically pointed (X)[X e Q-* Q e L(Wlx)]. Thus the hyperarithmetic recovery of 0 from XeQ does not utilize X itself but only some ordinal recursive in X It follows from 2.11 and 2.12 that every hyperarithmetically pointed P e L(a) can be contracted to an absolutely pointed Q e L ( a ) . The local forcing proof of 3.3 is easily adapted to show: If' & is a sentence of -S?(a, 3~) and P is absolutely pointed, then there is an absolutely pointed OC P such that O H - " ^ Now the functions An | Pn and Xn | 5" can be made to satisfy some additional requirements: (3.8.3)

Pn is absolutely hyperarithmetically pointed;

(3.8.4)

Sn < «(»« (satisfiable by 2.11 and 2.12);

(3.8.5) the choices of Pn and S" are always the least possible with respect to Godel's standard -d, wellordering of L(«). Fix XeO (= f){P„in <w}) arithmetically pointed. Let

to see that 0 is absolutely hyper­

y„ = sUP{o/f" In <

w).

y0 < O / , thanks to (3.8.3). The definition by recursion of An | Pn and An | 8" took place overL(a) but by virtue of (3.8.4) did not involve any ordinals greater than y n . Hence, the recursion can be visualized as taking place over L(u>/) rather than L(a). By (3.8.5) the results of the recursion over L(a) and L(cu/) are identical, since the standard J , well-

242 230

GERALD E. SACKS

orderings of L{<x) and L(mxx) are compatible. Thus Xn \ Pn and Xn | on are 27a over L{w/) as well as L(a). Since w^ ii sdmissible, Q eZ,(o//)) 3.9. The Relation hH-4 The line of argument pursued in 3.8 succeeded only because the assumption A = L(a) made it possible to force with absolutely pointed P's. Then the recovery of O ( = f] {Pn\n < o>}) from XeO could be accomplished by a manipulation of ordinals rather than reals. The recovery of Q when A is arbitrary is based on a uniqueness feature of Hf" expressed by 3.10, a feature that makes it possible for ordinals to do all the work in the recovery of Q even when A # L(a). According to 3.2, there is a A0 formula 3&(u, v, y) such that PWA3F

iff A \={Ey)S8{P,3?,y),

(3.9.1)

whenever j F is a Sx sentence of if(a, .T). The proof of 2.5 shows 38(u, v, y) can be chosen so that it has no infinite parameters; thus it is absolute in nature. A useful interpretation of @{P, S? y) is: y is a proof that P WA P. A more extended notion of proof will be needed in (3.9.8) below. The A0 predicate, y proves (P f-H #"), is gegerated by four rules. (3.9.2)

If @(P, ST, y), then y proves (P hH ^ ) .

(3.9.3)

If j proves (P hH ^ ) and P DO, then j proves (Q YrA 3F).

(3.9.4) Suppose £ = f) {Q, I» < w] and {.F, 11 < cu} is an a-finite sequence of 2^ sentences. Ifj t - proves 0 , hM ^ for each » < w, then Ai | yt proves ( 0 r H &«^i). (3.9.5) Suppose Q C P, 0 WA &t and d is constructed drom m via the local forcing method o f 3.3. If y encodes the construction, then y proves (O H-" &)■ Define (P,SJrr^. where JF is 2?, and S < a as follows: (A)[A e P ^ P = {8}*} *]niform myperarithmetic pointedness ss as 2.9), (3.9.6) PH-F, r

(Ey)[y proves (P H-' i ) & (A')(A e ? — e i ( ( S + 1, A'))].

(3.9.7) (3.9.8)

243 ORDINALS AND HYPERDEGREES

231

Clause (3.9.8) will turn out to be less restrictive than it appears for two reasons: The construction of a P intended to force some Zx SF will always be encodable by a y that is hyperarithmetic in P, hence hyperarithmetic i n l e P; and the hierarchy L ( « / , X) has an inherent repetiveness that can be exploited (as in 3.11) to juggle the relative locations of y and P. The y of clause (3.9.8) includes a proof that P is uniformly hyperarithmetically pointed via some ordinal; that ordinal is always assumed to be the 8 of clause (3.9.6). It follows from (3.9.1)-(3.9.5) that there is a J 0 formula ®*{u, v, y) such that for all X and S Xe{b)*k{{S}*,S)ti-AP iff L(S + \tX)

(= {EyM) @*({$}x, SF,yM).

(3.9.9)

The formulas ®v{u% v, y) and the minimal forcing relation {P,y)WrA* are defined simultaneously by recursion on y < a for all EJ #"'s. Fix y. Assume for all X and 8 < y: JTe{S}*&({§}* 5 ) ^ - 4 ^ , iff L(8 + 1, X) \= (£y6+1) &,{{&}*, SF, j 6 + 1 ),

(3.9.10)

where SSt(w, a, ^) is a formula all of whose parameters are ordinals less than 8, and all of whose quantifiers are of rank at most 8. Define (P, y) H+M & by: (P.yjH-'.F,

(3.9.11)

(P,y) H-^(S)5
(3.9.12)

It follows from (3.9.9) that there exists a formula #„(«, v,y) such that (3.9.10) remains true when 8 is replaced by y. The proof of 3.11 will clarify the definition of htf" Proposition 3.10 singles out the property of hl+ A needed for the proof of 3.14. PROPOSITION

3.10.

For each X and I, J r , there is at most one (P, y)

such that XePk(P,y)

H+/I^r-

244 232

GERALD E. SACKS

Proof. By (3.9.12) and (3.9.10). LEMMA 3.11. Suppose Pff^, Then there exists aQCPandaX0
where & is a E1 sentence of &(*, JT). such that 0=hP and (Q, A0) H+" &'■

Proof. By (3.1.1) and (3.1.3) it is safe to assume that # " is ranked. The requirement that 0 be contained in P is of no real concern, since P can be thought of as" forcing the ranked sentence J f & J s P and consequently Q can be required to force ^ & J e P instead of &. The search for A0 is simplified by perturbing the standard wellordering of L(cV, X). Recall the description of {8}x given in 2.9. T h e standard wellordering is derived from an onto function/: « / -*L(
(3.11.1)

It follows from (3.11.1) and some further mild perturbations o f / that there exists a limit ordinal A < w / that satisfies (3.11.2)-(3.11.5). (3.11.2) If 8 < A, then there exists an onto map w: w -> S and a < A such that w = {/>}* for all X e P. w and P are easy to find since P every 8 < t o / is recursive in P and P is uniformly hyperarithmetically pointed. (3.11.3) If 8 < A and Z is first-order definable over L(8, X), then there exists a p < A such that Z = {p}x. Furthermore, p is independent of X, i.e., P depends only on 8 and the definition of Z. (3.11.4) There is a j3 < A such that (P, 0) hM JF. To ffnd jS, note that the rank of J^ is less than w / by (3.1.1). Consequently a proof j (in the sense of (3.9.2)) of P hM .F can be found in L(w/, P) by paying attention to the details of 2.5. Since P is uniformly hyperarithmetically pointed, there is a y < O / such that j eL(/3, Z ) for all p ^ y and all X e P. By (3.11.1), there is a 0 > y such that P = {£}* for all X e P. (3.11.5) {A}* is defined in two stages. The first stage yields the same onto function h: w -> A for rll X e P a a si (3.11.22 )bove. The eecond stage yields {A1/ itself. If X e P, then {A}* is the perfect set R =

245 ORDINALS AND HYPERDEGREES

233

0{Rn\n < w}. The functions Rn and y„ are first-order definable over L(A, X) uniformly in X e P. The dependence of Rn and yn on X is minor, since X is used only at the outset to define R0 = P; subsequently Rn+1 and yn are defined from Rn via the ordinals less than A: R0 = P

and

RnDRn+1,

(Kn+1,y„)eL(A,A'), (*»+x>y»)^~^"

or

(Rn+1,yn)H-Aa«,

(3.11.5a) (3.11.5b) (3.11.5c)

where * » is {Eyh»^)@hn{{hnY> ♦*", yhn+l)> h i s as in (3.11.5), and 3Sh is as in (3.9.10). The local forcing argument of 3.3 is applied to satisfy (3.11.5abc). Suppose Rn is given. Then, as in the proof of 3.3, there is an Rn+1 C Rn such that Pn+iH-A ~®n

or

Rn+1\j-A&».

(3.11.6)

The nature of local forcing allows Rn+1 to be first-order definable over anyZ,(8, X) that contains Rn , the ordinal hn, and a counting of An, since the rank of J" 1 is hn + 1. In addition, the construction of Rn+1 can be construed as a proof (in the sense of (3.9.5)) of (3.11.6). Hence, by (3.11.1 )-(3.11.3), there is a ( # n + 1 , yn) that satisfies (3.11.5abc). The splitting trick of 3.8 can be added to the definition of Rn to ensure that R = ft {Rn\ n < w} is perfect. To obtain the desired O and A0 first suppose there is an n such that (fl-i.yOH-**-. Let Z be the leftmost path through i? n + 1 . Then Z is recursive in and

(3.11.7) Rn+l,

L(X Z) fc= 38n

since the rsnk of So^ is less tnsn A But then z by (3.9.10). Thus, (Q,X0) is ({hnY,hn). Now, suppose there is no n that satisfies (3.11.7). Then, (3.11.5c) implies (R, A) H- " (8)6<^ ~ (Ey6+1) ^S({S}*", &, y+'), and consequently (Q, A0) is (R, A).

246 234

GERALD E. SACKS

LEMMA 3.12. Let .F(», x) be a formula of &(*, 3~) with no unranked quantifiers. Suppose P H-4 (n)(Ex)S?(n, x). Then there exists a QC P and a 8 < a such that O WA (n){Exs)^\n, x*).

Proof.

Clearly (»)(Q)FDO(ER)OORIR

Hence, by 3.11, for each n and

ti-A

(Ex) F(n,

*)].

OCP

(ER)(Ey)[(R, y) hff" (Ex) JF(n,x) k&~eQ].

(3.12.1)

The effect of R H-4 T e Q is intended to be the same as that of 0 D R. Strictly speaking ST e Q is not a ranked sentence of JS?(«, ^ " ) , however there is no difficulty in treating it as such. The argument of 3.2 shows that the relation R WrA 9, restricted to ZyTs, is £x over A. Since A satisfies Ix dependent choice, it follows from (3.12.1) that there exist functions An | Qn and An | yn in A such that Q0 = P, O0 is uniformly hyperarithmetically pointed via y0 , and for all n, * (Qn+i, y««) M-A (Ex) &{n,

x)k^eQ„.

Clearly, Qn D Qn+1 for all n. Let Q = f] {Qn I n < o>}. The splitting trick of 3.8 can be used to ensure that O is perfect. Let X £ Q to see that O is uniformly hyperarithmetically pointed. For each n, y„ < « / and~0n = {yn)x since X e O n . Proposition 3.10 makes it possible to define An | y„ and An | On by"a recursion Z1 over L ( w / , X): yn+1 is the least 0 such that X e {p}x and (W, P) Hfx (St) ^ ( « , *} & J" e Qn . Thus there is a p such that for all X e O, An | yn = {p}* and p < a , / , and consequently Q possesses the desired uniform pointedness. For each n, let 8 n be the least 5 such that Q ft* (Ex*) &■(*,&), S„ is well defined since 0 C On . An j S„ belongs to ,4 by 3.2. Then 8 = U {S„ I n < w} < a.~ 3.13. Sw/>/>05e 7 is ;«wr«r in ffc roue of ftA (3.5). Jnen « / = a flnrf raery ;ea/ in A is hyperarithmetic in T. Furthermore there exists no hyper degree less than that of T and greater than every one of A. THEOREM

247 ORDINALS AND HYPERDEGREES

235

Proof. By (3.6.1) every real in A is hyperarithmetic in T and so t o / > au For each 8 < a, (3.6.2) )mplies s 5 / has the same hyperdegree as T or some real in A. Hence, it suffices to show t o / < a. As noted in 3.7, it is enough to see that L(<x, T) satisfies the A0 bounding principle (3.7.1). Let &(«, c) be a formula of &(a, ST) with no unranked quan­ tifiers. Let D be the set of all P such that either P WA ~(n)(£x)^(n, x), or (£5)4<0[P^>X£*s)^(n,**)]. Lemma 3.12 implies D is dense (as in 3.5). By 3.2, D is ^-definable. Consequently, there i s a P e Z ) such that TeP, and so L(a, T) satisfies (3.7.1). COROLLARY 3.14. / / A is a countable admissible set that satisfies 2Z, dependent choice, then the set of all hyperdegrees contained in A has a minimal upper bound.

Proof. If the set H of hyperdegrees of all reals in A has no least upper bound, then 3.13 furnishes a minimal upper bound for H. COROLLARY 3.15. If pis a countable ordinal, then the set of hyperdegrees of reals in L(£) has a minimal upper bound.

Proof. Boolos and Putnam [13] call 8 an index ordinal if (L(S + 1) - L(8)) n 2 " # 0 . They show L(8 + l) contains a one-one correspondence between S and to whenevee r is sa nidex ordinal. Thhu it is safe to assume that either j3 is the llmit of index ordinals or j8 is the successor of an index ordinal. It follows that the hyperdegree of Y is an upper bound for the hyperdegrees of L(/3) if and only if t o / ^ /S. Let y be the least admissible ordinal > jS. Let a be the least P such that L(P) n 2 » = L{y) n 2". <x is admissible because if a < y, then a sits inside L{y) in very much the same way constructible aleph-one sits inside L. If a. < jS, then the sought after minimal upper bound exists by 3.14. Suppose a > //? Then a = y and there is an index ordinal S such that a > S ^ j8. It follows there is an Z e L ( a ) such that t o / > 0. Clearly the hyperdegree of X is an upper bound for the hyperdegrees of L(j8). -V is in fact a least upper bound. Suppose every real in L(jS) is hyperarithmetic in Y. Then, t o / > 0 and XeL(a) C L ( t o / , Y), since t o / is admissible.

248 236

GERALD E. SACKS

COROLLARY 3.16 (cf. 4.26). <x is a countable admissible ordinal greater than w if and only if there is a T C w such that a = w,7".

Proof. Let K be generic in the sense of H-„ (2.14). By 2.18, L[a, K) is a countable admissible set satisfying 27, dependent choice. In addition, the least ordinal not countable in L[a, K] is a. Let T be generic in the sense of hM with A = L[a, K]. By 3.13 «/ = a. Let F be a type n object (» ^ 2). F is said to be normal if " £ (the equality predicate for objects of type < w) is recursive in F in the sense of Kleene [17]. jScF is the set of all reals recursive in F and was christened the 1-section of Fin [17]. COROLLARY 3.17. If F is a normal type n object, then the hyperdegrees of the reals in the 1-section ofF have a minimal upper bound.

Proof. It was observed in [18] that l S c F = 2 " n ^ l countable admissible A that satisfies 2T, dependent choice.

for some

COROLLARY 3.18. Let A be a countable admissible set such that the hyper jump of each real in A also belongs to A. Then the hyperdegrees in A have a minimal upper bound.

Proof. Let HCA be the set of all members of A hereditarily countable in the sense of A. The standard arguments that show that HC (the set of all hereditarily countable sets) is an admissible set satisfying I", dependent choice also show that HCA has the same properties. The point to remem­ ber is that the closure of HCA under hyperjump implies that every linear ordering of w in HC A is a wellordering in the sense of HCA if and only if it is an actual wellordenng. THEOREM 3.19. Let A be a countable admissible set. Then there exist reals S and T such that for all real X: X 6 A if and only if X < „ S and X <,, T.3

Proof. By a type omitting argument similar to the one given in Keisler [3, p. 58]. Let at be the least ordinal not countable in A. Define Ax to be HCA if a e A, and A if a $ A. Ax is admissible as in the proof of 3.18, because HCA is closed under hyperjump when < e A. (It is not 3 The oripinal version of 3.19 assumed A satisfied Zl dependent choice and was proved by pointed forcing (cf. 5.2 and (5.4.1)).

249 ORDINALS AND HYPERDEGREES

237

known if HCA is admissible for every admissible A.) Also a is the least ordinal not in A1. Let V be the following collection of sentences: SCw,

X recursive in S (X e A n 2"),

ty)[yei~Vy-§],

<«<«).

fi<6

Clearly if is consistent and 2X over Ax . Since ^ is admissible, the argument of Keisler cited above shows y has a model that "omits" a. Thus there is a real S such that every real AT in A is recursive in 5 and

and ST simultaneously, models that "omit" a, and by interweaving some addi­ tional steps to ensure {8}s = {p}r-+{Sf eA, (3.19.1) for all S and p less than a. Suppose that only finitely many steps in the Henkinesque constructions of models of y and 5 " have occurred. Then, (3.19.1) can be satisfied for a fixed S and p as follows. Assume 8 is such that {8}s must be the characteristic function of a real. Case (i). There exists an n such that each of the sentences, {S}-(n) = 0 and {§}s(n) = 1, is consistent with the decisions made up to now in the construction of the S" model. Choose an m so that {p}T{n) = m is con­ sistent with the construction up to now of the ST model. Then {8}s(n) # m can be added consistently to the construction of the Sf model, and so {S}* ^ {p)T is guaranteed. Case (ii). There is no n that fits the needs of Case (i). Thus for each n, only one of the statements, {8}s(n) = 0 and {8}5(«) = 1, can be added consistently to the construction of S. Define a real B by: n e B if and only if {8}5(«) # 0 is a logical consequence of all the decisions made up to now in the construction of the & model. That set of decisions is a finitely generated extension of ST, and hence, is 27, over HCA . Con­ sequently, the hypothesis of Case (ii) implies B is Ax over HCA , hence a member of HCA , and so {S}* e A is assured. COROLLARY 3.20 (cf. Theorem 5.2). Let A be a countable admissible set. If the hyperdegrees contained in A have a least upper bound, then, that least upper bound is contained in A.

250 238

GERALD E. SACKS

Proof. The 5 and T of 3.19 force the least upper bound, if it exists, into A. Richter [19] denned and studied the following hierarchy of hyperdegrees associated with iteration of the hyperjump. A(0) is the hyperdegree of the empty set. h(y + 1) is the hyperjump of % ) . k(X) is the least upper bound of {%) | y < A}, when that least upper bound exists; otherwise, h{X) is undefined. Richter defined S to be the least A such that h(X) is undefined. He showed Hr > § ^ wf», where Ex ii she Souslln operator and «fi ii she least ordinal not recursive in E1. A more current characterization of «f» is: the least admissible ordinal that is a limit of admissible ordinals. COROLLARY 3.21.

S =

wf>.

Proo/. Let a be wf1. Some of Richter's results [19] are equivalent to: (1) for each y < a, % ) is defined and belongs toL(tx); (2) each hyper­ degree in L(<x) is less than h(y) for some y < a. It follows from 3.20 that {%) ! V < <*} has no least upper bound, since it has none in L(a).

4. MINIMAL SOLUTIONS OF « =

o>/

a is a countable admissible ordinal. In Section 3 a solution X of a = w / was developed in two steps. First a /C was chosen generically using the finite conditions of 2.14 so that L[<x, K] would be an admissible set in which every ordinal less than a was countable. Then ii was seen in Section 3 that every T generic in the sense of H-", with A = L[ac, K], was a solution of a = u>%*. Thh eurppse ef Section n is to find d minimal solution, an X such that no real of lower hyperdegree than X is a solution. Again the argument is in two steps The second step resembles that of Section 3 but the first differs markedly. K is now constructed generically via perfect conditions that lend themselves to the splitting arguments needed for the desired minimality. The principal technical lemmas a r e

4.13 and 4.21. Both of them deal with some of the thorny problems encounterpH

in s p l i t t i n g a r m i m p t i t s w h e n t h e n o t i o n s of p o i n t p H

tinrl

251 ORDINALS AND HYPERDEGREES

239

4.1. Perfect Conditions on K From now on K is always a map from (« - w) x » into <x such that for each )S e (« - w), Am | /£(£, m) is a one-one map of w into /3 with the following property: if L(«) |= [j8 is a cardinal & cofinality £ > m],

(4.1.1)

then the range of Am | K(fi, m) is an unbounded subset of p. Clearly, every ordinal less than a is countable inL[ot, K]. Let 5, r.... be finite conditions on K as in 2.14. Let H, I, /... denote a-finite perfect conditions on K. A typical H is given by an cx-finite partial function Atcr | t* (i < u> and a e «') that meets requirements (4.1.2)-{4.1.7). (4.1.2) If s* is defined, then sj is a finite condition on K; i„* is said to belong to H (in symbols s* e / / ) . (4.1.3)

V i s defined.

(4.1.4) If 4 + 1 is defined and {j)t<^7J = oj), then t,< is defined and *.♦< is extended by **+» (*.* ^ 4 + 1 ). (4.1.5) Suppose 5„> is defined. Let hj be the set of all r e «*+» such that 4 + 1 is defined and (/)Ki(r/ = oj). {i*« I T e V } is called the set of immediate successors of t* in ff. All the immediate isuccessors of sj have the same length /„* and the same domain rjj as defined in 2.14. The relevant domain dj is the set of all /9 e -qj such that /S has property (4.1.1) above. (4.1.6) Let dj- be {pk \ k < *}. For each y e 77{0t | A < z}, there is an <m^> i<2 and a T e bj such that ( « " U ^ - , "fl >

ttj-

(4.1.7)

The requirement imposed by (4.1.6) and defined by (4.1.7) is called diversification and is needed for the proof of complex minimization (4.21). {^+1 [ T e bj} is said to be diversified. A path through H is a sequence Xi \ s*ei such that j * , is defined and 4* 3* '&+« for all i. K satisfies H (in symbols AT e H) if there is a path At I »L through H such that tf satisfies si for all i.

252 240

GERALD E. SACKS

4.2. The Forcing Relation 77

\ya&

Let JF be a sentence of the language i$?[a, j f ] as in 2.13, and let 77 be an a-finite perfect condition on K. The forcing relation HH-0&is similar to that of 3.1. (4.2.1) ? is ranked. In this case, H Wa JF iff L[«, K] h & true for every K e H. (Keep in mind the convention concerning K established at the beginning of 4.1. The definition of H H-0 & for ranked JF refers only to those K's in 77 that obey the convention. The existence of such K's in H follows from (4.1.7).) (4.2.2) 77 ^ „ ^ .

JF & # is not ranked. 77 H-„ & & ST iff H H-„ ^

and

(4.2.3) (E*")#♦(*») is not ranked. 77 H-. (£V)jF(* B ) iff fl H-. JF(C) for some c 6 tf(/3). (4.2.4)

H H-, (£*)#"{*) iff H H-„ ^ ( c ) for some c £ #X«).

(4.2.5) <§ is not ranked. 77 H-. ~ ^ iff it is not the case that / H - . 9 for any 7 C 77. (7 C 77 means 7 extends 77, i.e., every K that satisfies 7 also satisfies 77.) AT is said to be generic if for each sentence JF of i f [a, JT] there exists an 77 such that K e 77 and either 77 H-„ ^ or 77 (♦+-„ ~ # \ A splitting lemma (4.4) is needed to establish the existence of generic K's as well as the validity of A0 bounding (with x e K as an additional atomic predicate) in L[a, K] for all generic TTs. LEMMA 4.3. Ex over L(<x).

The forcing relation 77 H-a 3F, restricted to £% JFV, is

Proof. Suppose S < a and X is any real such that S ^ w/. The set of all 77's that are perfect conditions on K, and that belong to L(8), can be construed via indices as a set of numbers 77^ in X. The indexing of the 77's is a slight modification of the indexing of the ordinals less than Wlx by elements of Ox. Similarly the set of all J^'s of rank less than 8 is 77^ in X. Thus, the predicate P6(77, JF), defined by 7/eL(S)&rank.jF < 8 & # H - - . F , is 77^ uniformly in any A' such that 8 ^ w,*, since the universal quantifier on A' in clause (4.2.1) of the definition of H-a can be regarded

253 ORDINALS AND HYPERDEGREES

241

as a universal quantifier on reals thanks to the countability in Uwtx, X) of the set of all finite conditions in H. Let Xb encode a Levyesque collapse of 8 to w generic over L(<x). For example Xs could be {<m, n> I K(8, m) < K{h, «)}, where Am | K(S, m) is a one-one map of w onto 8 generic over L(«) in the manner of 2.14. Then, PS(H, 3?) is 77^ in Xs, and consequently is Zx over L(«, Jf,) by 2.5. To be precise, there is a S1 formula Z(S, H, P) such that P6(tf,JF)

iff L(«,I1)M(8,ff,n

But (4.3.1) holds for all generic Xt\ PS(//,JF)

iff

(4.3.1)

hence

0^s~~Z(S,//,^),

(4.3.2)

where 0 is the null forcing condition, and H-4 is the Levyesque forcing relation designed to collapse 8 to w generically over L(a). Since Z is 2X and the collection of forcing conditions needed to collapse 8 is a set in L(a), it follows that the right side of (4.3.2) is 27, over L(<x) uniformly in 8. Hence flff/, restricted to ranked .F's, is 27, over L(«), since it is equivalent to (E8)P6(H, JF). LEMMA 4.4. Lrt {JFf\i< w} 2>e an «-_/im*e *e?uen« o/ 27, sentences of &[a, JT\. Suppose H is such that W)nAE])n>AJ JAwi rAere exisfc a JCH

H-.^].

such that (i)[J H-« ^ ] .

Proo/. Partial functions /„% *„', and e.« are defined by recursion on 1 for all i < to and some a e «« Their definitions require certain choices to be made. All such choices are the least possible with respect to the standard J j wellordering of L(a) as in (3.8.5). It will follow from 4.3 that all three partial functions are a-finite. Let r0° be the null condition on K, and let 70° be H. Fix i < w and o 6 a1'. Assume t* and /.* have been defined so that tj e I,* and Ij is a perfect condition on AT contained in H. Choose V , rj+1 and PT+1 in accord with the following requirements.

254 242

GERALD E. SACKS

(4.4.1) {*;+1 | T 6 bj] is a diversified set of immediate successors of t,* inside IJ. tl+1elt+1CI„\

/ r

(4.4.2)

(t)[teli+1-^ii+l>t],

(4.4.3)

1

(4-4.4)

^ ^ ,

forallreV-

The choice of 7;+1 is the least possible in the following sense. By 4.3 there is a 27: formula R(I, J^) such that L(<x)h=R(I,^)

iff

Ift-.&i.

Suppose R(I, f§ is (Ex)P(x, I, 3FX), where P is a A0 formula. Let (A:0, 7°) be the least pair (*, 7) that satisfies P(x, I, Jtft. Then 7° is said to be least among those 7's that force J ^ . Let J be the perfect condition given by the a-finite partial function Aia ! t*. Suppose &■< is (Ex)9t(x), where x is the only unranked variable in 'Six), to see ) H-« ^ . Let b be the a-finite set of all r e ai+l such that 7<+1 is defined. Then for each r e b, 7'+1 H-« &AO for some c7 e %?(a). By 4.3 cT can be construed as an a-finite function of T. Hence, there is a 0 < a such that {<:T | T e 6} C ^(/S). Consequently

for all T e b, and so / H-. (Ex)9i(x), LEMMA 4.5.

since / C (J {7<+1 ) T e 6}.

G F ) ( f l X $ / W 7 ^ „ J^ or J fra ~ j F ] .

Proo/. By induction on the complexity of SF. Clause (4.2.5) makes it safe to restrict attention to ranked JF's. Let (ExB)9(xB) be a ranked sentence. If there is an 7 C 77 such that 7 (f. 9(c) for some c e
~9{c)l,

since 9(c) is less complex than (Ex)9(x) for all c e <€(&).

(4.5.1)

255 243

ORDINALS AND HYPERDEGREES

L e t / be a constant of i f [at, X] that denotes a map of w onto <>) J^* 8 , .y) /or'^omf /5 < a. Then, there exists a J C H such that

/H- a (^)(^)^(.r s ,r) /or sowe y < ex. Proof.

Clearly (iW(<W<*»(£/)n/[/

H-. (Ey) &(c, y)].

Let / be a constant of i f [a, j f ] with the property claimed in (4.5.2). Then, as in the proof of 4.5, there is a / C H such that

(0[/^„(£y)^(/'-v)]It follows from 4.3 that there is an a-finite h: w - * a such that

(017 *.(£>* W/»,y")]. Let y be sup{/» | i < u>). Then, / H-„ (xB)(Ey*)&(xB,

y).

256 244

GERALD E. SACKS

THEOREM 4.7. Suppose K is generic in the sense of 4.2. Then L[«, K] satisfies (x) (x is countable) and A0 bounding (with x e K as an additional atomic predicate). Furthermore L[a, K] has a J , (in K) wellordering.

Proof. Lemma 4.6 implies A0 bounding holds in L[<x, K], just as 3.12 does in the proof of 3.13. The Ax welloidering ofL[«, K] is derived from the standard order in which the elements of L[a, K] are constructed from K via ordinals less than a. For the remainder of Section 4 assume K is generic in the sense of 4.2. By 4.7 L[ac, K] is a countable admissible set that satisfies Z1 dependent choice, and a is the least ordinal not countable in L[«, K]. Hence the pointed perfect forcing argument of 3.13 is applicable to L[«, K]. Thus if T is generic (in the sense of VrA with A = L[a, K\), then c/ = <*. The primary objective of Section 4 is to prove that T has the following additional property: (X)[X
w,x < «].

Of course the proof will draw heavily upon the fact that K was construct­ ed from the perfect conditions of 4.1 rather than the finite Levy condi­ tions of 2.14. On the other hand it is quite possible that finite conditions suffice. 4.8. K-Uniform Pointedness Let P e L[a, K] be perfect as in 2.1, and let 9 be a term of &[a, Jf] that denotes P. 0> can be viewed as a function that takes K's as arguments and perfect P's as values; thus P = 0>(K). Only a bounded part of K is needed to evaluate &(K)\ thus P = 0>(KB), where j5 is the rank of ^ as a term of J?[«, Jf"]. P is said to be AT-uniformly hyperarithmetically pointed via 0> and S if for all TeP: /3<
and and

P = ^(/Cfi),

(4.8.1)

X« = {8}r.

(4.8.2)

Clearly ^-uniform pointedness implies uniform pointedness in the sense of 2.9, but the converse is false. However forcing with /C-uniform P's is the same as forcing with uniform P's, since each one of the latter is readily extendible to one of the former with the help of 2.3. The forcing relation P H-A' &, where P is A'-uniformly hyperarith­ metically pointed and & is a sentence of if (a, S~), is defined in the same

257 ORDINALS AND HYPERDEGREES

245

fashion as the forcing relation hM of 3.1. The only difference between the two is in the notion of P each employs. LEMMA

4.9.

The relation P hhA' JF, restricted to Ex F's,

is Sx over

L[«, K]. Proof. Similar to that of 3.2. Again, most of the work is done by 2.5. 4.10.

Complex Forcing

By 4.9 there exists a formula 0 * R-*" P that expresses P H-* & as follows: (4.10.1) ^ 6 H- jr «F is a 2^ formula of J?[«, X] with ^ , 8, and J^ as free variables. (4.10.2) If L[a, K] h [^6 ^ * " # ] , then ^ is a term of J^fo, J f ] that denotes a P that is iC-uniformly hyperarithmetically pointed via & and S, . F is a S1 sentence of &(<*, ST) and P \±K 3?. (4.10.3) If & is a Tj sentence of J2?(«, F ) and P hr-*^, then L[ac, K] \= [0>s H-*" &\ for some ^ and S such that P is if-uniformly hyperarithmetically pointed via 9 and S. Formulas such as 0* Yrx & are characteristic of iterated forcing situations. Define the complex forcing relation (#,$»)

H- B jr

by H H - . [ ^ H - ' ^ ]

(4.10.4)

for every JG, sentence . F of JSf(«, .T). (The relation # H-. 9 was defined in 4.2.) Since & is ^ , the meaning of (#, ^ 5 ) H-B ^ is: for every KeH it is the case that 0>{K) is if-uniformly hyperarithmetically pointed via 0> and 5, and that L(a, J ) (= JF for every P e &>(K). A pair (tf, 9s) as in (4.10.4) is called a complex forcing condition. 0>B is called the virtual part of (H, 9s). The advantage held by a virtual 9s over an actual £»(£♦) is membership in 1(a), an advantage used heavily in the splitting arguments of 4.13 and 4.21. From now on the ordinal superscript in (H, P6) will be omitted for the sake of notational simplicity. Thus (H, 9) will denote a typical complex condition. LEMMA 4.11. The complex forcing relation (H, 9) Vra J r , restricted to E1 &ys, is E1 over L(a).

258 246

Proof.

GERALD E. SACKS

By 4.3 and (4.10.4).

PROPOSITION 4.12a. Suppose P is K-uniformly hyper arithmetically pointed and XeL[a,K]. Then there exists a K-uniformly hyperarithmetically pointed O C P such that X is hyperarithmetic in O.

Proof. Assume P is ^-uniformly hyperarithmetically pointed via 0> and S. Thus P = 9(K% where p is the rank of 0>. Choose y ^ p so that X is hyperarithmetic in X*. (Strictly speaking K* was denned to be that part of K that collapses all infinite ordinals < y to <♦», but there is no harm in thinking of K* as a relation on to.) Construct Q from P and Kv as Q was constructed from P and 7 in 2.3. Let J be a term of jSf[«, J f ] that composes 9 with a description of the construction of O from P and 2£*. Then Q = £(#»), since P is hyperarithmetic in A>.~Fix TeQ to see 0 is suitably pointed. As in 2.3, O C P and K- is recursive in P, T, hence hyperarithmetic in T since r~e P. The rank of J ( = y) is less than a,,7", and Kv = {P}T for some p < w/ determined by y and 8. ♦X" ^ * Q by the concluding argument of 2.3. Let U, V, W,... be complex forcing conditions. U is extended by V (in symbols U D V) if U = (H, &>), V = (I, M),HDI a n d / H-„ & D 2.. The relation U D V is 2?, overL(o<) by 4.3. PROPOSITION 4.12b. Agpow (//, 0») « a complex forcing condition and § < a. Then there exists a complex forcing condition {H, 2) C (H, 9) such that /3 < rank 2.

Proof. Define 2 as in the proof of 4.12a. Thus rank M = y; and for each K e H, 2(K") is if-uniformly hyperarithmetically pointed. LEMMA 4.13 (Complex Splitting). Let {Pt \ i < w} be an a-finite sequence of Ey sentences of if(a, ST). Suppose U is a complex forcing condition such that ('W)uo y{EW)yDW[W H-0 #%. Then there exists aWCU

such that {i)[W hh3 &%.

Proof. The splitting of 3.8 and 4.4 are combined, a-finite partial functions t*, /„*, 2[<\ and r « are defined by recursion on i for all i < u>, some a e a1 and all; < 2 \

259 ORDINALS AND HYPERDEGREES

247

Suppose U is (H, &). Let 70° be H, 2%-° be &, let r0° be the null finite condition on K, and let r°-° be the null finite condition on T. Fix i and a. Assume t0\ I/, M™ and r*'* have been defined for all j < 2 \ Choose the rj+1's as in 4.4. (All choices are the least possible with respect to the standard J , wellordering of L(«) as in (3.8.5).) Thus the fj+1's constitute a diversified set of immediate successors to tj inside I,*. Choose P+l, 2\+1-> and ri+1^ (j < 2 t+1 ) to meet the following requirements. (4.13.1)

n+1e/:+1Ci0'.

(4.13.2)

(t)[te £■*-+$» +1

(4.13.3) I\ H-. [ri of r" inside - 2 ^ ] .

+12i

>*].

and r' +1 - 2j+1 are incomparable extensions

(4.13.4)

/; + 1 H-. [r\^

(4.13.5)

Same as (4) with 2/ replaced by 2; + 1.

(4.13.6)

( / r 1 ,
e 1 ^ ^ C <%*\. for a l l ; < 2*+l.

(4.13.7) Rank j2i+1-> > & for a l l / < 2 i + 1 , where ft is thesupremum of all ordinals occurring in the definitions of t™, I„m, jg»-' and r™>> for all m < i and p e «< for which they are defined. Requirement (4.13.7) is satisfied with the help of 4.12. The desired W is (J, Si). J is given by Aw | tj as in 4.4. For each KeJ, M(K) is given by A?; j r*j», where oi is the unique T such that K satisfies rT\ By (4.13.6) 9(K) H-. ^ for alli and all KeJ.lt remains only to show that (J, 3$) is a complex forcing condition, i.e., to find a S such that for all KeJ, S?(K) is ^-uniformly hyperarithmetically pointed via 0t and S. Fix K e J and Te@{K) to clarify the definition of 8. (The value of S, of course, will not depend on the choices of K and T.) For each r, let i:

=

\){i\-i\i<2'}.

(lr\ «T«) is a complex forcing condition since it is a finite union of complex conditions that are pairwise disjoint by (4.13.3) and (4.13.4). Again, let at be the unique T such that A' satisfies tT\ Since (/j,., J&) is a complex condition and T e J ^ X ) , it follows that c u / > rankJ*,.

260 248

GERALD E. SACKS

But then by (4.13.7), <*>/ is greater than every ordinal occurring in the definitions of ta\ Ja\ 2.\-\ and r««*. Consequently these four a-finite partial functions are in fact w^-finite, because all choices made in the course of their definitions were actually made within L(«/), (Keep in mind that the standard A1 wellorderings of L(«) a n d L ^ ) a r e compatible; see the last paragraph of Subsection 3.8.) Thus the entire construction outlined in (4.13.1)-(4.13.7) is an ^/-finite object. Hence rank St < « / . In addition there is an ^/-finite partial function ST* such that tf»nk*T' = {ST'}r0

(4.13.8)

for all K0eITl and T0 e £S(KQ), i.e., J2/(J£0) is uniformly /^-pointed via A* and if for all K0 e IrK Equation (4.13.8) suggests the following ^/-finite definition of \i \ ai by induction on i a0 = 0

a(i + 1) = unique r such that K™*3i< e (*+1 e 7*,.

(4.13.9)

The correctness of (4.13.9) needs to be checked. Clearly o{i + 1) is the unique T such that K e ri+1 e Poi. By 4.12b it is safe to assume rank 2L is greater than any ordinal mentioned by any member of/',.. Hence ^rank
Since AT is generic, there is an H such that K e H and ffrK&PH-*"*-].

(414.1)

261 ORDINALS AND HYPERDEGREES

249

Formula (4.14.1) makes it possible to extend the complex forcing relation of 4.10 to all sentences of jSf(«, ST). Let U be a complex forcing condition. Then U H-a JF if U = (H,&) and (4.14.1) holds. To make the definition of complex forcing precise, observe that for each n ^ 1 there is a Zn formula of JSP [a, Jf] that expresses P H-* & for all ZnF's of JSP(«, JT). PROPOSITION

4.15.

Suppose o < * and (tf,^)H-0{8}^Z.[a,jr].

Then there exist m, (I, »J), and (1,13)

such that (H, &>) D (I, »J) and

(/.•JB)H-.{«H«)»=w /or a// e < 2. Proo/. There is no harm in assuming {S}^ denotes the characteristic function of a subset of w. Fix a generic # e H and let P = ^ ( ^ ) . Then

As in the proof of 3.4 there exist Q, R, and m such that PDQ,

PDR

£?H-.{SH*) = o. and otherwise {S}7- would be hyperarithmetic in P and hence belong to L[«, /C] for every l e P . Thus / / H-. (£m)(F>^)(£1^)(t;)1,<2({8}^(m) = v & ^ D *J). Hence there exist / C H and m, °J2, *J such that (H, &) D (/, «jg) and (VJ)H-0{5}» = . for all v < 2. LEMMA 4.16. Let &{x, y) be a 27, formula of jS?(a, .T). Sw#>ose /S < a and U H-. {x*){Ey)&{x», y). Then there exists aV C U and ay < a such that

vwa (*»)(%") ^(**,y).

262 250

GERALD E. SACKS

Proof 4.16 follows from 4.13 and 4.11 in the same manner that 4.6 follows from 4.4 and 4.3. 4.17.

Unbounded h's and Inconsistent Formulas

Assume 8 < a. S is said to be unbounded via the complex forcing condition (H, 0>) if (£f,$»)H-.«l W '" = *Suppose 8 is unbounded. Let if(a, {8}-*") be the language obtained by replacing every occurrence of ST in the language £?(*, 3T) by {8y. There exists a 2^ formula jT6(y, n) = p of &{<x, {8}*) such that for each yea — w, {H, 0>) H-. ~~[An ; Jfe(y, n): w -±=±- y].

(4.17.1)

Xn | Jf6(y, n) is simply the first counting of y constructible from {8}^ via an ordinal less than a. A set 5 of sentences of i f (a, {8}*) is said to be pairwise inconsistent if the conjunction of any two distinct members of S is false in L(a, {S}7) for every T. The next three propositions isolate the splitting of finite conditions on K needed for the proof of complex minimization (4.21). PROPOSITION 4.18. Suppose 8 is unbounded via (H, 0>), t e H and S? is an ^-finite diversified set of immediate successors of t in H. Then there exist OL-finite sets {sD|pe b}, {J^ | | £ e} and {(Hp, 0>,) | p e e} such that: {sD|pe b} is a diversified subset of Sf\ {^D\Peb} is a set of pairwise inconsistent ranked sentences of £(«, {8}*~)) ands„ e HD, (H, &) D (Hp, @D) and (//„, ^„) H-. F, for all p e e.

Proof. Recall from 4.1 that all immediate successors of * in H have the same relevant domain d. If d is empty, then the conclusion of 4.18 can be satisfied trivially by setting {s.|Pe b} equal to a singleton {s} for some t e $f. Thus suppose d is {ft | i < *}, where z > 0. As in 4.1 £,(«) (= [£, is a cardinal & coffnallty 0, > w],

(4.18.1)

for all i < z. y is diversified, hence for each y e /7{ft | i < z}, there is an x.- and a i u e ^ such that

(>),<Mr,> «.") > nl

(4-18.2)

263 ORDINALS AND HYPERDEGREES

251

It follows from (4.18.1) that <«£v>1<2 can be chosen independently of y; i.e., there exists an <«,>< y j .

(4-18.3)

Call two members of Sft say *, and *2 , equivalent if *x(ft , n,) = J2 (& , «,) for all i < z. Form > 0 C y by choosing just one member from each equivalence class of ST. The advantage of S?a over & is that distinct members of S^ correspond to distinct members of J7{& | i < z}. (s corresponds to (s(Bn , «„),..., s(B, , , n. ,)>•) Thus, y„ = f j j p e c), where c C 71{ft | » < *}. ^ is diversified thanks to (4.18.3). For each pec there is an <m/>f<s and an (H,, ^„) C ( # , £?) such that s„ e H, and ( « , . &.) !+o (»W*"i(ft - «,') =

ft].

(4-18-4)

where X„ is the 2^ formula of (4.17.1). By 4.11 m» can be regarded as an tx-finite function of p. Hence the same reasoning that established the existence of <«<> in (4.18.3) also establishes the existence of an <m<> such that for each y e c, there is a p e c such that ('•),<>[/>■ >y,&™, D = 0 -

(4.18.5)

Let b be the set of all P's that satisfy (4.18.4). For each p e b, let 3F„ be

(i)i
«i) = P.]-

PROPOSITION

4.19.

Sup/ww* teH

and

(ff^H-.E^--] /or a// ; < i. 7%«i, rAerc e*«< a-/imle *er* {*„ | p e c}, { ^ | p e c} anJ) | p e c) {] as the set of all immediate successors of t in H. Assume that i > 1, and that the a-finite sets {i, | p e c], {&? \ p e c} and {(//„, »i) \Pec}(j < i) satisfy the conclusions of 4.19. It is safe to assume {s0 \ P e c} has the same form as ^ in the proof of 4.18. Thus c C J7{ft \ i < z} and s0 is

264 252

GERALD E. SACKS

the unique member of {s„\Pe c} with the property that *,(& , «;) = Pi for all i < z. For each pe c there is an < > / > i < s and an (7 0 , ^ / ) C {H., &1) such that *,e/„ and

(J„, ^.0 rM0<<.[*;(ft ,*«■)= ft]. where jfa is the Sx formula of (4.17.1). As in the second half of the proof of 4.18, there is an <wi> such that for each y e c, there is a /> e c such that (i) £ <,[p i >y < &m/ = « J .

(4.19.1)

Let b be the set of all p'$ that satisfy (4.19.1). For each Peb, let J " be (i) t < i [jr^/5,, »%) = p j . Then the a-finite sets {sp\Pe b}, {&j \peb) and {(/„, 9j) | P e b) (j < i + 1) satisfy the conclusions of 4.19. PROPOSITION

4.20.

Suppose teH

{H, 0>>) H-a h l 5 1 ^

and = a & {S}^ $L[OL,

X]],

for all] < i. Then there exist a-finite sets {sD\Pe c}, {Jy \j < i & p e c} am* {(#„, ^>/) | j < i & p e C} such that: {s„\Pe c) is a diversified set of immediate successors of t in H; {&,' \j< i &c p € c} is a set of pairwise inconsistent ranked sentences o/JSf(a, {S}^); ands, e H,, (H, &) D (Hp, 0>J) and (H„, &/) H-« &jfor all j < » and p ec. Proof Consider the a-finite sets supplied by Proposition 4.19. They satisfy all the claims of 4.20 save one: pairwise inconsistency of the members of {&j \j < i & p e c}. But 4.19 does provide inconsistency for the members of {P? \ p £ c} for each ; < i. Hence the missing inconsistency can be realized by developing a set {& \j < i) of pairwise inconsistent sentences of i f (a, i&V), and then replacing each J ^ ' bv ) in such a wav thaf (s I p £ b\ is a diversified set of immediate successors of *in H; and ,. e / / a n d

(J,, J„) H-. &J & 9", for all; < i and p eft. Each AT* is of the form &{3y' | k ^ j St, k < i}. The ^ / s are defined by means of a process with (J) stages. A typical stage is devoted to choosing a single pair (&„?, <S?) of inconsistent ranked sentences of JS?(«, {8}^).

265 ORDINALS AND HYPERDEGREES

253

For the sake of notational simplicity, only the first stage is described; it has all the features of the later stages. The first stage is a search for 9* and S y ; their existence will follow from the premise that (H,0») H-. [{&y t L(at, K)]. By 4.15, for each pec, there exist m° and (/„ , «2,) C (H,, &>p°) such that

for all v < 2. As in the second half of the proof of 4.18 there is an m and z b0Cc such that {s„ | p e b0} is a diversified set of immediate successors of t in H and m' = m for all p £ b0. (4.18 relied on the fact that if a diversified set is divided into countably many parts, then one of those parts is diversified.) For each p e b0 there is a V < 2 and an ( I , , 9.) C (/„, ^ V ) such that (/,,«JH-.{8H«)««*. As in the previous paragraph there is a tu < 2 and a 6, C i 0 such that {*. | p e i x } is a diversified set of immediate successors of t in H and ©» = to for all pe bx. Let ^ be [{S}^(m) = 1 - w] and let SF0» be [{Sy(m) = to]. Then each p e 4, has the following properties: # „ D I,,

and

Each of the remaining Q - 1 stages has the same pattern of contractions of forcing conditions and diversified sets as the first stage. LEMMA 4.21 (Complex Minimization). (H, &) Wa W?r

Suppose

= afc {8}^ *L[«, •#-]],

TTien rfcere «ctrtJ a (/, 3?) C (//, ^ ) such that

Proo/. The stratagems of 3.4 (Case 2) and 4.13 are combined. (J, Si) will be defined so that {8} is one-one in the following sense: suppose

266 254

GERALD E. SACKS

Kt e J and Ttem(JQ (*' < 2); if (K0 , T0) # (J^ , 7\), then {8}r° / r {8} '. The one-one-ness of {8} will make it possible to recover (K""**, T) from {BY whenever K e J and T e # ( * ) . a-finite partial functions *.<, /„*, J « , r<-\ and . F ^ ' are defined by recursion on i for all i < w, some a e «< and a l l ; < 2*. The definition proceeds as in 4.13 with no change in (4.13.1)-(4.13.5) and (4.13.7), but with (4.13.6) replaced by (4.13.6*) below. (It*1,1?1J)

H-. ft?1-*,

(4-13.6*)

where {9**U |y < y+i & T g bj) is a set of pairwise inconsistent ranked sentences of JP(«, {S}^). Requirement (6*) is satisfied with the help of 4.20 and 4.15. By 4.20 there exist (J1*1, &/) and 0™ {j < 2i and T £ bj) such that: r*+1 e / T i+1 ,

(/„',££•') 3 (jr1.^*). and j F ^ ' s constitute a set of pairwise inconsistent ranked sentences of jBf(«, {8}^). By 4.15, there exist 7'+1, :«*+*•« J«**'* +1 , and m« (; < 2 i and r e i 0 < ) such that: rj + 1 £/; + 1 , (;;+1,^-')D(/i+1,^+1^),

(4.21.1)

(/;+1, *«•«):+»{8}^(m^) = o,

(4.2i .2)

and (4.21.1) and (4.21.2) with 2; replaced by 2; + 1 and 0 by 1. Then, 3**.w i s [ j r « & {S^wJ.i) = 0] and grJ+».W+» is [ J F ^ & {8}^(<->) = 1]. Define (/, M) as in 4.13. The argument given in 4.13 to show (/, SI) is a complex forcing condition still holds. Fix K e J and T e J>(/Q to see how to recover ( K r a n k * T) from {8}r by a uniform method encoded by an ordinal less than <x. K™™ corresponds to a unique path through / . Thus there is a function ai such that

Similarly there is a function hi such that {T} = fl {^/"(Krank*)

j i < «}.

Thanks to (4.13.6*) a simple recursion on i yields c« and hi: o0 =

267 ORDINALS AND HYPERDEGREES

hO = 0; (<j{i + 1), h(i + 1)) is the unique (r,j) j < 2i +1 and

255

such that r e b^ ,

The functions oi and hi belong to£(a, {§}r)> because the entire construc­ tion of (/, 0t) can be viewed as an element of L(a), and because just one of the ranked sentences <S\+^ (T £ bj a n d j < 2 i+1 ) is true in L(«, {S}r). LEMMA 4.22. Let K be generic in the sense of 4.2. Suppose that the set of all p < a such that

L(a) f= IP is a cardinal] is not bounded below a. Then O J / < <xfor every X eL[oc, K]. Proof. Suppose not. Then there is a term c of rank y < <x such that

In addition there is an H such that K e H and # H-« w^ = <*.

(4.22.1)

Let |3 be a cardinal in the sense of 1(a) such that c, H eL[fi). A sequence {/„ | a < co} of perfect conditions on K is defined by recursion on n. I0 is H. In+1 is the "least" / such that TnDl

and

7|+ 0 [7ieO c ]

if there is such an /; if not, 7n+1 is In.[ne Oc] is a 27j sentence of JSP[«, jf] that says n is a notation for an ordinal recursive in w].

268 256

GERALD E. SACKS

When Z* is not bounded below «, Theorem 4.26 follows fairly directly from Lemmas 4.21 and 4.22, but when Z* is bounded, a modification of 4.21, namely 4.25, is required. Suppose the supremum of Z * is ft, < a. Clearly ft, is anZ,(«)-cardinal. If ft, e Z*, define Z 0 * = {ft,}. If ft, * Z*, define Z 0 * = {ft | i < to}, where Xi \ pi is an ct-finite sequence of members of Z * whose supremum is ft, . Z0* is used to simplify the convention on K stated in 4.1 as follows: K is a map from (Z 0 * - to) x to into ft, such that if p e Z 0 *, then the range of Xm | K(fi, m) is an unbounded subset of ft The new convention on K implies AT eL[«, X ] as well as L[a, AT] |= (*)(* is countable).

When working with the new convention, it is convenient to reduce the collection of forcing conditions H,I, /,..., of 4.1 to those that mention only those fts in Z„*. Since Z* is countable, no difficulty is caused by requiring that each forcing condition H mention every member of Z 0 *. Thus each path through H is the complete graph of a map from (Z 0 * — to) x to into ft, . Let {n}* be a term of JS?[«, J f ] that denotes a map from (Z 0 * - to) x to into ft, . The n is said to be unbounded via / / i f H H-, [Am | {H}-*"^, M): to

'"*>♦ j8],

for all 0 e Z*. Let J8?[«, {«}•*"] be the language obtained by replacing every occurrence of X in the language i f [a, Jf] by {n}*. A set 5 of sentences of i f [a, {»}*"] is said to be pairwise inconsistent if the conjunction of any two distinct members of S is false in L[a, {»}•*'] for every K. LEMMA 4.24.

Suppose n is unbounded via H as in 4.23, * e H and &>

is an oc-finite diversified set of immediate successors of t e H. Then there exist oc-finite sets {s„\ Pe b), ffi, \ p e b] and {//„ | p e b} such that: {s„\ pe b} is a diversified subset of S>\ {&, \ p e b} is a set of pairwise inconsistent ranked sentences of i f [a, {n}*~]) and speHp, H D H,, and H,W, ^o, for all p eh. Proof Same as that of 4.18 with Jf6 replaced by {«}•*', and (//, 9) by H. Thus (4.18.4) becomes

and W„ becomes ( i W W

4

^ , «,) = ft].

269 257

ORDINALS AND HYPERDEGREES

LEMMA 4.25. Suppose n is unbounded via H as in 4.23. Then there exists a JCH such that

/ H-« p r e £.[<*, {«}•*"]). Proof. Similar to, but much simpler than, the proof of 4.21, with 4.24 taking the place of 4.20. J is denned so that n is 1-1 on / , thereby making it possible to recover K from {n} jr when K e / . As in 4.21, a-finite partial functions IJ, r„\ and P* are denned by recursion on t for all i and some a e a<. I0° = H and r0° G / 0 ° . The t\+1's constitute a diversified set of immediate successors to t* inside I/. In addition the following requirements are met with the help of 4.24. t?1 el™ CIW',

(4.25.1)

{t)[tel™^t™^t},

(4.25.2)

lMYr^,

(4.25.3)

where the JF»t in (4.25.3) form a set of pairwise inconsistent ranked sentences of ^{a,{n}^\. Let J be the perfect condition given by Xio \ ta\ Fix K e J to see how to recover K from {«}•*' by a uniform method encoded by an ordinal less than a. K corresponds to a unique path through / . (Keep in mind that the conventions on K and J established in 4.23 hold throughout 4.24 and 4.25.) Thus there is a function m such that K = (J {tU \ i < <,}. A simple recursion on i yields oi. a{i + 1) is the unique T such that L[a, {«}■*-] (= Fj. THEOREM 4.26. Let <x be a countable admissible ordinal greater than w. Then there exists an X C w such that

(i) a , / = a; (ii) (Y)[Y
<«].

There are four cases of which the third is the most trouble­

270 258

GERALD E. SACKS

Case I. There is an XeL(*) such that e o / = a. Every such X sstisfies (ii), since every real in L(«) is hyperarithmetic in every Y with the property that « / ^s a. Owe 2. Z* (defined in 4.23) is empty and the hypothesis of Case 1 does not hold. Thus L(a) satisfies (*) (* is countable). Let T be generic in the sense of 3.5 with A equal to L(<x). The argument of 3.8 shows wj < «. By (3.6.1), w/ > a. Suppose Y
Let n ^ 1. An ordinal a is said to be Zn admissible if £,(a) satisfies Rep(fi), the replacement axiom schema of ZF restricted to 27,, formulas.

271 ORDINALS AND HYPERDEGREES

259

For each real T, u>l is the least <x such that L(x, T) satisfies Rep(s). A real 5 is En recursive in T if SeL(wTr , T). Two reals have the same En degree if each is En recursive in the other. THEOREM 5.1. Let a be a countable En admissible ordinal greater than o>. Then, there exists a T such that:

(i) (ii)

a>£ = <*; cu£ < <xfor every S of lesser En degree than T.

For n > 1 a T with property (i) was first discovered by Jensen. A sketch of the proof of 5.1 was given in [20]. That sketch should not be difficult to complete after a reading of Section 4 of the present paper. The appropriate notions of pointedness are as follows. P is En pointed if

(T)[TeP-^PeL(wl;T)). P is uniformly En pointed via 8 if (r)[reP-*P

= {8}r&8
The proof of 2.10 can be modified to show: Each En pointed P contains a uniformly En pointed Q of the same En degree as P. Note that 2.2 and 2.3 hold for En pointedness. P is if-uniformly En pointed via & and 8 if for all T £ P: P < wln 8 < «,£_

and and

P = &{KB), KB = {8}T.

When n > 1, the proof of Lemma 4.13 (complex splitting) needs only one substantial change to make it work for En ,F/s and complex conditions whose virtual components are En pointed. Requirement (4.13.7) is augmented by

W)
(5.1.1)

The forcing relation hK restricted to En formulas is En , hence (5.1.1)

272 260

GERALD E. SACKS

implies that the (7*+1, J * + w ) chosen inL(a) to force ^^ could just as well have been chosen in L(su P i ft). Thus, as in 4.13, the entire construction of (J,&) is an wTr ffinite object when Te&(K) for some KeJ, since in that event wT£ " ^ sup,- ft . Note that the predicate L(j3) < „ _ ! L(a) i s r n overL(a). Call a transitive set ^ Zn admissible if it is closed under the operations of pairing and set union, and satisfies the An separation and bounding schemas of ZF (cf. 2.4). THEOREM 5.2. Let A be a countable Zn admissible set that satisfies En dependent choice. Then, the En degrees contained in A have a minimal upper bound; if they have a least upper bound, then, that least upper bound is contained in A.

The proof of 5.2 is similar to that of 3.14. The only difference is the use of an elementary chain argument as in 5.1. THEOREM 5.3. Let M be a countable standard model of ZF and V = L. Suppose JT is a regular uncountable cardinal of M. Then there exists a TCw such that:

(i) M{T) is a model of ZF in which X is the least uncountable cardinal; (ii) if S e M(T) and M(S) has the same properties as M(T) in (i), then M(S) = M(T). The proof of 5.3 for Jf = a>2 is due to Prikry. A T with property (i) was given by Jensen and Solovay in [21]. The proof of 5.3 is similar to that of 4.26. The appropriate notions of pointedness are as follows. P is constructibly pointed if (T)[TeP-+

PeL(T)].

P is uniformly constructibly pointed via 8 if (T)[TEP^P

=

{S}T&8<^w]>

where wf""1 is the least uncountable cardinal of L{T). P is X-uniformly constructibly pointed via 0> and S if for all T e P: P = 3»{KB)

and

£ < wf{T),

KB = {S}r

and

8 < cuf'7"'.

Of course, P, ft and S belong to M.

273 ORDINALS AND HYPERDEGREES

261

5.4 Questions (5.4.1) Suppose A is a countable En admissible set such that the En degrees contained in A have a least upper bound. Does that least upper bound belong to A ? If it is assumed that A satisfies En dependent choice, then the forcing argument of 5.2 provides an affirmative answer. The compactness argument of 3.20 does not require En dependent choice but appears to succeed only when n = 1. It is tempting to think that there exist compactness theorems for En admissible sets (n > 1) sub­ stantially different from Barwise's for the E1 case. (5.4.2) Does every countable set of hyperdegrees have a minimal upper bound ? ACKNOWLEDGMENT The author wishes to thank B. Dreben, G. Kreisel, A. Nerode, and H. Rogers for their unfailing encouragement.

REFERENCES 1. H. FRIEDMAN AND R. JENSEN, A note on admissible ordinals, m "Lecture Notes in Mathematics 72," pp. 77-78, Springer-Verlag, 1968. 2. T. GRILLIOT, Omitting types; applications to recursion theory, / . Symbolic Logic 37 (1972), 81-89. 3. H. J. KEISLER, "The Model Theory of Infinitary Languages," North-Holland, 1971. 4. G. E. SACKS, Forcing with perfect closed sets, in "Proc. Symposia Pure Math. X I I I , " pp. 331-355, Amer. Math. Soc, 1971. 5. G. E. SACKS, "Degrees of Unsolvability," Princeton Univ. Press, 1963; 2nd ed., 1966. 6. H. FRIEDMAN, Minimality in the J , 1 degrees. Fund. Math,, to appear. 7. S. C. KLEENE AND E. L. POST, The upper semilattice of degrees of recursive un­ solvability, Ann. Math. 59 (1954), 379-407. 8. R. PLATEK, Foundations of recursion theory, Ph.D. Thesis, Stanford Univ., 1965. 9. J. BARWISE, Infinitary logic and admissible sets, / . Symbolic Logic 34 (1969), 226-252. 10. S. KRIPKE, Transfinite recursions on admissible ordinals (Abstracts), J. Symbolic Logic 29 (1964), 161-162. 11. C. SPECTOR, On degrees of recursive unsolvability, Ann. Math. 64 (1956), 581-592. 12. G. KREISEL, The axiom of choice and the class of hyperarithmetic functions, Indag. Math. 24 (1962), 307-319. 13. G. BOOLOS AND H. PUTNAM, Degrees of unsolvability of constructible sets of integers, / . Symbolic Logic 33 (1968), 497-513. 14. C. SPECTOR, Hyperarithmetical quantifiers, Fund. Math. 48 (1959), 313-320. 15. S. FEFERMAN, Some applications of the notion of forcing and generic sets, Fund. Math. 56(1965), 325-345.

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GERALD E. SACKS

16. R. GANDY AND G. E. SACKS, A minimal hyperdegree, Fund. Math. 61 (1967), 215-223. 17. S. C. KLEENE, Recursive functional and quantifiers of finite type. Trans. Amer. Math. Soc. 91 (1959), 1-52; 108 (1963), 106-142. 18. G. E. SACKS, The 1-section of a type n object, in "Generalized Recursion Theory," pp. 81-96, North-Holland, 1974. 19. W. RiCHTER, Constructive transfinite number classes, Bull. Amer. Math. Soc. 73 (1967). 261-265. 20. G. E. SACKS, F-recursiveness, in "Logic Colloquium '69," pp. 289-303, NorthHolland, 1971. 21. R. B. JENSEN AND R. M. SOLOVAY, Some applications of almost disjoint sets, in "Mathematical Logic and Foundations of Set Theory," pp. 84-104, North-Holland, 1970. 22. S. SIMPSON, Minimal covers and hyperdegrees, Trans. Amer. Math. Soc, to appear. 23. C. JOCKUSCH AND S. SIMPSON, A degreetheoretic hierarchy for ramified analysis, An.. Math. Logic, to appear.

275

THE K-SECTION OF A TYPE n OBJECT.' By GERALD E. SACKS.

1. Introduction. This paper includes a proof of the plus-one theorem for the case k> 1. The case k = 1 was covered in [1]. Suppose n> k> 1 and F is a normal type n object. Then there exists a normal type k+1 object G whose k-section equals that of F. PLUS-ONE THEOREM.

The ^-section of a finite type object H consists of the type k objects recursive in H; it was first defined and studied by Kleene [2]. H is said to be normal if the equality predicate for objects of type less than that of H is recursive in H. The plus-one theorem states that fc-sections generated by normal finite type objects are in fact generated by normal type k+1 objects. Thus for each k>1, very little about the type of a normal object can be inferred from its fc-section. The role of normality in the plus-one theorem is uncertain, because there exist substantially nonnormal objects whose fc-sections are the same as those of normal objects. But at the least it may be noted that the normality of H imposes a hierarchic structure on the computations associated with recursion in H, and that hierarchies are often amenable to forcing and stability arguments. The proof of the plus-one theorem appears to split into two irreconcilable cases: *= 1 and * > 1 . The difference between the two merits more reflection than a brief introduction allows. Perhaps it will suffice to draw attention to the most immediate difference between the 1- and 2-sections of a normal type n object "H. Each member of the 1-section of 2H owes its existence to a computa­ tion with the property that it and all its subcomputations are encoded by members of the 1-section. However more than one member of the 2-section of 3 H derives from a computation such that it but not all its subcomputations are encoded by members of the 2-section. It is loose yet not misleading to say the 2-section of 3H exhibits gaps. The presence of gaps is to be expected in the study of effective computations of uncountable length. Manuscript received November 22, 1974. 'The preparation of this paper was partially supported by NSF contract GP-29079 and by the Institute for Advanced Study, Princeton, New Jersey. The author is indebted to A. Kechris and L. Harrington for many discriminating suggestions. American Journal of Mathematics, Vol. 99, No. 5, pp. 901-917 Copyright © 1977 by Johns Hopkins University Press.

901

276 902

GERALD E. SACKS.

Section 2 reviews the elements of recursion in a normal object via the hierarchic approach, introduces the notion of admissible set with gaps and shows it to be exemplified by the it-section of any normal object of type greater than k, defines a notion of reflection for recursion in a normal object, and proves the lemmas of simple and further reflection, the latter lemma a central result due to Harrington [3]. Section 3 introduces the notion of countable abstract it-section (Jt>l), a combination of admissibility and reflection, and illustrates it by means of the it-section of any normal object of type greater than k, devises a version of forcing to show each countable abstract Jt-section is the it-section of some normal type lfc + 1 object, and ends with the concept of stable counting, equivalent to but simpler than the concept of countable abstract ^-section. Section 4 sketches further results, some actual, others potential, by stating Harrington's plus-two theorem [3] for it-envelopes (defined by Moschovakis [4]), commenting on an extended plus-one hypothesis, and formulizing a plus-three problem.

2.

Recursion and Reflection.

2.1. Finite Types. Let m be a nonnegative integer. Tp(m) is the set of all objects of type m. Tp(0) = co, and Tp(m+1) is the set of all functions from Tp(m) into w. Standard, effective coding devices make it safe to view an object of type m as an object of type q for any q>m, to construe a finite sequence do,. •.,a, of objects of types at most m as a single object (a 0 ,...,a r > of type m, to regard a function from Tp(m) into Tp(m+1) as an object of type m + 1, and to think of an arbitrary object of type m + 1 as a set of objects of type m. Sometimes it will prove convenient to equate Tp(m) with its set-theoretic counterpart R(u+m). (fl(0) = 0, R(8 + l) = 2H(e\ and R (X)= u {R (8)\8<\) when A is a limit ordinal.) Fix n > 0 in order to establish some conventions which will on occasion be violated. The objects of type n + 2 are denoted by F,G,..., and are called functionals. The objects of type n + 1 are denoted b y / , g , . . . , and are called functions. The objects of type at most n are denoted by a,&,..., and are called individuals. The set of all individuals is denoted by /. A typical function is a map of / into w. Subsets of / are denoted by R,S,.,., and are called sets. The objects of type less than n, together with the nonnegative integers, are denoted by r,s and are called subindividuals; the set of all such is denoted by SI. These conventions make it possible to state certain results about objects of finite type without making any explicit references to the type involved.

277 fc-SECTION OF A TYPE n OBJECT.

903

£ is the equality functional for sets:

£ R S

< - >-I° * s ; l

(£ is equivalent to Kleene's n + 2 £.) F is said to be normal if F is of the form where F' is a functional of the same type as F More generally F is said to be normal if £ is recursive in F, since in that event F and (F,E) have the same degree; i.e. each is recursive in the other. The assumption of normality renders the study of sets recursive in F amenable to methods associated with the hyperarithmetic and constructible hierarchies. Every F in this paper is normal unless otherwise indicated. 2.2. The L(F) Hierarchy. Let F and / be as in 2.1. Harrington [3] found the following hierarchy of sets constructible relative to F to be useful in the study of recursion in F. L(0,F)=/=fi(w+n)=rp(n). XGL(o + l,F) iff XCL(CT.F) and X is first order definable with parameters over M(a,F). M(a,F) = . L ( \ , F ) = u { L ( 6 , F ) | 5 < X } when X is a limit ordinal. L(F)=u{L(a,F||o}. Note that if n = 0 and F = 2 £, then L(F) = L. Let £(ff) be a first order language suitable for dissecting L(F). £ ( f ) is essentially the language of ZF augmented by ranked variables x\y',... for each a, set constants to name the elements of L(F), and a function symbol f. L(a,F) will be of primary interest only if a is recursive in F in the sense of 2.3. If F is 2 £, then the ordinals recursive in F turn out to be the familiar recursive ordinals, but if n>0 then certain complexities occur, the simplest of which is the existence of gaps between some of the ordinals recursive in F. 2.3. Set and Ordinals Recursive in F. The use of hierarchies to develop Kleene's notion of recursion in a normal F goes back to [5], [6] and [7]. The following version, devised by Harrington [3], stays close to the idea of defining the hyperarithmetic reals by iterating the Turing jump. Let X C /. The F jump of X, denoted by ;X, is the effective disjoint union of certain definable subsets of /. Let {e)xT be the e-th function (e-th according to some standard Godel

278 904

GERALD E. SACKS.

numbering of formulas) from / into by means of a first order formula whose only parameters are nonnegative integers, and whose atomic parts are of the form yEz and yE%, where y and z range over / and % is a set constant corresponding to X. (Here 7 is of course equated with fl(u+n).) Let Wexbe{a\aelb{e)x„(a) = 0}. Then jX is the set of all triples <e,a,0> such that aGWex, and all triples (e,a,y + l) such that F({e}<x-">)« y. The first part of jX encodes subsets of / first order definable using only integer parameters; the second part encodes the F-images of subsets of I definable using arbitrary individuals as parameters. If F is 2E, then /0 is equivalent to the effective disjoint union of the arithmetic reals. OF,\ \F:0F^On, and H0F (aGrange | \F) are defined by a simultaneous recursion. (1) For all a e l , < l , a > E O F , ||F = 0, and HF = Q. (2) If (e,a)eOF and |<e,a>| r =o, then |<2 e ,a>|GO F , |<2e,a>| = a + l ,

and HaF+ , - / £ t f .

(3) Suppose (m,a)eOF, \(m,a}\F=a, and W * > c O F Then <3mF m e F 5 ,a>GO and |<3 -5 ,a>| = A, where X is the least limit ordinal greater than \b\F for all b e WeU {<m,a». In addition HF is the set of pairs such that e

bEOF

&

\b\F=o<\

&

cGHF.

Let KF be the least ordinal that does not have a notation in OF. The hierarchy {HCF\O for all aGl. {e}F(a) is said to be defined and equal to x if e = <e0,e1>, < e 0 , a ) £ O f , ]<e 0 ,a>| F =a, and { e , } f ( o ) - x . / is recursive in F (in symbols / < F) if there is an e such that / » = {e} F (a)foralla. A set R of individuals is recursive in F (in symbols R < F) if the character­ istic function of R is recursive in F. An ordinal a is constructive in F if a = |<e,0>|F for some <e,0>6O F . (In short o has an integer among its notations.) a is recursive in F if a is the height of a prewellordering of 1 recursive in F. Every a constructive in F is recursive in F, and each ordinal recursive in F is less than some ordinal constructive in F.

279 K-SECTION OF A TYPE M OBJECT.

905

The hierarchy {Hf\aefl/+j

(4)

for all a. A set R is recursively enumerable in F if there is an e such that for all a, aGR~{e}F(o)

is defined.

(5)

Further properties of the hierarchy are based on Grilliofs selection princi­ ple proved by Harrington and MacQueen [8]: there exists a uniform method for selecting a nonempty recursive (in F) subset of a nonempty recursively enumer­ able (in F) set of subindividuals. To be more precise, there exists a recursive number-theoretic function t such that for all e: if

(£r)[{e}F(r)

then and

(r)[{te}f(r) F {r)[{te} (r)

is defined], is defined] = 0^{e)F(r)

(6) is defined].

Moschovakis [9] has shown that when n > 0 there exists a set of individuals recursively enumerable in F with no nonempty subset recursive in F. Gandy's selection principle, which antedates Grilliofs, is also useful. It provides a uniform method for selecting an integer from a nonempty recursively enumer­ able (in F) set of integers. For each o < K F , HF and M(a,F) convey roughly the same information, so it is not surprising that they are equivalent in a sense made precise in Lowenthal [10]. The f/„F's are needed only to see which ordinals are construc­ tive in F. Once those ordinals are known they can be used to read off the sets recursive in F from the M(o,F)'s. The equivalence of Kleene's notion of recursion in F via schemes with the above is proved in detail in [10]. It must not be forgotten that the equivalence holds only for normal F*S, and consequently that nothing in this paper casts any light on Klpcne recursion in nonnormal objects.

280 906

GERALD E. SACKS.

2.4. Admissible Sets with Caps. It is not hard to verify that a set of integers is recursively enumerable in F iff it is I , definable over M (K0F,F) by a formula of £(£) with integer parameters only. When n = 0, M(K^F) is an admissible structure, but when n>0 it fails to satisfy AQ collection for reasons associated with the existence of gaps between some of the ordinals recursive in F. The failure can be circumvented by invoking the notion of admissible set with gaps, a slight modification of Barwises notion of admissible set with urelements [11]. A set A is admissible with gaps if A is closed under the operations of pairing and union, and satisfies the principles of \ , separation and collection when the quantifiers occurring in those principles are suitably interpreted. An unrestricted quantifier ranges over A, a restricted one over the transitive closure of A. Consider a typical instance of A0 separation: (Ex)(y)[yex~yeb

&

§(y)].

b and all parameters of the \ formula §(y) belong to A. The proposed interpretation requires x to be a member of A and y to range over all elements of x, not merely those in A. AQ collection is treated similarly. A typical instance is: (x)xeb(Ey)§{x,y)-+(Ez)(x)xf;b(Ew)wez§(x,w), where b and all parameters of the A,, formula § (x, y) belong to A; y and z must be members of A, while x can be any member of b. is admissible with gaps means A remains admissible with gaps when the \ formulas are augmented by 5 r (x)=y and the latter is interpreted as F(x)=y. The ^-section of F, denoted by kscF, is the set of all objects of type k recursive in F. It can be viewed as a set theoretic object, in fact an admissible set with gaps when k< n + 1, by interpreting objects of type k as codes for sets hereditarily of cardinality at most 3k_v (Recall 3o = w a n ^ H P + i = 2 3 -.) Let m be the map that takes a code b to the set mb coded by b. The relation b

is a code

& mb = x,

(1)

where b: Tp(k- l ) - » y * ) and x is hereditarily of cardinality at most H _ „ is defined by induction on the rank of x: b is a code for {m(br)\rE Tp(k-l}}. The set of all codes of type k is recursively enumerable in lc+1£ when fc=,, and recursive in * + 1 £ when fc>l, since a code of type k is essentially a reflexive wellfounded binary relation on T (k-1). It is important to note that a coding relation need not be antisymmetric. Thus an ordinal o recursive in 3 £ is in

281 fc-SECTION OF A TYPE fl OBJECT.

907

general coded by a prewellordering, rather than a wellordering, of the reals recursive in 3 £, since there may not exist a wellordering recursive in 3 £ of height a. PROPOSITION 2.5. Let b and c be codes and § (y) a A0 formula of ZF whose parameters are coded by c. Then the predicate P(d), defined by

(Er)[d=br & 8(md)], is recursive in *+1E,b,c. Proof. The restricted set quantifiers of § (y) correspond to quantifiers ranging over Tp(k-1), and the latter are equivalent to * + I £. For each Z c Tp(k) let 9HZ be the set of all sets with codes in Z. There is no harm in dropping the distinction between Z and 9R.Z (e.g. to say Z, rather than 3ltZ, is an admissible set with gaps) when each member of 9IIZ is coded by some member of 9HZ. PROPOSITION 2.6. Assume k is admissible with gaps (cf. 3.2).

Proof. To verify \ separation, let b,c6kscF be codes and §(y) a A0 formula whose parameters are coded by c. It suffices to find a code b*GkscF such that (y)[yEmb*~yemb

&

§(y)]

holds in <9H(Jk«cF),F>. By 2.5 the desired b* exists and is a map from Tp(k-l) into Tp{k) whose range is the set of all d that satisfy the predicate P{d) of 2.5. Note that the presence of F in § (y) causes no difficulty, because only part of the graph of F is relevant to yemb

&

§(y),

and that part is coded by an element of kscF determined by b,c. To verify AQ collection, let b,cGkscF be codes and §(x,y) a A0 formula whose parameters are coded by c and such that (x)semb(Ey)§(x,y) holds in 91t( t scF). For each r e Tp(k-1) {e)F is a code and

let Qr be the set of all e such that

§(m(br),m{e)F)

282 908

GERALD E. SACKS.

holds in 9lt{ k scF). Since Qr is recursively enumerable in (F,r~), Gandy's selection principle (2.3) yields a function t such that t is recursive in F and tr(=Qr for all r. It follows from the bounding principle (2.3) that there is an ordinal o recursive in F such that {tr}F is one-one reducible to H0+1 for all r. Since HB + 1 is recursive in F, a code dEkscF can be extracted from Ha+, with the property that

2.7. OdinaZs r/iaf fle/fert to < • The notion of reflection (or stability) is needed to describe some further properties of kscF, properties that stem from Grilliot's selection principle rather than Gandy's. In section 3 it will be shown that the notions of reflection and admissibility with gaps suffice to characterize fc-sections of normal objects of type greater than k for all k>l. For each i < n let KF be the supremum of all ordinals with notations in 0F of type i; i.e. O|F for some ( e \ a > E O F such that aGFp(i). Thus « * = « / . An ordinal r is said to be reflecting if for every 2 , sentence S of £"f) with integer parameters only, it is the case that M(T,F)r= g

implies

M(KF,F)\=

§.

The 2t sentence (£a)[eGH/] is typical, since each HF+1 encodes all first order facts about {HaF\a < y} in a positive way. Thus to say T is reflecting is to say: an Hf that stands in a A0 relation to some integer is developed before stage T only if one is developed before stage y for some y recursive in F. Note that if y is recursive in F and K,F (otherwise KF would be of no interest); on the other hand Harrington [3] has shown KF is much greater than «F_, in a sense whose precise meaning is not relevant here. In order to prove the plus-one theorem it is enough to know Kf > KF_ „ an inequality equivalent to the stability result stated at the end of [1]. 'Kechris and Harrington (3] have found an intrinsic definition of K,F, i.e. a definition in terms of the notion of recursive enumerability in F rather than in terms of a hierarchy generated by F in a somewhat arbitrary fashion.

283 fc-SECTION OF A TYPE fl OBJECT.

LEMMA 2.8. (Simple Reflection).

909

K,F > KF_ ,.

Proof. Fix e and assume eGHF is satisfied by some a
& (Ea)[o<\s\F

& eGHaF].

(1)

Let S be the set of all s that satisfy (1). By Grilliot's selection principle (2.3 (6)) there is a nonempty recursive (in F) TcS. But then the bounding principle (2.3) yields a y recursive in F such that | s | F < y for all s G T. Clearly e G HF for some a< y. The next lemma will be needed in section 3 to prove the converse of the result that implies the plus-one theorem whenfc= 2; i.e. to show the 2-section of a normal type 3 object is an abstract 2-section. LEMMA 2.9. (Further Reflection, Harrington [3]). If R is a recursively enumerable (in F) set of subindividuals, then K,F > «£•*>. Proof. Fix e and assume e G H / is satisfied by some o with the intent of showing it is satisfied by some a < KF. Thus there is a subindividual s such that seO

& (£5)[6<s5|F'',>

&

eGH/].

Let W be the set of all individuals c such that c is a weUordering of all subindividuals. Note that W is recursive in F because n > 0 and F is normal. For each cG W, let \c\ be the ordertype of c. Call an ordinal y subconstructive (in F) if Y = |s| F for some subindividual s G O F . Let A be the ordertype of the subconstructive (in F) ordinals. A splits W in a curious fashion: (A) {c | |c| < A) is recursively enumerable in F. (B) If \c\ > A, then e G HF for some o < 4F-C\ For each sG Of, let |«| be the ordertype of the set of all subconstructive (in F) ordinals less than \s\F. To prove (A) observe that

(Es)[sGOf

& |c|<|s|]

is recursively enumerable in F, since Grilliot's selection principle implies the recursively enumerable (in F) predicates are closed under existential quantifica­ tion over subindividuals ([12]). To prove (B) assume |e| > A. Then there is an initial segment of c, call it c„ such that |c s | = A, where cs is either c or that part of c below some s in the field

284 910

GERALD E. SACKS.

of c. R is recursive in (F,c,s), since the recursive (in F) enumeration of R comes to an end as soon as {HaF\a < KF_J} is developed, and the latter occurs as soon as Of has been enumerated to the point where the subconstructive ordinals have ordertype equal to \cs\. Consequently K ^ f >< K<*f, and so there is a o < #e<£f> such that e&Hf. It follows from 2.8 relativized to c that (EO)[O
& e£H/..

(1)

The set of all c that satisfy (1) is recursively enumerable in F, since an ordinal is constructive from (F,c) iff it has a notation in OF of the form (m,c) for some integer m. (A) and (B) make it possible to enumerate all of W recursively in F as follows: enumerate c if |c|<X or if c satisfies (1). (Gandy's selection principle implies that the union of two sets recursively enumerable in F is recursively enumerable in F.) Since W is recursive in F, the bounding principle implies the enumeration of W is complete by some O0Sections. 3.1. Abstract k-Sections. Assume k> 1. A set A is said to be locally of type k if Tp(k- 1)6 A and each member of A is coded as in 2.4 (1) by some element of Tp(k)n A. A pair (A,B) of sets is called stable if A e B , and A < X B (i.e. A CB and each 2j sentence of ZF (with parameters in A) is true in (B,e) only if it is true in 1) if there is a B such that: (1) (2) (3) (4)

A and A and A and (A,B)

B are locally of type k. B are admissible sets with gaps. B are closed under recursion in is stable.

k+1

E.

The above definition has not been chosen with much respect for economy (e.g. in the presence of (4), (3) is equivalent to something much weaker), because in all practical cases the problem of showing A is an abstract k-section quickly reduces to finding a B such that (A,B) is stable. As a rule if there exists a B satisfying (4) then there exists one which can be developed by some inductive definition over A. Clause (3) is not superfluous: there does exist an admissible set with gaps which is locally of type 2 but not closed under recursion in 3 £. Thus the gaps do

285 fc-SECTION OF A TYPE n OBJECT.

91 1

make a difference, since every admissible set (without gaps) which is locally of type k is closed under recursion in t + 1 £ for all k> 1. But the only substantial difference between an abstract it-section ( * > l ) and an abstract 1-section ([1], pg. 84) is the stability imposed by (4). An abstract Jc-section A is said to be countable if the B of (l)-(4) has the additional property that HA is countable).

(5)

Suppose Z C Tp(k). Z is said to be a countable abstract it-section if 9ltZ satisfies (l)-(5) (cf. 3.12). LEMMA

3.2. Assume Jfc< n + L Then kscF is a countable abstract k-sec-

tion. Proof. Fix k > 1. Let P be the complete recursively enumerable (in F) set of integers, and let B be <m,((sc(F,P)), Clearly B is closed under recursion in l+1 £ since k < n + 1 and F is normal. By 2.6 B is an admissible set with gaps. The presence of P in B implies H9lt( t *cF) is countable). The stability of 9Ti( t scF) in B follows from the reflection of K to K0F. Simple reflection (2.8) suffices if n > l , since in that case F e F p ( n - l ) . But further reflection (2.9) is needed if n = 1. 3.3. The Forcing Relation ffz. Assume Z is a countable abstract Jc-section for some it> 1. The construction of a type k+lG whose Jc-sections is Z is given in forcing theoretic terms chiefly for the sake of future developments, since the construction utilizes little of the power of forcing. p,q,r... denote forcing conditions, p is a forcing condition if: (1) p:T + 1-»Tp(k), where T is an ordinal; and (2) there is a normal G G Tp{k+ 1) such that T is constructive in G and (a)[aPCT = / / 0 c ] . The C of clause (2) is unique in the following sense. Let Gp be the restriction of G to those R<=Tp(k) such that (£e)(£a)o
& fl- {e}^].

(3)

Then Gp is uniquely determined by p, and moreover is equivalent to p, because

286 912

GERALD E. SACKS.

the generation of {tf„ G |a
Proof. An immediate consequence of the fact that Gp (defined by 3.3 (3)) decides whether or not p is a forcing condition, and if so, whether or not p|f z mEOs Since A may not be a transitive set, it is worth noting that if pEA and p | f z w G O c , then Gp forces m to be a notation for some a GA. 3.5. Genericity for |fz. Assume C = * + 1E on Pp(fc)- Z. G is said to be generic with respect to (fz if for each sentence DC of the form meO3

or

R<§,

(1)

where fl6Z, there is a p such that either

pfrz%

or pff z ~X,

and the graph of Gp is contained in the graph of C. Since Z is countable, generic G's exist. As usual a sentence about a generic C of the form occurring in (1) is true if and only if it is forced by some p such that Gp C G. LEMMA

3.6.

If C is generic, then Z c kscG.

Proof. Suppose R:Tp(k- 1)—»u belongs to Z. Fix p. Since G is generic it suffices to find a
<WTc,a> is not first

287 Jc-SECTION OF A TYPE n OBJECT.

913

order definable over

(Tp(k-)),e,(Hnc,a)) for any o < T, because it encodes all such definitions. Consequently C p «H T c , f l >) is undefined for all a. Extend C p to Gq by setting

Gq((HSa>) = Ra for all a. Let e0 be a trivial index; i.e. {e0}* = X for all X. Then Ka= t/*-<e0, a, y + 1>E H / ^ forallaGTp(fc-l). 3.7. B-Genericity. Since Z is a countable abstract fc-section, there is a B such that (?TtZ,B) satisfies 3.1 (l)-(5) with 91LZ for A. Assume G is generic in the sense of 3.5. Note that the restriction of G to Z (denoted by G \Z) is readily encoded by a type k object since Z is countable, and that G is equivalent to <*+ '£, G fZ >. The existence of a generic G such that (G f Z) G B follows readily from the countability of L!)HZ in B and the admissibility (with gaps) of B. It is not difficult to find a generic G whose /c-section exceeds Z. 3.8.

If G is generic and ( G f Z ) e B , t h e n ^ s c G C Z .

Proof. Suppose R E(kscG)-Z for the sake of a reductio ad absurdum. Thus for some T and m, R is one-one reducible to WTG and T = |TTI|C. There is no p such that Gp C G and ppmEO§; if there were such a p, then fi would belong to -?ILZ because T would. Consequently there is a
(1)

Observe that f/TC G B, because ( C t Z ) G B , G = <' ( + 1 £,CrZ> and B is closed under recursion in t + 1 E. Define r by ro=H,C for all o < T. Clearly r6fi,
and

rfrzr7lGO^

(2)

288 914

GERALD E. SACKS.

It follows from 3.4 and the stability of 9ltZ in B that some r in 911Z satisfies (2), an impossibility according to (1). THEOREM 3.9. Z is a countable abstract k-section iffZ is the k-section of some normal type k + 1 object.

Proof. Assume k > 1, since the case of k = 1 is covered by 4.4 of [1]. If Z is the k-section of some normal type k+1 object, then Z is a countable abstract fc-section by 3.2. Suppose Z is a countable abstract Jc-section. Let G be generic as in 3.7. Then Z^scC by 3.6 and 3.8. COROLLARY 3.10. (Plus-One Theorem). Suppose n>m>k>l and F is a normal type n object. Then there exists a normal type m object G such that kscF=kscG.

Proof. Assume f>l, since k=l is covered by [1], Let Z = (m _ 1) scF. By 3.2 and 3.9, there exists a normal type m G such that (m- 1 ) $ cF=(m- 1)S cG. But k<m, so Tp(k)cTp(m-1), 3.11. Stable Countings. to be a counting of Z if:

and consequently

kscF=kscG.

Assume k>l and ZcTp(k).

<( =
is one-one;

(0:9HZ^OJ

and 9LZ|= §t(b)

iff

tj(Mob)=0,

(1)

where g,(r) is the e-th \ formula of ZF and 6G9HZ. Note that in (1) the restricted quantifiers of §e are interpreted as in 2.4, since 9HZ may not be transitive. Strictly speaking t is a type k object because t0 is, but it is more to the point to regard t as a type 1 object, since most of the information carried by t is carried by tv Thus P, the complete recursively enumerable (in i + 1 E ) set of integers, can be regarded as the natural counting of ksc(k+lE) if the usual correlation between sets recursive in k+1£ and their Godel numbers is kept in mind. A counting t of Z is said to be stable if ^LZ
+

%t))).

289 fc-SECTION OF A TYPE n OBJECT.

915

Further reflection (2.9) implies P is a stable counting of ksc(k + 1E). The next proposition is easily verified. PROPOSITION 3.12. Assume k> 1 and Z c Tp(k). Then Z is a countable abstract k-section iff Z has a stable counting.

Proposition 3.12 clarifies what is needed for an application of Theorem 3.10. Let Z2 be the collection of all lightface A2 sets of reals. The natural counting of Z2 is given by Pf, the complete 22 set of integers. Whether or not Pf is stable seems to depend on circumstances. If V = L , or if the axiom of determinateness holds, then Pf is stable. At this writing it is within the realm of possibility that Pf is not stable yet Zf possesses some unnatural stable counting. The stability of Pf is equivalent to: for every wj predicate Q, (E2x)[Q(2x)

& *x<%P*]-*{&x)[Q{x)

&

2

xGZ2],

where h is a type 2 variable. 4. Further Results and Open Questions. Moschovakis [4] defined the fc-envelope of F (denoted by kenF) to be the collection of all X 6 Tp(k) such that X is recursively enumerable in F. He showed that if F is a normal object of type n and n > k +1 > 2, then the it-envelope of F is not the ^-envelope of any normal object of type k+ 1. Thus there is no plus-one theorem for ^-envelopes, but there is a plus-two theorem for them discovered by Harrington [3]. PLUS-TWO THEOREM. Suppose n>k + 1>2 and F is a normal type n object. Then there exists a normal type k + 2 object C such that kenF = kenG.

It is tempting to think that the plus-two theorem is a consequence of the plus-one theorem (3.10) by the following false argument: By 3.10 there is a type k + 2 C such that k*iscF

= k+iscC;

(1)

(2) But then kenF=henG because /c-envelopes are determined by (k+ Insections. Step (2) is plausible because kenC is 2j over

0?H( (t+ „«rC),G>.

(3)

((3) is an admissible structure with gaps as in 2.6.) In short the ^-envelope of G is determined by the restriction of G to its (fc + Insection, but not by its (*+ l)-section alone. A slight modification of the arguments of section 2 yields a type Jfc + 2 C which satisfies (1) but whose fc-envelope is different from that of F.

290 916

GERALD E. SACKS.

On the other hand Harrington [3] points out that the proof of the plus-two theorem is readily altered so as to produce a type k + 2 G whose (* + lnsection and /c-envelope are the same as that of F. Moldestad [13] defines the (U)-section of F (denoted by ^scF) to be the collection of all XeTp(k) such that X is recursive in for some YE Tp(i). (Thus °kscF=kscF.) Harrington [3] remarks, and Moldestad [13] verifies, that the ideas occurring in the proof of the plus-two theorem suffice to prove a strengthened plus-one theorem: if n > c>2 and F is a normal type n object, then there exists a normal type k+1 C such that

*-*kF=k~\G. Nonetheless the following hypothesis remains unproved. EXTENDED PLUS-ONE HYPOTHESIS. Suppose n> k>\ and F is a normal type n object. Then there exists a normal type k+l object G such that k -\F = k-\G. h

~\F is called the extended /c-section of F. Note that the extended /c-section of a normal type (k + l) object G is equivalent to the hierarchy {Hac\ok + 2>3 and F is a normal type n object. Does there exist a normal type k + 3 object G such that kskG = kskF? kskF denotes the /c-skeleton of F, a concept as yet undefined. Does there exist a concept of ^-skeleton such that the answer to the plus-three problem is yes, and such that the ^-skeleton of a normal type k + 3 object cannot be the /c-skeleton of any normal object of type k + 2 or k+ 1? A suitable concept of ^-skeleton might justify the study of recursion in normal type 5 objects. At the moment every phenomenon associated with recursion in a normal type 5 object can be exemplified by recursion in some normal object of lower type. But surely

•It is also true when It = 3 if the axiom of determinateness is assumed.

291 fc-SECTION OF A TYPE n OBJECT.

917

the transcendence of type 5 over lower types can be made manifest by some concept accessible to recursion theorists. HARVARD UNIVERSITY AND MASSACHUSETTS INSTITUTE OF TECHNOLOCY.

REFERENCES.

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

G. E. Sacks, "The 1-section of a type n object," Generalized Recursion Theory (North-Hol­ land, 1974), pp. 81-93. S. C. Kleene, "Recursive functionals and quantifiers of finite type," Trans. Amer. Math. Soc. 91 (1959), pp. 1-52; and 108 (1963), pp. 106-142. L. A. Harrington, "Contributions to recursion theory on higher types,' Ph.D. thesis, Massachusetts Institute of Technology, 1973. Y. N. Moschovakis, "Structural characterizations of classes of relations," Generalized Recur­ sion Theory (North-Holland, 1974), pp. 53-79. J. R. Shoenfield, "A hierarchy based on a type 2 object. Trans. Amer. Math. Soc. 134 (1968), pp. 103-108. T. Grilliot, "Hierarchies based on objects of finite type," Jour. Symb. Log. 34 (1969), pp. 177-182. G. E. Sacks, "Higher recursion theory,' (Notes by D. B. MacQueen of lectures at the Massachusetts Institute of Technology, 1971-72). L. A. Harrington and D. B. MacQueen, "Selection in abstract recursion theory," Jour. Symb. Log., to appear. Y. N. Moschovakis, "Hyperanalytic predicates,' Trans. Amer. Math. Soc. 129 (1967), pp. 249-282. F. D. Lowenthal, "The Minimal Pair Problem for Higher Type Objects," Ph.D. thesis, Massachusetts Institute of Technology, 1974. K. J. Barwise, "Admissible sets over models of set theory," Generalized Recursion Theory (North-Holland, 1974), pp. 97-122. T. Grilliot, "Selection functions for recursive functionals," Notre Dame Jour. Form. Log. 10 (1969), pp. 225-234. J. Moldestad, Report of a Recursion Theory Seminar (Chapter IV), University of Oslo, 1973.

292 FUNDAMENTA MATHEMATICAE CIII (1979)

Effective bounds on Morley rank by

Gerald E. Sacks * (Cambridge, Mass.) To the Memory of Andrzej Mostowski

Abstract. Effective bounds are obtained on the Morley rank of a 1-type, the Morley rank aT of a totally transcendental theory T, and the Blum density number dT of a quasi-totally transcendental theory T by means of type-omitting and absoluteness arguments over admissible sets. The above restrictions on T imply that aT and dr arc ordinals recursive in T. Every theory T is seen to have a universal domain hyperarithmetic in the hyperjump of T.

1. Introduction. This paper might better have been titled: On the Absolute Character of the Morley Derivative. For the bounds given below on Morley rank, and on Blum's density number for quasi-totally transcendental theories, are derived from some absoluteness properties of Morley's analysis of 1 -types. Let T be e aount­ able theory of first order logic. Assume Tis complete and substructure complete (]) in order to smooth the application of Morley's rank-and-degree machine to T. (The details of his machine will be reviewed in Section 2. A full account was given in [12].) Suppose s* is a substructure of a model of T, and p is a 1 -type over sf (in symbols s/ e Jf(T) and p e &/). If p has a Morley rank, then that rank is denoted by rM(p), and the existence of that rank is indicated by the inequality: rM(p)
rM(p) =

p<<x>-+peN&.peN

Another aspect is the fact that the relation peSs/

&rM(p)^p

is X] over N. (1) implies a bound first obtained by Lachlan [8]: (2)

''M(/')
* This article is based on a talk given at the 1972 Orleans Logic Conference. Its preparation was supported in part by NSF grant MCS 76-10430. The author is grateful to L. Harrington and S. Simpson for several helpful comments. (') Substructure completeness is equivalent to admitting elimination of quantifiers. 3-

293 112

G. E. Sacks

since every Morley rank achieved by some p over some si is also achieved by some p over some countable si, and since each countable si belongs to some countable N. The notion of "next" admissible set, as in [1, p. 60], is needed for an effective version of (2). Let si* be the least admissible set N such that si e N. Define caf to be the least ordinal not in si*. An effective version of (2) is: (3)

peSsi

&rM(p)
+

&rM(/>)
If si is countable, then there exists a real number C such that si is coded by C and col = wf. (One such C is in essence a counting of si generic over si*.) The con­ clusion of (3) now becomes: p is (i.e. coded by a real) hyperarithmetic in si, and rM(p) is an ordinal recursive in si. The proof of (3), given in Section 3, is in the main an omitting types argument over a countable admissible fragment of £„u„. The argument succeeds thanks to some weak absoluteness properties of the Morley derivative extracted in Section 2 from the strong absoluteness of the Cantor-Bendixson derivative. The notion of universal domain also plays a part in the proof of (3). A universal domain °U has the advantage that Morley rank equals Cantor-Bendixson rank for all p e S%. (3) is applied in Section 5 to prove the existence of a countable universal domain <# for T with most of the effort directed towards holding down the ordinal needed to construct 11 ffom T. As a further application of (3), a bound on dT, Blum's density number, is obtained in Section 4. Blum [2] calls T quasi-totally transcendental (q.-t.t.) if the Morley ranked points of Ssi are dense in Ssi for every si e Jf(T). If T is q.-t.t., then dT is defined to be the least p such that for all si eJf(T), the points of Ssi of Morley rank less than p are dense in Ssi. It will be shown that dT
294 Effective bounds on Morley rank

113

where F(g,x) is a formula in the language of T with parameters a in A, the uni­ verse of st. Sf: S3-+Ss4 is the continuous onto map induced by / . If qeSSB, then F(g,x) e (Sf)q <- F{fg,x) e q. S is a contravariant, limit preserving functor from X{T) into the category of Stone spaces. DS is the Morley derivative of S. p e DSst iff p e 5 ^ and there exists an / : j * - . £ in X(T) such that (Sf)~l(p) contains a limit point of S3. DSf is the restriction of 5 / to JM0, and maps DS3S onto 0 S J * . D*S, the /?th Morley derivative of S, is denned by transfinite recursion. D°S = 5, i)' + >S=2>(Z>'S), and DkSsf = 0 {X>'5J/| P<X)

when A is a limit ordinal. DfS is a contravariant, limit preserving functor from X{T) into the category of Stone spaces. If there is a p (necessarily unique) such that peD'SJ-D^Ss*, then p is said to be a Morley ranked point of Srf of Morley fi. dSrf, the Cantor-Bendixson derivative of Ss/, is the set of all limit points of Ss/. d"Ss/ is denned by transfinite recursion. d°Ss* = SJ/, d'+1Ss
^ w = n Vw e

/?<;.}.

f+

If there is a /? such that p e d S^-d ' S J / , then p is said to be a Cantor-Bendixson ranked point of Srf of Cantor-Bendixson rank /?. The latter state of affairs is indi­ cated by: rCB(P) = 0
Proof. It suffices to show dfStf c D*Ss* for all § by transfinite induction. The latter is straightforward, because the identity monomorphism \A: sd-^sH belongs to X(T). ♦ Suppose # , r, if e 3f(T). i" is finitely generated if there exists a finite Ya V such that r is the least substructure of Y whose universe contains Y. Suppose peD^Sr and /: f c r is an inclusion, p is said to split in D'SiT if {DfSty\p) has at least two members. # is an a-universal domain for T if for all j3sS?/% for some finitely generated n T 0 c ^ . <W is a universal domain for T if * is an a-universal domain for T for all a. LEMMA 2.2. Suppose * fo on a-universal domain for T. Then D"S^ = d'SV for all (3
295 114

G. E. Sacks

system of all finitely generated substructures of <%. Then D"S<% is the inverse limit of the inverse system {D'SiT,-}. ♦ The existence of a countable universal domain for T is implied by Proposition 31.2 of [12]. Unfortunately there is a gap in the proof of 31.2. It will be filled in Section 5 by the type omitting argument of Section 3. The next proposition retains all that is correct in the proof of 31.2. It will be needed in Sections 3 and 4. card is an abbreviation for cardinality. PROPOSITION 2.3. Suppose sieJT(r) is infinite and cardasgcardsi. Then there exists a Ql^si such that c a r d ^ = cards/ and
n
1. ^0 = si. 2. For each 0 < a , each finitely generated -f c%, and each isolated p e DfS"T, choose a iT^p finitely generated over Y with the following property: if p splits in D^Str for some # " e Jf(r), then p splits in D'Sirvep. The existence of i T £ „ follows from the limit preserving property of D'S described in the proof of Lemma 2.2. Let ^ „ + 1 be a model of T which extends <#„ and every Wvf p, and which has the least possible cardinality. <% is a-universal, since each finitely generated substructure of
& rCB(p) = p.

(ii) PeN&peN&Nt[peSs/&

rCB(p) = /?].

In addition all values of the function g, defined by g(P) = {p\ peSs/&

rCB(p) = P} .

belong to N, and g is A, over N. Proof. Since si belongs to N, its complete first order theory also belongs to N. That theory is given by F u D i ( ^ ) , where Di(si) is the diagram of si, since it was

296 Effective bounds on Morley rank

115

assumed in Section 1 that 7" is substructure complete. A typical/, e Ss/ is a consistent set of formulas of the form F(x) with parameters in A. "Consistent" means consistent with respect t o T u Di(&/), and has the same meaning inside N that it has outside N F(x) has a Cantor-Bendixson rank )|

F(x)ep}.

Part I of the proof is an induction on jS intended to show the equivalence of (i) and (ii) for all ( l e f t The induction is performed simultaneously with a recursion on /? designed to show that the relation rCB(F(x))
(1)

is A, over N, Part II of the proof will show that rCB(p) implies rCB(p) e N. Part I. Fix /? e N and suppose (i) holds. Let G(x) isolate p in d*Ss/. Thus G(x)ep, and for all q, (2)

G(x)eq

^ q=

pvrCB(q)
Then p is generated by the set Z(G) denned by (3)

(G(x)} u {~H{x)\ rjfi{x))
;

that is p is the unique 1 -type containing all of Z(G). By induction (1) is J, over N, so Z(G), and hence /;, are members of N. Also by induction, rCB(p)>P and (2) are true in N. Consequently (ii) holds. Now suppose (ii) holds. Then p is generated in N by a set Z(G) as in (3). Since the concept of first order logical consequence is A0, Z(G) generates p outside N. By induction Z(G) insures that rCB(/i) = p. To see that rCB(F(x))<0 is J „ observe that rCB(F(x)) = 0 if and only if there is at least one p, but not infinitely many, such that F(x) e p and /? is generated by some Z(G) as in (3). It is a A0 matter to decide if Z{G) is consistent and complete (that is generates a 1-type), and if two Z(G)'s generate the same 1-type. Part II. Let a be the least ordinal not in N. Suppose p e Ss/ and r C B (pKa. Then there is a G(x) such that (4)

{G(x)}u{~H(x)\

rCB(H(x))
generates p. Since (1) is /I,; it follows that (4) belongs to N, and is equal to (3) for some fS)
297 116

G. E. Sacks 3. Bounding Morley rank.

THEOREM 3.1 (weak absoluteness of the Morley derivative, cf. Lemma 5.2). Let N be an admissible set, and sit a substructure of a model of T. Suppose s4 e JV. Tlien peSs/

& rM(p) = /? -

peN&PeN

Proof. By 2.1, rCB(p)< 0, and so by 2.4, p e JV. Let ZFC„ be a finite fragment of set theory strong enough to carry out all routine arguments of model theory. The following collection 9 of axioms is lx over JV, and talks about an end extension of JV in which ZFC„ is true. (i) ZFC„. (ii) The diagram of JV. For each ae JV there is an individual constant a and a sentence

00 [ye f l ~\J

(y = £)]-

(iii) o o > r M ( p ) > 5 . One such axiom for each ordinal 5 e N. (jv) j*<=<& and % is a countable 5-universal domain. One such axiom for each 5 e JV. Assume rM(p) * JV in the hope of a contradiction. For the moment assume JV is countable. Then 0 is countable, and by 2.3, p has a model, namely K, the real world. It follows that $F has a model Jf that omits a, the least ordinal not in JV. (For details concerning the construction of such an Jf, see Keisler [6, p. 58]). (iii) implies Jf is a proper end extension, because rM(p) must have a value in Jf, and that value cannot belong to JV. For each z e Jf, let tc(z) be the transitive closure of = with respect to eA, the e relation of Jf. If the restriction of e_K to tc(r) is well-founded, then z is said to be a standard set of' Jf Let St(Jf) be the set of all standard sets of Jf. St (Jf) is admissible, contains JV, and its ordinals are just those less than a (cf. [1], p. 60). Jf can be regarded as an end extension of St(Jf). The countability of * in Jf makes it possible to construe t as a member of St(J'). The universe of ^ becomes w, and the relations of <2r become sets of n-tuples of finite ordinals. From now on superscripts will be used to indicate in which universe ranks are being computed. Since si can be injected into 45T, there is a q e S% n Jf such that q is a pre-image of p and rM(g) = r^(p). Since 9 is an actual pre-image of p in the real world and p has a Morley rank, ^ must also have a Morley rank. It follows from 2.1 and 2.4 that 4'B(<7)
298 Effective bounds on Morley rank

117

set having si0 as a member., is countable, hence susceptible to the above argument for countable N. ♦ COROLLARY 3.2. Suppose si is a countable substructure of T and p is a \-type over si. Ifp has a Morley rank, then p is hyperarithmetic in si and its Morley rank is an ordinal recursive in si. 4. Bounding dT and aT. dT, Blum's density number, was defined in Section 1. THEOREM 4.1. If T is quasi-totally transcendental,

then dT is recursive in T.

Proof. Similar to that of 3.1. Let si be a substructure of a model of T. Define d$ to be the least 0 such that the points of Ssi of Morley rank less than p are dense in p. As in the proof of 3.1, the following set 9 of axioms is X, over T+, the least admissible set with T as a member. (i) ZFC„. (ii) T is quasi-totally transcendental. (iii) si is a countable substructure of'a model of T. (iv) d?>5.

One such axiom for each

b<m\,

Assume dT>w[ in hope of a contradiction. Then for each &<m\, there ii a count­ able si such that d?>8, and so each T+- finite ( 2 ) subset of <$ is modeled by the world. It follows that'S has a a o d e l JT that omits w]". (ii( requires d? to have a value in ^f, and (iv) implies that that value is nonstandard. As in 3.!, the countability of si implies si e St(^K), the standard part of Jf. Since 7" is quasi-totally transcendental, the isolated points of Ssi are dense in Ssi (cf. [12], page 200), and every isolated point of Ssi has a Morley rank. Let / be the set of isolated points of Ssi. By 2.4,

JeSt(jV). Fix pe / t o show rM(p)<«[. Suppose r£(/>) = c; by 3.1 c is a countable ordinal (possibly nonstandard) of JT By (i) and 2.3, there is a °U e ^ such that j/<=<^ and ^

Y\m is a countable, (c+l)-universal

domain].

Let q e S* be a pre-image of p such that r&p) = r&q). By 2.2, /•£,(?) = r£(9). It follows from 2.4 that rfg^) < o)[. It was noted in the proof of 3.1 that any assign­ ment of Cantor-Bendixson rank in StfyK) persists in JT. Consequently c<m\. It now follows that the least upper bound (in JT) of {>JM(P)\ P e 1}

(1)

is less than w[, since w\ i „r. But tiv) )mplies shat the least upper bound of ((1 )i nonstandard. ♦ COROLLARY 4.2.

dT<^.

aT, tth Morley rank of T, was sefined i i nection 11 Morley [[9 ]sowed that t 7 -
299 118

G. E. Sacks

Proof. Same as that of 4.1. Let ^ e JffT). Define a? t t ob eth eeast t? ?uch that every p e Sst has Morley rank less than /?. <S0 is the result of two changes in the axiom set <§ of 4.1: "quasi-totally" is replaced by "totally", and d% by a?. The part played by I in 4.1 is now played by all of Sat. The concluding contradiction concerns the least upper bound (in JT) of {r£(»| P e Sj*}- ♦ 5. Countable universal domains. The purpose of this section is not to prove the existence of a countable universal domain * for T, for rhat tact tollows easily ffom the Lachlan bound on Morley rank expressed by formula (2) of Section 1, but rather to estimate the least ordinal needed to construct such a * ffom T. Let WO be the set of all countable wellorderings of integers, and for each y e WO, let \Y\ be the ordinal represented by Y. Suppose P(X, /?) is a predicate with variable X ranging over the reals, and j8 ranging over the countable ordinals. P(X,0) is said to be l\ if YeWO-*P(I,|Y\) is l\. PROPOSITION 5.1. The predicate p e Ssi & rM(p)^P, is I\ ( 3 ).

restricted to countable

Proof. The predicate in question, call it P(p,jf,P),

at's,

is equivalent to

(1)a (1) implies />(/?, si, 0] by 2.1. Suppose P(p,st,P) holds. Extend at to 3S, a coun­ table, (£+1)-universal domain provided by 2.3. Let q e S3S be a pre-image of;? of the same Morley rank as p. Then rCB(q)>p by 2.2. Since the predicate q e S3 is arithmetic, it need only be verified that [rCB(q)^P] is I\. The proof of 2.4 shows that the restriction of (2)

y 6 W O -*

rrB{q)>\Y\

to any admissible set N is 77, over JV uniformly in N. The last phase refers to a /7j predicate independent of N; its only parameters are integers. Thus there is a A0 for­ mula R such that for all Y and q, (2) is equivalent to (3)

HYP(Y,q)t(v)R(v,

Y,q).

H Y P ( y , q) is the least admissible set with Y and q as members; each element of H Y P ( y , q) corresponds to a real hyperarithmetic in Y, q. A universal quantifier restricted to HYP( Y, q) is seuivalent to an unrestricted eeistential luantifier rcf. .[1], p. 418). Hence (3) is l). ♦ An admissible set N is said to be recursively inaccessible if for each h e e, there is a y e N such that xey and y is admissible. LEMMA 5.2. Let N be recursively inaccessible and st a substructure of a model of T. Assume at e N, Te N and N¥[T

is countable & (x)(x is wellorderable)].

(') Proposition 5.1 can be derived from Lemma 4.8 of [3] without any mention of universal domains.

300 Effective bounds on Morley rank

119

Then (i) and (ii) are equivalent for all p and all 0 < o o : (i) peSsi

& rM(p) = p,

(ii) PeN&peN&N¥[peSsi&

rM(p) = p].

Proof. If p, J* 6 N and p e Ssi, let r^(p) be the Morley rank assigned to p inside N. By 3.1 it suffices to prove (1)

rM(p) = /? « - rM(/>) = 0

for all si, p and 0 in N by induction on p. Since elementary arguments of model theory can be carried out in N, the value of /•£,(/>) is determined, as in the real world V, by the countable structures. Thus it is enough to prove (1) for .aTs countable in N. Fix P and assume rM(p) = p. By induction r^(p)^p. Suppose r^(p)>p. Then in N there is a @^si and a q=>p such that 9 is a limit point of DfS3g. In V there is a neighborhood Fof q that isolates 9 from all points of S3S save those of Morley rank less than p. But then by induction F isolates q in DfS@ in TV. Now assume /•£,(/>) = p. By induction rM(p)>p. Then in K the following is true for each n. There exist a countable 3Sn and distinct 9 o , ...,9„ such that (2)

sic3$n&.p^qieS@n&.rM{q^p

{i
Since r£B=g/•"(/>)< oo, it follows from 2.4 that /> is hyperarithmetic in si. Suppose /?<wf. Then 5.1 implies (2) is l\ with parameter si. By Kleene's basis theorem ([11], p. 420), &n can be assumed to be recursive in (9", the hyperjump of the real that codes si. Since N is recursively inaccessible, 0" e N. Let 38 x be the amalga­ mation in N of all substructures (of models of T) coded by reals recursive in G*. In V, p has infinitely many pre-images in 0X of Morley rank ^p, and by induction the same is true in N. But then r^(p)>p. Finally suppose p^wf. Since rM(p)^p, it follows as in the proof of 5.1 that there exist @ and q such that (3)

pczqeSa&r^q^cof,

The second half of (3) is equivalent to {m)[me(9M

-*

rCB(q)>\m\]

( H is the ordinal less than cof represented by m.) It follows from 5.1 that (3) is l\ with parameter si. As in the previous paragraph, (3) has a solution <<#, q} in N. By 2.4 and 2.5, ' C B ( ? ) = '«(?)

= 00 .

But then r^(p) = 00 >p. ♦ THEOREM 5.3. Let si be a countable substructure of a model ofT. Then si can be extended to a countable universal domain W hyperarithmetic in the hyperjump of si. Proof. A compound of 2.3 and the last part of 5.2. # is the limit of a chain [<W„\ n} defined by recursion on n. <*. is recursive in # „ + 1 uniformly in n. (This

301 120

G. E. Sacks

can be taken to mean that the even part of the code for # n + 1 is the code for <2f„.) Each «f, has lower hyperdegree than 0 ^ (in symbols <&n
{p\peSr&.rCB(p)
is n\ with parameter ■f. By 2.5 each member of (1) is hyperarithmetic in -V. Con­ sequently the members of (1) can be arranged in a sequence hyperarithmetic in 0* Let p be the /th element of that sequence. Assume ^ „ < G*. Since jtf is recursive in <#„, it follows that (2)

<" = <

and

0*^,0".

Consequently < < < and fl^** Assume /> e .D'SiT and that p splits in £>'$#' for some W e X(T). let ^ n + 1 be * , . ) The definition of <W,+1 has two cases. C a s e I. 0
and so 5.1 implies that the predicate

q0 # qx &<Wn<=ir & (0,<2[/><=<7, e J ^ W r ]

is l\ with parameter « , . By Gandy's basis theorem, (3) has a w\.
(If not,

By 3.2, rM(p) = oo. Hence there exist

g0 * qx&<%Kcif' &(i)f<2[Pcql&ru(qi)

9o, ? 1

solution

and # " such that

= oo] .

with parameter <#„, because [rM(g,) = oo] is equivalent to (m)[mee>w

-*

^ Z T W ] ,

and (5) is i j by 5.1. Now proceed as in case I. • (= U ( * . l " < " } ) 'S hyperarithmetic in 0*, because # „ < 0 * uniformly in 77. The last assertion is a consequence of several uniformities, two of which are worth mentioning, (i) The set of indices of solvable l\ predicates with parameter P is recursive in ep uniformly in P. (ii) An index for computing 0Y hyperarithmetically from (f can be obtained effectively from an index for computing Y hyperarithmeti­ cally from <9\ if it is assumed that Y
302 Effective bounds on Morley rank

121

of ff*. Suppose p is an iiolated point of D*Sr. Then rCB(p) < coo ,snce ^ 5 ^ c D'S-T for all y. Let p be the /th element of (1). Suppose p splits in D*S?^" for some rT e JT(r). If (i,;k it,I) is the nth member of E, then /> splits in £>%„+1, hence in some finitely generated substructure of <#. ♦ COROLLARY 5.4. T/ioy a countable universal domain that is syperarithmetic in tht hyperjump of T. 6. Open questions. 1. To what extent is the assignment of Morley rank absolute? Does Lemma 2.4 remain true when the Cantor-Bendixson derivative is replaced by the Morley deriva­ tive? Can the assumption of recursive inaccessibility in Lemma 5.2 be substantially weakened? 2. Does every countable theory T have a countable universal domain <% such that °U is hyperarithmetic in nT If not, can it at least tb erranged that a>w * =>co Added in proof. C. Calmer (Stanford Ph. D Thesis, 1978) has answered all of the above questions. In particular, Lemma 2.4 holds for the Morley derivative.

References [1] J. Barwise, Admissible Sets and Structures, Springer-Verlag, Berlin-Heidelberg-New York 1975. [2] L. Blum, Generalized Algebraic Structures: A Model Theoretic Approach, P h . D . Thesis, Massachusetts Institute of Technology 1968. [3] D. Cenzer, Monotone inductive definitions over the continuum, i. Symb. Logic 41 (1976) pp. 188-198. [4] V. H a r n i k and M. M a k k a i , Applications of Vaughl sentences and the covering theorem, J. Symb. Logic 41 (1976), pp. 171-187. [5] A. Kechris, Z\ derivatives with applications, MIT Logic Seminar Notes 1974. [6] J. Keisler, Model Theory for Infinitary Logic, North-Holland 1971. [7] G. Kreisel, Analysis of the Cantor-Bendixson theorem by means of the analytic hierarchy, Bull. Acad. Polon. Sci. 7 (1959), pp. 621-626. [8] A. Lachlan, The transcendental rank of a theory. Pacific J. Math. 37 (1971), pp. 119-122. [9] M. Morley, Categoricity in power, Trans. Amer. Math. Soc. 114 (1965), pp. 514-538. [10] W. Richter, Constructive vansfinite number classes, Bull. Amer. Math. Soc. 73 (1967), pp. 261-265. [11] H. R o g e r s , Jr., Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York 1967. [12] G. Sacks, Saturated Model Theory, Benjamin, Reading, Massachusetts 1972. H A R V A R D UNIVERSITY and MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Accepte par la Redaction le 20. 12. 1976

303 J. Barwise, H. J. Keisler and K. Kunen, eds., The Kleene Symposium ©North-Holland Publishing Company (1980) 201-222

Post's Problem, Absoluteness and Recursion in Finite Types Gerald E. Sacks* Massachusetts Institute of Technology and Harvard University, Science Center, 1 Oxford Street, Cambridge, MA 02138, U.S.A. Dedicated

to Professor

S. C. Kleene

on the occasion

of his 70th

birthday

Abstract: The unresolved character of the power set operation stymies the solution of elementary problems arising in KJeene's theory of recursion in objects of finite type. E.g. Post's problem for 3 E has a positive solution if V-L (NORMANN, 1975), and a negative if AD holds. Let S be the class of all sets R Q2" such that R is recursive in 3 E,fe for some real b. A forcing construction shows S is not recursively enumerable in 3 E when there is a recursively regular well-ordering of 2" recursive in 3E. It follows that the concepts of 2*. and weak 2*. definability differ. With the aid of the notions of indexicality and ordinal recursiveness, an absolute version of Post's problem for 3 E is devised, and studied when certain recursively enumerable projects are equal.

The problem raised by POST (1944) concerning the Turing degrees of recursively enumerable sets of numbers has persisted in the study of generalizations of ordinary recursion theory. A seemingly temperate ver­ sion of his problem in the context of KJeene's theory of recursion in objects of finite type (KLEENE, 1959) is as follows. Post's Problem for "E: Does there exist a type n object A such that A is recursively enumerable in "E; A is not complete; and A is not recursive in "E,b for any object b of type less than n? Kleene's theory of recursion in normal objects of finite type is reviewed several pages ahead. Until then make do with the following. The objects of type 0 are the natural numbers. An object of type n ( « > 0 ) is a collection of objects of type less than n. "E is the equality functional for objects of type less than n. Thus „ ,

,_ f0 ' '~\\

(X y

if x=y, ifx^y,

•This paper is based on an address given at the Kleene Symposium in May 1978. Its preparation was supported in part by National Science Foundation Grant MCS 7807344. The author is indebted to S. Homer, D. Normann and T. Slaman.

304

202

G. E. Sucks

where x and y are arbitrary objects of type less than n. Kleene's notion of reducibility for objects of finite type is written < , and read "recursive in". Thus A fC "E,B means A ii not rrcursive in "E,B. Let A be type n and recursively enumerable in "E; A is complete if B .Y is hyperanthmetic, where X is any type 1 object, e.g. real number. An affirmative answer to Post's problem faces two impedimenta. Both have to do with the computations developed in the course of recursively enumerating a collection of objects of finite type. The meaning of "computation" will have to await the promised review of Kleene's theory, but for now there is no harm in thinking of a computation as a well-founded tree that serves as a proof that some object belongs to some recursively enumerable collection. The two obstacles are: (I) a violation of parity between elements of recursively enumerable collections and computations; (II) lack of a suitable well-ordenng of computations. The first obstacle goes back to KREISEL'S early work (1961) on the foundations of metarecursion theory. A violation of panty is at the least a difference in type or complexity. There is a panty (and it is quite useful) between elements and computations in ordinary recursion theory. In that case a computation has the form of a finite tree, hence can be effectively encoded by a number, and consequently there is little difference between a complete recursively enumerable set of numbers and the set of all computations. The next proposition is a classic example of obstacle (I) at the type 2 level. Proposition 1. Let A be a set of natural numbers. If A is recursively enumerable, but not recursive, in 2E, then A is complete (SPECTOR, 1955). Proof. Since A is not recursive in 2E, the natural enumeration of A is cofinal with the natural enumeration of C, the complete, recursively enumerable in 2E subset of u. Hence w e C iff (En)[n(=A & m is enumerated in C before n is enumerated m A], and the latter is recursive in A,2E. Kreisel was the first to point out the violation of parity inherent in Proposition 1. A number is enumerated in A iff an appropriate computation exists. In this case a computation has the form of a countable tree

305 Absoluteness and recursion infinite types

203

encoded by a real recursive in 2 E. Kreisel's remedy is almost immediate. Let REC(2E) be the class of all reals recursive in 2 E. Enumerate subclasses of REC(2E) rather than subsets of u. The associated computations will still be encoded by elements of REC(2E), and so obstacle (I) will have vanished. The remedy has a further prescnption somewhat less immediate. The desire for parity requires some revision of the notion of completeness. Let A CREC( 2 E) be recursively enumerable in 2 E. A is said to be complete if B is recursive in A,2E on REC(2E) for every recursively enumerable (in 2 E) 5CREC( 2 E). To be more precise, there is a KJeene partial recursive functional {e) such that for all R eREC( 2 E):

ReB~{e}(2E,A,R)

= 0;

2

R$B~{e}( E,A,R)=\. Thus {e} correctly answers all membership questions about B from A, so long as those questions are restricted to REC( 2 E). To require {e} to do more is to violate parity, because computations involving arguments out­ side REC(2E) have little hope of being encoded by elements of REC( 2 E). Note that {e} is not required to be total. The next theorem attests to the efficacy of Kreisel's remedy. It is a positive solution to Post's problem at the type 2 level analogous to the FRIEDBERG (1957)-MUCHNIK (1956) solution at the type 1 level. Theorem 2. There exist A and B, each a recursively enumerable (in 2 E) subclass of REC(2E), such that A{B,2E and B £A2E on REQ 2 E). Proof. Implicit in Sacks (1966). Let Ax and Bx be hyperregular,
and

B= {HS\S e B , } .

Let ( c ) b e a Kleene partial recursive functional, and suppose B 2. Let A be a type (n-1) object. If A is recursively enumerable in "E, but not recursive in "E,a for any object a of type less than n-1, then A is complete.

306 204

G. E. Sacks

Proof. Same as that of Proposition 1. The Kreisel remedy is readily applied at all type levels beyond the first. Fix n > 2. An individual is defined to be any object of type less than n-1. By a set is meant any collection R of individuals such that R 3 , but is more easily grasped when n = 3. In accord with the definition of set given above, a set R is a collection of reals such that R < 3 E,a for some real a. A set R is said to be indexical' if there exists a nonempty set / of reals such that K3E,R and

aeI^R
0#/CuoX2",

and (e0,a) G 7—7? = (e 0 }( 3 E,a)

for all <e 0 ,a>G/.

'Cf. the definition of indexical given in Webster's Third International Dictionary. This appears to be the reason that NORMANN (1975) introduces >M.

2

307 Absoluteness and recursion infinite types

205

Gandy's selection theorem for recursively enumerable (in 3E,.R) classes of numbers makes it possible to pass effectively from an indexical set R to a unique e (denoted by eR) that witnesses the indexicality of R. Define IK={eK}(3E,R)

and

i[cA]=(J{IK\R£cA).

According to Wadge, AD yields a continuous function / satisfying either (1) or (2). (1)

x6i[c4]~f(x)EC

(2)

(for all realx).

x*C+*f(x)ei[cA].

Let b be a real that encodes/, that is b encodes rational approximations of / o n all rational arguments. Assume (1). Then REcA~j{IR]&/? = {e 0 }( 3 E,a)

&RecA, and (ii) holds. Two remedies have been found for impediment (II), one by HARRING­ (1973), and the other by NORMANN (1975). The first approach is to restrict attention to sets and computations that of necessity lend them­ selves to a suitable well-ordering. In the case of 3 E this means narrowing down to the class of sets recursive in 3E, namely REC(3E). For "E ( n > 3 ) the restriction is to the subrecursive sets, an uncountable, but bounded subfamily of the indexical sets (cf. SACKS (1977) for details). The second approach is to assume a suitable well-ordering of 2" recursive in 3 E. TON

Harrington's Remedy: There exist recursively enumerable (in 3 E) classes A,B CREC( 3 E) such that A fLB,3E and B£A,'E on REC(3E). Normann's Remedy: Assume that the cardinality c of the continuum is regular and that some well-ordering of the real of height c is recursive in 3 E. Then there exist recursively enumerable (in 3 E) classes A and B of sets such that for every set R, A*B,3E,R on the class of all sets.

and

BfiA^R

308 206

G. E. Sacks

The virtue of Harrington's remedy is its explicitness. No assumptions are made about 3 E. There is no reference, at least no direct reference, to an indeterminate continuum. All sets to be enumerated, as well as the associated computations, have numerical codes, and Post's problem is solved by an adaptation of the a-recursion theoretic methods that succeeded at the type 2 level. But a virtue pursued to the limit, particularly the virtue of explicitness, becomes a fault. Harrington's result says very little about recursively enumerable (in 3E) classes of sets. It avoids the problem of the continuum, a problem that Normann confronts more directly. His assumption concerning the regularity of the continuum and its computabihty from 3E is true in L and elsewhere. He in fact manages with a weaker regularity assumption made explicit just before Lemma 5 below. (This paper and its sequel deal with the singular case.) An unex­ pected consequence of Normann's regularity assumption is the nonrecursive enumerability in 3 E of the class of all sets (Theorem 9), one of the reasons for introducing indexical sets. The absolute approach to Post's problem for 3 E formulated below owes more to Normann's remedy than it does to Harrington's. It consists of staying as close as possible to the ordinals arising as lengths of computa­ tions generated by recursion in 3 E. These ordinals define sets that will take the place of the sets mentioned in Normann's remedy. Some of his sets are lost, sacrificed on the altar of absoluteness, but the loss will be seen to be less than one might think. The time of the promised review of Kleene recursion in 3 E is at hand. Recall how the hyperarithmetic sets of numbers are obtained by iteration of the Turing jump through the recursive ordinals. The class of all sets is similarly obtained by iteration of the 3 E jump through the ordinals recur­ sive in 3 E and an arbitrary real. Do not forget that "set" has been defined above to mean a collection of reals recursive in 3E,fl for some real a. The sets and ordinals in question are defined by a simultaneous recursion. The hierarchy developed is equivalent to an initial segment of L(2"). Thus the sets recursive in 3 E,a for some real a are obtained by starting with the continuum, iterating first order definability, and stopping at the first ordinal that cannot be justified from below in a certain predicative fashion. Let XC2". The 3 E jump of X, denoted by/(X), is an encoding of certain definable collections of reals. Let {*>}* be the definable over <2",e,X> by means of a first order formula whose only parameters are nonnegative integers, and whose atomic parts are of the f o r m ^ e z o r ^ E x . where / and z are real variables and x is a constant naming X. j{X) is the collection of all <e,a,w> such that {e)*(a)=m. 0 is the collection of notations for ordinals recursive in 3E,tf for some real a. Notations are reals a,b,c,...; if 6e0, then \b\ is the ordinal

309 Absoluteness and recursion infinite types

207

represented by b. H0 is the set obtained by a iterations of the 3 E jumpy. The simultaneous recursive definition of 0 and Ha has three clauses. (1) e6 and || = 0 f o r a l l a e 2 - \ And # o =0. (2) Suppose < c , a > e e and |<e,a>| = o. Then < 2 ' , a > e 0 , |<2',a>| = a+1, and H„+,=j(HJ. (3) Suppose <m,a>G0 and |<m,a>| = a. Define rV%={b\{e}"-(a>b)

= 0),

where a*b is a real code for (a,b). Suppose WeH-cC. Then <3 m -5',a> e © and |<3 m -5',a>| = \ , where \ is the least limit ordinal greater than a and greater than \b\ for all bG W*\. And Hx is the set of (b,c) such that fee©, |*|<«,} is equivalent to a hierarchy of all the computations that arise in the comput­ ing of partial functions from 3 E and real arguments. (e}( 3 E,a) is defined and equal to m if e = <e0,, <fl >ee, |<e 0 ,a>| = o and {«,}?♦(*) = m. Thus to compute {e}(3E,a) is to iterate the 3 E jump until <e0,a> is seen to be a notation for an ordinal o and then to pluck the value from Ha in a first order fashion dictated by ev A function/ is recursive in 3 E,a (in symbols / < 3 E , a ) if there is an e such t h a t / ( d ) = {| for some ( c , a ) 6 f l . y is recursive in 3 E,a if y is the height of a prewell-ordering of 2" recursive in 3 E,o. If y is constructive in 3 E,a, then y is recursive in 3 E,a. If y is recursive in 3 E,a, then y is less than or equal to some ordinal constructive in 3 E,a. Clause 3 of the definition of 8 and H implies a bounding principle: if /:2"->© is recursive in 3 E,a, then there exists an ordinal y constructive in 3 E,a such that \f(b)\
An index for a set R is (d,e,a}, where |<^,a>| = a and the above equation relating R and W is satisfied by o, e and a. From now on assume that indices for sets are encoded by reals. Note that if b is an index for R, then R<3E,b. Let S be an arbitrary collection of reals. Recursion relative to 3E,S is defined in the same hierarchic fashion as recursion in 3 E. The only

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difference is beginning with S rather than 0 . Thus the simultaneous recursive definition of 0 s and Has has three clauses identical with those given for 0 and HB save that H$ = S. K,5 is of course the supremum of \b\ for all beOs, and may exceed *,. {e}(3E,S» is defined and equal to m if e = <e0,e,>, <e o ,a>E0 s , |<e 0 ,a>| = a and {e]}f(a) = m. 0 and {#><*,} can be imbedded effectively in 0 5 and {H?\a
is defined

(for every collection S of reals). As in Normann's remedy given above, interest tends to center on recursively enumerable classes of sets. Let A be a recursively enumerable (in 3E) class of sets for the sake of sharpening the intuitions behind Kreisel's remedy for impediment (I) to the solution of Post's problem. A set R is put into A if and only if there exists a computation Z of {e}(3E,R). The existence of Z is equivalent to the existence of some Hf with some first order property specified by e. Since R is a set, there is an a such that R < 3 E,a. Then Z, if it exists, will be recursive in 3 E,a, and so will be a set located in the Ha hierarchy not far from R. Thus recursively enumerable (in 3E) classes of sets are enumerated via computations which are themselves sets. To carry the parity between elements of recursively enumerable classes and computations one step further, let A be a class of sets recursively enumerable in 3E,R0, where R0 is a set; that is there is an e such that REA^{e)(3E,R0,R)

is defined

(for every set R). As in the previous paragraph, R lands in A if and only if there exists a computation Z to that effect. As above, if such a Z exists, it will be recursive in 3E,R0,R, and so will be a set. Now the requirements of parity have been fully met: the elements of A, the parameter R0 that defines the procedure for enumerating^, and the associated computations all live on the same level of complexity; they are all sets, that is sets located in the Ha hierarchy for recursion in 3 E.

311 Absoluteness and recursion in finite types

209

But there is an unexpected fly in the ointment. The class of all sets is not recursively enumerable in 3 E when 3E satisfies Normann's regularity assumption. Let < be a well-ordering of 2". -< is said to be recursively regular if there is no function/< 3 E,a for some real a such that the domain of / is a proper initial segment of < and the range o f / i s unbounded in «<. NRA: Normann's Regularity Assumption. There exists a recursively regular well-ordering -< of 2" such that -< is recursive in 3 E. 3 Lemma 5. Suppose Normann's regularity assumption is satisfied by -<. Let A C2U be recursively enumerable in 3E,a for some real a, and let B be a proper initial segment of <. Then An B is recursive in 3E,b for some real b. Proof. An immediate consequence of the main theorem of (1977).

NORMANN

The proof below that the class of all sets is not recursively enumerable in E (when Normann's regularity assumption holds) is largely a proof of the existence of a non-indexical set. This notion of set, defined just before Proposition 4, is in need of some clarification. Let R be a set. Thus R<3E,a for some real a, and R is first order definable over (2",c,/f 0 >, where o= |<e\a>| for some e. Let aR be the least such a. In short aR is the first level of the {Ha\o
Proposition 6. Let R be a set. R is indexical iff aR is recursive in 3E,R. Proof. First suppose R is indexical. Thus there is a nonempty IQ2U such that K3E,R and 3 (a)[ael^R<3E,a]. (a)[ael^>R< E,a].

It follows from Gandy selection and bounding that there exists an ordinal y recursive in 3E,R such that (fl)[ae/-»|o|<7]. (fl)[ae/-»|o|<7]. Consequently aR
Analogous to the assumption in a-recursion theory that a* is a regular o-cardinal.

312

210

G.E. Sacks

of all sets obtained by iterating the 3 E jump through all ordinals recursive in 3 E,a for some real a. To say R is indexical is to say R is located at some ordinal recursive in 3E,R. Proposition 7. (i) Every Hg is indexical. (ii) The class of indexical sets is recursively enumerable in 3 E. Proof, (i) L e t / b e { a | | a | = o } . T h e n / < 3 E , / / 0 , and//„ < 3 E, a for all a £ / . (ii) By Proposition 6, R is indexical iff there is an e such that <e,0>eG* &Ris first order definable over //K,,<»|„Observe that every recursively enumerable (in 3E) class A of indexical sets has a natural enumeration tied to that of G. The members of G are enumerated in K, steps. At step o, all a 6 0 such that |a| = o are enu­ merated, and also all R GA such that aR = a. Some such enumeration of A seems to be an essential part of priority arguments in the study of recursively enumerable classes of sets. This is why NORMANN (1975, p.8) restricts his attention from M to 'A/. If A were allowed to have some non-indexical member S, then the reason for putting S into A might not become evident until S was located at stage as of the Ha hierarchy. By then it would be too late to put S in A, because as exceeds all ordinals recursive in 3 E,S\ and the latter are the only ordinals allowed to serve as the height of a computation that puts 5 in a recursively enumerable class. Proposition 8. Assume there is a well-ordering W of2" recursive in 3 E. Then (i), (ii) and (iii) are equivalent. (i) R is an indexical set. (ii) There exists a real a such that R<3E,a and a < 3E,R. (iii) There exists a be6 such that aK = \b\ and b < 3 E,R. Proof. Note that if S is a nonempty set, then the least member of 5 (least with respect to W) is recursive in 3 E,S. Assume (i) to show (ii). Thus there is a nonempty K3E,R such that R <3E.b for every be I. Let a be the W-least member of /. Then a satisfies (ii). Assume a satisfies (ii). Then aR is recursive in 3 E,a. The set of all b that satisfy oR = \b\ is recursive in 3 E,a. The least such b is recursive in 3E,R. (i) follows from (iii) by Proposition 6. Theorem 9. Assume 3E satisfies Normann's regularity assumption. Then the class of all sets, namely {R\R C2" & ( E f ) ( a e 2 " & R < 3 E,a)}, is not recursively enumerable in 3 E,c for any c£2"

313 Absoluteness and recursion infinite types

211

Corollary 10. Assume 3E satisfies NRA. Then there exists a non-indexical set. Proof. Theorem 9 and Proposition 7(ii). Corollary 11. Assume the cardinality of the continuum is regular. Then there exists a normal type 3 object F such that {R\R C2U & (Ea)(ae2u

&

R
is not recursively enumerable in F,b for any b(E2u. Proof. Let W be a well-ordering of the continuum of minimal height. Define F=< 3 E, W) and imitate the proof of Theorem 9. NORMANN (1975, p. 9) asks if the concepts of 2* and weak 2* definabil­ ity coincide. Let A be a class of sets. Call A weakly recursively enumerable in 3 E if there is an e such that for every set R,

ReA~(a)[R<3E,a>{e}(3E,?,a)

is defined]. 3

Clearly every class recursively enumerable in E is also weakly recursively enumerable in 3 E. For 3 E, (weak) 2* definability is the same as (weak) recursive enumerability. Corollary 12. Assume 3 E satisfies Normann's regularity assumption. Then {R\R C2U & (Ea)(aG"»

&R<3E,a)}

is weakly 2* definable, but not 2* definable, in 3 E. The Ramified Language £(^,5). The proof of Theorem 9 is a forcing argument that ranges over the Ha hierarchy of sets associated with recur­ sion in 3 E. The result of adjoining a generic TC"U to { # > < « , } will be studied. The resulting structure, {//„ r |o<»c l }, is of greatest interest when K, r =K|. The language £(K„!>) contains just what is needed to analyze {/f/|a
|?2y=t{

and

/2GW(e,t„H^),

is an ordinal term, and t2 and t3 are real terms. formula V has one or more Godel numbers of the form am,b] bn): e specifies the logical form of f; ai specifies a real a, occurring in 9; and b, is a notation from 0 that specifies an

314 212

G.E. Sacks

ordinal constant |b,| occurring in §. There is no need to choose a unique Godel number for ff, since each is recursive in every other via 3 E. Thus the 3 E degree of the Godel number of <5 is unique. Let l"f 1 denote a Godel number of f. The collection of all Godel numbers is recursively enumer­ able in 3 E. 9 is said to be a bounded formula if every ordinal quantifier in 9 is bounded (by some ordinal less than *,). Every formula $ has a rank based on the range of its ordinal quantifiers, its ordinal constants and its logical complexity. It is a routine matter to define a rank function whose range is well-founded with the following properties: (rl) rankf r (a)| 5 =a has lower rank than || T =i3 if a<0. In addition the graph of the rank function can be taken to be recursively enumerable in 3 E, since the rank of 5" can be effectively obtained from a Godel number of <5. The Forcing Relation. Assume NRA holds. Thus < is a well-ordering of 2" recursive in 3E and recursively regular. Forcing conditions are denoted by p,q,r,.„, and specify a small part of the characteristic function of T. A typical forcing condition is a function f:{c\c for some real b. A condition/? is encoded by a pair <e,6> such that p = {e}(3E,b). p is extended by q (in symbols p>q) if graph pQ graph?. The forcing relation p [f ff is defined by recursion on the rank of UJ. Each clause of the definition either parallels some clause of the definition of {///>(a)=l. (^is//q-~[q^\. (f3)^H-(Eo)S"(o) iff pfr9(a) for some <*<«,. (f4)pff(Ex)5r(x) iff P^(2L) for some aG2". (f5)p(f|<2 f ,a>k=a + l iff pff|<e,a>|„-=a. (f6) Let A| 5 =A iff (f6.1)-(f6.3) hold for some (f6.1)pH-|<m,a>k = a . Let W be W(e,a,//J). (f6.2)pH*XEo).|x|5-o].

a<\.

315 Absoluteness and recursion in finite types

213

w (f6.3)/>lK(7)e = <»]. (f7) The above clauses indicate, it is hoped, how to force b £ W(e,a,//J). Recall that H^e, a,//J) denotes a first order definable subset of <2*\e,//J> whose definition is built up from the atomic predicate x e / / J , real variable quantifiers, and logical connectives. Thus clauses such as (f2) and (f4) can be used to reduce the forcing of reals into W(e, a, //„5) to the forcing of reals into H^ and the latter situation reduces to events occurring below level a.

Proposition 13. There exists a recursive (in 3E) set of codes for the forcing conditions. Proof. Call a collection R of reals a small set if it is contained in { c | c < a } for some real a, and is recursive in 3E,6 for some real b. A forcing condition is in essence a small set. According to Lemma 1 of NORMANN (1977), every small set occurs in the Ha hierarchy for 3 E prior to level | < | , the ordertype of < , an ordinal recursive in 3 E. Thus there is a recursive (in 3 E) set of codes for the conditions defined at level | < | . From now on assume that the forcing conditions are codes as in Proposition 13. It will be convenient to allow p,q,r,... to denote both forcing conditions and their codes. The ordinal rank of a bounded formula 9 is the greatest ordinal constant that occurs in 9, either as a strict upper bound on some ordinal variable or as a parameter in some predicate. Sublemma 14. Ifa€6,

then the relation

pfr9, restricted to ff's of ordinal rank < \a\, is recursive in 3 E,a (uniformly in a). Proof. By effective transfinite induction on the rank of f. Suppose by way of illustration that f is (o)0pT means (jBW«WEr)?[r|rff§(j3)]. The ordinal rank of % 0 ) is at most that of 9. The rank of § 0 ) is less than that of 9\ Proposition 13 justifies the induction step up to 9 from sentences of lower rank. 3

The next lemma will establish bounds on the ordinals recursive in E,7,a when T is generic. It will be seen that ifpH-[|a| 5
thenp|f|a|5<^

316 214

G.E. Sacks

for some 0 recursive in 3E,p,a. The proof of bounding seems to require the notion of weak forcing. Say^fff (readp weakly forces S) if there is no q such that p >q and q$~f. Note that Sublemma 14 remains true when (fis replaced by |f*, thanks to Proposition 13. Lemma 15. Suppose />P(Eo)[|a|g-=a). Then there exists a q and a y such that

p>q,

«H-H«r-% ^< 3 E.P. fl

and y is recursive in 3E,p,a. Proof, q and y, regarded as partial recursive functions of 3 E, p and a, are defined by effective transfinite induction on the rank of <.p,a). A rank can be assigned to , when (1)

/>P(Eff)[|«j 5 -ff],

by appealing to the inductive definition of ((-*. (1) becomes (2)

(^ > 9 (Er) ? > r (EY)[r[f|a| ? r =Y].

The pair , if rH-|a| 5 =y, of course has rank y. The rank of ,«>, if (1) holds, is the supremum of the ranks of the pairs needed to verify the truth of (2). Logical complexity also plays a part in the assignment of rank to (p,a). For example, suppose a is (2e,b) and (1) holds. Then ,a> has higher rank than ,<>,£>> because it has greater logical complexity. The logical difference has to be considered since (p <2' b)) and {p <e b}) may have the same ordinal rank. Thus a pair <£«>, if (1) holds, will have an ordinal rank less than or equal to K\, and a logical rank developed in accord with clauses (f5)-(f7) of the definition of ff" To see how q and y are defined, let a be <3 m ,5',c> and assume (1) holds. Then />P(Ea)]|<m,c>| 5 =o]. By induction there exist q0 and y0, each computed from 3 E,p,a, such that p>q0 and 9oH-K«n.c>|5=YoLet H^be W(e,c,H1t). Then

q0fr'(x)(Eo)[xeW^\x\5=o], and so for each

bel",

(q) a

Fix b and q
317 Absoluteness and recursion infinite types

215

that q >r and r(fb€ W, then r(q,b) is the least such r (least with respect to <), and y(q,b) = 0. If there is no such r, then <7tr(Ea)[|b|y=a]. By induction there exist r(q,b) and 7(9,6), each computed from *E,q,b such that q>r(q,b) and r(, a,. It follows that y(q,b) is bounded, for all q , by some y, recursive in 3E,p,a. Hence

(3)

*oH-1»k
The desired q and y can now be obtained from (3) by an application of Sublemma 14 similar to the one above. Lemma 16. Suppose pUx)(^.o)[xeW(e,2i,Hy)-^\x\?=o). a« a
Then there exists

Proof. Let W be rV(e,a,Hy). Then for each b, (?),>f(Er),>,[i-HEa)(bGW-*|btg-c)]. Now proceed as in the last part of the proof of Lemma 15 to obtain the bound on a. Let TQ2" and 9 be a sentence of £(*„!!"). r is said to be generic over <3 if there is a/> such that 7 satisfies/>, and e i t h e r p ^ orp^—^. Tis said to be generic if T is generic over ff(o)[|b|r*a]. Then there exist a q, a real c and a bounded sentence % with the following properties. (1) p>qandql<5. (li) «?H-(o)[|c|5^a]. (iii) For all TC2": if <J is a true statement about T and beeT, then c(EST and \b\T>\c\T.

318 G.E. Sacks

216

Proof. The definition of q, c and 9 has three cases, only the last of which is of interest. Case I: b is not of the form <2',a> or <3 m -5',a>. Let q=p,c=band 9 be 1 = 1. Case II: b is of the form <2',fl>. Let q=p, c = <e,a> and f be 1 = 1. Case III: b is <3 m -5',a>. Si^care III,: ^H-(aX|<m,a>| 5 ^a]. Let , c = <w,a> and f be 1 = 1. Sufccow III2: otherwise. Thus there is a q0|?r=Y. Suppose (for the sake of a contradiction) that (1)

q0Hx)(Eo)[xe W(e,z,H?)^\x\^<,].

Then by Lemma 16, q0Hx)(Ea)a|?r=Y & c £ W(e,z,H?). Lemma 18. Su/>/>0«> />H-(o)[|b|«r*fe<»]- Then there exist a q and a bounded sentence 9 such that (l) p>qandqVt<¥; and (li) for all TQ2", if 9 is a true statement about T, then b^0T. Proof. MOSCHOVAKJS (1967) showed that every co-recursively enumerable (in 3E) predicate P(x) is expressible in the form (Ey)R(x,y), where R(x,y) is recursively enumerable (in 3 E). The intuition behind R is the same as that behind the desired 9. To say b$eT is equivalent to saying any attempt to recursively enumerate (in 3 E, T) b in 0 r must fail because the associated computation tree is not well-founded. 9 specifies an infinite descending path through that tree. The path is obtained by repeated application of Lemma 17. Let/><, = /> and b0=b. Assume p, and b, satisfy the initial supposition of Lemma 17; let (pi+l,b,+v%+i> bear the same relation to (prb,) that (q,c,9} bears to ,*>> in Lemma 17. Let q be U { « ! ' < " } . and let 9 be a_bounded sentence logically equivalent to the infinite conjunction of the «j/s. (The ordinal rank of 9 is the supremum of the ordinal ranks of the 9,'s, and 9 has a real parameter d such that (rf), is a Godel number of $ for all i >0.) Clearly <7|p7. Fix Tc2". Suppose (for the sake of a contradiction) that 9 is a true statement about T and that fcG0r. By property (iii) of Lemma 17, b^G7 for all f, and \b,\T (i<w) is an infinite descending sequence of ordinals.

319 Absoluleness and recursion in finite types

217

Lemma 19. For each p there exists a T that satisfies p and is generic on all bounded 5"'s. Proof. Let % be the bounded sentence whose Godel number is a. A function P will be defined so that for each a, if a is the Godel number of a bounded sentence, then either P(a)^u7a or />(a)[r—0Ja. The characteristic function of T will be U {P{a)\a&2"). P is defined by transfinite recursion on <, a well-ordering of 2" that is recursive in 3 E and recursively regular. Fix a. Let P0{a) be the union of the given/? and the P(b)'s for all b0(a) is a union of compatible forcing conditions, hence it is the characteristic function of a set of reals. It need only be shown that the domain of P0(a) is bounded above (with respect to <) by some real d. For each b. Then S < 3 E , a for some a if and only if b&6s. By Lemma 18 there is a forcing condition p such that either (B) or (C) holds. (B) /?H-|b|g-=Y for some y and is generic on all bounded sentences. Let S be any set recursive in 3 E,a for some a. Clearly 0 H _ g > S , where 0 is the null forcing condition. It follows that 7, is not recursive in 3 E,a for any a. Hence b$8T', and so (C) must hold.

320

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G. E. Sacks

Let O0<»c, be a limit ordinal greater than the ordinal rank of % (mentioned in (Q), and greater than the ordinal needed to compute < recursively from 3 E. Repeat the construction of Lemma 19 from the vantage point of o0 to obtain a T2 that satisfies p, is generic on all bounded sentences of ordinal rank less than o 0 , and is first order definable over H„. Thus 72< 3 E,fl for any a such that a<E0 and |a| = o0, and so b&6TK But (C) implies 6 € 0 * . Indexicality is one of two notions needed for the absolute formulation of Post's problem. The other is ordinal recursiveness. Let R C2". R is said to be ordinal recursive in 3 E if there exist an integer e and //-sets H\.,.,Hm such that

R=ee}(3E,H\...,//"). By an //-set is meant any set in the hierarchy {//><«,}. A typical ordinal recursive set is a set first order definable over some HB by means of a formula without real parameters. Suppose S is recursive in 3E,b. The real b plays a double role. First b0eeT and HMT is the universe over which S is defined. Second 6, is the real paramete/needed in the definition of S. Thus every set recursive in 3E,b for some real b is ordinal recursive if and only if every real is ordinal recursive. Proposition 20. Suppose there exist a well-ordering W of 2" such that W is recursive in 3 E. Then every set recursive in 3E,bc for some 6=2" is ordinal recursive in 3E. Proof. Let y be the height of W. Let / be the unique function that maps {H„\o) for each b, and so each real is ordinal recursive in 3 E. The next lemma clarifies the connection between //-sets and ordinal recursive sets. Two sets of reals are said to have the same 3E-degree if each is recursive in the other and 3 E. Lemma 21. (i) Let H' and H2 be H-sets. Then there exists an H-set / / 3 such that / / ' X / / 2 has the same 3E-degree as H\ (ii) R is ordinal recursive in 3E ;/ and only if R is first order definable over some Ha by means of a formula whose only parameters are integers. Proof, (ii) follows from (i). The proof of (i) is an effective transfinite recursion that defines functions / and g with the following properties:

321 Absoluteness and recursion infinite types

219

//„X// T is first order definable over /// ( O T ) via the formula (with integer parameters only) whose Godel number is g(o,T). Fix (y,S) and assume/ and g are already defined for all 'S below . Define/(y,8) to be the strict least upper bound o f / ( O , T ) for all less than . The definition of g(y,S) has two cases. First suppose there is a p is the least pair of //-sets whose product is not first order definable over Hp for any p
on the class of all //*-sets? The class of all //*-sets is recursively enumerable in 3 E. The natural enumeration of the //*-sets parallels the development of the ordinals less than K, by iteration of the 3E jump. It follows there is a useful well-order­ ing of the objects allowed to occur as members of recursively enumerable classes even when there is no well-ordering of 2" recursive in 3 E. And if such a well-ordering of 2" does exist, then Proposition 20 implies that every real is an //--set, and consequently that the incomparability of A and B asked for in the absolute problem is the most possible. Thus parity

322 220

G. E. Sacks

is maintained between the objects put into recursively enumerable classes and the objects that index reduction procedures without any weakening of incomparability requirements. The axioms of set theory leave open whether or not every real is ordinal recursive in 3 E, hence a positive solution to the absolute Post problem for 3 E leaves open how much incomparability is achieved, but nonetheless renders that incomparability as definite as the ordinals. A last word about the choice of //'-sets as the objects to be enumerated. An //"-set has all the advantages of indexicality and ordinal recursiveness. The principal argument in favor of indexicality was given just before Proposition 8, and the one for ordinal recursiveness in the previous paragraph. But there is still another point to consider. Why H* rather than / / ? The //-sets are of course indexical and ordinal recursive, and in addition they form a recursive (in 3E) class. Why go to //*? One answer is suggested by Lemma 22, namely that the //*-sets constitute the largest class of objects suitable for the absolute study of Post's problem relative to 3 E. A second answer, perhaps more practical than the first, is that the //■-sets have useful closure properties exemplified by. the statement and proof of Lemma 21(i). In order to state a morally certain conjecture concerning the absolute Post problem, it is necessary to define two notions of recursively enumer­ able projectum relative to 3 E. The greater projectum is denoted by p. It is the least a < K, such that there exists an //-set / / ' with the following property: every //-set is recursive in 3 E,//,,// T for some r«T. The lesser projectum is denoted by 17. It is the least a
Yes, it does. Cf. SACKS (1980).

323 Absoluteness and recursion in finite types

221

corresponds to the case when p is less than K, and is a recursively regular cardinal. A quite different state of affairs ensues when p is singular; that case is handled with the aid of Kr*, the greatest reflecting ordinal relative to <3E,6>, pinned down by HARRINGTON (1973) in his Ph.D. thesis. The supposition TJ = p is analogous to a fundamental truth of a-recursion theory: an cr-recursively enumerable subset of an ordinal less than a*, the 2, projectum of a, is a-fmite. A closely related supposition, termed adequacy, is made by FENSTAD (1979). (2) Does mere exist a generic extension of L in which the absolute Post problem has a negative answer? in which every subset of T recursive in 3 E and a real is indexical? The more appealing conjecture, in both cases, is yes. (3) All the results of this paper, save for Proposition 5, lift to "E when n>3. Some changes in definitions and proof are necessary. They are designed to take into account the difference between the Gandy and Grilliot selection principles, a difference manifest only if n>3. (4) Is there an absolute version of Theorem 9? One that does not assume some well-ordering of the reals is recursive in 3 E? It might be possible to show that the class of all sets ordinal recursive in 3E is not recursively enumerable in 3 E,/?, for any ordinal recursive set R, by assuming that P < K , and p is recursively regular. References FENSTAD, J. E.

[1979] General Recursion Theory (Springer-Verlag, Berlin), to appear. FRIEDBERG, R.

[1957] Two recursively enumerable sets of incomparable degrees of insolvability, Proc. Nat. Acad. Sci., 43, 236-238. HARRINGTON, L.

[1973] Contributions to recursion theory in higher types, Ph.D. Thesis, M.I.T. KLEENE, S. C.

[1959] Recursive functionals and quantifiers of finite type, Trans. Am. Math. Soc,, 91, 1-52. [1963] Recursive functionals and quantifiers of finite type, II, Trans. Am. Math. Soc, 108, 106-142. KREISEL, G.

[1961] Set theoretic problems suggested by the notion of potential total­ ity, in: Infinitistic Methods (Pergamon Press, Oxford), pp. 103-140. MOSCHOVAKJS, Y. N.

[1967] Hyperanalytic predicates, Trans. Am. Math. Soc, 138, 249-282.

324 222

G. E. Sacks

MUCHNIK, A. A.

[1956] On the unsolvability of the problem of reducibility in the theory of algorithms, Dokl. Akad. Nauk. USSR, 108, 194-197. NORMANN, D .

[1975] Degrees of functional, Preprint Series in Mathematics No. 22, Oslo. [1977] A note on reflection, preprint, Oslo. POST, E.

[1944] Recursively enumerable sets of positive integers and their deci­ sion problem, Bull. Am. Math. Soc, 50, 284-316. SACKS, G.

E.

[1966] Post's problem, admissible ordinals and regularity, Trans. Am. Math. Soc. 124, 1-23. [1977] RE sets higher up, in: Logic, Foundations of Mathematics and Computability Theory (D. Reidel Publ. Co., Dordrecht). SIMPSON, S.

[1974] Post's problem for admissible sets, in Generalized Recursion Theory (North-Holland, Amsterdam). SPECTOR, C.

[1955] RecurS1ve wellorderings, J. Symbolic Logic, 20, 151-163.

325 SOUTHEAST ASIAN CONFERENCE EN LOGII

ON THE NUMBER OF C O U N T A B L E

MODELS

G e r a l d E. S a c k s * Harvard University M a s s a c h u s e t t s I n s t i t u t e of T e c h n o l o g y U. S. A.

1.

Introduction Let C be a PC

c l a s s of c o u n t a b l e s t r u c t u r e s , and let n ( C ) U • , Lu

be the n u m b e r of i s o m o r p h i s m

types of e l e m e n t s of C.

Morley

[5]

proved : n ( C ) < 2" ~ n ( C ) < », .

(1 )

Each c o u n t a b l e s t r u c t u r e M has a rank r ( M ) , a c o u n t a b l e o r d i n a l measures

the l e n g t h of the c a n o n i c a l

in S e c t i o n 2. structures

Scott s e n t e n c e for M, as d e f i n e d

Let n ( C , 8 ) be the n u m b e r of i s o m o r p h i s m t y p e s of

in C of rank s. n(C)

Morley obtained

< 2U

-

(B)B
(1) by

[n(C,fs) I

showing

«]•

H. F r i e d m a n ' s e x a m p l e F s h o w s M o r l e y ' s r e s u l t cannot be F is P C „ , M , n ( F )

n(C) < 2 .

improved.

»,, and n ( F , B ) £ u for e v e r y c o u n t a b l e 8. that n ( C ) < tu if C is EC

V a u g h t has c o n j e c t u r e d U

that

A s t r o n g e r c o n j e c t u r e a l l o w s C to be EC

and j that

C is the class of all c o u n t a b l e m o d e l s of some c o m p l e t e theory some c o u n t a b l e f r a g m e n t of L

. Qi*

p o i n t out that PC

The p u r p o s e of this p a p e r is to behave quite differently with res-

p e c t to V a u g h t - t y p e c o n j e c t u r e s .

The d i f f e r e n c e

S

is a c o n s e q u e n c e

The F r i e d m a n e x a m p l e F has the p r o p e r t y

that (BL

in

> LO

and EC

of c o m p a c t n e s s a r g u m e n t s .

is,

<<"1

[n(F.B) < 2 ]

326 186

G.E. SACKS

yet n(F)

u>-,.

On the other hand, Theorem 4.1 below implles

(2)

(6).

if C is EC

[n(C,B) < 2] - n(C) < u

Note that (2) does not assume N(C) < 2 U .

.

Throughout the paper T is a complete theory in some countable fragment of i.

.

Section 2 reviews Scott sentences and the fact M that the rank of a structure M is at most u, + 1, and ends with a

a

1

thata he rae eofsaestr ct re Mtisia mmost for T. andtend 3 ithlembes n t, eexistencen ff nha omictmode mforlT of e tion 3(d,s)<2 u cri esl counenume a.

nho rthelco nt bletmode s ofd 3 hen combine<

fo Sactico n ab ebtain (e) asu tse foSectiog r2sand. Theorem 4.2.

Suppose the number of countable models of T is less

than 2 U .

Assume that the canonical Scott rank of each countable M model " of T is at most to ■. . The n T has only countably many coun ¬ table models. The above theorem suggests that the main obstacle to proving Vaught's conjecture has to do with structures M whose Scott rank is the greatest possible, namely m® + 1. Section 5 raises the question of a bound on the Scott ranks of the countable models of T when n(T) < «. special cases.

SSom bouund aar ooffere ii

For example, the proof of Theorem 4.2 shows that

every countable model of T has rank at most ^,

where *]

is the

least lo admissible ordinal relative to T.

2.

Scott Rank Scott [1] discovered that each countable structure M can be

characterized up to isomorphism by a single sentence F of L to -I , Ul

that is

» M F and

[ O F] - N ~ M

327 On the number of countable models for every c o u n t a b l e s t r u c t u r e N .

A canonical

187

c h o i c e for F, d e n o t e d

by F„, is d e v e l o p e d from M by an i n d u c t i v e p r o c e d u r e that at a c o u n t a b l e ordinal

r ( M ) , the Scott rank of M.

Let 1.0,M) be the L respond M.

l a n g u a g e w h o s e p r i m i t i v e symbols c o r -

to the r e l a t i o n s , f u n c t i o n s and d i s t i n g u i s h e d

F o r each limit ordinal

x, l ( x , M ) is u U ( 6 , M ) | 6

6, let T ( 6 , M ) be the c o m p l e t e

e l e m e n t s of

< A } . F o r each

theory of M in the l a n g u a g e

And let 1.(6+1,M) be the least f r a g m e n t of L and s a t i s f i e s

terminates

1..6,M).

that i n c l u d e s

1(6,M)

the f o l l o w i n g c l o s u r e c o n d i t i o n .

if n < u and p(x-,,...,x n ) is a n o n - p r i n c i p a l of T ( 6 , M ) r e a l i z e d

n-type

in M, then A < F ( x ) | F ( x ) e p}

b e l o n g s to the f r a g m e n t . Scott showed there exists a c o u n t a b l e 6 such that 1.(6 + 1,M) 1..6,M). model

Let r ( M ) be the least such 6.

Then M is the unique

of T ( r ( M ) , H ) , and F M , the c a n o n i c a l

a s e n t e n c e that a s s e r t s

atomic

Scott s e n t e n c e f o r M, is

■ "I am an a t o m i c model

of T ( r ( M ) , M ) . "

S c o t t ' s proof that r ( M ) exists and is c o u n t a b l e is in e s s e n c e a counting argument.

If 1.(6 + 2,M) ^ 1 ( 6 + 1 , M ) , then there exist two

n-tuples of H that are e q u i v a l e n t with r e s p e c t to all n - p l a c e

for­

m u l a s of L ( 6 , M ) , but i n e q u i v a i e n t with r e s p e c t to some f o r m u l a of 1(6+1,M).

Once an i n e q u i v a l e n c e o c c u r s , it becomes

permanent.

Since M is c o u n t a b l e , there can be only c o u n t a b l y many

initial

o c c u r r e n c e s of i n e q u i v a l e n c e . The set of i n e q u i v a i e n t end result of a *, m o n o t o n i c as Nadel

i n d u c t i v e d e f i n i t i o n , and c o n s e q u e n t l y ,

[2] o b s e r v e d ,

r(M) < J* + 1.

(1) (ui

n - t u p l e s of M can be c o n s t r u e d as the

is the least ordinal

not r e c u r s i v e

in M.

In (1) M a m b i g u o u s l y

328 188

G.E. S A C K S

d e n o t e s a c o u n t a b l e s t r u c t u r e M and some real

that e n c o d e s M . )

For the r e m a i n d e r of the p a p e r , let T be a c o m p l e t e , c o n s i s ­ tent t h e o r y in some c o u n t a b l e f r a g m e n t K of L

. I n this

setting,

lil-r * y

" c o m p l e t e n e s s " m e a n s every s e n t e n c e of K or its n e g a t i o n to T , and " c o n s i s t e n c y " r e q u i r e s that no i n f i n i t e

belongs

disjunction

belong to T unless o n e of its c o m p o n e n t s also b e l o n g s . Define n ( T , B ) to be the number of ( i s o m o r p h i s m types o f ) m o d e l s of T of Scott rank B . L t: Ml Ml C*

2 . I •

S u [J [Jos e

'' ( T , e )

2

I VI

all

couIItaUIe

B .

TJlCl I

T

llOO

aIJ

a Proof.

It suffices to show S n ( T ) , the set of n - t y p e s of T , is

c o u n t a b l e for all n < u.

In that e v e n t , an a t o m i c model of T is

c o n s t r u c t e d by o m i t t i n g all n o n - p r i n c i p l e

n-types.

Fix n and suppose S n ( T ) is u n c o u n t a b l e . fect kernel

Then S (T) has a p e r ­

Z whose c a r d i n a l i t y is 2 " . L e t T a m b i g u o u s l y d e n o t e a

coding of T by a r e a l .

The coding of T by a real gives rise to a

coding of Z by a real Z Q and a coding of n-types by r e a l s . r e l a t i o n p e Z Q is a r i t h m e t i c .

According p,Z0,T

there exists a p e Z Q such that u.,

Thus the

to the Gandy basis

theorem,

Z„,T

*:°

. A slight

variant

of G a n d y ' s a r g u m e n t y i e l d s a real Z. that codes a p e r f e c t subset of Z such

that (p)[p e Z 1 - w ^ ' Z l ' T

(3)

u^''1] .

For each p e Z,, a Henkin c o n s t r u c t i o n p r o v i d e s a model M arithmetic

in p , T and r e a l i z i n g p .

Then

Mp p.T P>zl>T •l £ "i £ "i and c o n s e q u e n t l y

Z, ,T r ( M p ) < u,,1 +1

by

of T

Z,.T Wi

(2) above.

The s e t {M | p e Z1 }

329 On the number of countable models

has cardinality of T. 3.

189

2 " , and gives rise to 2 " isomorphism

It follows from

(3) that n ( T , B )

U

2

types of m o d e l s

for some 6 £ m^'

♦ 1.

r. Enumeration of Models Let T be a complete theory in some c o u n t a b l e fragment of

L

Assume n(T,e) < 2 U for all c o u n t a b l e 8 .

.

carr.es Lemma

2.1 leads to a natural

enumerate

Scott sentences of the countable m o d e l s of T.

T h e reasoning

that

of the c a n o n s l

A tree T of c o u n t a b l e ,

c o m p l e t e , consistent theories extending T is defined by r e c u r s i o n . Each countabl e mooci OT T is tns atonii c model o* some theory on the

a Level 0 of T : theory T. Level * :

S occurs on limit level

x iff there exists a

strictly expanding sequence { T 6 | « < x and J& occurs on level 6} whose union is S. Level 6+1 :

Suppose S occurs on level 6 , and S is a complete

theory in some countable fragment

Kofi

. I f every model

of S

ui i ■ d)

l

is a t o m i c , then S has no successors on level wise.

If n ( S , B )

2 " for some countable B , then n ( T , y )

some c o u n t a b l e if > B. the proof of Lemma able.

6 + 1 . Suppose

1

Hence n ( S , B ) < 2

2 U for

for all countable B.

As in

2 . 1 , 2 , the set of n-types of S (n < u ) is count­

Let W be {X||cr 2 and X is the set of non-principal

of S realized

other­

in some model of S ) .

n-types

If some coding of K by a real is

fixed, then W can be regarded as a boldface r^ set of r e a l s .

Con­

sequently W is either countable or of c a r d i n a l i t y 2 . If the l a t t e r . then n ( S , B ) 2*" for some countable B as in the proof of Lemma 2 . 1 . Thus W is c o u n t a b l e . For each X e W, let K v be the least fragment of L includes

K and all formulas

A ( F ( x ) | F ( x ) e p} for all p e X.

that

330 190

G.E. S A C K S S' is a s u c c e s s o r of S on level

6+1 iff S' is a c o m p l e t e ,

c o n s i s t e n t theory in K„ for some X e W. Let M be a model of T.

End of d e f i n i t i o n of F.

M d e f i n e s a c o u n t a b l e path P H

through

T that c o r r e s p o n d s to the Scott a n a l y s i s of M. PM(°)

T.

If P M ( ' ) is defindd roa 1ll < X ,t theP P M ( x i

ss { { P „ (y) | y < \ } .

S u p p o s e P M ( « ) is defined and o c c u r s on level 6 of T, and M (= P u t * ) -

If M is the atomic model of P M < 6 ) , then P M ( 6 + 1 ) is

undefined.

Suppose otherwise.

Then the d e f i n i t i o n of T i m p l i e s

P M ( 6 ) has a u n i q u e s u c c e s s o r S' on level 6 + 1 of F such that M f= S' ' It f o l l o w s that P M ( r ( H ) ) fined.

T h u s each canonical

c o u n t a b l e level of T. model

by Lemma 2 . 1 .

PC

Scott t h e o r y e x t e n d i n g T o c c u r s on some

C o n v e r s e l y , each theory on T has an a t o m i c Hence T is an e n u m e r a t i o n of all the c a n o n i c a l

Scott t h e o r i e s e x t e n d i n g T. enumeration.

T ( M , r ( M ) ) , and P H ( r ( M ) + l ) is u n d e ­

Note that there are no gaps in the

The e n u m e r a t i o n can be said to be c o n t i n u o u s .

A

class of c o u n t a b l e s t r u c t u r e s of c a r d i n a l i t y less than 2U

need not be c o n t i n u o u s Lemma

3.1.

Let Tu

in this

sense.

be the r e s t r i c t i o n of T to l e v e l s b e l o w

S u p p o s e L ( a , T ) is E-, a d m i s s i b l e .

u.

Then Ta is a £.,(L(<x,T) ) s u b s e t of

L(o,T). Proof.

(Sketch)

T u is defined by a I.,(L(a,T) ) r e c u r s i o n .

the s u c c e s s o r step in the d e f i n i t i o n of T. t]

r e l a t i v e to some c o u n t i n g s of Z and S.

Recall

It was noted that W is Hyperarithmetic

theory

[ 3 ] , s t a r t i n g with any c o u n t i n g s of Z and S, y i e l d s a c o u n t i n g W hyperarithmetic

in those of Z and S.

of

C o u n t i n g s of Z and S can be

added g e n e r i c a l l y to L ( o , T ) w i t h o u t v i o l a t i n g E,

admissibi1ity.

Then a c o u n t i n g of W belongs to every such g e n e r i c e x t e n s i o n , and so

331 On the number of countable models

191

does a counting of each S'a s u c c e s s o r of S.

But then S1 b e l o n g s to

L(a,T).

in L ( a , T ) . )

(Of course S' need not be c o u n t a b l e

passage from S to S' is given by a £ , ( L ( a , T ) )

4.

Main

The

formula.

Results

Let T be a c o m p l e t e

theory in some c o u n t a b l e f r a g m e n t of

L T h e o r e m 4.1.

Suppose T has at most o n e model

each c o u n t a b l e ordinal 6. It was remarked PC

of Scott rank 6 for

Then T has only c o u n t a b l y many m o d e l s .

in S e c t i o n 1 that there e x i s t s an u n c o u n t a b l e

class K of c o u n t a b l e s t r u c t u r e s such that at m o s t one member

of K has Scott rank B f o r each c o u n t a b l e ordinal 4.1 d i s t i n g u i s h e s

between PC

and EC

U)-i > U)

an aspect of V a u g h t ' s

6.

c l a s s e s with r e s p e c t to

conjecture.

Section 3, and s o m e t y p e - o m i t t i n g [4].

Let

a

According

argumentation

e n u m e r a t i o n of in the style of

be the least y such that L ( Y , T ) is z.

For the sake of a c o n t r a d i c t i o n , ter than „ .

assume T has a model

a, is ^ ( a , T ) .

a

3 . 1 , T , the r e s t r i c t i o n

It follows that T

Q

of T

n-type of T Q .

Q

grea­

it T .

of T to levels

below

is A , ( L ( a , T ) ) .

Two m o d e l s of T of rank a+1 are c o n s t r u c t e d . is a model

admissible.

of rank

Then T has a unique theory on level a ; call

to Lemma

Theorem

tO-i » W

The proof of 4.1 relies on T , the natural

Barwise

Thus

The f i r s t , M , ,

atomic over the r e a l i z a t i o n of some

The s e c o n d , M ? , is a model

non-principal

of T Q not atomic over any

n-tuple of its e l e m e n t s . C o n s t r u c t i o n of M, :

Since T has a model

than o, Ta has a n o n - p r i n c i p a l in some fragment

Ka.

Let r

n-type p.

of rank

greater

T Q is a c o m p l e t e

be the least fragment

theory

c o n t a i n i n g K„

332

192

G.E. SACKS

and

new i n d i v i d u a l

constants

£-| , . . . , c _ .

Then

T; = T u { F ( c ) I F ( x )

e p}

Q

is

a complete

follows

theory

that

<M, . c 1 1

.

a sentence

non-principal principal the

c

{

Thus

0

A{F(x)|F(x)

the Scott

realized

i n M-j i s a t o m i c

i t s complete

axioms If

be l e s s t h a n

asserting

of T a

the r e a l i z a t i o n

r(M») > a .

h a s an

atomic

a, b e c a u s e

over

f o r each o f some

theory

over

The l a n g u a g e

p implies

in

of

i n the language

that

type

L ( a + 1,M-|)

each

n-type

of T

Q

model

set A of

2

with

power t o

r(M-j) = a + 1.

exists a I (L(a,T))

of s e t theory

includes

i s an a t o m i c

Hence : There

6 < a,

M-| r e a l i z e s a t y p e n o n -

The f a c t t h a t

1

i t

the following

properties:

V (= A, t h e n

The is

By Lemma 2.1

M, i s a m o d e l 1

analysis of M .

Construction

< 2 f o r a l l 6 < w-| ,

a

e p} and s o h a s e n o u g h e x p r e s s i v e

complete

of

n(T ,e)

r(M-|) f a, s i n c e

over T .

over T .

formula

Since

> .

n

r(M-|) c a n n o t

includes

f i + 1

.

n(Tg,B) 5 a f o r a l l B < «»j.

a t o m i c model

T

in

(i)

V i s a proper

(ii )

L ( o , T ) < V , and

(iii)

i n V there

axioms

end e x t e n s i o n

of

L(a,T),

1

of A that

A-|(L(a,T)).

i s a model

insure

The 1

2

of T . a

a r e z (L(a

( i ) and ( i i i )

, T) ) , s i n c e

1

flavor of A i s derived

s o l e l y from

T

o

(ii).

Ha

Let otherwise To

V be a model a would occur see t h a t

a t o m i c model of

M

that

2

i n V.

n-tuple. 2

omits a.

Hence r ( M j )

r ( M ) = a+1, suppose

o f T„.

0

f i s z (L (a , M ) ) , 2

2

that since

<_ a,

because

<_ a + 1 by ( 2 ) o f S e c t i o n a.

2

L e t f be a f u n c t i o n of K

Then w,

r(M )=

2

(n < u ) t o a f o r m u l a

TI (L (a ,M ) ) , 1

of A that

that

defines

Then M

2

c a r r i e s each

2.

i s the n-tuple

t h e a t o m r e a l i z e d by

the notion

a c o n s e q u e n c e o f t h e A-| d e f i n a b i l i t y

o f atom i s of T

a

over

L(a,M ). 2

333 On the number of countable models

The

I

definition

2

to

property

in

L(S,M)

r(M )

o f f , when

( i i ) o f A.

I t follows

f o r some 6 < a,

< °, a p o s s i b i l i t y

2

Similar

reasoning

e M^.

<M

n

c-|,...,c >

2>

levels

below

Q

Theorem 4 . 2 . than

2

W

already also

thanks

shows

that

t h e range

f again

that

M

from

V.

But then

i s not atomic

2

Note

through

that

thanks

of f i s contained

out i n the d i s c u s s i o n

argument goes

o f M-| .

over any

with

the Scott

analysis of

i n V as i t i s i n t h e r e a l

to property

Suppose

. Assume

ruled

o f M,,.

i s t h e same

n

that

i n V, y i e l d s

because a i s omitted

The a b o v e

<M , C|,...,c > i n p l a c e 2

interpreted

193

world

a ta l l

(ii).

t h e number o f c o u n t a b l e the canonical

Scott

models

rank

of T i s less

o f each

countable

M model able

M o f T i s a t m o s t u-| .

By M o r l e y

countable strong 3.1,

ordinal

the tree

o f T.

regular

many

count-

o f rank

at least

c a r d i n a l o f L(a,T)

2

set

( L ( B , T ) )

6 . and l e t T

B

e

of M

V i s a proper

(ii)

L ( , T ) < V , and

(iii)

i n V there

below a, i s

be a t h e o r y

on

since

that

level

6 i sa

L(a,T). i n the proof

2

By Lemma

Assume

2

B

o f ZFC

of L(a,T).

i s r. a d m i s s i b l e ,

A o f axioms

(i)

B

of a fragment

to levels

i n the sense

and T

i n the construction E

i s a model

2, restricted

g

i sa

=

L(a,T)

L e t a be a

out the following manipulations.

T of Section

V be a m o d e l 2

that

The s t r u c t u r e L ( B , T )

As

r(M )

such

<_ a f o r a l l s < u* ,

L e t 6 be a l e p h - o n e

T has a model

there

[ 5 ] n(e,T)

enough t o c a r r y

I,(L(a,T)).

Let

countably

models.

Proof.

6

Then T h a s o n l y

such

end e x t e n s i o n

that

o f Theorem 4 . 1 , : i f

V

(=

A,

then

of L ( B , T ) ,

1

i s a model

of A that

a + l now shows

omits

r(M) = 6+ 1.

M o f T*,

8.

The a r g u m e n t

And

given

<_ 8 , s i n c e

in 4.2

B | V.

that

Hence

334 194

G.E. SACKS J^ + 1 .

r(M)

But by assumption r(M) <_ J*.

5. Further Results and Question Suppose n ( T ) , the number of countable models of T, is countable.

Then there exists a countable ordinal B such that r(M) < 8

for every M a model of T.

Let the least such 6 be r ( T ) . Clearly r(T) is t>- relative to

What can be said about r(T)? T.

Is there a sharper bound?

The argument of Section 4, suitably

modified, yield the following bounds. '•

I f (6)B
[n(B>T) < ] ] , ther r(Ti " 2\ - 1 .

2 M J it ehe

least r2 admissible ordinal relative to T.) II.

If n(T) < 2U and r(M) < J*

for every M a model of T,

then r(T) < «I + 1 . III.

If n(T) < 2" and r(M) < u" for every M a model of T,

then r(T) < „J. Can the bounds in (I)-(III) be improved? Ramas Salam [6] has extended invariant model theory.

(III) to the domain of Vaught's

Can the same be done for (I) and

(II)?

If n(T) < a, then every model of T must complete its canonical Scott analysis on some finite level of T, and so r(T) < also follows that each model of T has a presentation in T.

T. Millar

a

.

It

hyperarithmetic

[7] has shown that "hyperarithmetic" cannot be im-

proved to "arithmetic" in the previous sentence. Suppose ( B ) B < u

[n(B,T) < u ] .

Does it follow that n(T) <

The proof of Theorem 4.1 can be stretched to show (B)„

[«(B,T) < 3J * nT)) - „.

u

?

335 On the number of countable models

195

References

1.

D. S c o t t , Invariant Borel

s e t s , Fund. M a t h . 5 6 ( 1 9 6 4 ) , 1 1 7 - 1 2 8 .

2.

M. N a d e l , Scott s e n t e n c e s and a m i s s i b l e s e t s , A n n a l s of M a t h . Log. 7 ( 1 9 7 4 ) , 2 6 7 - 2 9 4 .

3.

H. R o g e r s , Theory of R e c u r s i v e F u n c t i o n s and E f f e c t i v e

Computa

bility, McGraw-Hill, 1967. 4.

K. B a r w i s e , A d m i s s i b l e Sets and S t r u c t u r e s ,

Springer-Verlag,

1975. 5.

M. M o r l e y , The number of c o u n t a b l e m o d e l s , J o u r . S y m b .

Log.

35(1970), 14-18. 6.

R. Salam, On V a u g h t ' s c o n j e c t u r e in i n v a r i a n t model

t h e o r y , in

preparation. 7.

T. M i l l a r . R e c u r s i v e theories with n o n - a r i t h m e t i c P r o c e e d i n g s of the 1982 AMS Summer

models,

Institute in R e c u r s i o n

Theory,

to a p p e a r .

* This paper is based on a talk given at the 1971 m e e t i n g of the I n t e r n a t i o n a l C o n g r e s s of L o g i c , M e t h o d o l o g y and P h i l o s o p h y of S c i e n c e in B u c h a r e s t .

Its p r e p a r a t i o n was s u p p o r t e d in part

by NSF G r a n t M C S - 8 1 0 4 2 6 6 .

336 Proceeding!! of Symposia in Pure Mathematics Volume 42, 1985

Post's Problem in ^-Recursion GERALD E. SACKS1

Abstract. In the setting of £-recursion a positive solution to Post's problem is given for a class of structures that includes all £-closed, initial segments of L. The solution, based largely on fine structure and reflection properties, does not give rise to injuries when L(<) is 2j inadmissible; i.e., each Friedberg-Muchnik requirement is attempted at most once.

1. Introduction. The partial £-recursive functions were discovered by Normann [1], and independently by Moschovakis. They constitute a generalization of KJeene's theory of higher types recursion from normal objects of finite type to all sets. Thus they give a meaning to {e)(x) for all x based on computations. The fundamentals of £-recursion are reviewed formally in §2. They are dealt with intuitively in the present section in order to provide a rough idea of the proof of the main result of the paper. By a positive solution to Post's problem is usually meant the construction of two incomparable r.e. sets A and B. The meanings of "incomparable" and "r.e." depend on the choice of underlying structure £, of which A and B are subsets, and the choice of recursion theory on i . In this paper a?is always £-closed; that is, f(x) e
( )

If £(2") *= [2" is well ordered and of regular cardinality], then there exist two subsets of £(2"), each £-recursively enumerable on £(2"), such that neither is £-reducible to the other on £(2").

1980 Mathemaiics Subject Classification. Primary 03D65; Secondary 03E45. The preparation of this paper was supported in part by National Science Foundation Grant MCS-81-04266 The author is grateful to Theodore Slaman for his generous assistance 1

© 1985 American Mathematical Society 177

337 178

G E SACKS

The proof of (1) bears only superficial resemblance to the positive solution of Post's problem given in classical recursion theory [3, 4] or in a-recursion theory [5]. The results of the present paper originated in the desire to circumvent the assumption of "regular cardinality" in (1). In order to clarify the consequent of (1), consider subsets A and B of some £-closed transitive set g. A is £-r.e. on g if there are a partial £-rec. function f(x, y) and a/> e
A = {x\x e g&f(x,

p) converges}.

The schemes for computing £-recursive functions can be relativized to an arbitrary class C, thereby giving a meaning to {e }c(x\ by adding the scheme (3)

g(x) =

xnc.

A is £-reducible to B if there are a partial function/, £-rec. relative to B, and a p e
338 POSTS PROBLEM IN E-RECURSION

179

The central positive result of the paper is: (4)

If L ( K ) is £-closed, then there exist two subsets of L(K), each ^-recursively enumerable on L ( K ) , such that neither is £-reducible to the other on L ( K ) .

A curious feature of the proof of (4) is the presence of a Friedberg-Muchnik scheme of requirements without injuries when L(K) is not 2 , admissible. At most, one positive attempt is made to satisfy each requirement, and no negative requirement is injured. Further results, such as Slaman's density theorem [7], do necessitate Friedberg-Muchnik injury arguments. The reasons that r.e. degree theory on £-closed structures is less troubled by injuries than its counterparts in classical or a-recursion theory have to do with reflection phenomena. There is no hope in any recursion theory worthy of the name of effectively enumerating all divergent computations. But something very close to such an enumeration can be realized in many £-closed structures. To define the concept of reflection in £-recursion it is helpful to make a connection between £ ( x ) and L(x). Say y is £-recursive in x if y = (e}(x) for some e. Define KX0 = sup{ y|y is an ordinal £-recursive in x} and K" = s u p { ( c ^ > e T C ( x ) } . (TC(x) is the transitive closure of x.) Then £(X) = L(K*,TC({X})).

y is said to be x-reflecting if [L(S,TC({*}))

t=^] -

[L(K0*,TC({*}))

^ ]

F

for every i^a 2f sentence with x as parameter. Define K* = sup of all x-reflecting ordinals. Clearly, < is the greatest x-reflecting ordinal. It can be shown that K* < K\ More important is the fact that: (5)

If L(K) is not 2 , admissible, then < < K for all x e L ( K ) .

If x is a set of ordinals, then K* turns out to be large enough to decide whether or not {e}(x) converges for all e. If {e}(x) converges, the computation of {e }(x) is a well-founded tree whose height is an ordinal recursive in x, hence less than KJ. If {?}(x) diverges, then any infinite, descending path through its computation tree is called a Moschovakis witness to the divergence of (e}(x). The connection between reflection and divergence needed for the solution to Post's problem is: (6)

If x is a set of ordinals and {<)(x) diverges, then a Moschovakis witness to the divergence is first order definable over L(K;, X).

339 180

G E SACKS

It follows from (5) and (6) that: If £(K) is not 2 , admissible, x G L(tc) and {e}(x) diverges, then some Moschovakis witness to the divergence belongs to

(7)

UK).

It is (7) that makes possible a positive solution to Post's problem by a Friedberg-Muchnik scheme without injuries. A requirement is satisfied in §4 below by waiting for a certain computation to converge or diverge. If it converges, then the usual steps are taken to create and preserve an inequality. If it diverges, then (7) implies there is a stage of the construction by which the divergence is apparent. At that stage a witness to the divergence is chosen and its role as a witness is preserved forever. If £ ( K ) = £ ( « , ) , then the requirements can be indexed by countable ordinals, and each can be satisfied in its turn as in the previous paragraph. For an arbitrary 2 , inadmissible £ ( K ) , an indexing of requirments has to be chosen short enough to allow all requirements to be met before time runs out. In the case of £(<0j) = L ( K ) , any countable set of requirements can be satisfied by stage o for some o < K. In general, the requirements are indexed by ordinals less than p, the greater r.e. projectum of £ ( K ) . P = M T « . « * / ) [ / i s a pariial function, £-rec. on L(K), from 7 onto £ ( * ) ] . It is shown in §3 that: If x < p and p e L(K), then sup{ K?-y\y < x) < «.

(8)

It follows from (8) that any set of requirements bounded below p can be satisfied by stage o for some o < p. (There is an additional complication having to do with the r.e. cofinality of p and a technique similar to Shore blocking, but it can wait until §§3 and 4.) The proof of (8) has two parts. Define the lesser r.e. projection 7, =

M

YT<,(^)Ue22-L(K)&,4ii£-r.e..nL(K)].

A condensation argument given in §3 shows: (9)

If L(K)is£-closed, thent; = p.

The proof of (8) consists of proving (8) with r, in place of p and then invoking (9). The problem of extending the solution of Post's problem from initial segments of £ to other effectively well-orderable £-closed structures is discussed in §5. The only substantial difficulty resides in the proof of (9), a Godel type condensation argument that appears to be useless when £ ( K ) is replaced by £[K. A) for some A c L ( K ) . An important special case is £(2"). Normann's result (1) can be redone as follows. In the first half of his proof he shows: (

'

If £(2")1= [2" is well ordered and of regular cardinality], then TJ = p for £(2").

340 POSTS PROBLEM IN E-RECURSION

181

Then the second half of his argument can be replaced by the proof given in §4. Thus Normann's assumption of "regularity' in (1) was needed only to show i, = p. Slaman [8] has shown: There exists a model M of ZFC in which £(2") t= [2" is well { ' '! ordered and of singular cardinality] and ij = p for E(2U). It follows from the argument of §4 that in Slaman's model M, £(2") admits a positive solution to Post's problem. Thus the study of Post's problem for £(2"), when 2" is well orderable in £(2"), appears to turn more on the equality of ij and p than on the regularity of 2" The existing technology of forcing over £-closed structures [8-10] gives no hint of how to create an £(2") in which 2" is well ordered and for which TJ # p. 2. Review of ^-recursion. The partial £-recursive functions are defined via Normann's schemes [1]. {e} is the partial £-recursive function with index e. x „ . . . ,x„ are arbitrary sets.

(1)

{«}(*x....,*.)-*,

(2)

{e}(x1,...,xn)

(3) (4)

{e)(Xl

x„)^[J{{c}(y,x2,...,xn)\yex1}

{e}(x,

x„)=

{e}(c,Xl,...,x„,

{c}{{dl)(xl tie v,

ym)=

if e = (2, ife =

{e}(x1,...,xn)={x„xJ)

(5) (6)

= x,-xJ

ife = { l , n , i ) . n,,,j). (3,n,i,j). if * - <4, « , c>. x„)

{dm)(x, =

{c}(x1,..,,xj

xn)} (5,n,m,c,dl,...,dm).

ife = < 7 , n , m > .

{e}(x)i means {*•}(*) is defined or converges, and {e}(x)T means {e}(x) is undefined or diverges. In scheme (4), {e}(x1,...,xjim{c}(y,x2,...,x„)l for all.y e xr An important derived scheme, called bounding, is

(7)

{ 2 - - 3 " } W « { { » } 0 0 | , 6 {«}(*)}.

Implementation of the schemes generates the universal computation tree > v. Each node of the tree is of the form (e,xu... ,x„>, or more simply (e, x), and is called a computation instruction, a >v b is read: 6 is a subcomputation instruc­ tion of a, and is defined by: There exists a finite sequence b0,...,bn such that a = fc0, />„ = b, and 6 j + ! is an immediate subcomputation instruction of b, (;<«). The inductive definition of immediate subcomputation instruction has a clause corresponding to each Normann scheme. It is reasonable to regard (7) as typical, because it combines (4) and (5). The corresponding clauses are: (8) (m, x) is an immed. subcomp. mstruc. of (lm ■ 3", x). ^

If {m)(x)[ and v e {m}(x), then (n, y) subcomp. instruc. of (2"'• 3\x).

is an immed.

In addition: If c is not the index of a Normann scheme, then ( c , x) is an ^10) immed. subcomp. instruc. of (c, x).

341 182

G E SACKS

The tree-like object consisting of (e, x> and that part of > „ below (e, x> is called the computation of (e)(x). Note that {e)(x)l iff the computation of {e)(x) is well founded. Let A be a class. A is E-recursively enumerable in x if, for some e, A=

{y\{e)(x,y)l}.

If x = 0, then ^ is £-recursively enumerable. 5 is said to be £-recursively enumerable in x on C if B = /« n C for some .4 £-recursively in x. The relation { e } ( x ) l is £-recursively enumerable, and consequently the rela­ tion "6 is an immediate subcomputation instruction of c" is £-recursively enumerable. Note well: The subcomputation relation >v is not £-recursively enumerable. If {e)(x)l, let \{e)(x)\ be the ordinal height of the computation of {«}(*)• |{e}(x)| is a partial £-recursive function of e and x. A set.y is said to be £-recursive in a set x ify = {
and

K* =

\J{K*0<°\aeTC(x)}.

TC(x) is the transitive closure of x. A set z is £-closed if xBz&{e}(x)i

-

{e}(x)ez

for all e < w. Define £ ( x ) , the £-closure of x, to be the least £-closed z such that r 2 T Q { x } ) . Then £(X) = L(K*,TC({X})).

5 is said to be x-reflecting if [L(8,TC({X}))^]-[L(K0*,TC({X}))^]

for every 2 j sentence JFof ZF with parameter x. Define < to be the supremum of all x-reflecting ordinals. Thus K? is the greatest x-reflecting ordinal. Note that Kr originated in an application of forcing [11] to the theory of Kleene recursion in finite types, where it was necessary to know that

Later Harrington [12] showed K?" > pcf, and more importantly, gave a characteri­ zation of K? that inspired Lemma 2.1. If { e ) ( x ) T , then there exists an infinite descending path below (e, x ) in >„. Any such path is termed a Moschovakis witness to the divergence of {*}(*). They were introduced by Moschovakis [13] to show £ ( 2 " ) is not 2 , admissible. Observe that the relation "z is an A/-witness to {e}(x)T" is £-recursively enumerable. Let and x e
342 POSTS PROBLEM IN E-RECURSION

183

LEMMA 2.1 [9). Suppose some well-ordering of TC(x) is E-recursive in x. If {e)(x)1, then some Moschovakis witness to {e}(x)T is first-order definable over L«TCC({x))). Harrington [12] showed icf •"•* > K?-° for all o, 6e 2". But Slaman [14] found sets x and y such that < > < K*. It follows from Lemma 2.1 that the function \x\«.xr, restricted to x c On, is £-r.e. LEMMA 2.2 [9]. Suppose some well-ordering of TC(x) is E-recursive in x. Then < ' > K* for ally, and < < K* for all z < £ x. LEMMA 2.3 [9]. Suppose x is a set of ordinals, and L(K, X) is E-closed and 2 t inadmissible. Then <•>' < K for ally e L(K, X). THEOREM 2.4 [9]. Let x be a set of ordinals. Then the following are equivalent. (i) Kxr-y € £(x)for ally e E(x). (ii) £ ( x ) arfm/M Moschovakis witnesses. (iii) £ ( x ) is 2t inadmissible. THEOREM 2.5 [9]. Suppose x is a set of ordinals and L(K, X) IS E-closed. If L(K, X) is not 2 j admissible, then L(K, X) admits Moschovakis witnesses. Slaman [14] has devised an £-closed, 2 , inadmissible set that does not admit Moschovakis witnesses. Let
(11)

= Anx,

iff-<6,«,i>.

B is said to be £-reducible to A if for some e, B(x) = {e}A(x) for all x. Note well the difference between (i) x is £-recursive in y, and (ii) x is £-reducible to y. In (i) y occurs as an argument or parameter, and in (ii) as an additional predicate. For example, suppose x and y are infinite subsets of
foraUxe^

and

A = {x\x e
p) = 0}.

A is £-recursively enumerable on ,f if there exist e and p e
p)i).

Finally, A is £-reducible to B on e i such that {e}fl(x,

p)i

forallxe,?

and

^ = {x\x e ^& {e} 8 (JC, p) = 0 } .

343 184

G E SACKS

The notion of £-reducibility is not, in general, transitive. Call A c
and

sup{\f(x)\\z

e d } = K,

where K is the least ordinal not in S. Suppose B is t n dom(Xx|{e}(*, />)). Then, for all x e
a

The proof of (iii) is essentially the same as Spector's proof [15] that every nonhyperanthmetic n{ subset of u has the same hyperdegree as Kleene's 0. Let A and B be ^-recursively enumerable on
344 POSTS PROBLEM IN E-RECURSION

185

(ii) By (i) it is safe to assume A is "complete" Thus it suffices to find a regular, r.e. C such that A ^e C. The original construction of C is a transfinite recursion that applies some of the thinking behind the regular sets theorem of a-recursion theory. A much shorter proof was subsequently found by Slaman [7]. (2.7 is not needed in this paper, but is used in the proof of Slaman's density theorem [7].) D Let A be a class. All of the definitions and results of this section through Theorem 2.4 relativitize to A. {e]A(x) has already been defined. The computa­ tion tree for [e}A differs from that of (e) only by nodes corresponding to scheme (11). A set y is f-recursive, relative to A, in a set x if y = {e)A(x) for some e; in symbols, y ^A x. Define «o ; * = U { Y | y « E * }

and KAx =

\J{<xa\o^TC(x)}.

A set z is £-closed relative to A if, for all e, xez&{e)A(x)l

-

{e}A(x)eZ.

Define EA(x), the £-closure of x relative to A, to be the least z 2 TC({x}) that z is £-closed relative to A. Then EA(X)

(12)

=

such

L[KA\A;TC({X})].

The right side of (12) is the set of all sets constructible via ordinals less than KA-* with TC({;c}) as the starting set and A as an additional predicate. (In addition to the usual Godel operations, A n x is allowed.) 8 is x-reflecting relative to a if [ L [ 5 , ^ ; T C ( { x } ) ] NJT] _ {L[KA\A;TC({X})}

t= j r ]

for every 2 , sentence.Fof ZF with parameter x and additional predicatey e A. Let *;"'* be the greatest x-reflecting ordinal relative to A. The treatment of Kr and Moschovakis witnesses in 2.1-2.5 relativizes to A without any surprises. For example, Theorem 2.5 becomes THEOREM 2.9. Suppose x is a set of ordinals and (L[K, A; x], A) is E-closed, but not 2 , admissible. Then L[K, A; x] admits Moschovakis witnesses. 3. Fine structure. Assume L(K) is £-closed. This section concentrates on several structural properties of L(tc) needed in the next section to establish a suitable indexing of Friedberg-Muchnik requirements. Define P = MY,<«(£/) [/is a partial £-rec. (on L(K)) map of y onto L ( K ) ] and V = My*AEA)[A

£ 2 ' -

L(K)&AhE-T.e.onL(K)}.

LEMMA 3.1. i7 = p.

PROOF. First suppose that (1)

L(K) is 2 , admissible and equal to £ ( / } )

345 G E SACKS

186

for some 0 < K. By Proposition 2.7, £-r.e. coincides with 2 , ( L ( K ) ) , and p and T, are equivalent definitions of the 2 , ( L ( K ) ) projectum of L(K). (2)

Now suppose (1) is false.

To see TJ < p, let / be a partial £-rec. map of p onto L(K). If p < TJ, then ( d o m / ) e L ( K ) , and consequently (range/) e L ( K ) . To see TJ = p, fix y < p and let A c y be £-r.e. in some p e L(K) via Godel number e with the intention of showing A e L(K). It is safe to assume p is an ordinal, since each member of L(K) is £-recursive in some ordinal less than K. Let g be a universal, partial £-recursive function. Define H =

g[yU{p}}.

(His said to be the partial £-recursive hull of y U { p}.) Suppose H is bounded below K, i.e. H c L(S) for some 6 < K. Then for>- < y,

and so,4 e L(K). Suppose tf is unbounded in L(K). Observe that (3)

z e / / - 0{z) e / / ,

where 0 is Godel's order of constructibility function. If z e //, then z = g(x) for some x e y u (p), and g{x) is first-order definable over L(x + \g(x)\ + u). Since / / is £-closed, \g(x)\ e H. The unboundedness of H implies H admits A/-witnesses. Fix JC e //. It follows from (2) and Lemma 2.3 that K* < K. By (3) the supremum of all ordinals in H is »c. Choose 8 e H so that < < fi. Then K* < £ 5, x, and so < e H. Consequently, H admits M-witnesses by Lemma 2.1. It follows from (3) that every element of H is well ordered in H. Thus H is extensional, hence isomorphic, to a transitive set H0 via the collapsing map /. (3) also implies H0 = L(/3) for some 0 « K. The partial £-recursive function g maps y U {/(/>)} onto L ( 0 ) . Hence/? < K, since y < K. Since / / admits M-witnesses, the map / preserves divergence, as well as convergence, facts. Hence A is first-order definable over L (fi). D PROPOSITION 3.2. Let p be an ordinal and A a set of ordinals. Suppose the function \X\K?X does not attain a maximum on A. Then the range of \X\K?X is bounded on A by sup{ *£•' -'|.v, .* e A). PROOF. Suppose not. Then there exist c, d e A such that (4)

Kr'>KrJ>sup{KP':\y,z^A).

It follows from Lemma 2.2 that K^'J > K?d. The desired contradiction is: «>•*•'' is a
346 POSTS PROBLEM IN ^-RECURSION

187

THEOREM 3.3. Assume L(K) is E-closed and p e L(tc). Further assume L(K) IS either 2 , inadmissible or not of the form E(B)for any 8 < K. Ify < p, then SUP{.C,'-*|*

PROOF. AS in the proof of 3.1, p can be assumed to be an ordinal. First observe that

(5)

sup^'l* <*}<«.

Define/(e, 8) = \{e)(p, 5 ) | . / i s a partial £-rec. map from
x,y < y - . (ES)(S

< y&x < c S & y < £ 5 ) .

Consequently, the domain o f / c a n be construed as a subset of y. It follows from Lemma 3.1 that (dom / ) e L(K), hence (range/) e L(K), and so (5) is proved. Suppose \8\K?S does not attain a maximum on y. Then Proposition 3.2 yields sup{Krs\8
(7)

The theorem now follows from (5) and (6). If \8\K?-S does attain a maximum on y, then that maximum is less than K by Lemma 2.3. a Cofinality considerations figure prominently in the solution to Post's problem. Let X < K. Define r.e. cf(X), the r.e.-on-L(ic) cofinality of X, to be Hy(EA)[supA

=\&A is £-r.e. on L(K) &ordertype of A is y ] .

LEMMA 3.4. r.e. cf(ic) = r.e. cf(p).

PROOF. Assume L(K) is either 2 , inadmissible or not of the form £(/?) for any p < K. To show cf(K) < cf(p), let A be an unbounded subset of p and £-r.e. on L(K). Let/? be the parameter needed to define a partial E-rec./from p onto L(K). As in the proof of 3.1, p can be assumed to be an ordinal. Define xr=sup{K>'|>>S:;C} r

and

Ar = {x'\x

e A }.

Theorem 3.3 implies A' c K. A is unbounded in K, since K^ < «,"•' by Lemma 2.2 if f(y){. \Ar\, the ordertype of A', is at most \A\. The predicate v = «wr, restricted to w e On, is £-rec. by Lemma 2.1. It follows that Ar is £-rec. on L(K). To show cf(p) < cf(K) let B be an unbounded subset of K, £-r.e. O I ) I ( K ) , and of ordertype cf(x). B can be replaced by an £-rec.-on-L(»c) set with the same ordertype as B. Assume (a) no such set exists. Define S° to be the set of all a such that: a is the length of a computation that put some x into B greater than any y put into B by any computation of length less than o. Clearly 5 ° is £-rec. on L(K). | 5 ° | < | B | since B° is "strictly increasing" If B° is bounded below K, then there exists a a0 such that computations of length at most a0 put an unbounded set of ordinals into B; but that contradicts assumption (a). Thus it is safe to assume is the set that replaces B.

347 G E SACKS

188

B° divides K into blocks. Let 60 < 8, < • • • < ST < ■■■ (T < |<°|) be b eisting of B° Define block T to be [«T. 5T+1) . The blocks are needed to describe the enumeration of A0, a subset of d o m / . AQ will be unbounded in p, and its ordertype will be at most |5°|. A° is the range of a partial function g defined by a partial recursion on |5°|. g(r)= ux [x is enumerated in dom / via a computation of length in block T and (B)$ T0. But ( ( d o m / ) - 2) <£ L ( K ) , SO° some ,x e ( d o m / ) - r must be enumerated by a computation of length greater than 8,, and conse­ quently g is defined for some r > T0. JC is recursively enumerated in X0 as follows. First enumerate x in d o m / . Suppose the length of computation used to put x into dom / belongs to block T. Computeg(B) for all B < T. Enumerate x in A0ifg(r) = x. The above instructions are effective largely because 5° is £-recursive on L(K). a

PROPOSITION 3.5. Suppose y = r.e. cf(\) for some X <
T2 <

X —■ | ( T j , g ( T , ) > | < \{T 2.

g(T2)>|.

PROOF. The proof is similar to the second half 01 the proof of 3.4.

►

•#. oOiUiion 10 Posi s problem.

THEOREM 4.1. Suppose L((c) is E-closed. Then there exist two subsets of L(K), each E-recursively enumerable on L(K), such that neither is E-reducible to the other on L(K). Assume L(K) is either 2 , inadmissible or not equal to E(B) for any B < *. Otherwise Proposition 2.7 implies that the positive solution of Post's problem for L(K) provided by a-recursion theory [5] is also a solution in the sense of £-recursion theory. Thus L ( K ) has certain structural properties described in §§2 and 3. In particular, K? < K for every x e L(K). The subsets of L(K) to be constructed are A and B. All requirements are of the form A * {y}B or B * {y} *, and are indexed by ordinals less than p. Let / be a partial function, £-rec. on L ( K ) , from p onto L(K). Then requirement 2x is A # {f(x)}B

and

requirement 2x + 1 is B *{ f(x)}

A

348 POSTS PROBLEM IN ^'-RECURSION

189

Each requirement belongs to a block labeled by an ordinal less than y, the r.e.-on-L(K) cofinality of p. By Proposition 3.5, with p in place of A. there exists a function g suitable for dividing p into blocks. Assume g(0) = 0. Define (req. 2x)e

block 2T and

(req. 2x + 1) e block 2T + 1.

i f g ( r ) < x < g(r + 1). The construction has the form of an effective transfinite recursion on K. At stage a (a < K) all decisions are made by examining the set of all computations of length at most a. (It will be convenient to assume that the "length" of a computation exceeds all ordinals mentioned in the computation.) Let A <0 = set of all elements put in A before stage a and A' = A <0 U (elements put in A at stage a }. Each requirement is acted on at most once. Requirement 2x is acted on at stage a if (i)-(iv) hold. (i) req. 2x has not been acted on prior to stage a. (ii) (5, g(8)> has been computed for all 5 < T prior to stage or. (Req. 2x) e block 2 T . / ( X ) has been computed prior to stage a. (iii) Every requirement in block y, for all y < 2-r, has been acted on pnor to stage a. (iv) Let <J(2T) be the first stage after all the activity of (ii) and (iii) has occurred. Then either (iv)(a) or (iv)(b) holds. (iv)(a) { / ( X ) } < < < , ( 2 ( | ( < ° ( T T ) , / ( X ) » converges via a computation of length at most a. (Let (<J(2T), f(x)) be an ordinal that encodes the ordered pair (CT(2T), / ( * ) ) and exceeds both of its elements.) (b) A Moschovakis witness to the divergence of {/(X}S B < °' : "«O2TT), / ( X ) » is first-order definable over L(a). Suppose req. 2x is acted on at stage o. If (iv)(a) holds and {/U)}8<°""((a(2T),/(x)»

= l.

then ( O ( 2 T ) , / ( X ) > is put in A at stage a. Requirement 2x + 1 is handled similarly. Simply replace 2T + 1, and exchange A and B in the above instructions.

2xby2x+1,2Tby

PROPOSITION 4.2. Suppose every requirement in block z is acted on. Then there exists a a such that every requirement in block z is acted on prior to stage a. PROOF. Suppose z = 2 T and (req. 2x) e block 2-r. Let a2x be the unique a such that req. 2x is acted on at stage a. Let p be an ordinal that encodes (B"a
by Lemma 2.1. And supM \2x

e block2r)

< K by Theorem 3.3.

►

349 190

G E SACKS

PROPOSITION 4.3. Assume z < r.e. cf(p). Suppose every requirement in block y, for ally < z, is acted on. Then there exists a a such that every requirement in block y, for ally < z, is acted on prior to stage a. PROOF. If u < u, then no requirement in block v is acted on until every requirement in block u has been acted on. Consequently, if 7 is a successor ordinal, then the desired a exists by Proposition 4.2. Assume z is a limit. For each >•<--. let T( v) be the least a such that every requirement in blocks is acted on prior to stage o. r(y) < K by Proposition 4.2. Since T( V)

< a(y + 1) < r(y + 1)

for all v < 7, it remains only to show the supremum of P-

{o(y)\y
is less than K. ^ i s £-r.e. on L(K), since the graph of XX\c* is £-r.e., as was noted after Lemma 2.1. The ordertype of y i s at most z, since X^|a(/) is nondecreasing. By Lemma 3.4, z < r.e. cf(«), hence sup y < K. ► Now the proof of Theorem 4.1 is easily completed. Fix x < p in order to see req. 2x is acted on. Suppose (req. 2x) e block2T. Assume for the sake of an induction that every requirement in blocks, for ally <2T, has been acted on. By Proposition 4.3, O(2T) is well defined. Suppose (1)

{/(*)}

B<

"2"««T(2T),/(;C)»

converges. Then req. 2x is acted on at some stage o > O(2T), clause (iv)(a) holds, and A'((O(2T), / ( X ) » disagrees with (1). The construction insures that

A°((o(2r), f(x)))

= A((o(2r),

f(x)));

and (2)

5 " 2 " - B° = B n o.

Consequently^ * {f(x))B Suppose (1) diverges. Then Theorem 2.5 implies req. 2x is acted on at some stage o > O ( 2 T ) and clause (iv)(b) holds. Thus there is a Moschovakis witness w to the divergence of (1) first-order definable over L(o). It follows from (2) that w also witnesses the divergence of {f(x))B((o(2r),f(x))). Thus again A * {/(*)}* To complete the induction observe that the above argument shows every requirement in block 2T is acted on. It remains only to show A and B are £-r.e. on L(K). Suppose z, a typical member of A, is put in A at stage o. The decision is based on the set K„. the set of all computations of length at most o. Since K,<Eo (uniformly in o), it suffices to show o < E 7.

350 POSTS PROBLEM IN E-RECURSION

191

Let z be (O22T), / ( * ) ) a>d assume clause (iv)(a) of the definition of acted on holds. Then o < E z, B<<{2r), and B<"^) < E Ka(lr) < B z. D 5. Further results and open questions. As in §1, a positive solution to Post's problem consists of two "incomparable" r.e. sets. The method of §4 sheds a little light on Post's problem for structures other than initial segments of L. Assume ^ c .. Let L[K, A] as usual be the set of all sets constructible via ordinals less than K with x e A as an additional predicate. (Simply add A O y to the nine Godel operations that define constructibility.) The structure { L [ K, A}, A) is said tobe£cllosedif {e}A{x)e L[K, .4] when* e L[K, A]. As described in §2, the definitions of K0 and K, relativize straightforwardly to A; {e} is replaced by {e} A throughout. Similarly ij, p and r.e. cofinality relativize to (L[K, A], A). The proof of Theorem 4.1 shows:

(1)

Assume (L[K, A], A) is £-closed. If r, = p and KAX < K for a l l x e L[K,/4], then (L[K, /4], /<) admits a positive solution to Post's problem.

It seems unlikely that TJ = p in general. An immediate corollary of a selection theorem of Griffor and Normann [16] states:

(2)

If L[K, A] N [there exists a greatest cardinal and it is regular], then7, = p f o r ( L [ K , ^ ] , ^ ) .

The assumption in (1) on K, can be related to inadmissibility as in Theorem 2.9. If (L[K,A],A) ( '

KAX


is £-closed, but not 2 , admissible, then eL[K,A].

The proof of Theorem 4.1 also applies to an arbitrary £-closed structure S. Let K be the least ordinal not in
(4)

Assume <^is £-closed and effectively well orderable. If i\ = p for S, and
The proof of Theorem 4.1 yields a property stronger than subgenericity for those K where blocking is unnecessary, that is, when r.e. cf(p) = p. The simplest example is: (

'

E{ux) admits a positive solution (A, B) to Post's problem such that for all S < « „

a

351 192

G. E SACKS

A set A c £(u>,) with property (6) is said to preserve the K0-spectrum of £ ( u i ) It is not known if the K0-spectrum can be preserved when p fails to be r.e. regular. A consequence of (4) and Theorem 2.5 is: (7)

Suppose x is a set of ordinals, and L(K, X) is £-closed but not 2 , admissible. If ij = p for L(K, X), then L(K, X) admits a positive solution to Post's problem.

At this writing virtually nothing is known about situations where Post's problem has a negative solution, or where r, * pp (8) (

Does there exist a set x of ordinals such that L(K, x) is £-closed and 2 , inadmissible, and does not admit a positive solution to Post's problem for £-recursion theory? Does there exist a set x of ordinals such that L(K, x) is £-closed and 2 , inadmissible, and TJ * p for L(K, X)1

According to (7) a negative answer for (8) implies a negative answer for (9). It might be easier, however, to deal with (9) directly. Both questions are good candidates for forcing arguments. Forcing over £-closed structures has been extensively studied in [9, 10 and 14], but no technique so far developed appears to be of any use. Normann's result on Post's problem for £(2"), quoted in §1, leads to the following terminal question: Does there exist a model of ZFC in which (10)

£(2")t=

[2" is well ordered]

and £ ( 2 " ) does not admit a positive solution to Post's problem? REFERENCES 1 D. Normann. Set recursion. Generalized Recursion Theory II, North-Holland, Amsterdam 1978, pp. 303-320. 2 Degrees offunciional!. Preprint Series in Math., No. 22, Univ. of Oslo, Oslo, 1975. 3 R Fnedberg, Two recursively enumerable sets of incomparable degrees of unsolvabilitv. Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 236-238. 4. A Muchnik. On the unsolvubility of the problem of reducibihtv In the theory of algorithms, Dokl Akad. Nauk SSSR 108 (1956), 194-197. 5. G Sacks and S. Simpson. The a-fmite injury method, Ann Math. Logic 4 (1972), 323-367. 6. E Post. Recursively enumerable sets of positive integers and their decision problem. Bull. Amer. Math. Soc. 50(1944). 284-316. 7. T. Slaman. The E-recurswely enumerable degrees are dense, these PROCEEDINGS. 8. The extended plus-one hypotheses—a relative consistency result, Nagoya Math. J. 92 (1983) (to appear). 9 G Sacks. The limns of E-recursive enumerability. Nagoya Math J. 92 (1983), 107-120. 10 G Sacks and T. Slaman. Inadmissible forcing, Ann Pure Appl Logic (to appear) 11 G Sacks, The k-seclion of a tvpe n object. Amer. J. Math.. 99 (1977), 901-917. 12 L. Harrington. Contributions lo recursion theore on higher types, Ph D Thesis, M I T . . Cam­ bridge. Mass.. 1973 13 Y. Moschovakis, Hyperanalyticpredicates, Trans. Amer Math. Soc. 138(1967), 249-282.

352 POSTS PROBLEM IN E-RECURSION

193

14. T. Slaman, Forcing and reflection in E-recurston, Ann. Pure and Appl. Logic (to appear). 15. C. Spector, Recursive wellordenngs, J. Symbolic Logic 20 (1955), 151-163. 16. E. Griffor and D. Normann, Effective cofinalities and admissibility in E-recursion, preprint, University of Oslo, 1982 17 R. Shore, The recursively enumerable a-degrees are dense, Ann. Math Logic 9 (1976), 123-155. DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY, CAMBRIDGE, MASSACHUSETTS 02138 MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE, MASSACHUSETTS 02139

353 Reprinted from Annals of Pure and Applied Logic .11(1986) 87-120 with kind permission of Elsevier Science - NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands

87

THE LIMITS OF ^-RECURSIVE ENUMERABILITY Gerald E. SACKS* Harvard University, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Communicated by A. Nerode Received 1 September 1984 A classical result of Kleene is: 2 T 1 H Y P is recursively enumerable in 2 £. Restated in £-recursion theory, it says: 2 T i £ ( c u ) is £-recursively enumerable in to. Forcing, compactness and fine structure techniques decide various possible extensions of KJeene's result. Suppose L(K) is £-closed but not X, admissible. Let cf and gc denote cofinality and greatest cardinal in the sense of L(K). Then (i) <- (ii). (i) 2 8 C , " n L(K) is £-recursively enumerable in some a£L(K). (ii) w
1. Introduction This paper studies the question: is every naturally enumerable set recursively enumerable? At the lowest level, where 'set' means set of integers, Church's thesis seems to yield an affirmative answer. At many higher levels the main theorem (9.1) below appears to imply a negative answer. Examples of-naturally enumerable sets abound in Godel's L. Perhaps the simplest one beyond to is 2 T I L(a)f), the set of all hyperarithmetic reals. Its natural enumeration is tied to the iteration of the Turing jump through the recursive ordinals. A classical result of Kleene [1] is: (1) 2 T i L(cof) is recursively enumerable in 2£. The Kleene result, restated in the language of £-recursion, is: (2) 2
354 88

G.E Sacks

It was shown in [3] that (3) is false if (4) £(2^) 1= [2W is wellordered & and card(2"0 is regular]] On the other hand (3) is a consequence of the axiom of determinateness. Another generalization of (2) is: (5) 2«C(K) n L{K) is £-recursively enumerable in some a e L(K). gc(ic) is the greatest cardinal in the sense of L(*r), if there is one, and there is one if L(K) is not X, admissible. It is shown in Section 6 that (5) is false if L{K) is £-closed and not X, admissible, and gc(jc) is a regular cardinal in the sense of L{K). The assumption of I, madmissibility is inspired by the fact that £(2<") is 21, inadmissible, a result of Moschovakis [4]. Theorem 9.1 provides a complete resolution of the problem of enumerability in the inadmissible case. Theorem 9.1. Suppose L(K) is E-closed and not I, admissible. Then (i) and (ii) are equivalent. (i) 2gc(K) n L{K) IS E-recursively enumerable in some a e L(tc). (ii) L(K) t [(x) < cofinality(gc(/0) < gc(#r)]; or L(K) b [w = coffnality(gc(jc))] and every amenable subset o/gc(xr) belongs to L{K). (x E gc(/c) is amenable if (x n 6) e L(gc(*)) for all 8 < gc(jc).) The admissible case of the problem of enumerability is still under study. Some partial results are given in Section 9. One such is: (5, Section 9) If 2 is a set of ordinals and E(z) is I, admissible, then £(2) is £-recursively enumerable in some a e £(2). It is reasonable to regard (5, Section 9) as a more suitable generalization of (2) than (3) or (5). Some evidence in favor of this view is the selection theorem of [5], which implies £(2) is X, admissible if 2 is a set of ordinals and £(2)t=[cofnality(gc(K)) = a;]. The proof of Theorem 9.1 breaks into three cases defined by the cofinality of gc(*r) inside L(K). (a) L(K)t[gc(K) is regular]. In Section 6 it is shown that 2 ^ ' n L(K) is not £-recursively enumerable in any a e L{K). The argument is based on a forcing technique developed in Section 5, and geared to producing £-closed, generic extensions of £-closed structures. (In general set-generic extensions of £-closed, initial segments of L are not £-closed. For contrast recall that set-generic extensions of X, admissible sets are always X, admissible.) Section 4 is devoted to the verification of a fine structure fact needed to complete the forcing argument of Section 6 when L(K) is uncountable. (b) L(K) t[w< cofinaHty(gc(K-)) < gc(jc)]. ln Section 8 it is shown that 2 ^ K | n L(K) is £-recursively enumerable in some a e L(K). The work of Silver [6] and S. Friedman [7, S] on singular cardinals of uncountable cofinality is applied to show that Godel's order of constructibility function O, restricted to 2pc,A) n L(K), is £-recursive in some d e L(K). (c) L(tf}t[« = cofinality(gc(A-))]. In Section 7 it is shown that 2^MnL{K) is

355 77i
89

not E-recursively enumerable in any a e L{K), if there exists an amenable subset of gc(x-) outside L(K). The proof is inspired by Green's compactness and type-omitting theorems [9] for structures such as the X, admissible closure of Beth,,, The principal result of Section 3 is the existence of Moschovakis witnesses (to divergence) in L(K, X), if x is a set of ordinals and L(K, X) is £-closed but not I, admissible. The witnesses are needed for the forcing arguments of Sections 5 and 9. Section 10 lists several open questions.

2. £-recursion The partial £-recursive functions are denned by the Normann schemes [2]. {e} denotes a partial ^-recursive function with index e. xlt xn are arbitrary sets. (1) {e}(xl,..,,xJ=xiife=(Un,i). (2) {e}(xv ..., xn)=x,-xl if e = (2, n, i, j). (3) {e){x„ ...,x„) = {x„ x,} if e = (3, n, i, j). (4) {e}(xu..,,xn)^U{{c}(y,x2,...,xn)\yex1} if e = ( 4 , n, c). (In (4) {e}(xu . . ., xj is not denned unless {c}(y, x^ . . ., xn) is defined for all y ex,.) (5) {e}(xv . . . , xn) = {cXI^X*,, . . . , x„), .... {dm}(xv ..,, x„)) if e = (5, n, m,c,du . , . ,dm). (6) {e}(c, xv...,xn, ><„...,y m ) = {c}(x„ . . ., xn) if e = (6, n, m). In brief the Normann schemes are the rudimentary schemes for partial functions and an enumeration scheme. If {e}(x) is defined (i.e. has a value), then {e}(x) is said to converge, in symbols {e}(x)l. Otherwise { u . Each node of the tree is of the form (e, xt, . . . , xj, or more simply, (e, x), and is called a computation instruction. a>nb is read: b is a subcomputation instruction of a, and is defined by: there exists a finite sequence bQ, . , bn such that a = b'Q, b„=b, and bl + l is an immediate subcomputation instruction of b,(i
356 90

G.E Sacks

each Normann scheme, but as promised above, only the bounding scheme will be considered. (8) (m, x) is an immediate subcomputation instruction of (2 m • 3", x). (9) If {m}(x)i and y e {m}(x), then (n,y) is an immediate subcomputation instruction of (2m • 3", x). (10) If c is not the index of a Normann scheme, then (c, x) is an immediate subcomputation instruction of (c,x). The tree-like object consisting of (e, x) and that part of > u below (e, x) is called the computation of {e}{x). It is readily verified that {e}(x)i iff the computation of {e}(x) is wellfounded. Let A be a class and x a set. A is said to be E-recursively enumerable in x if for some e, A= {y\{e}(x,y)i}. (A is £-recursively enumerable if x =0.) B is ^-recursively enumerable in x on C if J3 = A n C for some A £-recursively enumerable in x. The method of definition by effective transfinite recursion yields a partial £-recursion function / such that for all e and x, if {e}{x)i, then f(e, x)i and equals the computation tree of {e}(X). In addition there exists a partial £-recursive function |{e}(;t)| such that if {e}(x)i, then |{e}(x)| is defined and equal to the ordinal height of the computation of {e}(x). Brief review of the method of definition by effective transfinite recursion: Let <„ be a wellfounded relation and / a total recursive function from w into w. Suppose for all e < w, and all x in the field of u is £-recursively enumerable. Assume {2m ■ 3"}(x) converges for the sake of considering the definition of |(2^" • 3", JCC> in detail. \T" ■ 3", * ) | = a if (11)

(£y)y<0[|(m,jc)|

= y],

(12)

(y)(£r) T < 0 [ye{m}(*)-»

(13)

(T) r< „(£y)b' e {m){x) &\(n,y)\^

\(n.y)\

= T]. and r].

A set y is said to be £-recursive in x if y = {e}(x) for some e. Thus [(e, A ) | is £-recursive in x when {e)(x)[. In symbols, y^tx. *•; is the supremum of fhe set of all ordinals recursive in x. K* is the supremum of the set of all ordinals y such that y is recursive in x, a, where a e TC(A:), the transitive closure of x.

357 The limils of E-recurswe enumerabil.ty

91

Suppose {e}(x)i and \{e}(x)\ = 6. It is not difficult to check that the computation tree for {e}(x) is first-order definable over L(6,TC({*})). Consequently each set recursive in x belongs to E-closed if xeZ&

{e}(x)l->

L{K'0,TC({X})).

A set Z is

{e}(x)eZ

for all e < w. Define E(x), the E-closure of x, to be the least £-closed Z such that TC({x})eZ. If y is a first-order definable subset of L(y), then y is ^-recursive in y and the parameter from L(y) occuring in the definition of y. It follows that (14)

E(X) = L{K*. T C ( { X } ) ) .

The above equation not only exposes the absolute character of ^-recursion; it also makes possible a straightforward definition of K*r, the greatest ^-reflecting ordinal. An ordinal 8 is said to be x-reflecting if L(8, TC({JC})) 1= f -» L{KX0, TC({JC})) t

*

for every 2 , sentence 9 of ZF with parameter x. Clearly the supremum of x-reflecting ordinals is x-reflecting, and so K% is welldefined. K, had its origin in the application of forcing to Kleene recursion in finite types, e.g. [10] where it was observed that

Then Harrington [11] showed «r?*>jr|" and more importantly, gave a charac­ terization of K?♦" (a € 2W) that inspired the results of the next section. The dynamic meaning of tc, is as follows. Let /lcTC({x}) be £-recursively enumerable in x. Suppose some b is seen to be an element of A via a computation of height less than K*. Then some computations of height less than K'0> hence less than some 8 £-recursive in x, must establish some elements of A. The shortest such computations, since they are first-order definable over L(8, TC({;c})) with x as the only parameter, must be £-recursive in x. Thus A has a nonempty subset £-recursive in x. <•" denotes *•<"•">, and similarly for K™ Note that if a e TC(JC), then £((a, x)) is £(jr) and K*-° = K'. Proposition 2.1. (i) K™ =£ *•' /or all a e TC(.r). (ii) // E(x) is not X, admissible, then K',M < K' for some a e TC(x).

Proof, (i) Suppose K*-">K* Then E((a, x)) e L(K>■-. TC({*})), and by reflec­ tion, £«jt. a» e £ ( A - J " , T C ( { X } ) ) , an impossibility since K> 3= id.-'

358 92

G.E Sacks

(ii) Suppose Kxr" = K' for all a eTC(JE),

and

E{x)t(u)U€C(Ev)D(u,v,p), where D is a A0 formula of ZF with parameter p. It is safe to assume p=jtand c c T C ( x ) . For example, let p = {e}(x, b) for some b e TC(x). Then D(u, v, p) is equivalent to the following X, formula: (Ey){Eb)beTClx)[\{e}(x,

b)\ = y&D{u, v, {e}(x, &))].

By reflection, for each aeccTC(x),

there is a 6*£Ex, a such that

L(<5,TC({x}))t(Ev)D(a,

v,x).

by the remarks following Lemma 2.2 (Gandy selection), 8 can be taken to be a partial £-recursive function of x and a, defined for all a e c. The bounding scheme implies 6 is bounded above by some y=s£x, c as a ranges over c. Hence there is a d e £(x) such that E(x)t(u)utc{EvledD(u, since y e E(x).

v,x),

D

Lemma 2.2 (Gandy Selection). There exists a partial E-recursive function (e,x)l^

{e}(
The proof of Lemma 2.2 is the same as it is in the setting of Kleene's theory of recursion in normal objects of finite type (e.g. Fenstad [12]). Suppose P(x, y) is an ^-recursively enumerable predicate and {x)(Ey)[y is £-recursive in x & P(x, y)\. Then (x)(En)[{n}(x)i

& P(x, {n}(x)].

Gandy selection allows n to be construed as an £-recursive function of x. Thus there is an £-recursive/such that (x)P(x, f(x)).

3. Existence of Moschovakis witnesses The universal computation tree >u was defined in Section 2, where it was noted that {e}(x)l iff > u below (e, x) is wellfounded. If {e}{x)l then any infinite path in > u below (e, x) is called a Moschovakis witness to the divergence of {e}(x). They were introduced by Moschovakis [4], who used them to show, among other things, that £(2' u ) is not Z, admissible. An £-closed structure W. is said to admit M-witnesses if for all e < w and x e %: if {e){x)\, then there exiiss an M-witness

359 The limits of E-recursive enumerabiliry

93

to the divergence of {e}(x) in M. E{2W) admits M-witnesses, but £(ICX implies Kx0b>Kx. Otherwise I, statements about x would reflect from < b down to < . Thus it suffices to find a b By -A such that jefs**§ 6 Now suppose no such b exists. Then

The predicate JCJ < jcf* is £-recursively enumerable in x, since it is equivalent to (E6)[6 £-rec. in x, b & 6 is not x-reflecting], which is the same as (Ee)[{e}(x, b)i & \{e}(x, b)\ is not x-reflecting], and the latter is £-recursively enumerable in x by Lemma 2.2. (A standard corollary of Gandy selection is: if R(e, b) is £-recursively enumerable, then so is (Ee)R(e, b).) Hence for each bey there is a convergent computation c (£-recursive in b, x) such that either c puts b in A or the height of c exceeds KX. C can be construed as an £-recursive (in x) function of b by Gandy selection. The bounding scheme, (7) of Section 2, provides an upper bound y on the height of c (for all b ey) such that y is £-recursive in x. Since y < JCS=£ KX, it follows that c always puts b in A. But then v - A is empty contrary to hypothesis. D Proposition 3.2. Suppose some wellordering of TC(x) is E-recursive in x. If (c,y) is an immediate subcomputation instruction of (e,x), then some wellordering of TCCy) is E-recursive in y, x. Proof. Suppose e = 2 " - 3 " and c = n. Then {m}{x)[ computation tree for {m}{x) is first-order definable over (1)

and y e {m}(x).

The

L(6.TC({*})),

where 6 is any ordinal 3s]{W}(x)|. The only parameter occurring in the definition is x. Let w be a wellordering of TC(jr) £-recursive in x. Fix 5 so that 6 is £-recursive in x, 6»\{m}(X)\, and w belongs to (1). Then w gives rise to a wellordering v of (1) definable over (1). (1) is £-recursive in *, hence the restriction of v to TC(j) is £-recursive in v, x D

360 94

G.E. Sacks

Lemma 3.3. Suppose some wellordering of TC(x) is E-recursive in x. If {e}(x)1, then some Moschovakis witness to the divergence of {e}(x) is first-order definable over L«,TC({x})). Proof. Related to arguments of Moschovakis [4] and Harrington [11]. Both rely on the effective equivalence of x and x , true if x = 2 , but false in the present context. The witness Xt\(e ,x,) is denned by recursion on t<w. {e , x ) is (e, x). Assume (e , x ) has been denned so that w

W

t

t

(1) {e,}(x,)l (2) x eL(K* ,TC({x})), t

(3)

0

0

t

and

T

K ^ ^ K ? * .

For simplicity assume e = 2 • 3" Suppose {m}(x )1. Then (e , is (m, x ). Assumption (3) implies K ^ K ^ . Consequently {m}(x )i iff there is a convergent computation for {m}(x ) in L{K , TC({X})). Suppose {m}(x )i. It follows from Proposition 3.2 that there exists a wellordering w of {m}(x,) such that w is E-recursive in x , . . . , x . Then {e x ) is (n, y), where y is the w-least w such that (4) ue{m}{x )&\{n}{u)\^K*. It has to be shown that (5) {n}(y)1 & K "3= By Lemma 3.1 there is a z e {m}(x ) with the properties ascribed to y by (5); let z be the w-least such z. It sufficies to show y = z . Clearly y =s„,z , since m

t

t

t+l

t

X

t

X

t

t

Q

t+v

t

t+1

t

X O

t

0

0

\{n}(z )\ =

0

™>K . x

0

Suppose y< z . Consider, as in the proof of 2.1, the following £-recursive (in x , . , , x z ) enumeration of {z | z < , z ) : (I) If {n}(z)l, then enumerate z. (II) If K ^
Q

r

p

Q

0

X O

M

XO

X

0

Z

By the bounding scheme there is a y recursive in x , . . . , x , z that bounds the height of all computations occurring in (I) or (II). It follows that (II) never applies to any z < z ; otherwise 0

lv

t

0

0

is a consequence of (5) with z in place of y. 0

Hence (I) applies to every z < z , and in particular to y, so |{«}(y)| =£ y. But w

\{n}(y)\ & < > K

X

j

0

K l °

»>

X

K

y

according to (4) with y in place of u, (3), and (5) with z in place of y. 0

•

361 The limits of E-recursive enumerability

95

The notion of left-most path can be used to clarify the statement and proof of Lemma 3.3. Here 'left-most' refers to the fact that *, + 1 is the tv-least u satisfying (4). Let 7 ^ 5 be that part of the universal computation tree > u that lies below (e,x). Define >% by replacing the notion of convergence in the definition of > u by convergence via computations of height less than y. Let T\ex) be that part of >l that lies below (e,x). Restatement of Lemma 3.3. Suppose some wellordering of TC(x) is E-recursive in x. If {e}(x)i, then Tfe,x) is not wellfounded and its left-most, infinite, descending path is first-order definable over L(KXT, TC({*})). The restatement of 3.3 leads to yet another ordinal associated with an arbitrary set z. Define B' = ny(e)[{e}{z)1 -» T\e^ is not wellfounded]. Some early results of Harrington [11] for recursion in 3E are equivalent to a

for all a, be V. Slaman [13] has found sets x and y such that K*> d* and *f->
Lemma 3.4. Suppose some wellordering of TC(x) is E-recursive in x. Then for all uandv. (i) A:" 2=0"

(ii)0"^0". (iii) icj = 6*. (iv)

K*U^K*.

Proof, (i) Suppose {e}{u)\. Proceed as in the proof of Lemma 3.3 with u in place ofx. Let u,+ 1 be a y such that U

(«}(y)t
'2=K-^

•"■>■

The existence of such a y again follows from Lemma 3.1. Now the M-witness It | (e„ u,) need not be first-order definable over L{ic^TC({M})), but it is still an infinite, descending path through Tfru). (ii) Let * have the property that {e*}(w, v) = {e)(u) for all e, u and v. Then Ty 1

<**.<«,»)) -

Ty

' (
for all y. (iii) There exists a recursive function h with the following properties: (a) {e}(x)i ~ {A(0)(x)i. (b) If {h{e)}{x)l then r a w . , > hha s arn^ue iifinitee descending path. Intuitively, if {e}(x)l then T{hir)x) consists of the left-most infinite, descend­ ing path of Tu,x} and everything in TUx) to the left of that path, h is defined by

362 96

G.E. Sacks

not allowing the Normann schemes to continue beyond the left-most path. Suppose e = 2m ■ 3". The immediate subcomputation instructions of (h(e), x) are as follows. (c) (h{m),x) is an immediate subcomputation instruction of (h(e), x). (d) Suppose {h(m)}(x){. Let <„ be the wellordering of {h(m)}(x), Erecursive in x, as developed in the proof of Proposition 3.2. If {h{n)){z)l for all z<wy, then (h(n), y) is an immediate subcomputation instruction of (h(e), x). Assume {e}(x)l Then Tf„ie%x) has a unique infinite, descending path t that witnesses the divergence of {h\e)}(x), and hence by (a), the divergence of {e}(x). The wellfounded part of Te{'h(e),x) to the left of t has rank at most dx It follows that t is first-order definable over L(8X, x). Now assume 6" < KX. Then by reflection there is a witness to the divergence of {e}(x) that is £-recursive in x. Thus {c | {x)(x)l), the complete £-r.e. subset of w, is £-rec. in x, a contradiction. (iv) KXU5= dx" ^dx = Kx by (i), (ii) and (iii). D Lemma 3.5. Suppose x is a set of ordinals, and L(K, X) is E-closed but not I, admissible. Then < • ' 5= K for some y e L(K) with the intent of showing L(K, X) I, admissible. Let L(K,x)t(u)u€d(Ev)&{u,

v),

where 9 is a A0 formula. For simplicity assume 5? has no parameters. For the moment assume (1)

Kx-y-b^Kx-y

Then for each bed,

for all feed. there is a c e L(Kxyh,x)

such that

a

by reflection there is such a c in L(y, TC({x, y, b))) for some y^Ex, y, b. By Gandy selection y can be taken to be a partial £-recursive function / of x, y and b defined for all bed. The bounding scheme implies f(x, y,b)^6 for some 6^Ex, y, d, as b ranges over d. Hence L(ic,x)^(u)tted(Ev)v^(u,v), where p is L(d,TC({x, y, b})). Now for the proof of (1). By Lemma 3.4(iv) it suffices to find a wellordering of TC({x, y}) £-recursive in x, y. Since x is a set of ordinals and KX^K, there is a wellordering of TC({;t, >>}) in L« v ,x). By reflection there is one £-recursive in x, y. D Theorem 3.6. Let x be a set of ordinals. Then (i) «-> (ii) ^ (iii).

363 The limits of E-recursive enumerabiliry

97

(i) x*t<> e E(x) for all y e E(x). (ii) E(x) admits Moschovakis witnesses. (iii) E(x) is I, inadmissible. Proof, (iii) -> (i) by Lemma 3.5. Assume (i) to prove (ii). Suppose y e E(x) and {e}(y)]. There exists a set w of ordinals such that w e E(x), y^Ew and K*,W < K. By Lemma 3.3 there is an M-witness to {e}(y)T first-order definable over L(KX,W, TC({X, w})). Assume (ii) holds and (iii) fails. For each e < w and a g TC(x), let /(e, a, x) be the least y such that: |{e}(a, x)| < y or there is an M-witness to the divergence of {e}(a, x) first-order definable over L{y, TC({x})). / is I, over E(x), hence bounded by some y0 e E(x). But E(x) c L(y0> TC({x})) e £(x)• D Theorem 3.7. Suppose xtsaiW of ordinals and L(K, X) is E-closed. If L{K, X) is not X, admissible, then L{K, X) admits Moschovakis witnesses. Proof. Suppose v e L(K, X) and {e}(y)T- Proceed as in the proof of (iii) -* (i) of Theorem 3.6. ► Slaman [13] has constructed an £-closed, I, inadmissible set that does not admit Moschovakis witnesses.

4. Amenable singularities In this section a five structure fact is derived straightforwardly from the work of Jensen as presented in Devlin [14]. The fact is needed in Section 6 to circumvent the difficulty of finding a generic extension of L(K) when K is uncountable. Let A be a limit ordinal. A singular sequence through A is a cofinal subset of A whose ordertype is less than A. A set 5 c A is said to be amenable if (S n 6) e L(A) for all 6
364 98

G.E. Sacks

6
Either j8 = A or

J„ C [A is a cardinal]. Either way (Snd)eJx. Case II: p^0- Then there exists a Partia] ^ ( ( ^ r 1 ' -1 /Ip- )) map from wp"e onto J p .-.. Consequently there is a partial lMJPy, A}'1)) map g from a 0 into A cofinally such that o0=s wp"p. Assume o0 is as small as possible. Thus a 0 = r i ( ( 7 p r l > Ay1))

cofinality of A.

For the moment assume (1) a 0 « ZiUJf, A^)) cofinality of opfK It follows that (2) x
JOT X,< VetJ;(x),be sop Tmgfi8JS~ Vet 5 ,be the rangeTo;5 UJ <*■•then (2) implies (5 n <5 e J p; , Hence (S n 6) e / , and as in case I (S n 5) eJx. To prove (1) let a denote the right s,de of (1), and let r be a partial 2i«V1' V >J " ^ / T a ' m t o W < V cofinally For each y < a let g0(y) be the supremum of all g(x) such that, * < a g(x) and r(y) are: defined, g(*) € 5 tv) and the existential witness needed for the ZMJp;->, Ap ' ) ) definition of g(x) belong.to % , g0(y)
5. Countably closed forcing Assume L(K) is £-closed and X, inadmissible. A well known condensation argument shows there exists a greatest cardinal in the sense of L{K); denote it by gc(A-). Let G c gc(jf). The connections made in Section 2 (following the definition of KX) between convergent computations and the L hierarchy imply that L(K, G) equals {{e}(a, G)\a e L(K) & \{e}{a, G)\< K}. Consequently the forcing language j?(#r,
365 The limits of E-recurswe enumerabiiiry

99

when G is generic, need only describe the computation of {e}(a, G) when a€L(/c) and \{e}(a, G)\ q means that domain p c domainq and q \ (domp) equals p. The set 9> of all forcing conditions belongs to L{K), since the usual Godelian argument shows each bounded subset of gc(ic) in L(K) is constructed via an ordinal less than gc(jsr). Thus & is £-recursive in gc(jr). Three objects are defined simultaneously by recursion on O
(EY)r
(2)

9-{p, 2 ■ 3", a, %o) =

(3)

m

y)&se{m}(a,

<S)^\{n}(s)\

= r}.

and

<£) & \{n}(s)\^ T].

{{n}(s)\se9'(p,m,a,%y)},

q\h{n}(s)e{2 3"}(a, <£) iff p » q, s e Sr(p, m, a, % y) ?Ir-[*€{m}(0,9)4(£T) T < a (|{ll}W| = T)].

and

Quantifiers and prepositional connectives are forced in the usual manner, p forces (Ex)P(x) if p forces ^(/) for some appropriate instance up forces ~S? if no q « p forces $\ and p forces (jc)S^ if p forces ~{Ex) ~ 9. Some clarification of (l)-(3) is in order. Suppose p forces {2m -3"}(a, ») to converge via a computation of height o. Then by definition of forcing, and in a manner consistent with the assignment of ordinals to computations (Section 2, (11)-(13)), p forces m(a, <§) to converge via a computation of height y for some

366 100

G.E. Sacks

y
<§)-+ \{n){s)\ = x\

if it is understood that x ranges over the terms in °f(p, m, a, % y). Since, as already noted, quantifiers and connectives are forced in the usual manner, no further analysis of (lb*) is needed, (lc) is dealt with similarly. With respect to (2), if the elements of 3~{p, m, a, % y) suffice eo oame eht elements sf {m}{a, G), then certainly the elements of °T(p, 2m ■ 3", ", % o) suffice for thrse of {2 m -3"}(a, G). There are two points worth noting in definition (3). First q is required to extend p (p &q) so tnat q will force {2m ■ 3"}(a, ») to tonverger ana hence to be b eets Second q is required to force {n}{s} to converge, and hence to be a set, and to converge via an ordinal r < a for the sake of the recursion. The above approach to the definition of Ih is termed 'from the top down'. Another approach, 'from the bottom up', can be found in Slaman [15], where the hierarchial view is maintained throughout. If t is a term of jf(jc, S), then p\\-\{e}(T)\ = a means p Ih \{e,}(a„ fy[-o, where {e,}(a„ <8) is the normal form of {e}(t). Similarly ST(p, e, t, 0) = g-{p, e„ a„ % o). Lemma 5.1. The relations, p \h\{e}(t)\ = a and q\\-se{e}(t), and the set ST(p, e,t,o){p^* and s e J(p, e, t, o)), are E-recursive in a, gc(jr) (uniformly in o). Proof. By effective transfinite recursion on 0. gc(jr) ii sneeed because S ^i £-recursive in gc(jc). D The weak forcing relation, p\\-* 9, is given by p II ~ ~ 9. The ention oo weak forcing plays a curious, but essential, role in the computation of a bound on |{e}(f)| when p forces |{e}(/)| to be an ordinal less than K. The computation procedure succeeds only if a bound is sought on |(e}(r)| when p weakly forces (Eo)[\{e){t)\ = o]. (Note that an unbounded quantifier on o ranges over the ordinals less than K.) The relation > v is a forcing-theoretic counterpart of > u , the universal computation tree. > v is termed the tree of possibilities, because it encompasses all values of > u for all generic G's. A node of > v is of the form (p, e,t), where p is a forcing condition and t is a term of #(*:, <S). A preliminary definition of (p.e.t)>v{q,n,s) isp^q

and q\V* {e, t) >u(n,s).

To complete the definition of > v , ii need only

367 101

The limits of E-recursive enumerability

be indicated how the forcing of atomic sentences about > u can be reduced to the forcing of sentences from $(K, <$). Typical details parallel the definition of > u given in Section 2, (8)-(10). (8V) p II- <2" ■ 3", /) > u (m, t) (all p and /). (9V) Up\\-\{m}(t)\ = 7, 5 e Sr(p, m, t, y) and p \\-s e {m}{t), then p\y(2m-3",t)>Jn<s). (10v) If c is not the index of a Normann scheme, then p II- (c, t) > u (c, t) (all p and t). > v will turn out to give global properties similar to those of > u . Assume that gc(K) has uncountable cofinality in L(K). It will be seen that (11) the immediate predecessor relation in > v is ^-recursively enumerable in gc(*) on L{K), and (12) if plt-*(£0)[|{c}(Ol = a], then that part of > v below (p, e, t) is wellfounded and £-recursive in (p, e,t), gc(*). It should be noted that (11) and (12) are not true for every set of forcing conditions that belongs to L(K). A counterexample can be obtained by taking L(K) to be £(w,) and Levy-collapsing w, to a>. (11) and (12) are true in this paper because 9 is countably closed: (13) (n)\pn^pn+l]-> {Eq){n)\pn^q). Of course<3>is countably closed in the present context because gc(ir) has uncountable cofinality in L(K). The part played by (13) is discussed further in [5]. It is (12) that gives rise to the desired bound on |{e}(/)| when p weakly forces (£<7)[|{e}(/)| = a]. The bound is roughly equal to the height of > v below {p, e, t). Lemma 5.2. Suppose p\V*{Eo)[\{e}(t)\ = o). If > v (p, e,t), then

is wellfounded below

/»tt-'|{e}W v below (p, e,t). The only interesting case is e = 2m ■ 3" By recursion there is a 6-recursive in p, t, gc(*) such that />lr-'|{«}(i)|**, since (p, 2m ■ 3", t) > v (p, m,t). Consider the set K defined by: (p; o)eK^p^p'

&o*s6&p'\h\{m}(t)\

= o.

K is the set of all possible outcomes for \{m){t)\ consistent with p. By Lemma 5.1, K is £-recursive in p, t, gc(x-).

368 102

G.E. Sacks

Fix (p', o) e K and 5 e 3~(p", m, t, a) in order to bound |{n}(s)|. Then p'll-*(£j8)[5€{ffi}(r)-»

\{n}(s)\=p].

Let J be {9 | p ' 5=? & q \Ys e {m)(t)}. By Lemma 5.1, J is E-recursive in p', s, t, o, gc(jf). For each q eJ, there is by recursion a p such that <7lh*|{«}(5)|«p and p is E-recursive in q, s, gc(jc). Gandy selection (cf. remarks following Lemma 2.2) makes it possible to think of p as a partial E-recursive function of q, s, gc(x). As ^ varies over 7, p is strictly bounded above by some p, E-recursive in /, s, gc(jc). This last is a consequence of the bounding scheme. As s varies over ST(p'',m, t, o), pj is strictly bounded above by some p ^ E-recursive in p\ t, o, gc(#r), since 5"(p', m, *, o) is E-recursive in p', /, a, gc(jc) by Lemma 5.1. As (p', CT) varies over X, psr is strictly bounded by some pK E-recursive in p, t, gc(jc). Thus plh*(5)[5e{w}(f)^|{n}(5)|«px]. Let y = sup+(<5, px).

►

Lemma 5.3. If p\h*(Eo)[\{n}(t)\ = a], then > v is wellfounded below (p, e, t). Proof. The proof of Lemma 5.2 is largely a description of an E-recursive procedure for computing an ordinal y from p, e, t, gc(*r). The procedure is welldefined whether or not > v is wellordered below (p, e, t). Thus Lemma 5.2 can be rephased as follows. There exists a partial E-recursive (in gc(jc)) function {g} such that: if p\\-*(Eo)[\{e}(t)\ = o] and > v is wellfounded below (p, e, t), then {g}(p, e, t% and pll-*|{e}(Ol*{g}(P.e.

0-

The proof of the present lemma is in brief a comparison of > u below (g, (p,e,t)) with > v below (p,e.t). If {g} converges on (p, e,t), then > v is well founded below (p, e,t). This is because {g} assigns an 'ordinal rank' to the nodes of > v below (p, e, t). The value of {g} on each w^v (p, e,t) is a strict upper bound on the values of {g} on the predecessors of w in > v . Now suppose {g} diverges on (p, e, t). By Theorem 3.7 there is a Moschovakis witness 2 in L{K) to the divergence. Thus z0= (g, (p, e,t)) and 2„> u z„ + ! for all n. There must be a zm such that z„, = (g.{q,n,s))

and

(p, e, t)>v{q,

n,s).

The existence of zm follows from the description of {g} given in the proof of 5.2.

369 The limits of E-recursive enumerability

Since {g}(p, e, r)f, it must be that {g}(q, n, s)f predecessor of (/», e, j) in > v . (Note that predecessor of z0 in >„, since the computation includes some rudimentary steps apart from the precessors of w in > v . ) Similarly there is an m' > m such that zm.= {g,W.

"'.*■))

and

103

for some (q, n, s) an immediate zm need not be an immediate of {g} at a given node w of > v computation of {g} at immediate

(q,n,s)>v{q',n',s').

Thus z includes a built-in, infinite descending sequence 2' in > v . If 2 is defined as in the proof of 3.3, then 2' is seen to be defined simultaneously with 2. Thus 2' € L(K). Let Zo=(p, e, t) and z'n = {p„, en, t„). Then (Pn,en,t„)>v(pn

+ ven + vtn + l)

for all n. Since 9 is countably closed ((13) following 5.1), there is a q such that pn 5= q for all n. Then a

Thus 9 weakly forces the existence of a Moschovakis witness to the divergence of {«}(*), an impossibility since 9 *£p and p weakly forces {e}(t) to converge. D Lemma 5.4, If p\\-*(Eo)[\{e}(t)\ = a], then there exists a y E-recursive in p, t, gc(jc) such that

pB-*Ke>(0t«TProof. Immediate from 5.2 and 5.3.

D

The next lemma implies that L(K, G) is £-closed when G is sufficiently generic. Lemma 5.5 (bounding). Suppose p \\-\{m}(t)\ = p, and p \r(s)(Er)[s e ^ 0 7 , m, t, p) & s e {m}(r)-*|{n}(s)| = T]. 7/ien p !h |{2 m • 3"}(/)| = 6 /or some 6 E-recursive in p, t, gc(xr). Proof. By Lemma 5.4, p =s y0, where y0 is £-recursive in p, f, gc(jr). It follows from Lemma 5.1 that p, ST(p, m, t, p) and the relation s e {m}(t) (s e ST(p, m, t, p)) are £-recursive in p, t, gc(*r). For each s e 3~(p, m, t, p), the set of all q such that (1)

p»q&q]rS€{m}(t)

is £-recursive in p, t, gc(ic), s. For each such q, a

for some y £-recursive in q,s, gc(x-) by Lemma 5.4. According to the remarks

370 104

G.E. Sacks

following Lemma 2.2 (Gandy selection), y can be taken to be a partial £-recursive function of q, s, gc(*r). Two applications of the bounding scheme supply an upper bound <5 on y, for all s e 3~{p, m, t, p) and all q satisfying (1), such that 5 is f-recursive in p, t, gc(«r). ► Let 9 be a sentence of £{*, <S) and G a subset of gc(*r). G satisfies p, in symbols G ep, if p is an initial segment of the characteristic function of G. G is said to be generic on 9 if there exists a p such that G ep and either p W9 or p Ih ~9. G is said to be sufficiently generic if (i)-(iii) hold. (i) G is generic on all sentences of the form (2)

(Eo)[\{e}(t)\ = o}.

(ii) Up\r\{m}(t)\

= p, and G ep, then G is generic on

(3) (s)(ET)[s€Er(p,m,t,p)&se{m}(t)->\{n}(s)\ = T]. (lii) If C is generic on 9, then G is generic on every instance of every subformula of 9. Explication of (hi): |{e}(r)| = y is an instance of a subformula of (2). For each s e Tip, m, t, p), (£T)[J € {m)(t) -» |{n}(s)| = T] is an instance of a subformula of (3). Theorem 5.6. If G is sufficiently generic, then L(K, G) is E-closed. Proof. Suppose that e<w and a e L(K) are such that {e}(a, G)i and \{e}{a, G)\ =£ K. As usual the only interesting possibility for e is 2m ■ 3" Since G is sufficiently generic, there i s a p a n d a a O c such that G ep, p\h\{m}(a,

9)| = o,

and p\t-(s)(ET)[se9(p,

m,s, %o)&se{m}{a,


By Lemma 5.5 there is a 6 ^-recursive in p, t, gc(/r) such that plh|{2"-3-}(a, 9)1 = 6. Then |{e}(fl> C)| = 6. (Note that 6
= a], is E-recursively enumerable (in

371 The limits of E-recurswe enumerability

105

Proof. p\h*(Eo)[\{e}(t)\ = a] is equivalent to (1)

{g){p. e, t)l &plh*[{e}(t)\ « {g}(p, e, t),

where {g} is the partial f-recursive (in gc(jc)) function central to the proof of Lemma 5.3. By Lemma 5.1, (1) is ^-recursively enumerable (in gc(#r)) on L(K).

D Lemma 5.8. Let a e L(K). Suppose p does not weakly force (Eo)[\{e}(a, V)\ = a). Then there exists aq^p such that q forces the existence of a Moschovakis witness (to {e}(a, <$)])first-orderdefinable over L(Karp^K), <S). Proof. Recall {g}, the partial ^-recursive (in gc(jr)) function occurring in the proof of Lemma 5.3. Observe that if {g}(p, e, t)l, then p weakly forces |{e}(r)| to be at most {g}(p, e, t). This can be verified by transfinite induction on > v , since > v below {p, e, t) is wellfounded when {g}(p, e, t)[ (as was shown in the proof of 5.3). Suppose {g)(p, e, {a, <8))t According to Lemma 3.3 the divergence of {g} is witnessed by some z first-order definable over £,(«£•*•**>, a). As in the proof of 5.3, z has a built-in, infinite descending sequence z' in > v . In fact the definition of z over L(KpT-'-tc(-K\ a), as performed in the proof of 3.3, includes a definition of 2' over L(K''-*<**\a). z'0= (p, e, (a, »)>, z'n = (p„, e„, tn) and (Pr,,en,tn)>v(pn

+ ven + vtn

+ l)

for all n. Let q*pn. Then q weakly forces A„ | (en, t„) to be a Moschovakis witness (to {e}(a, 9)f) nrst-order definable over L«"' ♦«««>, «). D Proposition 5.9. There exists a c such that for all x: {x}(x)1 and there is no Moschovakis witness (to {c}(x)f) E-recursive in x. Proof. Let t be a recursive function such that for all e and x: {t(e)}(x)i iff there exists an M-witness (to (e}(x)f) ^-recursive in x. (The existence of t follows from Gandy selection, Lemma 2.2.) Let c be a fixed point of t. Thus {c}(x) « {t(c)}(x). If {c}(x)i, then there exists an M-witness to {c}(x)l Hence \c}(x)l and so there is no M-witness (to (c}(jr)t) £-recursive in x. D Theorem 5.10 (T. Slaman [15]). Assume a e K. If a

then 6 <

K?P.*««K

Proof. Let {c} be as in Proposition 5.9. Then p cannot weakly force {c}(a, <S)l

372 106

G.E. Sacks

It follows from Lemma 5.8 that there is q ^p such that q forces the existence of a Moschovakis witness w (to {c}(a, ^)f) first-order definable over £(*•***•>, 9). By 5.9, 9 weakly forces that no M-witness to {c}{a, S)T is £-recursive in <«,»>. Hence 9 weakly forces w not be a member of /,(«?•», »). (By reflection, if there is an M-witness before jt r , then there is one before JC0, hence an ^-recursive one.) Therefore q weakly forces jcf-* to be at most j c j ^ ^ * ' . ► All the arguments of the present section go smoothly for any notion of set forcing that is countably closed. A general treatment of countably closed set forcing over £-closed structures is given in [5], where a special case of countably closed, class forcing is also discussed. In the proof of Lemma 5.3 above, it was shown that if the attempt to compute a bound on |{e}(r)| fails, then some q forces a Moschovakis witness (to { v , is of no use. The results obtained are similar to, but stronger than, Lemma 5.4 and Theorem 5.10. Lemma 5.4 becomes: if there is a q s p and a y < * such that q I\-\{e}(t)l = y7 .hen nome euch q and y are £-recursive inp, t, 9>. (§>= (P, s=) )i ste forcing relation.) Theorem 5.10 becomes: The null condition weakly forces «?•**£ *?•*. The bounds on K«* for countably closed, and countable chain, forcing suggest it may be difficult to force K, to behave wildly. 6. Non-enumerability via forcing Recall from Section 1 if L(K) is Ix inadmissible, then there exists a greatest cardinal in the sense of L{K) denoted by gc(jc). In this section the forcing technique of Section 5 and the fine structure fact of Section 4 combine to show: Theorem 6.1. If L(K) is E-closed but not X, admissible, and L{K) t [gc(jc) is regular], then 2 ^ ) n L(K) is not E-recursively enumerable in any a e L(K). Proof. Suppose in hope of a contradiction that 2^")nL(k)={x\{e}(a.x)[}.

373 The limits of E-recursive enumerability

107

By Theorem 5.6, the null condition 0 weakly forces L(K, <§) to be £-closed. It follows from the absolute nature of the proof of Theorem 3.7 that 0 weakly forces L(K, <§) to admit Moschovakis witnesses. Thus either (1) or (2) holds. (1)

W(Eo)[\{e){a,<S)\

= cc).

(2) There exist q, z, and y < K such that q weakly forces: z is an M-witness to {e}(a, 9)f first-order definable over L(y, <§). If (1) holds, then there will b e a C c gc(xr) such that {e}(a, G)| but G 4 L{K). If (2) holds, then there will b e a G c gc(*r) such that G e L(x) but {e}(a, G)f. Assume that (1) holds. By Lemma 5.4 there is a limit ordinal A < K such that (3)

cpfr!{*}(«,<S)|
.%(9), the set of all sentences of if(jc, <S) of rank < A, is defined as follows. (As usual only the bounding scheme is considered; e = 2m - 3".) (4)

(i) (Eo)a<x[\{e}(a, H-[|{2"•3',}(/) = p], then (5)(£r) I

{«}(/) -»- |{*»}(s)| = T],

e y(p, m, r, p) c4 5 e {m}(r) A |{n}(j)| & T],

and (£a) o

p0 = 0.

pY = U {pp I p < y}

if y is a limitt

qb = least q >pd such that q H-f6 or q Ih ~/ A .

^1 (6) (7)

Pd

=

to (0)

if

6e5,

L < 1 > if ^ 5 .

e SP, and

{pr|y^.}^ESn6,gc(K),X,f,6

(uniformly in 6).

374 108

G E. Sacks

When <5 is a limit, then (6) holds because gc(«r) is regular and S is amenable. When 6 is a successor, then (7) follows from Lemma 5.1, since (5 D <5) e L(K). Thus G is well defined. By (3) and (4), {e){a, G)\. G i L(K) because S t L(K) and (8)

S=s £ G, gc(jc), A,/.

(8) is a consequence of (9)

(pd, S D 6) =££G, gc(jc), A,/, 6

(uniformly in 6);

and (9) is proved by recursion on 6. (The coding of S into G via (5) implies S and G have the same £-degree (modulo gc(jr), A,/).) Now assume (2) holds. Thus q weakly forces: there exist c, d and 8 < K such that (10)

(i) {c}(Y, V)l and {
By Lemma 5.4 there is a A < K and aqQ =s q such that q0 forces all computations mentioned in (10) to have height less than A. Repeat the construction of G given in (5) with p0 = q0 and 5 = 0. Then G is generic on every sentence of ♦%(¥), and so {e}(a, G)t by (10). It follows from (7) that G=sEgc(jc), XJ. Hence Ge L{K). 

7. Countable cofinalitv It was noted in Section 1 that L(K) is enumerable when cf(gc(*)) = w and L(K) = E(a) for some a e L(K). In this section compactness and type-omitting arguments of Green [9] are adapted to obtain the non-enumerability of L{K)CM^K) when cf(gc(*)) = w, L(K) is 2\ inadmissible, and some amenable subset of gc(K) exists outside L(K). Recall that Z c A is amenable if ( 2 n t5)eL(A) for every <5
Then (i) and (ii) are equivalent. (i) There exists an amenable subset of gc(jf) outside L(K). (ii) L(K) n ? c | " is not E-recursively enumerable in any a e L{K).

375 The limns of E-recursive enumerability

109

Assume that L(K) satisfies the three initial suppositions of Theorem 7.1. If L{K) is countable, then (i) holds, and so L{K) O2^{K) is not £-recursively enumerable in any a e L{K). If L(K) is uncountable, then the truth value of (i) is uncertain. If there exists a non-constructible real, then (i) holds. If (in V) the cardinality of gc(*) has countable cofinality, then (i) holds. If V = L, then there exists an uncountable K such that (i) fails (and the initial suppositions of 7.1 hold). Proof of 7.1. Assume (i) fails. Then (ii) fails because 2gc(,t> D L(K) equall she eet of all amenable subsets of gc(«r). Assume (i) holds. Suppose in hope of a contradiction that L(K)D2^^={x\{e}(a,x)i}, where a e L{K). It is safe to assume a c gc(jc), since any member of L{K) can be encoded as a subset of gc(jc). Then £({gc(*r), a}) is I, admissible. This last follows from a selection theorem proved in [5, Corollary 5.2]. (It also follows from an earlier selection theorem of Moschovakis (18), studied in Kirousis [19], if it is noted that the Moschovakis argument, given for a = 0, also works when a belongs to L(K) n 2*C n L(K) encode a map of gc(jc) onto KV Then £({gc(/r), b}) is L(AT2), where K2 is Z} admissible and K, < K2< K. Also K*2 = gc(*r). There exists an infinitary sentence & e &{K) that describes the conssruction over L(KV G) of an M-witness to the divergence of {e}(a, G). The initial clauses of 9 are: ^ c g c i f ) ; z e £(K, + 1, «); and z is an M-witness to {e){a,
376 110

G.E. Sacks

Let if ambiguously denote the language of if and the set of all sentences in the language of if. if is a 4f subset of A A truth assignment v is any function whose range is contained in {true, false}, v is acceptable if v does not give any sentence of if the value false, and if v respects the propositional connectives. (In particular, if u(\J, %) is true, then v{%) is true for some I) Assume that if and if have been amply provided with Henkin constants and axioms. Thus the problem of finding a model of if is in essence the problem of finding an acceptable truth assignment for if. Since A is the Z\ admissible closure of R(w + w), there exists a Sf, one-one map from A into R(w + w). For each n, define X"=f-l[f[<e]nR(w+n)]. if"eA, hence if n if" has a modell Consequently there exists an acceptable truth assignment whose domain is if". Note that if v is a truth assignment with domain if", then v = w °/, where w is some truth assignment with domain R(w + n). Let ST be a tree with w levels such that the nodes on the n-th level are all the truth assignments with domain R(m + n). A path through ST is a sequence {z„\n< w} of nodes such that for all n: z„ is on the n-th level and z „ c z„ + 1. A model for if is equivalent to an acceptable path z, namely a z such that for all n, z„ ° / is an acceptable truth assignment. The existence of an acceptable path follows from an appropriate syntactical notion of consistency. An ^-finite set K of sentences is consistent if no contradiction can be derived from K via proofs in A generated by the usual axioms for if, u and some additional algebraic laws based on the presence in A of the power set operation restricted to elements of R(w + co). For example, (1) is now an axiom.

(i)

V,A^(„> t>e2

it!

where / e R(w + w), {% \ i: € /} e A, and gf w is % if g(i) = 0, and ~% if g(i) = 1. All of the additional algebraic laws are true in every if„ w structure, hence the existence of a model for K establishes the consistency of K. An acceptable path {zn \ n < m} is defined by recursion on n. Suppose z„ has been chosen from level n. Identify zn with the sentence /\{<4\<§e!£"&{zn

°nCS) = true}

A A { - M l « e - S P " * ( ^ • / ) ( » ) = false}} Suppose every ,4-finite subset of y u { z j is consistent. Let C be the set of all nodes on the {n + l)-th level of ST. Foo rz „ + ,coose some eeC such that tvery ,4-finite subset of if U { z j U {c} is consistent. Suppose there is no such c. Then for each c e C, there is an ,4-finite ^ c c y such that ifcU{z„}U{c} is inconsistent. The relation of ifc to cislf, since the notion of proof alluded to above is If. Since A is 21, admissible, there exists an ,4-finite if0cif such that for

377 The limits of E-recunive enumerability

111

all c€ C, y„U {z„} U {c} is inconsistent. But then %U{;„}U{V{c|ceC}} is inconsistent. But axiom (1) implies \f {c\ceC} inconsistent. Recall that a\ the X" projection of a, is

is provable, hence SfQ U {z„} is

My(£/W r [/:*^y  Also B is an amenable subset of A if (Bfl«)e
= =.

Suppose SfisaZf set of sentences such that each a-finite subset of if has a model encoded by an amenable subset of a' Then SP has a model first-order definable over L{a) and encoded by an amenable subset of a* Proof. Same as that sketched above for Green's compactness theorem. A becomes L(a), and R(co + w) becomes L(a*). Let {r„ [ n < w) be a sequence of L(or)-cardinals whose limit is a*. R(w + n) becomes L(r„). Axiom (1) is now interpreted to mean / e L(a*) and 2' is the power set of / in L(a). An a-finite set K of sentences is consistent if no contradiction can be derived from K via proofs in L(a) generated by the usual axioms of SL^ and additional algebraic laws such as (1). All of the additional laws are true in every structure encoded by an amenable subset x of a* because x n L(r„) belongs to L(r n + 1 ). Hence the existence of an amenable model for an ar-finite K establishes the consistency of K. The nodes on the n-th level of ST are all lruth asssgnments.in L(a) wiih domain r„. Every path through ST is an amenable subset of a', hence every model of & provided by Green's argument is encoded by an amenable subset of a. The notion of consistency is jrf, hence there is a A% acceptable path through ST, and consequently a model of V first-order definable over L{a). ► Lemma 7.4. Assume L(xx) is II admissible, K{ < K, and L(*i)t=cofinality(jef) = a). Suppose K^K and there exists an amenable subset of K* not in L(K). Then there exists an amenable Gcjc*t not in L(tc) such that L{KV G) IS I \ admissible. Proof. Similar to that of 7.3. Some additional steps are needed to insure

378

112

G.E. Sacks

G i L(K) and L(K„ G) is X, admissible. Let if be the following 2%> set of axioms:

<S=M (d € 2*> n L(*i)) (x)[xea

** \J{x = b\bea})

(a e L(J,,)),

C>Y ( y < * ) ) , V is X, admissible. Any model of if is a 2\ admissible structure that properly end extends L{KX). Each Kj-fmite subset of if has a model of the form L(8, G0), where 8 < *-, and G0 e L(*r,), since new subsets of K;* are constructed at arbitrarily high levels of L(Kj). L(/c,, G0) can be encoded as an amenable subset of JCJ by application of/, a one-one Z*x' map of L(KJ) into K*. (Recall that every 2f subset of *•; is amenable.) By Lemma 7.3, ^ has a model M encoded by an amenable subset of K*. G is a distinguished element of M, hence G is amenable. G^L{KX). L(K„ G)cM, but L(KV G) + M. If K! ^ M, then L(jr„ G) is It admissible, since a failure of admissibility could be used to define r , over L(c, G), where c is any non-standard ordinal of M. The additional steps needed in the proof of 7.3 to omit if, from M are conventional. Suppose stage n of the construction of M concludes with the choice of node z„ on the tree 5" such that ifU{zn} is consistent. Let d be a Henkin constant such that ifVJ {z„} proves d is an ordinal. Suppose every *rrfinite subset of (2)

ifU{zn}U{d>y\y
is consistent. Let e be a new constant not occurring in ifU{zn}. xyfinite subset of (3)

Then every

y\J{zn}U{d>e>t\y
is consistent. If not, ifU{zn} proves d*ieve ^y0 for some y0
379 The limits of E-recursive enumerabiltty

113

(1) 9 is not defined by a path through W, where W is any narrow subtree of ST in L(K,). (For simplicity assume that incompatible nodes of :J7" always make incompatible statements about %) A subtree W of ST ii called narrow if the set of nodes of W has L(K-,)-cardinallty less than K\, hence at most at r„. Let |W| be the set of all paths through W. Then (2) (|W|n £.(*,)) isx-.-finite, since a path through W and in L(K,) corresponds to an w-sequence through r„ and in L(AT,), if r„ is the number of nodes in w. It follows from (2), and the existence of new subsets of Art at arbitrarily high levels of L{Kt), that each jr r finite subset of L(K), it follows that

GiL{K).

D

8. Uncountable cofinality In this section the works of Silver [6] and Friedman [7, 8] are applied to show 2^»nLW ii £-recursively eenmerable ei nome e e L(K), ii L(K) ii £-closedd

378

112

G.E. Sacks

G i L(K) and L(K„ G) is X, admissible. Let if be the following 2%> set of axioms:

<S=M (d € 2*> n L(*i)) (x)[xea

** \J{x = b\bea})

(a e L(J,,)),

C>Y ( y < * ) ) , V is X, admissible. Any model of if is a 2\ admissible structure that properly end extends L{KX). Each Kj-fmite subset of if has a model of the form L(8, G0), where 8 < *-, and G0 e L(*r,), since new subsets of K;* are constructed at arbitrarily high levels of L(Kj). L(/c,, G0) can be encoded as an amenable subset of JCJ by application of/, a one-one Z*x' map of L(KJ) into K*. (Recall that every 2f subset of *•; is amenable.) By Lemma 7.3, ^ has a model M encoded by an amenable subset of K*. G is a distinguished element of M, hence G is amenable. G^L{KX). L(K„ G)cM, but L(KV G) + M. If K! ^ M, then L(jr„ G) is It admissible, since a failure of admissibility could be used to define r , over L(c, G), where c is any non-standard ordinal of M. The additional steps needed in the proof of 7.3 to omit if, from M are conventional. Suppose stage n of the construction of M concludes with the choice of node z„ on the tree 5" such that ifU{zn} is consistent. Let d be a Henkin constant such that ifVJ {z„} proves d is an ordinal. Suppose every *rrfinite subset of (2)

ifU{zn}U{d>y\y
is consistent. Let e be a new constant not occurring in ifU{zn}. xyfinite subset of (3)

Then every

y\J{zn}U{d>e>t\y
is consistent. If not, ifU{zn} proves d*ieve ^y0 for some y0
381 The limits of E-recursive enumerability

115

The next observation follows from (b) and (c). (d) Suppose x e 2" n L(K) and d*sax. Then there exist n < w and y <,x"» such that y is a wellordering of a- and ord(y) = 0(x). Assume x€2". The computation of {e}(jc, d) proceeds as follows. Let xd be x©d, the disjoint, effective union of x and d. Thus x, rf =£„*,,, and 0(x)^0(xd). Let K, = {yt>£«4(E>i)(y*.x<->)}. AT, is £-recursive in x, d. One last observation is needed to further the computation of {e}(x, d). (e) Suppose v e L(K) and y is a linear ordering of a. Then y is a wellordering iff y has no infinite descending sequence in L(a). To prove (e), let {m}(y) be the natural computation of the ordinal height of y via transfinite recursion. Thus {m}(y)i iff y is a wellordering. By Theorem 3.7, L{K) admits M-witnesses. Every M-witness to {m)(y)} includes an infinite descending sequence through y. (arTlLJ c L „ , since a has uncountable cofinality in L„. Now let / ^ = {y I y £ £ , & y is a linear ordering of a &y has no inf. desc. seq. in L(a)}. K°x is ^-recursive in x, d. Define {e0}(x,d)^sup{ordy\yeK°}, {e}(x, d) = ^y[y « {e0}(x, rf)<5re Ly + 1 - L v ]. Assume x e 2a n L(/f). By (e), {^o}(x, ^) converges to an ordinal which is, by (d), sufficient to construct x. Theorem 8.2. Suppose L(K) is E-closed, not X, admissible and L(K) t w < cf(gc(jc) < gc(jc). Then 2^K) H L(K) is E-recursively enumerable in some a e L(K). Proof. 2gc(,t) n L{K) is enumerated as follows. Let e and d be as in Lemma 8.1. First try to compute {e}(x, d). If {e){x, d) converges to y, then x e L v + 1 - Ly. x is enumerated iff y
L(K) t ~(Ez)(x)x
y).

Then y ^ K iff L(y + l)t(Ez){x)x
y). ►

382 116

G E Sacks

9. Main theorem & extensions In this section the main theorem (9.1) is assembled, further consequences of the reasoning behind 9.1 are drawn for the admissible case, and a further result of Griffor and Normann [22] is stated. Finally, there is a non-enumerability result for E(x) when x is an arbitrary set of ordinals. Theorem 9.1. Suppose L(K) is E-closed and not I, admissible. Then (i) and (ii) are equivalent. (i) 2*c(,c) n L{K) ii E-recursively enumerable in some a e L(K). (ii) L{K) t w < cf(gc(*)) < gc(x)) or L(K) f( [w o cf(gc(*r)] ]nd dvery ymenable subset ofgc{K) belongs to L{K). Theorem 9.1 is an immediate consequence of Theorems 6.1, 7.1 and 8.2. A clearer result is available if countability is assumed. Corollary 9.2. Suppose L(K) is E-closed, countable, and not II admissible. Then (i) is equivalent to (ii). (i) 2 * « n L(K) is E-recursively enumerable in some a e L(K). (ii) L(K)tw
383 The limits of E-ttcursive emimembilky

117

Proof of (I). Let L{K) = E(b). b can be taken to be a set of ordinals, since in L(K) every set can be coded by a set of ordinals. By Theorem 3.6 there is a yeL(K) such that K\^ K. There is a z e L(K) such that z c K and y^Ez. By reflection there is such a z £-recursive in y. Since z=Ey, it follows that *rf, =/ff, and K- = fv-;. Thus K; > A-, and so by Lemma 3.4(iv) trr-x s= A;

for all x.

Suppose JC € 2^K) n L(K) with the intent of computing 0{x) from z, JC. O(X) IS the unique <5 such that 6
+

l-Ld.

By reflection 6 is £-recursive in z, x. It follows from Gandy selection that there is an e such that if x e 2^K) n £,(#), then {e}(z, x)l & {e}(z, x) = 0(x). Enumerate 2« c <"nL(ir) as follows. First try to compute {e}(z, x). If {e}(z, x)i, then enumerate x if £(6) is not a subset of L{{e}(z, x)). D The proof of (2) also establishes (5). (5) If z is a set of ordinals and £(z) is II admissible, then £(z) is £-recursively enumerable in some a e £(z). Proof of (2). Suppose gc(jc) < K and some amenable subset of gc(«r) lies outside L{K). The arguments of Section 6 show 2*c<"> n L(K) IS not £-recursively enumerable in any a e L(K). The only point to be checked is the existence of M-witnesses in L{K). Suppose x e L{K). It suffices to show K> d" K>K* since L(K) is not an £-closure. AT* 2= 6* by Lemma 3.4 and Proposition 2.1. Suppose gc(K) = K, a e L{K) and 2*«K) n L(K) = {x \ {e}(x, a ) | } . First assume /c is a cardinal of L. Then L(K) is a 2 , substructure of L. But (£y)[y e On &{e}(y, a)f]

(e.g. y = A)

holds in L, hence in L(AT). Now assume A is not a cardinal of L. Let p be a regular cardinal in the sense of L{K) such that a e L(p) and p exceeds all L-cardinals in £,(*). Let p + be the next £(A-)-cardinal after p. Then 2pn£(p+)={xUcpcfc{e}(jc,a)|}. The methods of Section 6 show 2"C\ L(p+) is not £-recursively enumerable in a. L(p+) admits M-witnesses because it is a 2 , substructure of L(A). Proof of (3). The proof of Theorem 7.1 applies after one minute change. In 7.1 it was argued that A 2 < K because L(A2) is 2\ admissible and L(K) is not. Now it is argued that K2< A because L(A-2) is an £-closure and L(K) is not.

384 118

G.E. Sacks

Although (4) defines an open case, the reasoning behind Theorem 9.1, (1), (2) and (3) yields a complete solution to the enumerability problem for £(y), where y is an ordinal. Let gc(£(y)) denote the greatest cardinal in the sense of £(y). Theorem 9.3. If y is an ordinal, then (i) <-> (ii). (i) £(y) is not E-recurswely enumerable in any a e E(y). (ii) £(y) is not Z{ admissible and £(y) t= [gc(£(y)) ii regular]] Proof. Left to the reader save for one point. Suppose £(y) 1= [w < cf(gc(£(y))) < gc(£(y))]. The proof of Lemma 8.1 describes a procedure P for computing 0(x) for all xe28c(£(r»n£(y). P can be extended to all y e E(y) by effective transfinite recursion on the set-theoretic rank of y. The extension succeeds because each 6 e £(y) can be mapped one-one into gc(£(y)) by some / £-recursive in 5, p. p is a fixed parameter from £(y) that maps y one-one onto gc(£(y)). ► Griffor and Normann [22] have extended the concluding argument of the proof of Theorem 9.3 with the aid of their notion of non-standard stage comparison. They show: Theorem 9.4 (Griffor & Normann [22]). If y is an ordinal, £(y) is not X, admissible, and £(y) t [w < cf(gc(£(y))) < gc(£(y))], then £(y) is E-recursive in some a e L(K). The next result is a partial generalization of Theorem 9.3. Theorem 9.5. Suppose x b a set of ordinals and E(x) 1= ~[co > cf(gc(£(jc))) < gc(£(jc))]. Then (i) «• (ii). (i) E(x) 1= [gc(£(x)) is regular] and E(x) is not I, admissible. (ii) 2gc(£(*» n E(x) is not E-recursively enumerable in any a e E{x). Proof. Suppose (ii) holds. By (5) above, E(x) is not 2\ admissible. Then £(x)t[cf(gc(£(x)))=/=w] according to the selection theorem proved in [5]. Suppose (i) holds. Then the proof of Theorem 6.1 is sufficient after a few changes. Suppose 2s<W» n E(x) = {y \ {e}(a, y)l}

385 The limits of E-recursive enumerability

119

for some a e E(x). It is safe to assume that a is a set of ordinals, and that x^Ea. By Theorem 3.7, E(x) admits Moschovakis witnesses. Hence the forcing technology of Section 5 applies to E(x) (in place of L(JC)). It follows that either (1) or (2) is true, as in the proof of Theorem 6.1. If (1) of 6.1 holds, then there is a G c gc(£(x)) such that G$E(x)

and

{e}(a, G)l

G is constructed as in 6.1 from a n X c g c ( £ ( x ) ) such that S i E(x) and (a)

(S n 6) e E(x)

for all 6 < gc(E(x)).

S is obtained as follows. The universal partial £-recursive function maps TC({x}) onto E(x). Consequently there is a partial £-recursive (in some b € £(*)) m a p / from gc(£(x)) onto E(X). Let S be domf. By bounding S^£(x). Clearly 5 is £-recursively enumerable in b. Since gc(£(x)) is regular in E(x), the GrifforNormann selection theorem [23] implies (a). If (2) of 6.1 holds, then the argument of 6.1 proceeds with no substantial change. ► The next corollary was first stated and proved in [3] in the language of Kleene's theory of recursion in finite types. Corollary 9.6. Suppose £(2<") t [2" is wellordered & card(2"0 is regular]. 2

Then 2 °'n£(2lu)

is not E-recursively enumerable in any a e £(2 l °).

10. Questions Which of the following are true? 1. There exists a set x of ordinals such that E(x) is not 2", admissible, E(x) t [to < cf(gc(£(x))) < gc(£(x))], and E(x) is not £-recursively enumerable in some a e E(x). 2. It is consistent with ZFC that £(2 w )l=[2"'is wellordered] and 22"H E(2a) is £-recursively enumerable in some a <= w. 3. If L(K) is £-closed and not I, admissible, and L(jr)t[«a then L(K) is £-recursive in some a e L(K). 4. There exist tc, x and y such that x is a set of ordinals, L(K, X) is £-closed and not X, admissible, y e L(K, X), and *.-; = K.

386 120

G.E. Sacks

Finally, is there a simple criterion for the ^-recursive enumerability of L{K) when L{K) lies in case (4) of Section 9? A criterion applicable to this case has been found by Griffor & Normann [22, §3].

References [1] S C Kleene, Recursive functionals and quantifiers of finite types I and II, Trans. Amer. Math. Soc. 91 and 108 (1959 and 1963) 1-52 and 106-142. [2] D. Normann, Set recursion, in: J.E Fenstad et al., eds , Generalized Recursion Theory II (North-Holland, Amsterdam, 1978) 303-320. [3] G.E. Sacks, Post's problem, absoluteness and recursion in finite types, in: J. Barwise et al.,eds., The Kleene Symposium (North-Holland, Amsterdam, 1980) 201-222. [4] Y.N. Moschovakis, Hyperanalytic predicates, Trans. Amer. Math. Soc., 138 (1967) 249-282. [5] G.E. Sacks and T.A. Slaman, Inadmissible forcing, Advances in Math., to appear. [6] J. Silver, On the singular cardinals problem, in: Proc. Int. Cong. Math., Vancouver, 1974. [7] S. Friedman, Negative solutions to Post's problem II, Ann. of Math. 113 (1981) 25-43. [8] S. Friedman, Uncountable admissibles II: Compactness, Israel J. Math. 40 (1981) 129-149. [9] J. Green. X, compactness for next admissible sets, J. Symbolic Logic 39 (1974) 105-116. [10] G.E Sacks, The ^-section of a type n object, Amer. J. Math. 99 (1977) 901-917 [11] L. Harrington, Contributions to Recursion Theory on Higher Types, Ph.D. Thesis, M.I.T., Cambridge, MA, 1973. [12] J.E. Fenstad, General Recursion Theory (Springer, Berlin, 1980). [13] T.A. Slaman, Reflection and Forcing in £-recursion theory, Ann. Pure Appl. Logic 29 (1985) 79-106. [14] K. Devlin, Aspects of Constructibility, Lecture Notes in Math. 354 (Springer, Berlin, 1973). [15] T.A. Slaman, Aspects of £-Recursion, Ph.D. Thesis, Harvard University, Cambridge, 1981. [16] G.E. Sacks, Selection and forcing in £-recursion, in preparation. [17] T.A. Slaman, The extended plus-one hypothesis: a relative consistency result, Nagoya Math. J. 92(1983). [18] Y.N. Moschovakis, On the Grilliot-Harrington-MacQueen Theorem, in: Logic Year 79-80, Lecture Notes in Math. 859 (Springer, Berlin, 1981) 246-267. [19] L. Kirousis, On abstract recursion theory and recursion in the universe of sets, Ph.D. Thesis, UCLA, 1978. [20] J. Barwise, Admissible Sets and Structures (Springer, Berlin, 1975). [21] G. Boolos and H. Putnam, Degrees of unsolvability of constructible sets of integers, J. Symbolic Logic 33 (1968) 497-513. [22] E.R. Griffor and D. Normann, The definability of E(a), Preprint, University of Oslo, 1982. [23] E.R Gnffor and D. Normann, Effective cofinalities and admissibility in E-recursion, Preprint, University of Oslo, 1982.

387 Repnnled from ADVANCES I N M A T H E M A T I C S

Vol 66. No I.October 1987

Inadmissible Forcing G. E. SACKS* A N D T. A.

SLAMAN

Harvard University, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

A structure is £-closed if it is closed under ail partial £-recursive functions from V into V, a set theoretic extension of Kleene's partial recursive functions of finite type in the normal case. Let L(K) be £-closed and £ , inadmissible. Then L(K) has reflection properties useful in the study of generic extensions of L(K). Every set generic extension of L(K) via countably closed forcing conditions is £-closed. A class generic construction shows: if L(K) is countable, and inside L(K) the greatest cardinal gc(io). has uncountable cofinality. then there exists a 7egc(k.) such that i,(K, D = £(7"), the least £-closed set with T as a member. A partial converse is obtained via a selection theorem that implies E(X) is Y admissible when X is a set of ordinals and the greatest cardinal in the sense of £(X) has countable cofinality in E(X).

'

1987 Academic Press. Inc.

1.

INTRODUCTION

The notion of partial recursive function with numerical arguments was extended by Kleene [ 1 , 2 ] so as to allow arguments of finite type. His extension was defined by schemes which give rise to infinitely long com­ putations. In the so-called normal case the equality predicate is treated as if it were recursive. Normann [3] and Moschovakis independently extended the normal Kleene theory so as to allow all sets to occur as arguments. Thus {e}(x), the eth partial recursive function with argument x, has a meaning for all xeV. Normanns definition is inductive, {e} is defined by interpreting e as the Godel number of a scheme. The schemes include rudimentary functions such as pairing and union, composition, effective bounding, and also enumeration (f(e, x) s {e)(x)). The attempt to compute (e}(x) via schemes produces a tree that is well founded iff (e}(x) is defined. Com­ putation trees are discussed in Section 2, but it is worth noting now that the unique source of infinitely long computations is the effective bounding

* The preparation of this paper was partially supported by NSF Grant MCS 7307344. The authors arc thankful for the assistance of E. GrifTor, D. Normann, and V Wiley.

1 Copyrigh! £ ; 1987 by Academic Press, Inc

388 2

SACKS AND SLAMAN

scheme: if {m}(x) is defined, and if {«}(v) is defined for all then {2"'-3"}(x) is defined and is equal to

ye{m}(x),

{{n}(y)\ye{m}(x)}.> If {e}(x) is defined, then {e}(x) said to convergence (in symbols {e}(x)l). Otherwise {e}(x) diverges {{e}(x)1). Let A be a transitive set. A is E-closed if xeA&{e}(x)l^{e}(x)eA. Every I , admissible set is £-closed but not conversely. The E-closure of a set*, denoted E(x)) ii she leass £-closed sse that contains ( x } u T C ( x ) . 2 The classic example of an inadmissible, £-closed set is £(2)u£(£(«)) u £(£(£(«))) u ■•■ Some of the results of [4] were obtained by showing certain generic exten­ sion of E{T) are £-closed. The language of [4] was that of Kleene [1 ], so for clarity it must be pointed out that £(2W) corresponds to recursion in 3 £ as follows: r e £ ( 2 ' u ) iff z is codable by a relation on 2™ recursive in 3 £, b for some b e 2'" Set forcing over a T, admissible structure poses few problems. It is well known that if A is £ , admissible and G is generic over A with respect to some set P (in A) of forcing conditions, then A(G) is £ , admissible. Class forcing (PsA) over ^ can be more intricate, as in Abramson [5] or Steel [6]. Set forcing over an £-closed, inadmissible structure poses a new problem. Let i be a term that refers to a generic object. What does it mean to force {
a ' For the sake of exposition the bounding scheme given here differs somewhat from that of Normann [3]. : TC(.v) is the transitive closure of .v.

389 INADMISSIBLE FORCING

3

The countable closure condition is not necessary for £-closed, generic extensions. The countable chain condition also suffices by a mode of argument [8] not employed in the present paper. The set forcing construction of [4] was extended to an arbitrary £-closed, inadmissible L(K) in [9]. The machinery of [9] is reviewed and amplified in Sections 2 and 3, and then applied to a class forcing construc­ tion in Section 4 to prove THEOREM 4.8. Let L{K) be countable, E-closed and X\ inadmissible. Suppose the greatest cardinal of L{K) has uncountable cofinality in L(K). Then there exists an Scwf 1 " ) such that L(K, S) is the E-closure ofS.

(According to a well-known fine structure fact, Proposition 2.5, L(K) f= [There is a greatest cardinal.], where L(K) is £ , inadmissible.) Theorem 4.8 resembles the following result of admissibility theory [ 7 ] : If L(K) is countable and £ , admissible, then there exists a F e > such that L(K, T) is the least £ \ admissible set with T as a member. A partial converse to 4.8 is obtained in Section 5 via a selection theorem. COROLLARY. 5.2. Let X be a set of ordinals. If in E(X) the greatest cardinal has cofinality w, ,hen E(X) )i £ , admiisible. If {e}(v) does not converge, then its computation tree has at least one infinite descending path. We call any such path a Moschovakis witness to the divergence of {e} at*. They were introduced in Moschovakis [10] to show £(2m) is not I , admissible, and have since proved essential to the study of recursion in higher type objects. A structure A is said to admit M. witnesses if for all e< to and xeA: if {e}(x) diverges, then some member of A is an M. witness for (e}(x). LEMMA 2.4 [9]. If L(K) is E-closed but not X, admissible, then admits Moschovakis witnesses.

L(K)

The above reflection property is precisely what is needed to make countably closed set forcing succeed over an £-closed structure. 3.5. Suppose A is E-closed and admits Moschovakis witnesses. Let PeA be countably closed in A. If G is P-generk, then A(G) is E-closed. The paper ends with some open questions and conjectures. THEOREM

390a

4

SACKS AND SLAMAN 2. EXISTENCE OF MOSCHOVAKIS WITNESSES

In this section it is shown that many inadmissible, £-closed structures admit Moschovakis witnesses. The proof extracts a reflecting ordinal from the failure of X, admissibility, and then draws on a characterization of reflection devised by Harrington [11] for recursion in objects of finite type. y is said to be E-recursive in x if there exists an e such that y= {e}(x). Whether or not an arbitrary z belongs to y can be decided by first com­ puting {e}(x) and then checking whether or not z belongs to {*}(*). The second step is legitimate according to the schemes of £-recursion . B is said to be E-recursively enumerable in x if there exists an e such that B={z\{e}(z,x)l}. From now on the prefix E will often be dropped. The enumeration scheme yields the usual Kleene recursion theorems and thereby the method of defining partial recursive functions by effective transfinite recursion. Let < K be a well-founded relation and / a recursive function from a into a). Suppose for all e < to and all x in the field of < R, {Ke)}(x)l whenever {e}(y)l for all y u is defined by reference to the schemes of £-recursion. Each element of the field of > „ is of the form <e, *> where e < w and xe V. a > v b is read: b is a subcomputation of a, and is defined by: there exists a finite sequence z0,..., zn such that a = z0, z„ = b, and z, + 1 is an immediate subcomputation of z, (i is an immediate subeomputation of <2 m -3", x). (2) If {!»}(*) is defined, and ye{m}{x), then (n, y) immediate subeomputation of < 2 " ' - 3 " , A > .

is an

The remaining clauses, save one, correspond to the unitary schemes and so give rise to finitary branching in > v. The final clause states: if c is not the Godel number of a scheme, then (r, x> is an immediate subeom­ putation of . It is readily verified that the relation "> is an immediate subeomputation of a" is recursively enumerable, and that {?}(x)j iff > u below <e,*> is well founded. The method of definition by effective transfinite recursion

391 INADMISSIBLE FORCING

5

yields a partial recursive function / such that for all e and x, if {e}(x)l, thence, x) is the set theoretic object consisting of > v below <e, JC>. Thus an £-closed set includes not only the end values of computations but also the computations themselves. (If { u below (e,x) the computation for {e}(x).) Similarly there exists a partial recursive function \{e}(x)\ such that if {e}{x)l then |{e}(.v)| is the ordinal height of > v below <e,x>. For example, \{2m■ 3"}(o)| =a if (1)

(£y)v<0[|{m}(a)|=>a

(2)

(x)(£t)T<,[xe{m}(ii)^|{n}(jc)|=T],and

(3)

(T)t<.(£,x)[(jte{m}(o)&|{«}(x)|>T)v|{i«}{fl)|-T].

An ordinal 5 is said to be recursive in x if there exists a c such that 5 = {C}(JC). Thus \{e}(x)\ is recursive in x if {c}(x)|. An essential result in ^-recursion theory is the Gandy Selection Theorem [19]: There exists a partial recursive function 4>(e,x) such that for all (En)„


~(e,x)l&

{e}(4(e,x),x)i.

Suppose P(x, v) is a recursively enumerable predicate and (x)(Ey)[y

is recursive in x& P{x, y)l

Then by Gandy selection there exists a recursive function / such that (x)P(x,f(x)). An £-closed set A admits Moschovakis witnesses if for all e < „ below (e,x) in A. Three more definitions are needed for the main lemma (2.3) on the existence of Moschovakis witnesses in £-closed sets. K0* is the supremum of all ordinals recursive in x. KX is the supremum of all ordinals y such that y is recursive in a, x, whereaeTC(.v). If {e)(x)l and \{e)(x)\=S, order definable over

then the computation tree for {e}(x) is firstL(S, {x}

uTC(x)).

Consequently each set recursive in x belongs to L(K0\ {X} uTC(x)), and E(x) = L(K\{X}

uTC(x)).

392a

6

SACKS AND SLAMAN

An ordinal y is said to be x-re/lecting if for every £ i formula <j>, L(y,{x}uTC(x))\=<j>(x) iff L(KX0,{X}VTC(X))\=(X).

K* is the greatest x-reflecting ordinal. To appreciate the definition of KX, let D(u, v) be a J 0 formula (with free variables u and v), and suppose L(KX, {x} vTC(x))

\=(Ev) D(x,v).

Then L(6, { x } u T C ( x ) ) | = (£i>)Z)(x,i;), for some S recursive in x. Suppose further that there is a well ordering > of T C ( x ) recursive in x. Then the presence of an existential witness for D below K; implies there is a witness recursive in x, namely the least one relative to the well ordering of L(<5, {x)uTC(x)) extending > . LEMMA 2.1. Suppose A is recursive in x and @ is co-recursively enumerable in x. If An® is nonempty, then there exists a ceAn® such that KX>KX' (Harrington-Kechris basis theorem [11]). Proof. First, observe that KX-C>KX implies KXC>KX. Otherwise £ , statements about x would reflect from KXC down to K*. Thus it suffices to find a c e A n B such that KX ^ K*-C. Now suppose no such c exists. Consider the following recursive (in x) enumeration of A: (1) (II)

If c e A - @, then enumerate c. If ce A and *•; < Kg-C, then enumerate c.

(Note that < can be effectively defined from x and any ordinal greater than K*r. Thus (II) applies when there is an ordinal recursive in x, c and greater than KX.) By effective bounding there is a y recursive in x that bounds the height of all computations occurring in the above enumeration of A. Since y. Then there is a well ordering of {y\y is an immediate subcomputation of (e,x)} which is (uniformly) recursive in <JC, z>. Moreover, if y is an immediate subcomputation of x then there is a well ordering of TC( >•) which is (uniformly) recursive in <.v, v, r>.

393 INADMISSIBLE FORCING

7

Proof. Assume e = 2m-3". If the only immediate subcomputation of <e, x> is <m, x> then the lemma is immediate. Suppose then that {m}(x)|. The value of {m} at x and the entire computation of {m}(x) from x is contained in L(|{m}(x)|, {x} uTC(x)), which is recursive in x. The <*, r)-recursive well-ordering of TC(x) induces a well ordering W of m{m}(jc)|, ( x ) u T C W ) . W restricted to {m){x) induces a well ordering of the immediate subcomputations of <>> is an immediate subcomputation of x and <e', >^> is not equal to <m,x} then e' = n and >-e {w}(x). So TC(j>) is a subset of L(|{m}(x)|, {JC} u T C U ) ) since this set is transitive and contains {m)(x). W restricted to the elements of TC(y) is a well ordering of TC(>>) and is recursive in <x, y, z}. | LEMMA 2.2. Suppose some well ordering of TC(x) is recursive in x. If {e}(.xr)T, then a Moschovakis witness for {e}(x) belongs to L « + l ,

(x}uTC(x)).

Proof Related to arguments of Moschovakis [10] and Harrington [11]. The witness h\(e„x,} is defined by recursion on t. (e0, x0) is O, x). Assume that (e„ x,> has been defined so that (a)

{*,}(*,)?.

x,eL(K;,,{i}uTC(jt)),

(b) (c)

Kxr^K;°-

(d)

there is a well ordering of TC(JC,) uniformly recursive in

%

<JC0..,.,JC,>.

For simplicity assume e, = 2 m -3". By the inductive assumption (d) and Lemma 2.1a let <„ be recursive in <*„,..., JC,> and well order {z\z is an immediate subcomputation of (e,,x,)}. If {w}(x,)T let 0 , + ) , x , + l > = <m,x,y. Assumption (c) on x, implies that *?>*§'. Consequently, {m}(x,)l if and only if \{m)(x,)\ to be <«, >♦> where <«, j > is the <„. least pair so that ue{«}(*,)&|{JI}(»)|>K?.

Since |{m}(x,)|
*<*;, >>eL«+l,

{.v} uTC(x)). It has to be

a

394 SACKS AND SLAMAN

8

By Lemma2.1 there is a ze {m}(x,) such that &Kxr<>- " ■ ^ K x r ° - ■"'•''

{n}(z)1

Let <«, z0> be the <„, least pair so that z 0 is such a z. It suffices to show that y = zQ. Clearly, <«, y} < „ <«,z 0 > since |{#i}(z0)l = °o > < • Suppose <«, ;>>> <„.<«, z0>. Consider (as in the proof of Lemma 2.1) the following recursive (in x0,.,.yx„z0) enumeration of W \ (n, z0> = <{«, z> | <w, z> <„. }. By induction, ff f is recursive in x0,..., x„ z 0 . (I) (II)

If {»}(*)|, then enumerate z. If K J * - " < K ^

*♦-", then enumeratez.

(Note that K ^ '»<*?>♦ *'- implies that *;* l'
a Hence (I) applies to every z so that <«, z> <„. <», z 0 >. In particular, if y^z0then \{n}(y)\*y. But | {«}(>)[

><>K|*"--4C'-8»>y

according to assumption (c). Thus z0 = y. Finally to complete the induction, Lemma 2.2 uniformly provides a well ordering of TC(>>) which is recursive in x0,..., x„ y. | LEMMA 2.3. Suppose L(K) is E-closed but not £ , admissible. Then K*
Proof. Let Kr* > K for some fixed x in L(K) with the object of showing that L{K) is X, admissible. oince any element oi Z*^KJ can De wen orocrco in /—^KJ,

£(K) N (£*'){. "

1S

a we'l ordering of TC(x)J.

By assumption KT ^ K so there is a y wnicn is recursive in x sucn tnat L(y) h (£H^)[W is a we" ordering of TC(JC)]. Thus the L-least well-ordering of W, of TC(x) is recursive in x. To prove that L(K) satisfies ^-bounding, suppose UK)\=

(u)uei(Ev)^(u,v),

395 INADMISSIBLE FORCING

9

where & is J 0 with parameter p. Let y be less than K SO that there is a well ordering W,,, of TC(, rf» which is recursive in /»,rf,y. Then there is a well ordering of TC«/>, 4 y>) which is recursive in p, d, y. To simplify the discussion, the parameters p, d, y together are referred to as p* For each bind

a This last follows from (l)-(3). (1) Let 0- be the least ordinal, if there is one, needed to construct (from z) Moschovakis witnesses for {e}(z) for all e such that {e}(z)T- Then 6:>K;. If not, then, by reflection, 81 would be recursive in z; this would make the complete recursively enumerable in z subset of w recursive, and absurdity. ( 2 ) o<x-r'>->> ^K<x.p'>.b f o r e a c n b in d by Lemma 2.2. (The parameter y was introduced to provide the well ordering needed to apply Lemma 2.2.) (3) For any u and o, 0 u sS0 u ", since a Moschovakis witness for a divergent u computation {e}(u) can be construed as a Moschovakis witness for a divergent <«, u>-computation {e*}(u, v). Thus tc<*<"'>•* > 0<Jt- "*>'* such that

^8*^K*^K.

L(K)

Hence for each 6 ed, there is a c

h^(b,c)

and c is recursive (by reflection) in x, p*, b. The set of all e such that {e}(x,p*,b)l

and

L(K) |= #-(b, {e}(x, />*, A))

is recursively enumerable in JC, />*, b. By Gandy selection (Sect. 2) one such e can be chosen by an effective method which is uniform in x, p*, b. It follows there is a partial recursive function/such that for each bed, f{x, p*, b)\

and

L{K) (= &{b,f(x,

p*, b)).

By effective bounding {f(x, p*, b)\bed) is recursive in x,p*,d must be an element of L(K) since L(K) is £-closed. |

and so

LEMMA 2.5 [9]. / / L(K) is E-closed but no! Y.i admissible, then admits Moschovakis witnesses.

L(K)

Proof. For zeL(/c) let y be the least ordinal so that there is a well ordering of TC(z) which is recursive in z,y. By Lemma 2.4, K;?
396 10

SACKS AND SLAMAN

Lemma 2.3, if {e){z, y) is divergent, then there is a Moschovakis witness to this effect in L{Kf+ 1 , T C ( : u y ) u {z, y}). In particular, the relevant Moschovakis witnesses for z are elements of L(xy' + 1, T C ( ( u y ) u { z j } ) which is a subset ofL(x). PROPOSITION 2.5 admissible. Then

(well known).

L(K)(=

Suppose L(K) is E-closed but not £ i

[There exists a greatest cardinal]

Proof. Let y be a cardinal in the sense of L(K). A standard downward Lowenheim-Skolem argument shows that L(y) is a Xi substructure of L(K). If/is £ i over L(K) and its domain and defining parameters belong to L(y), then its range belongs to L{y). | The greatest cardinal of I(K), when it exists, is denoted by gc(ic).

3. GENERIC EXTENSIONS OF £-CLOSED SETS

Let A be an £-closed set, and G a subset of some element of A. Define A(G) to be the set of all z such that for some e and a: z={e}(a,G),

aeA

and

\{e}(a,G)\

e A.

Suppose 0> is a partial ordering of forcing conditions such that 0> e A and is countable closed in A In this section, it will be shown that A(G) is £-closed, if G is ^-generic over A, and A admits Moschovakis witnesses. The forcing language if A has the power to describe the computation of {e}(a, G) when a and [{e}(a, G)\ belong to A. The primitive terms of e A be a partial ordering of forcing conditions />, 9 , r,.,.; ^ ^ ^ means p is extended by q (i.e., /> says less than q about G). The forcing relation /> hh ! { < } U , ^ ) | = d and the set ^(p,e,a^,a) of terms are defined simultaneously by recursion on aeA. Among the elements of ST{p, e, a, $, o) are terms that name the elements of {e}(a, <S) when ^ is generic, peG and p forces |{*}(«,#)| to be
397 INADMISSIBLE FORCING

11

the definition of < v, the universal computation tree of Section 2. $~{p, r M { m } U , * ) l = y ] ; p^(u)(ET)r
(T)T<JEU)[(U

= Ty,

e ST(p, m, a,
& n e { m } ( « , » ) & l { i i } ( u ) t > t ) v | { m } ( a , 8 r ) I = T], Define £-(/>, 2 m • 3", a, #, a) to be {{n}(s)\se3-(p,m,a,$,y)}. The quantifiers and prepositional connectives are forced in standard fashion. Thus p forces (Ex) &(x) if p forces jF(/) for some appropriate /, /> forces (.v) J^(.r) if/? forces ~ (£JC) ~ & ( x ) , and /> forces - i * " if P is not forced by any qHp. One detail above is the definition of

a when p^q and JG ^"(p, w, a, Sf, y). It is filled in by adding one more step to the recursion. Define qth{"}(s)e{2m

-3"}U, 9)

to be q H- {56 {m}(^, ST) & (£t) t < .(|{n}(*)| « T ) ] . The H-ea* /orcwig re/a/ion, p H- *F, is given by p f+- ~ ~ # \ /, H- |{,}(/)] = ff means /> H- I {«,}(*,. »)l = *. where {e,}{a„ 9) is the normal form of {e}(t). Similarly

3r(p,e,t,a)~F(p,enat,&,a). 3.1. pH~\{e}U)\ = a, % e , ! , « ) anrf ? H - * e {*}(/) (w*«« and se $~(p,e,t,a)) are recursive in a, 0> (uniformly in a).

LEMMA

p^q

Proof. Be effective transfinite recursion on a.

|

The relation > ,. is needed for the development of an effective bound on ! [e)(i)\ when p weakly forces \{e)(i)\ into A. An approximate version of

a

398 12

SACKS AND SLAMAN

is simply p^q and q |+- *< > v . t/ is the universal com­ putation tree of Section 2. A typical detail of the precise definition of > „ is ? H - < 2 m 3"./,)

>v
i f t h e r e i s a y e / f such that « H- |{m}(/,)| = y, t2e^(q,m,tl,y)

and

q [+- t2e {«}(/,).

> ,, is the forcing counterpart of > v. It includes all possible values of > v for all generic Cs. It will be seen that p ff- "Z> is an immediate subcomputation of a" is a recursively enumerable relation on appropriate triples

(,p,a,by, pVr*(Ea)[.\{e}(i)\ = a\ as a relation on p, e, and t, is recursively enumerable in 9\ and that > „ is well founded below (p, e, t) when /> weakly forces \{e}(t)\ into /4. These three facts are interwoven in an essential way with the existence of an effec­ tive bound on |{e}(/)|. LEMMA

3.2. Suppose > „ is welt founded below (p,e,t).

If

pH-*(E „. The case of maximum interest occurs when e = 2m • 3" By recursion there is a 6 recursive in p, t, 0> such that pYr*\{m){t)\^b. By Lemma 3.1 the set K, defined by

(p0,o}eK*-p^p0&p0

H- |{m}(/)|=a,

is recursive in p, t, 9. Assume (p0, <x> e K and se^(p0,

m, t, a). Then

p<,Vr*{Efl)lse{m}(t)^\{n}(s)\=n Suppose

Po

Z q. Then there is an r such that q > r and either r[^se{m}(t)

or

rtf-** {«}(')•

399 INADMISSIBLE FORCING

13

By Lemma 3.1 the set of all r^q that force se{m}(t) is recursive in q, s, /, a, i?. For each such r there is by recursion a p recursive in r, s, & such that rVr*\{n)(s)\^p. The effective bounding priciple and Gandy selection yield a strict upper bound y on p (for all relevant r and s) such that y is recursive in p, I, &. It is safe to assume 5 < y. | LEMMA 3.3. /towme /I arfw//5 Moschovakis witnesses and & is countably closed in A. If p\\-*(Ea)i\{e}{t)\=a], then > v is well founded below

(p,e,t)-

Proof. Lei {g}(p, e, t) (=y) be the partial recursive function defined in the proof of Lemma 3.2. If {g} converges on , e, t) then > v is well founded below ,• This is so because the value of {g} on any wHy(p,e,/> is a strict upper bound for the values of {g} on the predecessors of w in > y. Thus g assigns rank. Suppose {g}(p,e,t)l Let zeA be a Moschovakis witness for <£,.>. Thus 2 0 = < g , ,e, />>, and r„ >,,--„+, for all «. There must be a r m such that -'„,= <£. <9,*,5>>

and

> „ ♦

The existence of such a z„ is a consequence of the details of the definition of {g} by effective transfinite recursion on > , . The immediate predecessors of (g, (p.e, />> in > u need not have the form of :m, but every Moschovakis witness, as it winds its way down below (g, (p, e, r>>, must encounter elements of the form zm infinitely often. Thus each Moschovakis witness to the divergence of [g}(p,e,t] has a built-in, infinite, descending sequence in > , below - It follows there is a sequence (n>0) in A such that »«.'> and (P„.e„, /„> >,. „, ,.. Since J" is countably closed in A, there is a «y such that for all «, ^„5=
a

400 14

SACKS AND SLAMAN

Thus q weakly forces the existence of an Moschovakis witness to the divergence of {*}(»), and so p ( > 9 ) cannot weakly force {«}(?) to con­ vergence. I LEMMA 3.4. Assume A admits Moschovakis witnesses, and 0> is countably closed in A. If pth*(Eo)l\{e}(,)\=<jl then there exists a y recursive in p, t, 0> such that pVr*\{e}(t)\^y. Proof. Combine Lemmas 3.2 and 3.3.

|

Let G be a path through 0>, G is ^-generic with respect to a sentence jF if there is a p e G such that either p |+- J^ or p (+- ~ J*\ G is ^-generic over A if: (i)

G is generic with respect to every sentence of the form |{e}(/)l = <x

or

(£ff)[|{e}(0l=ffj.

(ii) Suppose there is a p e G such that p H- | { W } ( / ) | = C T . Then G is generic with respect to (u)(Ez)iue$~(p,m,t,o) 8iue{m}(t)^\{n}(u)\=z], and to every sentence of the form se{m}(l)

for all se3~(p, m, t, a).

THEOREM 3.5. Suppose A is E-closed and admits Moschovakis witnesses. Let &eA be countahly closed in A. If G is ^-generic over A, then A(G) is E-closed.

Proof. Let aeA. Suppose [{e}(f)|
\{m}(t)\=a

and Prr-(u)(Er)\_ue^(p,m,t,a)

&ue{m}(i)^\{n}(u)\

= r].

401 INADMISSIBLE FORCING

For each sef(p,

m, t,o) and q^p,

(i)

rU-se{m}(t)OT

(ii)

rH-st{m}(t).

15

there is an rsS^such

that either

The set of all r^q such that (1) holds is recursive in q, s, t a, 9 by Lemma 3.1. If r satisfies (i) then rYr*\{n\{s)\^p for some p recursive ciple, as in proof of and s) such that y is forces \{e}(t)\ to be

in r, s, & by Lemma 3.4. The effective bounding prin­ Lemma 3.2, yields a bound y on p (for all relevant r recursive in p, f, ^ . Since /f is £-closed, ye A. Thus p at worst max(<x, y). |

The proof of Theorem 3.5 shows: if/> forces |{e}(^, <S)\ < K , then/> forces |{e}U, «f)| to be less than * £ * * Further information can be obtained \l A is of the form L(K). For example, for each aeL(x) and p, there must be a such that q forces JK******** Such matters are discussed in [9] and [13]. The present section concludes with an application of Theorem 3.5 needed for the main result of the next section. Recall Proposition 2.5. The next result states that under suitable hypotheses, the greatest cardinal of an £-closed structure can be collapsed t o w , . CORROLARY 3.6. Assume L(K) is countable, E-closed and £ , inadmissible. Suppose gc(ic), the greatest cardinal in the sense of L(K), has uncountable cofinality in L(K). Then there exists an onto f: such that

L(K,))

a)fr*>->gc(K)

is E-closed, its greatest

cardinal is a>""\ and

Proof. Let 9 be the set of all pe L(K) of the form p:

<5->gc(K)

(<$<<"").

9 is an element of L(K) by a standard condensation argument, and 9 is countably closed in L{K), since cf(gc(»c))>co in L(K). By Lemma 2.4; L(K) admits Moschovakis witnesses. Let f:a)fM^gc{K) be .^-generic. Then / i s an onto map, and by Theorem 3.5, L(xJ) £-closed. OJ[ UI = wftK- n because 9 is countably closed. |

is

402 16

SACKS AND SLAMAN 4. £-POINTED PERFECT FORCING

Let L(K) be countable, £-closed and £ , inadmissible. By Proposition 2.9, L(K) has a greatest cardinal, namely gc(ic). Assume gc(»c) has uncountable cofinality in L(K). In this section it will be shown that there exists a 7"Egc(K) such that L(K, T) is the £-closure of T. T will be generic with respect to uncountable, £-pointed, perfect forcing conditions. The ideas underlying the proof of Theorem 3.5 will guide the proof that L(K, T) is £-closed. Some modifications are necessary because the collec­ tion of forcing conditions is now a "class" rather than a "set." Pointed, per­ fect forcine was aDDlied in T71 Iterated Derfect forcing was develoDed bv Baumgartner and Liver [14] for countable conditions and by Kanamori [i5] for uncountable conditions. By Corollary 3.6 there is a n / s u c h that UKJ) (= [There is only one uncountable cardinal.] and L(K, f) is £-closed. The forcing conditions will belong to L(KJ), and the desired Twill be generic over L(K,/), and the desired Twill be generic over L(K,f). Let to, denote the greatest cardinal of L(x.,f). Since/can be construed as a subset of w „ it follows that each bounded subset of to, (in L(K, /")) is constructible from a countable (in L(K, / ) ) initial segment of /"via an ordinal less than to,. Thus Seq, defined by a e Seq«- a: a - {0, 1} & a < at, & a 6 L(K, / ) , is a member of L(KJ). Suppose p£ Seq. Uaep, then a is said to split in p if i*<0> and -»*< 1 > belong to p. (If domain(o) = a, then domain(j*<0>) = a + 1 and ( 3 *<0>)(a) = 0.)/> is perfect if: (a) (b) (c)

(«)J«,(*)«d-<J>[(-r«)6pl («) J S ,<£*),«,[>£f& /splits in pi ( a ) a < „ » [ ( d o m a i n ( d ) = « & a is a limit & (/?),«,((*

(d) ( / ) w [ ( » > ) W ( K ^ / W c / ( m ) ) \Jnf(n) splits in />].

r/»)ep))

& („)(/(„) Spiits i n p ) ^

Clause (d) says that /> splits on a cte
a In short P can be computed from any branch through p.

403 INADMISSIBLE FORCING

17

Let P be the class of ail £-pointed, perfect ps in L(K, f). Note that P$L(K,f). Let p, q, r denote elements of P. p^q (read p is extended by q) if p 2 q. Define 2t to be , ^ >. PROPOSITION

4.1. ^> « countably closed in

L(K,f).

Proof. Let p„ (« < o>) be an infinite descending sequence in &, and let q be f){p„\n
and

y s£ E X, /?, /

PROPOSITION 4.2. 77i<>
Proof. Suppose p ^ t X, a, f for some a < w,, A Te [ > ] is defined by recursion on w,. If F fy does not split in p, then T(y) = 0 if 7" ry*<0>e/>, and = 1 otherwise. Assume T [y splits in p. Let $ be the order type of splits in p}.

{6\6<JSLT\6

Then

r(y) = {°

if

'6*

v

" [1 otherwise. r winds its way through P in the same manner that X does through 2,U| Consequently r <

e

p, X

and

X < £ /», T.

The initial assumption on * implies the degree of T is at most that of X. The £-pointedness of p implies the degree of X is at most that of T. | In the light of Proposition 4.2, an £-pointed p represents a cone of degrees whose vertex is the degree of p. If Te [ > ] , then p "forces" the degree of T to be at least that of p without "forcing" any bound on the degree of T. p is said to be E-pointed via & (8< w,) if (r)[Te

[/>]-»/» sS £ 7-,<5,/].

404 18

SACKS AND SLAMAN

4.3. Suppose YcWi and p is E-pointed via S. Then there exists a q such that p^q, q *S E p, Y, and Y < Eq,f, S. PROPOSITION

Proof, oeqis defined by recursion on the domain of a with the intent of coding Y into every branch of [>]. Suppose oeq and o does not split in p. Then «*<0>e? if 4*<0>e/>. Otherwise ^ * < 1 > 6 ? . Suppose J splits in p. Let r be the order type of {}• | y < domain^) & a \j splits in p). I f t = / r + l , then 4*<0>e
if / ? e r ,

4*e?

if / ? e 7 .

If T is a limit, then a splits in q. By construction /> ^ q. To see ], and so p < £ f, / , 6. By construction Y < £ T, p. Thus Y ^ET,f,S. q ^ E p, Y, so q ^ , 7", /, <5. The left-most branch of [f J is recursive in 9, hence Y ^Eq,f,6. | The language i?(ie,/, ,T) has the power to analyze the computation of (e}(x) when « y ( K , / F) and \{e}(x)\}(/,,..., /„) is a term. The language also includes: 6, | |, variables, quantifiers and prepositional connectives. The relations L(K,f, D h

\{e}(t)\=a,

j e J ( f , r, cr)and

«t,/,nH«{'}(') are defined simultaneously by recursion on a. 5"(e, f, t ) v | { m } ( 0 t - t ] . ^(2-

3",u)={{fl|(S)|(£?)!<.(«^(,}'))!;

405 INADMISSIBLE FORCING

19

and L(K,/,T)\=

{n}(5)6{2"-3*}(/)

Use2~(m,t,o) and L(K, f, T) (= se{m}(t). A sentence f is said to be ranked (and of rank at most a) if it is of the form |{e}(/)| = a. or S6)f}((), where se&~(e,t,o). The forcing relation p H-J* is defined by (T)lTelpl-*L(K,f,

T)\= 3F]

when & is ranked, and in a standard fashion when J*" is unranked (e.g., has an unbounded ordinal quantifier). The weak forcing relation p |+- *& is given by (?),>„(£>)„>,[> H-J 5 "]. The reason for defining forcing for ranked sentences'in term's of truth is to avoid quantification over a class of forcing conditions. The next lemma shows that the definition of |+- is workable. LEMMA 4.4. Suppose & has rank at most a. Let W be a well ordering of co, of height a. Then for each p, there is a q such that p^q, either q \\-SF or qYr ~ &\ and q s£ E>, W, r j F 1 (uniformly).

Proof. By Proposition 4.3 there is an r such that p>r, r < £ W, r ^ and <Wj&^y *ZEr,f,&0 for some 60 r is defined by recursion on the length a of reSeq. The range of h is the desired q. a = 0: h(l) is the shortest ser such that either sHi & or s H i ~ SF. (Thus h(0) decides whether q forces & or ~&.) ot = /?+1, where 0 is not a limit: let < 5 0 , J , > be the least pair of incom­ parable extensions of h(t f (0 + 1)) in r such that for each i < 2, either 5,-hfi-^

or

s,r+i~Jy

Define h((t [(j8+ 1 ))*<<>) = J,,

* = ;,

A(/) = U{A{/r?)lr<-i}.

«=A+I:

*(/ r/)*<'>=/'(' r/)*<'>.

406 20

SACKS AND SLAMAN

(Note that h(t \X) must split in r, because h(t ["(/?+ 1]) splits when P is not a limit.) With the aid of Lemma 3.1, q ^ E r, IV, rF\ hence q ^ t p, W, rP'< (uniformly), q is perfect because it is a homeomorphic image of Seq. To see q is ^-pointed, fix Te [ ? ] . Then Te [/?], and s o / > ^ £ , 7 , / , St for some ^ , < o ) , . Thus q ^ET,f <5,, r, 50. But Te [ r ] , so r ^ £ r , / , 32 for some 5 2 < O J , . Consequently q ^ET,f,50,6,,S2. | As in Section 3 the relation > „ is invoked to obtain an effective bound on | {
4.5. Assume

r 1

t Q&i.

Suppose

>„

is well founded below


= o},

then there exists {q, y, W(y)> recursive in p, T,fsuch

that

p^q,

*H-I{«}(/)l=y, and W(y) is a well ordering of a>, of height y. Proof. (q, y, W(y)) is computed by an effective transfinite recursion on the height of (p,e, t) in > v. As always assume e = 2 m -3". By recursion there exists (p0, y0, W(y0)> recursive in p, r/n,/such that p> p0, PoH-\{m}(t)\=y0, and W(y0) is a well ordering of co, of height y0. By Proposition 4.3 there is a pi such that p0>pit pi ^Ep0, W(y0), r / n , and <W(y0), r O < £ /?,,/, <50 for some<5 0 <w,. Let **%><,)= {/J a < O J , } be a well ordering of all terms that belong to Sr(m,t,y0)A straightforward definition of W°(y0) yields: W°iy0) ^ £ ^ ( 7 0 ) , "Y1; and V sco, for all B 1 . A contracting sequence { ? a | « « o , } of forcing conditions is defined by effective transfinite recursion on a. q0 = Pl, q. = (]{qJoi<X},

a The definition of q-, , has two parts. First, by Lemma 4.4, there is an r such that:

a

407 INADMISSIBLE FORCING

21

and (i) (")

rafrt,e{m}(i)0T r«H-r. *{«}(*).

If (ii) holds, then g; +, = r„. If (i) holds, then
>y ;

and so by recursion on > , there is recursive in r a ) V , / ( u n i f o r m l y by Gandy selection) such that r . > # + 1 ,

«: + ,i+-i{«}(Oi-7: +1 , and W(7, + 1) is a well ordering of height ya + I . Define 9«> = n{?«]*<<»j} and yx. = strict upper bound of a

Then ?«,«/>,, W(y0), r r , H^(r 0 ) < £ A r ' n , / To see qn is £-pointed, fix r e [ 9 a ] . Then Tepu hence /?, < £ , T,f,S1 for some countable 5, not depending on T. Thus qx < r , / , <5,, <50. To check that \_qx] is perfect, let C , = {/?| T f^splitsin^,} for all a<5„. Let 5u>=\J{6n\n
Hence 5a e C. Since q^ [+- |{*}(*)! < y » . »t follows from a local forcing construction as in Lemma 4.4 that there exists a q,

9rH2#WlM(')l-£l. and q ^eqv., W(y„), r ' n The desired y is the definite value of |{e}(/)| forced by q. W(y^) can be obtained effectively from the W(ya + Js. |

408 22

SACKS AND SLAMAN

A closer examination of Lemmas 3.2 and 4.5 will clarify the proof of Lemma 4.6. In both 3.2 and 4.5 a partial recursive function {g} was defined by effective transfmite recursion. It was shown by induction that if ,e,/> then {g}(p,e,t)[ and its value includes an ordinal y and a condition q

,e, 01, then < v below < g , ( p , c , ( ) ) is a well-founded relation closely associated with < v below , e, t). The arguments of 3.2 and 4.5 applied to < „ below > show that the value of U)(p,e,t) includes a y and a > is essentially the same as <, below , e, t). It is in the context of class forcing that further insight is gained. If & is a class, then the immediate predecessors of , e, /> is well founded with height less than K. The fusion construction of Lemma 4.5 supplies a q and a set version of < v below extracted from < y below < * , < * * , f) ). The set version of < v below ((]) e^ Is) is dense in the full class version of < y below . According to the second viewpoint the two cases that occur in the analysis of are {g}(p, e, t) converges or diverges, rather than < ^ is well founded or ill founded below , e, />. IfvC^(nxand

LEMMA 4.6.

p\+-*(Ea)[\{e}(t)\=ol then

{g}(p,e,t)i.

Proof. Virtually identical with that of Lemma 3.3. If {g}(p,e, r)t, then there is a Moschovakis witness to the divergence of {g)(p, e, t) in L(K, f) by Lemma 2.4. The witness yields an infinite descending sequence in < Y below (p,e,t)Since & is countably closed by Lemma 4.1, the infinite descending sequence contracts to a q below p that forces {e}(i) to diverge. |

LEMMA 4.7

FfrpcCol

and /7 4_«(£ CT )[|{e}(/)|=
409 23

INADMISSIBLE FORCING

then there exist (q, y, W(y)*> recursive in p, rP f such that

p^q,

*rH{«}(')l-y. and W(y) is a wellordering o/w, of height y. Proof. By Lemmas 4.5 and 4.6, and the intervening remarks.

|

Let 7"£ w,. T is ^-generic on a sentence SF of J5?(K,/, 3~) if there is a p such that Te [/>], and either p \\-& or p (+- ~ ^ F is ^"-generic if: (i)

r is ^"-generic on every i^ that is ranked or of the form

(£
(ii) if T e [ / 7 ] and/>r+-|{«}(0l = *. then there is a 9such that Te [/»] and 9 forces the following sentence or its negation, (u)(Ez)luz3-(m,uc)&.ue{m}(t)^\{n}(u)\=xl THEOREM 4.8. Let L(K) be countable, E-closed and £ , inadmissible. Sup­ pose the greatest cardinal of L(K) has uncountable cofinality in L(K). Then there exists an S^cof{K> such that L(K, S) is the E-closure ofS.

Proof. Let T be ^-generic as above. Then S is if T) encoded as a subset oftof"". By Proposition 4.3 and the genericity of T, each Yel^'n L(K, f) is recursive in T, f S for some countable S. Hence E(S) 3 L(K, f T). To check £ ( 5 ) £ L ( K , / , T), suppose r f < w , and L(K,fT))=

|{2"-3"}(r)|
Then there is a p such that Te [/>], p hh (£a)[|{m}(/)| = <JJ, and /> hh ( M ) ( £ T ) [ « 6 ^-(m, f, ff) & u6 {IM}(0

-|{«}(«)|=T]. By Lemma 4.7 there is „, y 0 , W(y0)> recursive in p, r / n , / s u c h that

/»oH-lM(OI-Vo. and W(y0) is a wellordering of co, of height y0. Now the fusion argument of Lemma 4.5 can be repeated to obtained qx and y^ such that Po^q^, <70oH-*l{2m33"}(/)|^yoc, v and yx^Ep, rJ. Thus q^ weakly forces |{2 m -3"}(OI to be less than K. The set of such q^s is dense in p, and so p weakly forces (£ff)[|{2 m -3"}(OI = ff]- (Note that p does not in general force a universal bound on | {2m ■ 3" }(/)| for all generic elements of [/>]• I

410 24

SACKS AND SLAMAN 5. SELECTION

In this section a selection theorem is proved to obtain a partial converse to Theorem 4.8. The theorem is inspired by a result of Y. Moschovakis [18] (also cf. Kirousis [20]). Let E(R(a)) be the ^-closure of the set of all sets of rank less than a. Moschovakis showed: if a has countable cofinality in E{R(a)), then £(J?(«)J is £ , admissible. Corollary 5.2 states: let A' be a set of ordinals; if in E(X) the greatest cardinal has cofinality m, then E(X) is £ , admissible. Corollary 5.3 is the intended partial converse to Theorem 4.8. Let X be a set of ordinals. Some fundamental facts concerning the structure of E(X) will prove helpful below. (R0) Let a, b, c,..., esup X. For all z, zeE(X) iff z ^ E X, a for some a. (Rl) Each element of E(X) can be coded by a subset of sup X. More precisely, if z e E(X\ then there exists a z* £ sup X such that z is recursive in z", and z* e E(X). z* is defined by a recursion on the length of com­ putation of z from x. (R2)

In E(X) there is a greatest cardinal.

Call it gc(E(X)). It is safe to assume gc(E(X)) is sup X. More precisely, there exists an X0 c sup X such that E{^0) = E(X) and gc(E(X0)) = sup X0. Let A £ sup X. Recall from Section 2 that A is said to be recursively enumerable in X if there exists an e such that for all a e sup X, aeA*-+{e}(a,X)l Each a corresponds to a node <e, > on the universal computation tree > v defined in Section 2. a belongs to A iff > „ is well founded beneath <> will be written a. Thus ae/f iff aj. Fact (Rl) makes it possible to think of each node below <e, {a,X)) as being of the form <e0, (b,X)) for some ^esupJf, or more simply b. Define min/4=min{|a||fle^}. \a\ is the height of > u beneath a. \a\ is defined by recursion to be the strict least upper bound (sup + ) of all \b\ such that b is an immediate subcomputation of a (as defined in Sect. 2). THEOREM 5.1. Let X be a set of ordinals. Suppose in E(X) there is an ascending sequence {K^JKCO} of cardinals (in the sense of E(X)) such that

supX = gc(£(X)) =

sup{K,/j
411 INADMISSIBLE FORCING

25

If A^ sup X is nonempty and recursively enumerable in X, then min A is recursive in X,{Kj\j<0)} (uniformly). Proof The predicate, b is an immediate subcomputation of a, is recur­ sively enumerable. The following modification, b is an immediate com­ putation of a via 0, is recursive. The idea is to let 0 bound the height of the computation needed to show b is an immediate subcomputation of a. For example: (1,)

<m, u) is an immediate subcomputation of <2m 3", «> via 0.

(2„) If {wi}(u)J, |{m>(«)|> is an immediate subcomputation of <2 m - 3", w> via 0. The theorem is proved by an effective transfinite recursion on min A, henceforth called the main recursion. There are countably many cases woven together at the finish by the Gandy selection principle. Case T-y. min/4 =

Suppose i<j and there is an aeAnK,

such that

min(/ln>c,)=|a|

= sup + {\b\ I b
412 26

SACKS AND SLAMAN

sufficiently large a, the second subcase holds. For each such a there is a b e Kj such that P(a)=\b\

and

P(a)>p(y)

for all y < a .

Each such b is associated with at most one a, but there are KJ+ , a's. Case 5'. Suppose min A = mm(A n K,). Let y(j, a) be the supremum + of all \b\ such that b < Kj, and b is an immediate subcomputation of a. y(j, a) is a partial function, and is defined iff every b < /c, (and an immediate subcomputation of a) converges. Suppose further that aeAnicl&\a\=mmA->yU,a)<min

A

for all 7 < co. In this case min^n*,) is computed by a recursion of length ie,+ 1 . Fix a < K I + , and assume p(y) has been computed for all y < a. Define /?-(a) = sup{/?(y)|y
first,

p~{a)+\>

^-(a)+l<min(/ln/c,), and /}(«) is computed as follows. Let y(j, a, /?) be the supremum* of all \b\ such that A < ^ and 6 is an immediate subcomputation of a via b. y(j, a, fi) is partial recursive in /, a, 0, and is defined iff every b
a, $'{«)).

Y, is nonempty by virtue of the suppositions that define Case 5'. Ya is recursively enumerable in p-(a),X,K„ hence the main recursion can be applied to compute p(a) =

min{y{j,atp-(a))\<j%a>eYa}.

It follows that {P(OL)\<X eco x Kt such that P(z) = y(j,a,p-(
and

/?(«)>/?(>-)

(y
Each such <;, a> is associated with at most two a's, but there are K, + , a's. For a given < La) the first a might occur at p0 when the only immediate

413 INADMISSIBLE FORCING

27

subcomputation of a via fi0 is <m, a, X), and the second at 0, when 0, > |<m, «.*>!

An effective procedure has been defined for each case q above. It con­ verges iff the suppositions of the case are true . If it converges, then it converges to min A. Let B be the set of all q such that the procedure for case q converges. Recall the Gandy selection principle from the proof of Lemma 2.3. B is a nonempty subset of w recursively enumerable in X, U,\i< w). By Gandy selection, an element of B can be computed from X,

{*,{«»}•

|

The proof of Theorem 5.1 establishes more than is stated. E(X) can be replaced by an arbitrary £-closed structure $ with an additional predicate. The only structural fact needed for the proof is somewhat weaker than the existence of a greatest cardinal of cofinality a in S. As above no assumption need be made about the power set operation inside
Proof. Let D(a, z) be A0. Suppose (b)btX(Ez):eElX)D(b,z) in the hope of bounding z. Then

(b)b€AEa)lieAEe)[{e}(X,a)l&D(b,{e}(X,a))l The set {\{e}(X, a)\ | {e}(X, a)j & D(b, {e}(X, a))} is recursively enumerable in X, b. Thus its min is recursive in X,b, {K,.KW} by Theorem 5.1. Consequently z is a computable function of b, hence bounded in E(X). | COROLLARY 5.3. Suppose L(K) is E-closed and not £ , admissible. If 5
Proof. Apply Corollary 5.2. COROLLARY 5.4. Suppose L(K) is E-closed and not £ , admissible. Let {a}^>\n
414 28

SACKS AND SLAMAN

Proof. Suppose 5 and 5 exist. Let gc(S) be the greatest cardinal in the sense of £(S). By remark (R2) at the beginning of Section 5, it is safe to assume S s g c ( S ) . It follows from Corollary 5.3 that gc(S) = < ( * ! for some positive n«a, and that << K * is regular in E(S). There is a teE(S) such that

a t will be used to define a violation of the £-closedness of E(S). Suppose ZECD£ , K > and ZBL(K). Z can be effectively coded by a coun­ table subset of wiM as follows. Let z(n) be the least triple „ is first order definable over L(y) via formula n with parameter p. A standard downward Skolem-Lowenheim argument shows e oi"*t. Thus {r(n)|n is uncountable and regular in £(S). By a well-known theorem of Hajnal, t(Y)eL(0, S) for some /?<£o£. Thus every subset of < t K ) in L(K) can be coded by a countable subset of «£"» in £(5), which is constructive from 5 below otf'"'. This process, reversed and slightly modified, violates the £-closedness of £(S). Let K be the set of all bounded subsets of »«*> in £(5), and let weK. Define A(w) as follows. If / - ' M encodes a subset of < , , c l that is the graph of a wellfounded relation, then h(w) is the ordinal height of that relation. Otherwise h(w) = 0. h is recursive in S, t, but maps Ke E(S) onto K. |

6. FURTHER QUESTIONS AND RESULTS

(Ql) Is the converse of Theorem 4.8 true? To make the question definite, let L(K) be countable and £ , inadmissible, and assume there exist 6
415 INADMISSIBLE FORCING

29

the methods of [7]. There it was shown that if a is countable, greater than co and £ , admissible, then there exists a n S s w such that (i)

E(S) = L(a, S) and

(ii)

for all ReE(S),

if S *£(/?), then L(a) £ £(£).

(Q3) Suppose L(K) is uncountable, £-closed and £ i inadmissible. Assume gc(ic) has uncountable cofinality in L(K). D O there exist 8 < K and 5 c ,5 such that L(K, S) = £(S)? S. Friedman [16, 17] has found a virtually complete answer to this question when K = L and cardinality of L(K) is 6. The analysis of E(X) made in the various sections of the present paper depends strongly on the assumption that X is a set of ordinals. Slaman [13] has shown: Let L(K) be countable and £-closed. Then there exists a countable X£2" such that (a)

L(K,X)

=

E(X);

(b)

L{K, X) does not admit Moschovakis witnesses; and

(c) if L(K) is £ , admissible but not the £-closure of any ZeLic), then L(K, X) is £ , admissible but the notions of £-recursive enumerability and boldface £ , do not agree on L(K, A').

REFERENCES 1. S. C. KLEENE, Recursive functionals and quantifiers of finite type I, Trans. Amer. Math. Soc. 91 (1959), 1-52. 2. S. C. KLEENE, Recursive functionals and quantifiers of finite type II, Trans. Amer. Math. Soc. 108 (1963), 106-142 3. D. NORMANN, "Set Recursion, in Generalized Recursion Theory, II," pp. 303-320, NorthHolland, Amsterdam, 1978. 4. G. E. SACKS, Post's problem, absoluteness and recursion in finite types, in "The Kleene Symposium," pp. 201-222, North-Holland, Amsterdam, 1980. 5. F. G ABRAMSON, Sacks forcing does not always produce a minimal upper bound, Advan. in Math. 31 (1979), 110-130. 6. J. STEEL, Forcing and tagged trees, Ann Math. Logic 15 (1978). 7. G. E SACKS, Countable admissible ordinals and hyperdegrees, Advan. in Math. 20 (1976), 213-262. 8 G. E SACKS, Selection and forcing, m preparation. 9. G. E. SACKS, On the limits of recursive enumerability, in preparation. 10. Y. N. MOSCHOVAKIS, Hyperanalytic predicates, Trans. Amer. Math. Soc. 138 (1967), 249-282. 11. L. HARRINGTON, "Contributions to Recursion Theory in Higher Types," PhD thesis, M I T , Cambridge, Mass., 1973. 12 J MOLDSTAD, "Computations in Higher Types," Lecture Notes in Math. Vol. 574, Springer-Verlag, Berlin, 1977. 13. T A SLAMAN, "Aspects off-Recursion," PhD thesis, Harvard, Univ. Cambridge, Mass., 1981

416 30

SACKS AND SLAMAN

14. J. BAUMGARTNER AND R. LAVER, Iterated perfect-set forcing, Ann. Math. Logic 17 (1979), 271-288. 15. A. KANAMORI, Perfect-set forcing for uncountable cardinals, Ann. Math. Logic 19 (1980), 97-114. 16. S. FRIEDMAN, Uncountable admissibles. I. Forcing, Trans. Amer. Math. Soc, in press. 17. S. FRIEDMAN, Uncountable admissibles. II. Compactness, IsraelJ. Math., in press. 18. Y. N. MOSCHOVAKIS, On the Grilliot-Harrington-MacQueen theorem, in "Logic Year 1979-80," Lecture Notes in Math., Vol.859, pp. 246-267, Springer-Verlag, Berlin, 1981. 19. J. E. FENSTAD, "General Recursion Theory," Springer-Verlag, Berlin, 1980. 20. L. KIROUSIS, A selection theorem (abstract), / Symbolic Logic 46 (1981), 38.

417 Reprinled from Annals of Pure and Applied Logic 81(1996) 171 -185 wilh kind permission of Elsevier Science - NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.

Effective forcing versus proper forcing Gerald E. Sacks' Harvard University. Massachusetts Institute of Technology, Cambridge. MA 01238, USA Received 1 July 1995 Communicated by A Nerode

Dedicated to the Memory of S. C. Kleene

Abstract 9, a notion of forcing over E{mt), the £-closure of L(w,), ,i said dt ob effective ei fvery sideways ^-generic extension preserves £-closure. There are set notions of forcing in £(co,) that do not preserve £-closure. The main theorem below asserts that & is effective if and only if it is locally proper, a weak variant of Shelah's notion of proper.

1.

Introduction

Effecting forcing alludes to forcing over an £-closed structure. The prime example is £(o>i), the £-closure of w,, the least £-closed structure with o>i as an element. E(uii) equals L(K) for some ordinal K less than the first I1 admissible beyond ©,. £-closed means: {«}(*) I —

{*}(*) € I(K)

for all x 6 L(K) and e < a). {e}(x) is Normann's extension of Kleene's concept of partial recursive function from objects of finite type to all sets x. In this paper £(JC) is always £ ( « , ) except in Section 2, where the elements of £-recursion are sketched. Let & 6 L(K) be a notion of set forcing. 9> is said to be effective if every .-^-generic G preserves ^-closure, that is, L(K,G)

is £-closed. (G can be assumed to be a subset

of mi, since every member of L(K) can be injected into w, in L(K).) Locally proper is a weak variant of Shelah's proper forcing. The main result below is the equivalence of effective and locally proper forcing. To force over f-closed structures is to force computations to converge or diverge. Forcing computations differs radically from forcing sets. Every set notion of forcing 1 The author is grateful to A. Kanamori and T. Slaman for valuable mathematical suggestions and other acts of friendship. Early support was provided by the NSF (DMS).

418 172

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Sacks I Annals of Pure and Applied Logic 81 (1996)

171-185

preserves X, admissibility. Not so for £-closure. If G collapses w, to w, then E(K,G) is not £-closed, because E(K,G) - E(K,A) for some AC en and E(B) is £ , admis­ sible for all fiCoi. The connection made in Section 3 between proper forcing and preserving £-closure is a consequence of how computations are forced to diverge. A Moschovakis divergence witness, denned in Section 2, is forced to exist by a construc­ tion that ranges unboundedly over a countable co-re substructure of L(K). (The latter is a weakening of the notion of X, substructure. Recall that one version of proper forcing uses countable elementary substructures and forcing conditions generic over those substructures.) Sections 4 and 5 contain imperfect results along the following lines. There exists a forcing relation that is locally proper but not proper, and another that preserves w, but not the £-closure of « , . £(a>i) Proviso. Almost everything below makes sense for L{K), an arbitrary £-closed, initial segment of L. On the other hand, this paper focuses on £(co,). From now on L(K), with no qualification, is arbitrary, and when £ ( w , ) is meant, some qualification such as (= £(C0])) will be inserted. / € L(K) IS a co-re substructure of L(K) if / is £-closed, and if for all e < w and m,a £ I: 3x[xemA{e}(x,a)1] (1.1)

—>3x[x€(mni)A{e}( X ,a)n

I is said to be pointed if 3b[b € / A K* £ I]. Kr, the greatest reflecting ordinal, is defined in Section 2. .9 e L(K) ( = £(co,)), a set forcing relation, is locally proper if for each p 6 P and a € L(K) there is a countable, pointed co-re substructure / 6 L(K) such that (1.2)

(i)p,a€l

and

(ii) 3q[q ePAp^qAq

generic over / ] .

(p^q is "p is extended by s -> 3r(r e D f l / A ^ n r ^ O ) ] . D is of the form {r\r \\- ? or r \\- n . ^ } , where 3? is (1.3)

(i) 3y[y € I A \{m}(a)\ = y], or (ii) Vb3y[b e c - y € / A \{n}{b)\*y)

(a,c G / ) .

The genencity of q will be seen to imply that p cannot force {e}(a) to converge outside /. For now the definition of proper forcing in £ ( w , ) is left open-ended. The following clauses certainly belong. There exists a closed, unbounded class C of countable, co-re

419 G.E Sacks!Annals of Pure and Applied Logic 81 (1996) 171-185

173

substructures that covers E(w,), and such that for rach / € C, (1.4)

Vp3q[p € / — q generic over/].

Familiar notions of set forcing in L(x)(= E(a>,)) are proper in the above sense. These include ccc, countably closed and perfect. It seems reasonable to require that C be £-recursive on L(K). 2. f-recursion revue {e}(x), the eth partial £-recursive function, is defined by the Normann schemes [4], A leisurely, but not overly detailed, account of ^-recursion can be found in [8]. From now on the "E" will be dropped as often as clarity allows, e.g., "recursive" in place of "£-recursive". If e is not the Godel number of a scheme, then {e)(x) diverges (f). One of several rudimentary schemes is {2}{x,y) = {x,y}. There are also schemes for bounding, composition and enumeration. The last is {l}(e,x,y)~{e}(x,y). The ~ symbol denotes equality for partial functions: their graphs are equal. A com­ putation instruction is an («+ l)-ruple of the form (e,jc0,...,*„_,). For simplicity let n be 1. (e,x) is an instruction to try and compute {
{m}(x)l

and

vy[y e M M — WO-) I]Scheme T (for typical) combines the most salient features of bounding and compo­ sition. From now on scheme T will be the only scheme considered. The concept of computation is derived from that of immediate subcomputation instruction. (2.2) (m,x)

is an immed. subcomp. instr. of (2m • 3",x).

(2.3) If {m}(x) I, then (n,y) is an immed. subcomp. instr. of {2m • 3",x) ffo all y e {«}(x). (2.4) If d is not the Godel number of a Normann scheme, then (d,z) is an immed. subcomp. instr. of (d,z). The relation, b is a subcomputation instruction of a, in symbols a > v b, is the transitive closure of the relation, b is an immed. subcomp. instr. of a. The latter is re, but the former is not. T{e,x) is the computation tree generated by (e,x); its top node is (e,x); underneath are all subcomputation instructions of (e,x). By transfimte induction, (2.5)

T{e,x) is wellfounded d— {e}(x) ] j

420 GE. Sacks/Annals of Pure and Applied Logic 81 (1996) 171-185

174

A class is £-recursively enumerable (or more simply, re) if it is the domain of a partial £-recursive function. The following predicates are re: (2.6)

\a\ < \b\.

(2.7)

\a\*k\b\ < oo.

oo is the length of every divergent computation. If A is a convergent computation instruction, then \b\, the length of b, is the ordinal height of Tb. Gandy selection says there is a partial recursive function s such that for all e,n < o>: (2.8)

M
j .

The proof of (2.8) is a fixed point argument based on the re-ness of (2.6) and (2.7); it assigns the least possible value to \{e}(s(e))\. Expressions (2.6) and (2.7) were initiated by Moschovakis, who aptly dubbed them stage comparison. Gandy selection is the most general selection principle available in £-recursion the­ ory. Note that selection fails for subsets of w, re in co,; otherwise £ ( « , ) would be Ij admissible. Perhaps this is why the study of forcing over £(co,) is so convoluted. In the absence of strong selection £-recursion makes do with reflection, y is an a-reflecting ordinal if L(y) (= S?

implies

£ « ) |= &

for every 27, formula & of ZF with a (and co,) as its only non-integer parameters. K°0 is the supremum of all ordinals recursive in a (and co,). 6 is recursive in a (and w,) if 5 = {e}(a, eoj). Since L(K) = E(ear), K- = s u p K | a < a > , } . Let < = sup{y|y

is a-reflecting}.

Clearly x g ^ < . Slightly less clear is < < K; it follows from the 27, inadmissibility of L(K). Much less clear is hf0 < < . An inequality of the last sort was proved in [5] in order to make a forcing argument work for the ^-closure of 2 2 ". Later Harrington [2] showed (2.9)

KJf < K*\

and it is his approach that leads to »cg < < in the setting of £(a>,). Recall the £(w,) Proviso of Section 1. In accord with it, from now on co, is a hidden parameter in computations of the form {e}(x) for x € E(a)j). For example, co, is a recursive ordinal. Kechns's basis theorem states: (2.10)

3b[b€mA

M(6,x)T]

, , -^3b[bemK{e}(b,x)]

H A s « * ^ .

421 G.E. Sacks! Annals of Purr and Applled Logic 81 (1996) 171-185

175

Harrington's approach to Kr in (2.9), together with (2.10), leads to: (2.11)

If {e)(a) T, then a Moschovakis witness to {e}(a) | is first-order definable over I « , w,).

The witness of (2.11) is a function on co defined by an co-recursion whose limit is K? and whose nth stage invokes (2.10). A variation on this cu-recursion is used to prove that locally proper forcing preserves £-closure in Section 3. Let 7 be a co-re substructure as defined in Section 1.1 is £-closed. Suppose {e}(a) ] for some a 6 / . A divergence witness for {e}(a) is definable over / as hinted in the previous paragraph. The simplest example of a co-re substructure (of L(K) = E(a>i)) is: !0 = {X\KXr^KrAxeL(Kr)}

(Kr = K?).

IQ is countable, and J0 fl to, = =b n co, for fore **, < co,. Slaman [9] introduced the Kr spectrum: Kr,0 = «r J

(2.12)

KrM1 = K/'; KrJ =sup{Kr.3\5

< A}.

KrJ is defined for all 6 < recf(K), the re cofinality of K (same as the i f K ' cofinality of K). Since L(K) = E(ia\), the only possible values for recf(ic) are to and w^. The Kr spectrum is so called, because for all 5 C e o , in L(K), (2.13)

3<5[<5 not a limit A K ? = KrJ].

Associated with each Kr,6 is a canonical co-re substructure (2.14)

Is = {x\**
There are of course other co-re substructures. For example, by forcing there is a co-re substructure ./ 6 L{K) such that Vx[x € J

> < « Kr) A J g L(Kr).

Forcing supplies an x such that < < K r but x 6 i ( « ) - /-(«,)• It turns out that for every co-re substructure / , the supremum of the ordinals in A is a member of the Kr spectrum. An arbitrary set forcing relation 9 e L(K) can be sketched as follows. A generic G is a subset of co,. Forcing conditions are coded by countable ordinals. A typical element of L(K,G\ for any GC co|, is of the form {e}(a,<7), where a 6 I ( x ) and |{e}(a,G)| < K. Among the formulas of the language ^ ( K , « ? ) are: (i) |{e}(<2,^)| =
422 176

G.E Sacks! Annals of Pure and Applied Logic 81 (1996) 171-185

( I ) and (lii) are ranked formulas. In (ii) the range of the existential quantifier is K. The meaning of (iii) will be clearer after $~{p,e,a,e\ a set of terms that name the elements of {e}{a,G\ is defined. A simultaneous recursion on a defines (2.15a)

p |h \{e}(a)\ = a

(2.15b)

("<$" is omitted for notational simplicity);

$~{p,e,a,o);

(2.15c)

q\rse{e}{a).

When a = 0, {e}(a) is <§. For all p, p \\- | £ | = 0. 3T{p,e,a,0) = {6\S < co,}. And q | h <5 e ^ iff q says so. A complete account of the recursion when a > 0 would include a clause for each Normann scheme. One clause corresponding to scheme T (2.1) will suffice to show how (2.15) is built up. Thus p\r\{2m

3"}(s)l = e

iff

(2.16a)

3y<£I> | h \{m}{a)\ = y],

(2.16b)

p |h V ^ 3 T < £ [ X 6 {m}(a) -» \{n}(x)\ = T],

(2.16c)

p | h V t < 2 [ | { « } ( a ) | = TV 3 x [ x € { m } ( a ) A | { / i } ( x ) | ^ T ] ] .

In (2.16b) and (2.16c), x ranges over ^ ( ^ m , ^ ; ) . F(p,2m

■ 3",a,a) = {{*}(0I> 6

^{p,m,a,y)}.

m

q ||- 5 € {2 ■ 3"}(a) iff for some will be a suppressed parameter, as is « , . Thus (2.15a) and (2.15c) are relations ^-recursive on L(K), and (2.15b) is a function ^-recursive on L(K). A set D of forcing conditions is said to be dense if

Vq3p[q>p€D]. Call G generic if for every dense D definable over L(K) there is a p g Z> such that G € p (i.e., G satisfies />). In Section 3 it is shown that only the T, D's matter. c,G) is £-closed. In short, there are no p and a such that p||-

|{e}( a )| = K.

The primary tool for studying &> is > r , the tree of possibilities. Each node of > of the form (p,e,s), where 5 is a term of i?(«,«?). (2.17)

(p,e,s)>v{q,f,t)

iff p^q

A q \\-' (e.s) >v (f,t).

>y anticipates all the possibilities for >v. P\r'

JF iff

\\-" is the weak forcing relation:

~1 3 * U » * A * ||- n f ] .

v

is

423 G.E Sacks/Annals of Pure and Applied Logic SI (1996) 171-185

177

To clarify (2.17) unpack it in the case of scheme (T). {p.2m-3n,s)>y

(q,n,t)

iff

P>
(some a < K),

{m}(s).

> , has the wellfoundedness property if >v below (p,e,t) is wellfounded whenever p |h 3(7[|{e}(/)| = a]. A countably closed P has wellfoundedness, but a ccc P need not. This is why the original proofs that countably closed and ccc preserve ^-closure are so different [6,7]. It might be possible to characterize all
3. Effective equals locally proper Keep in mind that L{K) is £(«,), and that w, and & are suppressed parameters. Theorem 3.1 Let 9 € L{K){= iff & is locally proper.

£(W,))

be a set notion of forcing. Then & is effective

Proof. Assume 0> is effective. Fix p,a,b,c and G so that G is generic and G € p. Choose q so that p^q and for some 6 q ||_ Kf.p.a.t.c = fl The reasoning behind (2.13) also shows 6 is Kr- 0. From now on the point is yet another suppressed parameter. It is safe to assume p,a,b,c are effectively equivalent to countable ordinals, i.e., have the same £-degree as a countable ordinal modulo the suppressed parameters. So p,a € h and VPA


,

Consider an arbitrary generic G0 € q. Let D be a dense set of forcing conditions. Then Go € r for some r 6 D. Suppose

(3.1)

r|(-K«}(a)|=7;

the set of all such r is computable from a,y. Let r0 be the least such r with G0 € r. Then '•o^£G 0 ,r,a.

424 G.E Sacks! Annals so Pure end Applied Logic c811996)

178

But K^a^Kr,s (3.2)

)71-185

because G0 G q. SS or e // Nott ehat the eet t o fll l ruch that

3 y [ [ g / A r | h |{m((a)| = y]

is re, because y, if there is one in /, has to be computable from r,a by reflection. Suppose r forces ~1(3.2). The set of all such r is co-re, since, by reflection, r does not force 1(32) iff 3s[s^Er

Ar^s

As

\\-(3.2)].

By Kechns's basis theorem (2.10), there is an r 0 such that Go € r 0 , r 0 ||- ~I(3.2), and *?*""*£ Kr> Again r 0 € / . Now suppose r forces (3.3)

Vi3y[A€c-.y

€ / A |{*}(£)|
In (3.3) y can be computed from b,c,G0 since

a But then y is bounded above by some yo computable from c,G0. Hence the least r, call it r 0 , that forces (3.3) is computable from c,G0. So r 0 6 / since KrG°'f $K r ,^. Note that the set of all r that force (3.3) is re. Finally suppose r forces ~~1(3.3). The set of all such r is co-re, since, by reflection, r does not force I ( 3 . 3 ) iff 3s[s^ErAr^s

As

\\-(3.3)].

Now assume 9 is locally proper. Fix p,a. Then there is a co-re substructure / such that p.a e / . In addition there is a 9 s£/7 such that

qCu(DDI), where Z) is any dense set of conditions as just before (1.3). Also / is pointed; there is a 2 € / such that tc\ $ I. z can nb taken nt ob e aelation no nrdinals, , s o* 1 g £ /or all x G /. From now on z will be suppressed as in o>,. Let sup / be the supremum of all ordinals in /. Suppose (3.4)

p||-\{e}(a)\&sup/.

The plan is to show that q forces a divergence witness for {e}(a), a witness first-order definable over L(mp/,G) for all generic G €
r|(-

|{/}(fc)|>sup/.

Then each generic G G q will belong to some r„ (with associated /„ and b„ as in (3.5)) on level n of the tree below (p,e,a), and the sequence / H ( / * A ) will define a divergence witness for {e}(a).

425 GE. Sacks/Annals so Pure end Applied Logic c81(9966 171-185

179

The r's for level n are found by recursion on n. At level 0, the p,e and a of (3.4) serve as the needed r,f and b. Fix «>0, and let r„,/„ and bn be associated with level n: rn,bn € / ; (3.6)

r„n9^0;

and

r»h !{/*}(£,)|> sup/. Assume /"„ is an instance of Scheme 7: /„ = 2" • 3". Let D be the dense set of conditions that force (3.7)

|{/,}&)l>«ip/

or force ~l (3.7). Then r„ng is covered by £>n/. The r's in DC\I that force (3.7) are in accord with the plan. Consider a n r e D n / that touches r„Dq and forces ~l (3.7). Then HI-{«}&,) = £ for some c € /. The plan requires that r n r„ n 9 be covered by conditions 5 6 / such that (3.8)

s||-3&[*ecA|{e}(&)|>sup/].

Let £ be the set of all s such that forces "3b..." of (3.8) or forces its negation. EM covers rnr,n?. Suppose (for a contradiction) that there is an s € E n / such that r n r„ n 9 n 5 g 0 and 5 forces the negation of "3b..." of (3.8). Thus (3.9)

5 |[- Vb3y[b G c —> y 6 / A |{»}(fc)| < y].

Recall the proof that effective implies locally proper. The y of (3.9) can be computed from 5,b,c by reflection. So s II- Vb[b € c —> \{n}(b)\ < y j for some y0 G /; hence

r n r „ n 9 r u | h |{/fl}(&,)|<sup/, contrary to (3.6). One last detail. The "s" of (3.8) is not of any use unless 5 € / implies some satisfies (3.8). Such a b is available because (3.10)

{b\becAs\\-

|{i>}(fc)|Sssup/}

is co-re and / is a co-re substructure of that sf-

|M(fc)|3*sup/

bel

Z.(K).

To check that (3.10) is co-re, observe

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is equivalent to (3.11)

3t3y[s^t At ||- |{y}(i)|$y<sup/].

Furthermore, t and y are computable from s,b by reflection; so (3.11) is re.

4. Preserving cardinals but not £-closure This section is devoted to an example of forcing that preserves co, but not E-closure. Once again L(K) equals £(toi). Assume the co-re cofinality of to, is w,. Theorem 4.1 There exist BCw, and & 6

L(K,B)

such that:

is E-closed; 9 preserves w^; but & does not preserve E-closure.

L{K,B)

Proof. If L(K,B) is £-closed, then w\M is preserved in L(K,B). Otherwise L(K,B) = E(A) for some A C w, and then K is I, admissible. B, as constructed below, is some­ what, but not quite, generic via conditions that are countable initial segments of co,. Forcing with such conditions is countably closed. It is not difficult to verify that countably closed forcing is locally proper, and so preserves £-closure by Theorem 3.1. B is designed to exclude the Kr spectrum in the following sense. Each tcFj has s aanoniica code ji = fiy(Kr,s

< £ >>,«,).

Note that y < w, since L(K) = £(o>,). js wiil be kept out of B for all 5. The forcing conditions in & are all countable, closed initial segments of o»] that are subsets of B. Suppose 0> preserved f-closure. Let H be ^-generic over L(K,B). Then L(K,B,H) is £-closed. By Proposition 4.3, the canonical code for *?•" belongs to H. But then it belongs to B despite its exclusion. First a standard argument to show ® preserves to,. Let / be a term in the forcing language for H over L(K,B) such that p ||- t: co —»co\

for some p € P. Let q0 = p. For each n, choose the least {q„+u5„} such hhat q„^qn+] and ,. Repeat further to generate yf (^ < wx). For successor J?, proceed as above. For limit 0, yf is the sup of everything previous. Thus {yp\P
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is a closed subset of co, in L(K,B). Part of the design of B below ensures that B is stationary with respect to all closed subsets of w, in L(K,B). SO some y„ 6 B. The definition of y^ implies there is a sequence An\r„ such that sup dom \Jnr„ = yp. Hence U„ r„ is a condition, and UBr„ ||- range(r)Cy Now the definition of B. B(= \J{pi,\5 < w,}) is constructed in w, stages. The ps's are as follows: (bl) dom(ps) =j&. PMUS) = 0. (Thus, j s $ B.) If 6 is a limit, then />* = U i ^ l r < &}• If « < ft then />, > />/,. (b2) {*,*}|«
(P < to, A B not a limit A B even)

be an enumeration of w x «, £-recursive in a>,. e, denotes

{«,}(*,/J,a»t).

p^ preserves K,.^, i.e., K/''' ^K,.^.

Of course pj ^ p5. In addition, either Pi lh ki| < »Cr,«.

or

/>* forces a Moschovakis witness to es T to be first-order definable over £(fcr,4>;0). (M) Suppose <5 is not a limit and odd. Let C be any closed subset of to, ^-recursive in Kr,^i,to,. Then pa(y) = 1 for some y € C. (Thus B n C ^ 0.) The object of (b3) is to make L(K,B) ^-closed. In order to see how (b3) is satisfied, recall the proof that countably closed forcing preserves £-closure as presented in [8] or [5]. That argument shows pj can be extended in one of two ways. The first produces a q and a 9 such that q,e^EPJ,dAg\\-

\e»\=6.

Then dom q <j6; otherwise j6<Eq,03\ (a < B < w, - a<£JJ,©i) and then pj does not preserve Kr> The second produces a q that forces some An/w(«) to witness the divergence of w, for all <5. Second, Lemma 4.2 implies: if C is a closed subset of w, £-recursive in K,,j_i,a>i, then ;a =

sup(Cn^).

428 182

GE. Sacks/Annals of Pure and Applied Logic 81 (1996) 171-185

Lemma 4.2 Suppose C is a closed subset of o>, in £(a>,). Let j be the canonical code for some element of the Kr spectrum. If C^Eb,a>\ Ab < j , then / = sup(Cny). Proof. Let / enumerate C in increasing order. Then / < £ C , c o i . It suffices to show

b
f(y) <j.

Assume b < y < j . Then j ^ £ y , a ) ] . So j 4£y,b,a)U /(y),6,co, and so / 3 ? £ / ( ? ) , » ] . Finally /(y) < / .

since 6<£y,co,. Hence / g

£

Proposition 4.3 Suppose L(K,B,H) « E-closed Assume wf(K) = cof M,W) anrf // w a c/oserf unbounded subset of wf(K) 77ien /Ae canonical code for «*♦* 6e/<™
(uniformly in <5).

Let; be the canonical code for xf-"'. Observe that V<5(c5 <j—*f(6)<

j).

Otherwise, for some 5 < / : / ( « ) » , and so j*€Ef(6>, /(y) = y and j 6 tf.

but then j^EB,H,6.

Hence

5. Locally proper but not proper The main outcome of this section is the existence of a set forcing 9 € L{K) such that 9 is locally proper but does not preserve X, stationary subsets of eof",. 5 C cof *> is

sa'd £& x 'stationary if5 touch"every * - wi,K) such that * jc , ose tl ,un!runf i

and rf ^Thus .t seems reasonable to assert 9 ,s not proper despite the fact that proper has not yet been denned. Marcus [3] offers a definition of proper and shows that proper forcing preserves I} stationary sets (Lemma 5.2 below). The re-connality of y^K is the least / such that for some W CL(K): W is £-re on

L(K);

sup w = r, ordertype of W = k. Since L(K) = E(w\{K)), recf{x) is either w or wf(,<). In this section, as in Section 4, recf(K) = o>fw Then recf(w\w) = ^K\ as are the l\{K) - cofinalities of W f"° and K.

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From now on, a>, is OJ\{K). In essence, & is the notion of forcing introduced by Baumgartner, Harrington and Kleinberg. 9 will be seen to preserve £-closure and so is locally proper by Theorem 3.1. o £ is an adaptation of Jensen's o to £(a>,). o£: There exists a sequence {Sa|a < ©,} in L{K) such that (l) S,Ca for all a < co,; (li) for each 2?"'-definable SCo),, the set {a|Sna = Sa} is ^-stationary; (n»{5 3 |ac)-least pair (S..C.) is chosen so that C, is a closed, unbounded subset of a, and does not touch {P\S.nfi = S,}. Suppose clause (ii) of cE fails for S. Then there is a closed, unbounded C C W | , 2 ? -definable, and not touching { a | S n « = 5,}. Let (S,C) be L(,c)-least. Let H be the X? hull of o,„5,C}. Let * map » isomorph.cally onto its transitive collapse Ho. Since the Z,< '-cofinahty of K is a,,, it must be that //, hence i/ 0 , belongs to L(K). Let 7r(co,) = a. Then TT(.S) = St, n(C) = C , S n a = 5 a , C n a = Ca and a € C. Theorem 5.1 There exists a set notion of forcing 0> € £(co,) such that & is locally proper but not proper. Proof. Let {Sx\a < to,} be a o£ sequence for £(co,) as above. Define 27, stationary sets Sto = {a|a = Sa},

and

5/, = {a|a - {0} = Sx}. Then StQ and its complement are stationary and definable over &>j); one of them, say Sto, contains an unbounded subset of {jt+i\* < to,}, the set of canonical codes for {>cr,a+l|<5 < to,}. The forcing relation 9 is designed to kill the stationarity of the complement of St0. A forcing condition p is the characteristic function of a countable closed subset of St0. A generic G is the limit of such ps, hence a closed subset of St0. Fix p e P and £ e to, to see & preserves ^-closure. It suffices to find g^p and 6 < K such that

a Choose c € S/0 and d < c so that p,b^Ed. c is the canonical code for some KrM] that will serve as 6. There is an e0 such that for all GC to,: (i){eo}(G,fc)T; (ii) «?•* is the least y such that a witness to the divergence of {e0}(G,b) is first-order definable over L(y,TC({G,b})) (cf. [8, p. 271]).

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184

0> is not countably closed; nonetheless the plan is to build q as if 0> were. Thus q is the limit of a sequence qn (w < co), where each qn forces the existence of the nth leg of a Moschovakis witness w to the divergence of {e0){^,b). Then q forces the existence of w. Some assembly is required to make (dom q) € St0; otherwise q is not a condition. Choose (kn\cn) 6 L(K) SO that (Jncn = c. The ?n 's can be chosen so that: (i) p0>q and qn^qn+x\ (ii) c„ 6 dom qn\ (iii) 9n^£Wi,cm for some m. Consequently dom q = c£St0. Observe that c is a limit point of St0; otherwise c<£x,w, for some x < c, since S/o^fW,. Lemma 5.2 (Marcus [3]) Suppose & e £(ft>i) (= Z.(K)) is proper, G is ^-generic and S G £(w\) touches every closed unbounded l\M subset of wh Then S touches every closed unbounded subset of w, in L(K, G). Proof. 3» is locally proper, so L(K,G) is £-closed by Theorem 3.1. The Kr spectrum of L(K,G) is a subclass of the Kr spectrum of L(K); also the canonical codes for the Kr spectrum relative to G are a subclass of those for L(K). TO be precise, consider K%t. There is a y such that Kr,J = K?S. The canonical codes are: jy = ptfilfi < W | A K r , r ^ £ ^,o)i];

aa Then jy =jf. A related fact is IrClf. Let D be a closed subset of wi in L(K,G). Suppose D^£b,couG. By Lemma 4.2 relativized to G, if y is a canonical code for some member of the KrG spectrum, then (5.1)

b<j—./eD.

. Recall that j(Kr,y) is the canonical code for Kr,y. For any p, let

C' = 0"<sup/)l(3/)LpeeA/eC]}. C is a closed unbounded if"' subset of co,, and so has an unbounded intersection with 5. Then there is a q ^ p and an / G C such that b < /(sup/)€Cpr\S, ^

and

is subgenenc over /.

Since 9 is proper, q forces sup I to be an element of the K? spectrum. Since G is ^-generic, G belongs to such a q, and so /(sup/) 6 D by (1).

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185

6. Further results and questions A trove of further information on Effective Forcing can be found in Marcus's Ph.D. Thesis [3]. She studies £-closed L(K)SS of the form £(y), where y is a regular cardinal of E(>). She makes explicit the definition of proper left implicit in the proof of Lemma 5.2. She also raises many questions. An ambitious reader might wish to verify directly that ccc forcing is locally proper by exploiting the tree-of-possibilities approach of Section 3. The example of Section 4 could stand some improvement. Surely there is a 3? € L(K) such that & preserves w, but tno ^-closure. .Beware ealse proofs oo fhis sast conjecture.) Slaman has asked for a forcing relation that is not re (cf. [8, 270) but is effective. Most likely there is one by the arguments of Sections 4 and 5. Farewell to higher recursion theory (but not to recursion theory; there is no way to say goodbye to recursion theory). References [1] K. Devlin, Constnictibility (Springer, Berlin, 1986). [2] L. Harrington, Contributions to recursion theory in higher types, Ph.D. Thesis, MIT, Cambridge, MA, 1973. [3] S. Marcus, Forcing in f-recursion, Ph.D. Thesis, MIT, Cambridge, MA, 1993. [4] D. Normann, Set recursion, in: Generalized Recursion Theory II (North-Holland, Amsterdam, 1978) 303-320. [5] G.E. Sacks, The i-section of a type n object, Amer. J. Math. 99 (1977) 901-917. [6] G.E. Sacks, Post's problem, absoluteness and recursion in finite types, in: The KJeene Symposium (NorthHolland, Amsterdam, 1980) 201-222. [7] G.E. Sacks, On the limits of recursive enumerability, Ann. Pure and App. Logic 31 (1986) 87-120. [8] G.E. Sacks, Higher Recursion Theory (Springer, Berlin, 1990). [9] T. Slaman, Aspects of £-recursion, Ph.D. Thesis, Harvard Univ., Cambridge, MA, 1981.