E-Recursion, Forcing and C*-Algebras

E -RECURSION, FORCING AND C*-ALGEBRAS

9154_9789814602631_tp.indd 1

30/4/14 3:41 pm

LECTURE  NOTES  SERIES Institute for Mathematical Sciences, National University of Singapore Series Editors: Chitat Chong and Wing Keung To Institute for Mathematical Sciences National University of Singapore ISSN: 1793-0758 Published Vol. 17 Interface Problems and Methods in Biological and Physical Flows edited by Boo Cheong Khoo, Zhilin Li & Ping Lin Vol. 18 Random Matrix Theory and Its Applications: Multivariate Statistics and Wireless Communications edited by Zhidong Bai, Yang Chen & Ying-Chang Liang Vol. 19

Braids: Introductory Lectures on Braids, Configurations and Their Applications edited by A Jon Berrick, Frederick R Cohen, Elizabeth Hanbury, Yan-Loi Wong & Jie Wu

Vol. 20 Mathematical Horizons for Quantum Physics edited by H Araki, B-G Englert, L-C Kwek & J Suzuki Vol. 21 Environmental Hazards: The Fluid Dynamics and Geophysics of Extreme Events edited by H K Moffatt & E Shuckburgh Vol. 22 Multiscale Modeling and Analysis for Materials Simulation edited by Weizhu Bao & Qiang Du Vol. 23 Geometry, Topology and Dynamics of Character Varieties edited by W Goldman, C Series & S P Tan Vol. 24 Complex Quantum Systems: Analysis of Large Coulomb Systems edited by Heinz Siedentop Vol. 25 Infinity and Truth edited by Chitat Chong, Qi Feng, Theodore A Slaman & W Hugh Woodin Vol. 26 Notes on Forcing Axioms by Stevo Todorcevic, edited by Chitat Chong, Qi Feng, Theodore A Slaman, W Hugh Woodin & Yue Yang Vol. 27 E-Recursion, Forcing and C*-Algebras edited by Chitat Chong, Qi Feng, Theodore A. Slaman, W. Hugh Woodin & Yue Yang

*For the complete list of titles in this series, please go to http://www.worldscientific.com/series/LNIMSNUS

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore

Vol.

27

E -RECURSION, FORCING AND C*-ALGEBRAS Editors

Chitat Chong

National University of Singapore, Singapore

Qi Feng

Chinese Academy of Sciences, China

Theodore A Slaman University of California, Berkeley, USA

W Hugh Woodin Harvard University, USA

Yue Yang

National University of Singapore, Singapore

World Scientific NEW JERSEY

•

LONDON

9154_9789814602631_tp.indd 2

•

SINGAPORE

•

BEIJING

•

SHANGHAI

•

HONG KONG

•

TA I P E I

•

CHENNAI

30/4/14 3:41 pm

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore — Vol. 27 E-RECURSION,  FORCING  AND  C*-ALGEBRAS Copyright © 2014 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 or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-4602-63-1 ISBN 978-981-4603-25-6 (pbk)

Printed in Singapore

April 7, 2014

17:18

WSPC 9 x 6: BC9154

front-matter

CONTENTS

Foreword

vii

Preface

ix

Selected Applications of Logic to Classification Problem for C*-Algebras Ilijas Farah Subcomplete Forcing and L-Forcing Ronald Jensen

1

83

E-Recursion 2012 Gerald E. Sacks

183

v

page v

May 2, 2013

14:6

BC: 8831 - Probability and Statistical Theory

This page intentionally left blank

PST˙ws

April 7, 2014

17:18

WSPC 9 x 6: BC9154

front-matter

FOREWORD

The Institute for Mathematical Sciences (IMS) at the National University of Singapore was established on 1 July 2000. Its mission is to foster mathematical research, both fundamental and multidisciplinary, particularly research that links mathematics to other efforts of human endeavor, and to nurture the growth of mathematical talent and expertise in research scientists, as well as to serve as a platform for research interaction between scientists in Singapore and the international scientific community. The Institute organizes thematic programs of longer duration and mathematical activities including workshops and public lectures. The program or workshop themes are selected from among areas at the forefront of current research in the mathematical sciences and their applications. Each volume of the IMS Lecture Notes Series is a compendium of papers based on lectures or tutorials delivered at a program/workshop. It brings to the international research community original results or expository articles on a subject of current interest. These volumes also serve as a record of activities that took place at the IMS. We hope that through the regular publication of these Lecture Notes the Institute will achieve, in part, its objective of reaching out to the community of scholars in the promotion of research in the mathematical sciences. January 2014

Chitat Chong Wing Keung To Series Editors

vii

page vii

May 2, 2013

14:6

BC: 8831 - Probability and Statistical Theory

This page intentionally left blank

PST˙ws

April 29, 2014

16:38

WSPC 9 x 6: BC9154

front-matter

PREFACE

The series of Asian Initiative for Infinity (AII) Graduate Logic Summer School was held annually from 2010 to 2012. The lecturers were Moti Gitik, Denis Hirschfeldt and Menachem Magidor in 2010, Richard Shore, Theodore A. Slaman, John Steel, and W. Hugh Woodin in 2011, and Ilijas Farah, Ronald Jensen, Gerald E. Sacks and Stevo Todorcevic in 2012. In all, more than 150 graduate students from Asia, Europe and North America attended the summer schools. In addition, two postdoctoral fellows were appointed during each of the three summer schools. These volumes of lecture notes serve as a record of the AII activities that took place during this period. The AII summer schools was funded by a grant from the John Templeton Foundation and partially supported by the National University of Singapore. Their generosity is gratefully acknowledged. May 2014 Chitat Chong National University of Singapore, Singapore Qi Feng Chinese Academy of Sciences, China Theodore A. Slaman University of California, Berkeley, USA W. Hugh Woodin Harvard University, USA Yue Yang National University of Singapore, Singapore Volume Editors

ix

page ix

May 2, 2013

14:6

BC: 8831 - Probability and Statistical Theory

This page intentionally left blank

PST˙ws

March 11, 2014

10:36

WSPC 9 x 6

singapore-final-r

SELECTED APPLICATIONS OF LOGIC TO CLASSIFICATION PROBLEM FOR C*-ALGEBRASa

Ilijas Farah Department of Mathematics and Statistics York University, 4700 Keele Street North York, Ontario, Canada M3J 1P3 and Matematicki Institut, Kneza Mihaila 34, Belgrade, Serbia [email protected] http://www.math.yorku.ca/∼ifarah Basics of Elliott’s classification program are outlined and juxtaposed with the abstract classification theory from descriptive set theory. Some preliminary estimates on the complexity of the isomorphism relation of separable C*-algebras are given.

Contents 0 Introduction 1 Operators on Hilbert Spaces 1.1 Subspaces and subalgebras of B(H) 1.1.1 Operator space 1.1.2 Operator system 1.1.3 Concrete C*-algebras 1.1.4 Non-self-adjoint subalgebras 1.1.5 von Neumann algebras 1.2 Exercises 1.3 Spectrum and spectral radius 1.4 Exercises 1.5 Normal operators and the spectral theorem 1.5.1 Types of bounded operators 1.6 Exercises 2 Preliminaries on C*-Algebras 2.1 Positivity, states and the GNS theorem a Partially

supported by NSERC. 1

3 4 6 6 7 7 7 7 7 9 9 10 11 11 12 12

page 1

March 11, 2014

10:36

2

4

5

6

singapore-final-r

I. Farah

2.2 2.3 2.4 2.5

3

WSPC 9 x 6

Exercises Continuous functional calculus Exercises Constructions of C*-algebras 2.5.1 Unitization 2.5.2 Direct sums 2.5.3 Direct products 2.5.4 Direct limits (also called inductive limits) 2.5.5 Matrix algebra over A 2.5.6 Stabilization 2.5.7 Minimal tensor product 2.5.8 Continuous fields of C*-algebras 2.5.9 Corners 2.5.10 . . . and so on 2.6 Exercises Local Theory of C*-Algebras 3.1 Polar decomposition 3.2 Stability 3.3 Exercises 3.4 Murray–von Neumann equivalence of projections 3.5 Exercises 3.6 Traces 3.7 Exercises UHF Algebras and AF Algebras 4.1 UHF algebras 4.2 Another look at the UHF algebras 4.3 Exercises 4.4 Bratteli diagrams 4.5 Exercises 4.6 AF algebras 4.7 Exercises The Functor K0 5.1 Computation of K0 in some simple cases 5.1.1 K0 of Mn (C) 5.1.2 K0 of B(H) 5.1.3 K0 of the Calkin algebra 5.1.4 K0 of the CAR algebra 5.1.5 K0 of other UHF algebras 5.1.6 K0 of a *-homomorphism 5.2 Exercises 5.3 Cancellation property 5.4 Classification of AF algebras 5.5 Exercises Elliott’s Program 6.0.1 Failure of cancellation 6.1 Exercises

13 14 17 18 18 18 18 19 19 19 20 21 21 21 21 23 23 24 26 26 28 29 32 32 32 34 35 36 38 40 40 42 43 43 43 43 43 43 44 44 44 46 48 49 51 51

page 2

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

7 Abstract Classification 7.1 Effros Borel space 7.1.1 Spaces of countable structures 7.1.2 Compact metric spaces 7.1.3 Separable Banach spaces 7.1.4 von Neumann algebras with a separable predual 7.1.5 Separable C*-algebras 7.2 Computation of the Elliott invariant is Borel 7.3 Comparing complexities of analytic equivalence relations 7.3.1 Relation E0 7.3.2 Essentially countable equivalence relations 7.3.3 Countable structures 7.3.4 Orbit equivalence relations 7.3.5 Turbulence 7.3.6 The dark side 7.4 Exercises 8 Estimating the Complexity of the Isomorphism of C*-Algebras 8.1 Turbulence: A lower bound for complexity 8.2 Below a group action: An upper bound for complexity 8.3 Cuntz algebra O2 8.4 Exercises 9 Concluding Remarks 9.1 The Borel-reducibility diagram 9.2 Selected open problems References

singapore-final-r

3

53 55 55 55 56 57 57 58 59 59 59 59 59 60 60 60 61 61 67 67 70 71 71 74 76

0. Introduction In recent years we have witnessed a number of applications of set theory to operator algebras. In the present notes I will focus on one very specific (yet particularly exciting) aspect of this development, applications of descriptive set theory (more precisely, theory of abstract classification) to Elliott’s classification program for nuclear C*-algebras. These lecture notes contain a plenty of exercises, many (but not all) of them fairly straightforward. They are meant as a bridge towards more advanced literature on classification of C*-algebras, a subject with an abundance of excellent literature. Hints to exercises often refer to the material covered at a later point in the notes and thus provide additional motivation for the introduced notions. The initial parts of the lecture notes, and §2 in particular, are very sketchy and are meant to outline the main ideas and provide a guide to the literature rather than to serve as a textbook. The basic premise is that every classical classification program in mathematics deals with analytic equivalence relations on Polish spaces. Moreover,

page 3

April 22, 2014

16:39

4

WSPC 9 x 6

singapore-final-r

I. Farah

classification invariants are usually coded by elements of a Polish space and the computation of invariants is given by a Borel-measurable function. I shall always start counting at zero. Therefore i < n means that i assumes n distinct values: 0, 1, . . . , n − 1. Also, an a ∈ Mn (C) is identified with its matrix entries aij for i < n, j < n. Suggested references For functional analysis, [64]. For the theory of C*-algebras, [62] and [12], and [68] for their K-theory. General theory of operator algebras is excellently surveyed in [4] and standard reference for the Elliott program as of 2002 is [70]. Very detailed notes on topics in operator theory starting with simple (partial isometries) to very advanced can be found at [76]. Classical reference for classical descriptive set theory is [48]. Somewhat dated surveys of some other applications of set theory to C*-algebras are given in [84] and [34]. Papers [33] and [32] contain more details and proofs of the results presented in §7 and §8.2. Survey of the latest developments in Elliott’s program will be available shortly in [88]. Acknowledgments These notes are based on two series of lectures given in 2012. At the Luminy Young Set Theory workshop in April 2012 I covered sections dealing with general theory of C*-algebras and Elliott’s classification of AF algebras by K-theoretic invariants. At the Asian Initiative for Infinity course in addition I covered this in more detail, and presented the abstract classification viewpoint of Elliott’s program. I am indebted to the organizers of both of these meetings for inviting me. I would like to thank to Asger T¨ornquist for kindly permitting me to use his writeup of Lemma 8.6, taken from the appendix of the original version of [33]. Finally, I would like to thank John Campbell, Boris Kadets, Vladislav Kalashnyk and Jiewon Park for noticing several mistakes in an earlier version of these notes. Last, but not least, I am indebted to Lai Fun Kwong from World Scientific for her outstanding help with editing the final version of these notes.

1. Operators on Hilbert Spaces We begin with a review of the basic properties of operators on a Hilbert space. Throughout we let H denote a complex infinite-dimensional separable Hilbert space, and we let (en ) be an orthonormal basis for H (see Example 1.1). For ξ, η ∈ H, we denote their inner product by (ξ|η). We

page 4

March 11, 2014

10:36

WSPC 9 x 6

singapore-final-r

Logic for C*-Algebras

5

recall that (η|ξ) = (ξ|η) and the norm defined by kξk =

p

(ξ|ξ).

The Cauchy–Schwartz inequality says that |(ξ|η)| ≤ kξkkηk. Example 1.1. The space n o X ℓ2 (N) = (αk )k∈N : αk ∈ C, kαk2 = |αk |2 < ∞ (sometimes denoted simply by ℓ2 ) is a Hilbert space under the inner product P (α|β) = αk βk . If we define en ∈ ℓ2 (N) by enk = δnk (the Kronecker’s δ), P n then (e ) is an orthonormal basis for ℓ2 . For any α ∈ ℓ2 , α = αn en .

Any Hilbert space has an orthonormal basis, and this can be used to prove that all separable infinite-dimensional Hilbert spaces are isomorphic. Moreover, any two infinite-dimensional Hilbert spaces with the same density character (the minimal cardinality of a dense subset) are isomorphic. Example 1.2. If (X, µ) is a measure space,   Z 2 L (X, µ) = f : X → C measurable : |f | dµ < ∞ /{f : f = 0 a.e.} 2

is a Hilbert space under Rthe inner product (f |g) = norm defined by kf k2 = |f |2 dµ.

R

f gdµ and with the

We will let a, b, . . . denote linear operators H → H. We recall that kak = sup{kaξk : ξ ∈ H, kξk = 1}.

page 5

March 11, 2014

10:36

WSPC 9 x 6

6

singapore-final-r

I. Farah

If kak < ∞, we say a is bounded. An operator is bounded if and only if it is continuous. We denote the algebra of all bounded operators on H by B(H) (some authors use L(H)), and throughout the paper all of our operators will be bounded. We define the adjoint a∗ of a to be the unique operator satisfying (aξ|η) = (ξ|a∗ η) for all ξ, η ∈ H. Note that since an element of H is determined by its inner products with all other elements of H (e.g., take an orthonormal basis), an operator a is determined by the values of (aξ|η) for all ξ, η or even by the values (aem |en ) for m and n in N. Lemma 1.3. For all a, b in B(H) we have (1) (2) (3) (4) (5)

(a∗ )∗ = a (ab)∗ = b∗ a∗ kak = ka∗ k kabk ≤ kak · kbk ka∗ ak = kak2

Proof. These are all easy calculations. For example, for (5), for kξk = 1, kaξk2 = (aξ|aξ) = (ξ|a∗ aξ) ≤ kξk · ka∗ aξk ≤ ka∗ ak, the first inequality holding by Cauchy–Schwartz. Taking the sup over all ξ, we obtain kak2 ≤ ka∗ ak. Conversely, ka∗ ak ≤ ka∗ kkak = kak2 by (3) and (4). Entries (1)–(3) state that B(H) is a Banach *-algebra (or a Banach algebra with involution * ) and (5) is sometimes called the C*-equality. 1.1. Subspaces and subalgebras of B(H) Before focusing on C*-algebras, I shall list some other structures based on B(H). Each one of these categories is replete with possible applications of logic. 1.1.1. Operator space is a closed linear subspace of B(H). In addition to its Banach space structure, an operator space X is considered with the Banach space structure on Mn (X), n × n matrices of elements of X identified with operators from H n to H n with respect to the operator norm. Morphisms

page 6

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

7

in this category are completely bounded maps—linear maps whose canonical extension to Mn (X) is bounded for all n. The standard reference on operator spaces is [17]. 1.1.2. Operator system is an operator space that is in addition closed under the adjoint. Morphisms are completely positive maps—linear maps that preserve positivity of operators and matrices. They are indispensable in the theory of C*-algebras, and more on this can be found in [63] and [5]. 1.1.3. Concrete C*-algebras are in addition closed under multiplication of operators. Morphisms are *-homomorphisms—maps that preserve all algebraic structure (+, · and ∗ ). Remarkably, such maps are automatically completely positive and completely contractive. We import the terminology from §1.5.1 wholesale and talk about operators in a C*-algebra that are normal, self-adjoint, positive, projections, etc. This will also apply to abstract C*-algebras once they are introduced in Definition 2.2. 1.1.4. Non-self-adjoint subalgebras are norm-closed and closed under + and ·, but not necessarily closed under ∗ . See [66] for more on this subject. 1.1.5. von Neumann algebras are C*-algebras that are closed in weak operator topology. An excellent introductory source is [46]. Theories of these categories have many connections between each other. For example, the study of tensor products of C*-algebras and their finitedimensional approximation is deeply steeped in the study of operator systems. In the present lecture notes I shall focus on C*-algebras. 1.2. Exercises 1.2.1. Prove that for all ξ, η ∈ H the following so-called polarization iden√ tity holds (note that i = −1) 3

(ξ|η) =

1X k i (ξ + ik η|ξ + ik η). 4 k=0

The right-hand side is a linear combination of norms of vectors ξ + ik η for 0 ≤ k ≤ 3. Consequentially, the scalar product on H is uniquely determined by its Hilbert space norm. The dimension of a Hilbert space is the least cardinality of an orthonormal basis.

page 7

March 31, 2014

8

18:14

WSPC 9 x 6

singapore-final-r

I. Farah

1.2.2. All orthonormal bases in a fixed Hilbert space have the same cardinality. Two complex Hilbert spaces are isomorphic if and only if they have the same dimension. Prove the following statements. 1.2.3. Prove that a Hilbert space is separable if and only if it has a finite or countable orthonormal basis. 1.2.4 (Inner automorphisms). Assume u is a linear isometry from H onto itself. Prove that Ad u : B(H) → B(H) defined by (Ad u)a = uau∗ is an automorphism of C*-algebra B(H). 1.2.5. If u : H1 → H2 is an isomorphism between Hilbert spaces, then (Ad u)a = uau∗ is an isomorphism between B(H1 ) and B(H2 ). The operator Ad u(a) is just a with its domain and range identified with H2 via u. 1.2.6. An operator u in a unital C*-algebra is a unitary if uu∗ = u∗ u = 1. Prove that Ad u is an automorphism of A. 1.2.7. Prove that all automorphisms of B(H) are inner. (Hint 1: As a warmup prove that all automorphisms of the Boolean algebra P(N) are “trivial.” You first have to replace the quotation marks with an appropriate definition of “trivial.” Hint 2 (for a better proof): Fix a unit vector ξ and let η be a unit vector such that Φ(pξ ) = pη (here pξ denotes the projection to the subspace spanned by ξ). With vξ,ζ denoting the rank 1 linear map sending ξ to ζ, let u(α) = β, where β is the unique vector such that Φ(vξ,α ) = vη,β . A *-polynomial is a term in the language {+, ·, ∗}. If P (x) is a *polynomial and a is an operator then P (a) is naturally defined (again, logic for metric structures provides the right setting for this; see [29]). For a set of operators X in B(H) or in a C*-algebra A let C ∗ (X) denote the C*-algebra generated by X. If X = {x1 , . . . , xn } is finite we may write C ∗ (x1 , . . . , xn ) instead of C ∗ (X). 1.2.8. Prove that C ∗ (X) is the norm-closure of {P (¯ a) : P (¯ x) is a *polynomial in n (non-commuting) variables with complex coefficients and a ¯ is an n-tuple in X for n ∈ N}. 1.2.9. Prove that C ∗ (a) is abelian if and only if a is normal.

page 8

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

9

An ideal in a C*-algebra is a two-sided, norm closed, self-adjoint ideal (there is some redundancy in this definition). 1.2.10. Prove that an ideal generated by an element a of a C*-algebra is equal to the closure of the linear span of {bac : b, c ∈ A}. 1.3. Spectrum and spectral radius We start with a useful lemma. Lemma 1.4. If kak < 1 then 1 − a is invertible in B(H). P n Proof. The series b = ∞ n=0 a is convergent and hence in B(H). By considering partial sums one sees that (1 − a)b = b(1 − a) = 1. The spectrum of an operator a in B(H) is sp(a) = {λ ∈ C : a − λI is not invertible}. The spectrum of a bounded linear operator is always a compact subset of C (Exercise 1.4.4), and it is moreover always nonempty (the latter fact follows from a clever use of Liouville’s theorem; see e.g., [64, Theorem 4.1.13]). Also, for a normal operator a we have that kak = r(a) (Exercise 1.4.8), where r(a) is the spectral radius of a defined as follows: r(a) = max{|λ| : λ ∈ sp(a)}. 1.4. Exercises 1.4.1. Prove that the spectrum of a finite-dimensional matrix is equal to the set of its eigenvalues. 1.4.2. Prove that the invertible elements form an open subset of B(H). (Hint: If ab = 1 = ba, find an ε > 0 such that kb − ck < ε implies kac − 1k < 1 and kca − 1k < 1. Then apply Lemma 1.4.) 1.4.3. Prove that sp(a) ⊆ {λ : |λ| ≤ kak} for all a. 1.4.4. Prove that sp(a) is compact for all a. (Hint: Use Exercise 1.4.2 and Exercise 1.4.3.) 1.4.5. Prove that kak ≥ r(a) for all a. 1.4.6. Find an example of a nonzero a such that sp(a) = {0}, and hence kak > r(a). (Hint: A well-chosen 2 × 2 matrix would do.)

page 9

March 11, 2014

10

10:36

WSPC 9 x 6

singapore-final-r

I. Farah

1.4.7. Prove that limn kan k1/n exists and is equal to r(a). (Hint: See [64, Lemma 4.1.13].) 1.4.8. Assume a is normal. Prove kak = r(a). n n (Hint: First prove that ka2 k = kak2 and then use Exercise 1.4.7. Prove this equality in the self-adjoint case first, using the C*-equality.) 1.4.9. Show that if λ ∈ / sp(a) then k(a − λ · 1)−1 k ≤ 1/ dist(λ, sp(a)). 1.4.10. Find a normal operator on a Hilbert space that has no eigenvectors. (Hint: Example 1.5.) 1.5. Normal operators and the spectral theorem In this section we introduce some distinguished classes of operators in B(H), such as normal and self-adjoint operators (cf. §1.5.1). Example 1.5. Assume (X, µ) is a probability measure space. If H0 = L2 (X, µ) and f : X → C is bounded and measurable, then mf

H0 ∋ g 7−→ f g ∈ H0 is a bounded linear operator. We have kmf k = kf k∞ and m∗f = mf¯. Hence m∗f mf = mf m∗f = m|f |2 . We call operators of this form multiplication operators. Recall that an operator a is normal if aa∗ = a∗ a. Clearly, all multiplication operators are normal. When H is a complex Hilbert space, normal operators have a nice structure theory. It is summarized in the following theorem, stated a bit prematurely since its proof involves Proposition 2.5. Theorem 1.6 (Spectral Theorem). If a is a normal operator then there is a probability measure space (X, µ), a measurable function f on X, and a Hilbert space isomorphism Φ : L2 (X, µ) → H such that ΦaΦ−1 = mf . Proof. For a proof see e.g., [2, Theorem 2.4.5]. Therefore every normal operator is a multiplication operator for some identification of H with an L2 space. Conversely, every multiplication operator is clearly normal. If X is discrete and µ is the counting measure, the characteristic functions of the points of X form an orthonormal basis for L2 (X, µ) and the spectral theorem says that a is diagonalized by

page 10

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

11

this basis. In general, the spectral theorem says that normal operators are “measurably diagonalizable”. An operator a is self-adjoint if a = a∗ . Self-adjoint operators are obviously normal. For any b ∈ B(H), the “real” and “imaginary” parts of b, defined by b0 = (b + b∗ )/2 and b1 = (b − b∗ )/2i, are self-adjoint and satisfy b = b0 + ib1 . Thus any operator is a linear combination of two self-adjoint operators. It is easy to check that an operator is normal if and only if its real and imaginary parts commute, so the normal operators are exactly the linear combinations of commuting self-adjoint operators. 1.5.1. Types of bounded operators The following distinguished classes of operators will play an important role. (1) (2) (3) (4) (5) (6) (7) (8)

a a a a a a a a

is is is is is is is is

normal if aa∗ = a∗ a, self-adjoint (or Hermitian) if a = a∗ , positive if a = b∗ b for some b ∈ H, a projection if a2 = a∗ = a, positive (or a ≥ 0) if a = b∗ b for some b, unitary if aa∗ = a∗ a = 1, an isometry if aa∗ is a projection and a∗ a = 1. partial isometry if both aa∗ and a∗ a are projections.

Note that a positive element is automatically self-adjoint. For self-adjoint elements a and b write a ≤ b if b − a is positive. Any complex number z can be written as z = reiθ for r ≥ 0 and |eiθ | = 1. Considering C as the set of operators on a one-dimensional Hilbert space, there is an analogue of this on an arbitrary Hilbert space. Theorem 1.7 (Polar Decomposition). Any a ∈ B(H) can be written as a = bv where b is positive and v is a partial isometry. Moreover, v can be chosen so that ker(v) = ker(a). This additional requirement makes v unique. Proof. See [64], but apply it to a∗ instead. 1.6. Exercises 1.6.1. The real and imaginary parts of a multiplication operator mf are mℜf and mℑf . A multiplication operator mf is self-adjoint if and only if f is real (a.e.). By the spectral theorem, all self-adjoint operators are of this form up to the unitary equivalence. 1.6.2. Two projections p and q commute if and only if pq is a projection.

page 11

March 11, 2014

10:36

12

WSPC 9 x 6

singapore-final-r

I. Farah

1.6.3. For projections p and q write p ≤ q if pq = p (equivalently, qp = p). Prove the following. (1) (2) (3) (4)

p ≤ q implies that p and q commute. This is a partial ordering on any set of projections. p ≤ q if and only if the self-adjoint operator q − p is positive. p ≤ q and kp − qk < 1 implies p = q.

1.6.4. If p and q are projections such that pq = q and (1 − p)(1 − q) = 1 − q then p = q. 1.6.5. Every self-adjoint unitary is of the form u = 1−2p for a projection p. 2. Preliminaries on C*-Algebras 2.1. Positivity, states and the GNS theorem Let X be a locally compact Hausdorff space. Recall that C0 (X) denotes the space of continuous complex-valued functions on X such that for every ε > 0 the set {x ∈ X : |f (x)| ≥ ε} is compact. It is considered as a Banach algebra with respect to +, · and adjoint defined as pointwise conjugation. If X is compact then we write C(X). By the following remarkable result, these algebras are exactly the abstract abelian C*-algebras. Theorem 2.1 (Gelfand–Naimark). Every abelian C*-algebra is isomorphic to C0 (X) for a unique locally compact Hausdorff space X. The algebra is unital if and only if X is compact. Space X is equal to the space of characters on A (see Exercise 2.2.2). A proof of this theorem can be found in e.g., [64] or [2]. In fact, the Gelfand– Naimark theorem is functorial: the category of abelian C*-algebras is contravariantly isomorphic to the category of locally compact Hausdorff spaces (cf. Exercise 2.2.6). Definition 2.2. An abstract C*-algebra is a Banach algebra satisfying the conclusion of Lemma 1.3. Concrete C*-algebras were introduced in §1.1.3. By the definition every concrete C*-algebra is an abstract C*-algebra. The fact that the converse is true can hardly be overestimated. Theorem 2.3 (Gelfand–Naimark–Segal). Every abstract C*-algebra A is isomorphic to a concrete C*-algebra.

page 12

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

13

From now on we will usually refer to abstract C*-algebras as C*algebras. An occasional concrete C*-algebra will also be treated in the abstract way, independently from its actual representation on B(H). To a logician, Theorem 2.3 states that C*-algebras form an axiomatizable class in an appropriately chosen logic. This fact was made precise and taken advantage of in [28] and [29]. 2.2. Exercises 2.2.1. Check the easy direction of Theorem 2.3. In the following three exercises we define the Gelfand transform and give a (very rough!) outline of the proof of Gelfand–Naimark theorem. Character of a C*-algebra A is a *-homomorphism φ : A → C. Let Aˆ denote the set of characters of A, and note that Aˆ = {0} for many C*-algebras A (e.g., all matrix algebras). 2.2.2. Prove that every character is continuous and has norm ≤ 1, and that Aˆ is a weak*-compact subset of A∗ . (Hint: For the first part one needs to check that the kernel is closed. For this apply Lemma 1.4. In the second part, by Alaoglu’s theorem (the unit ball of A∗ is weak*-compact) one only needs to check that the characters form a closed subset of the unit ball of A∗ .) 2.2.3. If A is an abelian C*-algebra then Aˆ is equal to the set of pure states of A. (Hint: This is a consequence of the Riesz Representation Theorem. See Exercise 3.17.) 2.2.4. If X ⊆ C is compact, then C(X) ∼ = C ∗ (ιX , 1), where ιX is the identity function on X and 1 is the constantly one function. (Hint: Stone–Weierstrass.) ˆ by 2.2.5. If A is a C*-algebra define the map Γ : A → C(A) Γ(a)(φ) = φ(a). Show that Γ is a *-homomomorphism. If A is moreover abelian, show that Γ is an isometric isomorphism. (Hint: For the second part, combine Exercise 2.2.3 and Exercise 1.4.8.) 2.2.6. Assume X and Y are compact Hausdorff spaces and Φ : C(X) → C(Y ) is a *-homomorphism.

page 13

March 31, 2014

14

18:14

WSPC 9 x 6

singapore-final-r

I. Farah

(1) Prove that there exists a unique continuous f : Y → X such that Φ(a) = a ◦ f for all a ∈ C(X). (2) Prove that Φ is a surjection if and only if f is an injection. (3) Prove that Φ is an injection if and only if f is a surjection. (4) Prove that for every f : Y → X there exists a unique Φ : C(X) → C(Y ) such that (1)–(3) above hold. Let A be a C*-algebra. A continuous linear functional φ : A → C is positive if φ(a) ≥ 0 for all positive a ∈ A. It is a state if it is positive and of norm 1. We denote the space of all states on A by S(A). 2.2.7. If ξ ∈ H is a unit vector, define a functional ωξ on B(H) by ωξ (a) = (aξ|ξ). Then ωξ (a) ≥ 0 for a positive a and ωξ (I) = 1; hence it is a state. We call a state of this form a vector state. 2.2.8. If A is unital then S(A) = {φ ∈ A∗ : φ = 1 = φ(1)}. 2.2.9. Prove that if A is a unital subalgebra of B then all states of A extend to states of B. (Hint: Exercise 2.2.8.) In the following X∗ denotes the Banach space dual of Banach space X. 2.2.10. Assume Φ : A → B is a unital *-homomorphism. Define Φ∗ : B ∗ → A∗ via Φ∗ (ψ) = ψ : Φ. (1) Prove that Φ∗ maps S(B) into S(A). (2) Prove that Φ∗ maps T (B) into T (A) (the definition of a trace and T (A) is given in §3.6). (3) Prove that Φ∗ is injective if and only if Φ is surjective. (4) Prove that Φ∗ is surjective if and only if Φ is injective. 2.3. Continuous functional calculus We are about to introduce one of the key tools in the theory of C*algebras building on Gelfand–Naimark theorem in Proposition 2.5 below. Following §1.3, a spectrum spA (a) of an element a of an arbitrary unital C*-algebra A can be defined as spA (a) = {λ ∈ C|a − λ · 1 is not invertible in A}. The reason one this notation is not in standard usage is contained in Lemma 2.6 below. Let us first prove its special case.

page 14

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

15

Lemma 2.4. Let a be a normal element in a unital C*-algebra A and let B = C ∗ (a, I) (the algebra generated by a and the identity). Then spA (a) = spB (a). Proof. Since B is a subalgebra of A, we have that spB (a) ⊇ spA (a). In order to show the converse inclusion, we need to show that an operator b ∈ B that is not invertible in B is not invertible in A. Assume the contrary, and fix a b ∈ B which is not invertible in B but has an inverse d in A. By Theorem 2.1, B is isomorphic to C(X) for a compact Hausdorff space X, and we can identify b with a function f on X. Since f is not invertible, we have f (x) = 0 for some x ∈ X. Fix ε > 0 pick an open neighbourhood U of x such that |f (y)| < ε for all y ∈ U . Now let g ∈ C(X) be a continuous function such that g(x) = 1, 0 ≤ g(y) ≤ 1 for all y and g(z) = 0 for z ∈ / U . Let c ∈ B correspond to g. Then kcbdk = kck = 1. On the other hand, kcbk = maxy∈X |g(y)f (y)k ≤ ε. Therefore, kdk ≥ 1/ε. Since ε was arbitrarily small and did not depend on d, this is a contradiction. Proposition 2.5. If a ∈ B(H) is normal then C(sp(a)) ∼ = C ∗ (a, I). The isomorphism sends function f ∈ C(sp(a)) to f (a), where f (a) is defined naturally in case when f is a *-polynomial. Proof. By Lemma 2.4 it suffices to prove that C ∗ (a, I) is isomorphic to C(sp0 (a)), where sp0 (a) denotes the spectrum of a as defined in C ∗ (a, I). Let X be a compact Hausdorff space such that C ∗ (a, I) ∼ = C(X), as guaranteed by Gelfand–Naimark theorem. For any λ ∈ sp(a), a − λ · 1 is not invertible so there exists φλ ∈ X such that φλ (a − λ · 1) = 0, or φλ (a) = λ. Conversely, if there is φ ∈ X such that φ(a) = λ, then φ(a − λ · 1) = 0 so λ ∈ sp(a). Since any nonzero homomorphism to C is unital, an element φ ∈ X is determined entirely by φ(a). Since X has the weak* topology, φ 7→ φ(a) is thus a continuous bijection from X to sp(a), which is a homeomorphism since X is compact. Lemma 2.6. Suppose A is a unital subalgebra of B and a ∈ A is normal. Then spA (a) = spB (a), where spA (a) and spB (a) denote the spectra of a as an element of A and B, respectively. Proof. Since an element invertible in the smaller algebra is clearly invertible in the larger algebra, we have that spB (a) ⊆ spA (a) and we only need to check that spA (a) ⊆ spB (a). Pick λ ∈ spA (a). We need to prove that a − λ · 1 is not invertible in B. Assume the contrary and let b be the inverse

page 15

March 11, 2014

10:36

WSPC 9 x 6

16

singapore-final-r

I. Farah

of a − λ · 1. Fix ε > 0 and let U ⊆ spA (a) be the open ball around λ od radius ε. Let g ∈ C(spA (a)) be a function supported by U such that kgk = 1. Then g = b(a − λ · 1)g, hence kb(a − λ · 1)gk = 1. On the other hand, (a − λ · 1)g = f ∈ C(spA (a)) so that f vanishes outside of U and kf (x)k < ε for x ∈ U , hence k(a − λ · 1)gk < ε. Thus kbk > 1/ε for every ε > 0, a contradiction. Note that the isomorphism defined in Proposition 2.5 is canonical and maps a to the identity function on sp(a). It follows that for any polynomial p, the isomorphism maps p(a) to the function z 7→ p(z). More generally, for any continuous function f : sp(a) → C, we can then define f (a) ∈ C ∗ (a, I) as the preimage of f under the isomorphism. For example, we can define |a| and if a is self-adjoint then it can be written as a difference of two positive operators as |a| + a |a| − a − . 2 2 √ If a ≥ 0, then we can also define a. Lemma 2.7 is an another application of the “continuous functional calculus” of Corollary 2.5. A remark about terminology is in order. It is customary among C*-algebraists to call 1-Lipshitz maps contractions. Recall that a map Φ is Lipshitz if d(Φ(x), Φ(y)) ≤ d(x, y) for all x and y. Although this terminology makes various fixed-point theorems falseb , I shall use it in order to be compatible with the standard terminology. a=

Lemma 2.7. Any *-homomorphism Φ : A → B between C*-algebras is a contraction (in particular, it is continuous). Therefore, any (algebraic) isomorphism between C*-algebras is an isometry. Proof. By passing to the unitizations, we may assume A and B are unital and Φ is unital as well (i.e., Φ(IA ) = 1B ). Note that for any a ∈ A, sp(Φ(a)) ⊆ sp(a) (by the definition of the spectrum). Thus for a normal using Exercise 1.4.8 we have kak = sup{|λ| : λ ∈ σ(a)}

≥ sup{|λ| : λ ∈ σ(Φ(a))} = kΦ(a)k.

b Outside

of the theory of operator algebras, contractions are usually required to strictly decrease the distance.

page 16

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

17

For general a, aa∗ is normal so by the C*-equality we have p p kak = kaa∗ k ≥ kΦ(aa∗ )k = kΦ(a)k, concluding the proof.

For subsets K and L of a metric space (X, d) and ε > 0 we write K ⊆ε L if and only if inf y∈L d(x, y) ≤ ε for all x ∈ K. Note that X and d are implicit in this notation. Lemma 2.8. Assume a and b are normal and ka − bk ≤ ε. Then sp(a) ⊆ε sp(b) and sp(b) ⊆e sp(a). Proof. It suffices to prove that for an arbitrary λ ∈ C such that dist(λ, sp(a)) > ε we have λ ∈ / sp(b). Fix such λ and let c = (a − λ · 1)−1 . By Exercise 1.4.9 we have that kck < 1/ε. Then c(b − λ · 1) = c(a − λ · 1) − c(a − b) = 1 − c(a − b). The righthand side is invertible by Lemma 1.4, and therefore b − λ · 1 is invertible as well. The following slightly amusing remark can be safely ignored. Consider K(C), the space of compact subsets of C, as a Polish space with respect to the Hausdorff distance dH (K, L) = max{inf{ε : K ⊆ε L and L ⊆e K}. Lemma 2.8 states that the map a 7→ sp(a) from normal operators in A into K(C) is a contraction (cf. the paragraph before Lemma 2.7). 2.4. Exercises 2.4.1. Prove the following are equivalent for all a ∈ A. (1) a = b∗ b for some b ∈ A. (2) a is normal and sp(a) ⊆ [0, ∞). 2.4.2. Assume a is a normal operator. Characterize a being self-adjoint, projection, positive, unitary, in terms of the spectrum of a. 2.4.3. A multiplication operator mf is invertible if and only if there is some ε > 0 such that |f | > ε (a.e.). Thus since mf − λI = mf −λ , sp(mf ) is the essential range of f (the set of λ ∈ C such that for every neighborhood U of λ, f −1 (U ) has positive measure). 2.4.4. If f is a continuous function on sp(a), prove that sp(f (a)) = {f (λ) : λ ∈ sp(a)}.

page 17

March 11, 2014

10:36

WSPC 9 x 6

18

singapore-final-r

I. Farah

2.4.5. Assume a is normal. Show that C ∗ (a) ∼ = C0 (sp(a) \ {0}). 2.4.6. Show that Theorem 1.6 is a corollary of Proposition 2.5. 2.5. Constructions of C*-algebras In many of the constructions of C*-algebras given below the fact that the algebraic structure determines the norm considerably simplifies the discussion. 2.5.1. Unitization Every C*-algebra A can be embedded in a unital C*algebra A˜ in a minimal way as follows. On A × C define the operations as ¯ and k(a, λ)k = follows: (a, λ)(b, ξ) = (ab + λb + ξa, λξ), (a, λ)∗ = (a∗ , λ) supkbk≤1 kab + λbk and check that this is still a C*-algebra. A straightforward calculation shows that (0, 1) is the unit of A˜ and that ˜ A ∋ a 7→ (a, 0) ∈ A˜ is an isomorphic embedding of A into A. 2.5.2. Direct sums Given C*-algebras A and B we define their direct sum A ⊕ B to be the set of all pairs (a, b) with the pointwise defined operations and norm defined by k(a, b)k = max{kak, kbk}. This is easily seen to be an abstract C*-algebra. If A and B are given with concrete representations on spaces H and K, respectively, then it is equally easy to represent A ⊕ B on H ⊕ K as block-matrices   a0 (a, b) = . 0b As is customary we write a + b instead of (a, b). One similarly defines a direct sum of any finite number of C*-algebras. Given an infinite family of C*-algebras Ai , for i ∈ I, the direct sum is defined as M Ai = {(ai : i ∈ I) : ai ∈ Ai for all i, and {i : kai k > 1/n} is finite i∈I

for all n}.

Operations are defined pointwise and the norm is the supremum norm. It is again easy to see that this is a C*-algebra. 2.5.3. Direct products Finite direct products coincide with finite sums. Given an infinite family of C*-algebras Ai , for i ∈ I, one defines Q i∈I Ai = {(ai : i ∈ I) : supi kai k < ∞}.

With operations and norm defined as in §2.5.2 this is a C*-algebra that has L Q i∈I Ai as an ideal. One can prove that i
page 18

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

19

2.5.4. Direct limits (also called inductive limits) Let Ai , for i ∈ Λ be family of C*-algebras indexed by a directed set. Also assume that for i < j ′ we have a *-homomorphism Fij : Ai → Aj and that these *-homomorphisms commute. Then Ai together with Fij form a directed system of C*-algebras. Then the direct limit of this system, A = limi Ai , is defined as follows. Consider the set of all (ai : i ≥ j) such that j ∈ Λ, aj ∈ Aj and ai = Fji (aj ) for all i ≥ j. This limit comes equipped with canonical *-homomorphisms Fi : Ai → A for all i which commute with all Fij . One should keep in mind that this is an abuse of notation, since the direct limit depends on connecting maps Fij as well as the algebras Ai (see the examples in §4.4). 2.5.5. Matrix algebra over A Given a C*-algebra A and n ∈ N we define C*-algebra Mn (A) as follows. Its elements are n × n matrices over A. The algebraic operations are defined to be the usual matrix operations. In order to define norm fix a faithful representation of A on a Hilbert space H. Now interpret each a ∈ Mn (A) naturally as an operator on the direct sum of n copies of H. We equip Mn (A) with the corresponding operator norm. By the automatic continuity (Lemma 2.7) the norm on Mn (A) is canonical. However, it is notoriously nontrivial to compute. For example, a deceivingly simple Anderson’s paving conjecture is equivalent to the positive solution to the central Kadison–Singer problem on extensions of pure states. 2.5.6. Stabilization Story goes that in the olden days, whenever encountered with a non-unital C*-algebra one would immediately unitize it. Nowadays, whenever encountered with a unital C*-algebra one stabilizes it and hence turns it into a non-unital C*-algebra. The motivation for this behaviour will become apparent in §5. Given A, define a direct limit as follows. Let An be Mn (A) and let Fn,n+1 : An → An+1 be given by adding the n + 1-st zero row and zero   a0 column to a, or in block-matrix notation Fn (a) = . 00 Then maps Fn,n+1 for n ∈ N define a commuting system of non-unital *-homomorphisms. The direct limit is a non-unital algebra called the stabilization of A. This algebra is just a special case of minimal C*-algebraic tensor product defined in §2.5.7 below. More precisely, Mn (A) is (isomorphic to) A⊗Mn (C) and the stabilization of A is (isomorphic to) A⊗K, where K denotes the algebra of compact operators on an infinite-dimensional, separable, complex Hilbert space.

page 19

March 11, 2014

20

10:36

WSPC 9 x 6

singapore-final-r

I. Farah

For future use in §5 we record a bit of notation. By M∞ (A) we denote n Mn (A), with the connecting maps as defined above. A C*-algebra is A stable if it is isomorphic to A ⊗ K. Since K itself is stable, A is stable if and only if it is isomorphic to B ⊗ K for some B. S

2.5.7. Minimal tensor product An algebraic tensor product of C*-algebras A and B is defined as a quotient of the linear span of elementary tensors a ⊗ b. It is customary to denote this algebraic tensor product by A ⊙ B. On this complex *-algebra one wants to define a norm satisfying axioms listed in Lemma 1.3 and take the completion. By the GNS theorem (Theorem 2.3) such completion is a C*-algebra. It turns out that in some cases there is no unique C*-norm on A ⊙ B; for example, this is the case with B(H) ⊙ B(H). This is even more remarkable in light of the fact that the tensor product of Hilbert spaces H and K is uniquely defined: If (eξ ) is an orthonormal basis of H and (fη ) is an orthonormal basis of K then eξ ⊗ fη is an orthonormal basis of H ⊗ K. I shall cut the corners and only describe construction of the so-called minimal C*-algebraic tensor product, without even explaining why is it minimal. As a matter of fact, I shall not even prove that it is uniquely defined (this requires showing a true, albeit not obvious, fact that A ⊗ B does not depend on the choice of representations of A and B). Assume A and B are unital C*-algebras. By the GNS theorem (Theorem 2.3) we can fix *-isomorphisms Φ : A → B(H) and Ψ : B → B(K). Without a loss of generality, we may assume these *-homomorphisms are unital. We can canonically identify B(H) with a subalgebra of B(H ⊗ K), by sending each a to the operator such that a(e⊗f ) = a(e)⊗f for all e ∈ H and f ∈ K. Similarly we identify B(K) with a subalgebra of B(H ⊗ K). This defines representations of A and B on B(H ⊗ K), and we identify A and B with their respective images. Then ab = ba for all a ∈ A and b ∈ B and we define A ⊗ B to be the C*-algebra generated by A and B. This product is sometimes denoted A ⊗min B. One can similarly define a tensor product of a family (finite or infinite) N of C*-algebras, i∈I Ai . If I is infinite then one needs to assume that all J but finitely many of Ai are unital and let i∈I Ai be the span of the set N of all elementary tensors i∈I ai where ai ∈ Ai and ai = IA for all but finitely many i. The assumption that A is unital was needed in order to have an isomorphic copy of B, 1 ⊗ B, inside A ⊗ B. The unitality of B is used in the analogous way.

page 20

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

21

Definition 2.9. C*-algebra A is nuclear if for every C*-algebra B there is a unique C*-algebra norm on A ⊙ B. It is not difficult to check that all finite-dimensional C*-algebras are nuclear and that the class of nuclear algebras is closed under taking tensor products and direct limits. Also, all abelian C*-algebras are nuclear. Therefore, all algebras (except B(H)) considered in these notes are nuclear and all tensor product norms used here will be uniquely determined. The theory of tensor products of C*-algebras is full of surprises and two of the most important classes of C*-algebras, nuclear and exact algebras, are defined by their behaviour with respect to the tensor products. This exciting subject is beyond the scope of the present paper and the reader may want to consult [5] for more details. See also Exercises 2.6.5 and 2.6.6. 2.5.8. Continuous fields of C*-algebras Given a compact space X and a C*-algebra A, let C(X, A) denote the algebra of all continuous functions f : X → A. The operations are given pointwise and the norm is the supremum norm, kf k = supx∈X kf (x)k. One can vary this definition by restricting the range of functions f to obtain more general C*-algebras. 2.5.9. Corners This is a special case of a hereditary subalgebra. Given a C*-algebra A and a projection p ∈ A, we can consider the subalgebra pAp = {pap : a ∈ A}. This is a unital C*-algebra, although it is typically not a unital subalgebra of A, even if A has a unit. 2.5.10. . . . and so on Some important constructions of C*-algebras, such as maximal tensor products, group C*-algebras (both full and reduced), multiplier algebras and coronas will not be used in these notes. 2.6. Exercises 2.6.1. Let X be a locally compact, non-compact, Hausdorff space. By Gelfand–Naimark theorem, the unitization of C0 (X) is isomorphic to C(Y ) for some compact Hausdorff space Y . What is the relation between X and Y ? 2.6.2. Prove that a direct product of infinitely many C*-algebras is nonseparable unless all but finitely many of them are isomorphic to C. 2.6.3. Assume A = limi Ai is unital. Prove that there is i0 ∈ λ such that for all i0 < i algebra Ai is unital and for all i0 < i < j the map Fij is unital.

page 21

March 11, 2014

22

10:36

WSPC 9 x 6

singapore-final-r

I. Farah

2.6.4. Check that Mn (Mk (C)) is isomorphic to Mnk (C). 2.6.5. Prove that Mn (A) is algebraically isomorphic to Mn (C) ⊗ A and that it carries a uniquely defined C*-norm. (Hint: To prove uniqueness of the norm use Lemma 2.7.) 2.6.6. Show that the CAR algebra can be identified with the unital direct N limit of algebras An = n M2 (C), which in turn can be identified with the N infinite tensor product N M2 (C). (Just like in the case of direct limits, C*-algebraic tensor product is the norm-completion of the algebraic tensor product.) 2.6.7 (Matrix units I). Prove that Mn (C) is the unique C*-algebra generated by elements (eij )i
page 22

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

23

This operator is in B(H ⊗ H) but it cannot be approximated by finite linear combinations of elementary tensors.) 2.6.14. Prove that the stabilization of A (§2.5.6) is isomorphic to A ⊗ K. 2.6.15. Prove that K ∼ = K ⊗ K. 3. Local Theory of C*-Algebras 3.1. Polar decomposition I now collect several somewhat technical (yet illuminating) results on the local structure of C*-algebras. By polar decomposition theorem (Theorem 1.7) every operator a in B(H) can be represented as a product of a positive element and a partial isometry, a = |a|v. The positive part of a is given by formula (aa∗ )1/2 and therefore belongs to C ∗ (a). In Exercise 3.3.2 we shall see that the partial isometry v need not belong to C ∗ (a). We now discuss to what extent this representation, analogous to z = ρeiθ for complex numbers, works in arbitrary C*-algebras. Lemma 3.1. If b is invertible, then b = cu for a unitary u and a positive c. Also, c = (bb∗ )1/2 . Proof. Since bb∗ is positive, c = (bb∗ )1/2 is well-defined by continuous functional calculus. Also, since b is invertible, so are b∗ , bb∗ and c. Then u = c−1 b satisfies uu∗ = (bb∗ )−1/2 bb∗ (bb∗ )−1/2 = 1 and u∗ u = b∗ (bb∗ )−1/2 (bb∗ )−1/2 b∗ = 1, and is therefore a unitary. Lemma 3.2. If 0 is not an accumulation point of sp((aa∗ )1/2 ) then there is a partial isometry v in C ∗ (a) such that a = |a|v. Proof. For a moment consider C ∗ (a) as a concrete C*-algebra on some Hilbert space and let a = |a|v be the polar decomposition of a. We have that b = (aa∗ )1/2 is in C ∗ (a) and it remains to prove that v ∈ C ∗ (a). Let f : sp(b) → C be defined by f (0) = 0 and f (t) = 1/t if t 6= 0. By our assumption f is continuous on sp(b) and therefore by the continuous functional calculus we have f (b) ∈ C ∗ (a). Since f (b) and b commute, f (b)b is a self-adjoint element whose spectrum is included in {0, 1} and it is therefore

page 23

March 11, 2014

24

10:36

WSPC 9 x 6

singapore-final-r

I. Farah

a projection. Denote this projection by p and note that p ∈ C ∗ (a). Since ker(p) = ker((aa∗ )1/2 ) = ker(a∗ ) we have pv = v. Finally, pv = f (b)|a|v = f (b)a and therefore pv ∈ C ∗ (a). 3.2. Stability If b is self-adjoint and ka − bk < ε then c = (a + a∗ )/2 is self-adjoint, belongs to C ∗ (a), and satisfies kc−bk < ε. This is an instance of the stability phenomenon: if an operator b belongs to a distinguished class of operators and ka − bk is small, then there is c ∈ C ∗ (a) in the same distinguished class as b such that kc − bk is small. Lemma 3.3. Assume p is a projection and ka − pk < ε with ε < 1/2. Then there is a projection q ∈ C ∗ (a) such that kp − qk < 2ε. Proof. We may assume p 6= 0 (otherwise take q = 0). By replacing a with (a + a∗ )/2 we may assume a is self-adjoint. By Lemma 2.8 we have that sp(a) ⊆ (−ε, ε) ∪ (1 − ε, 1 + ε). Since ε < 1/2 the function f on sp(a) that sends (−ε, ε) to 0 and (1 − ε, 1 + ε) to 1 is well-defined and continuous. By continuous functional calculus we have q = f (a) ∈ C ∗ (a) such that sp(q) = {0, 1}. Therefore q is a projection. A straightforward computation (Exercise 3.3.1) shows that kq − ak < ε and therefore q is as required. A straightforward modification of the proof of Lemma 3.3 gives the following. Lemma 3.4. Let F be a finite subset of C. Then for every ε > 0 there exists δ = δ(F, ε) > 0 with the following property. If b is normal and such that sp(b) = F and ka − bk < δ, then there exists a normal c ∈ C ∗ (a) such that kc − bk < ε and sp(c) = F . Although partial isometries are not necessarily normal, a result similar to the above still applies. Lemma 3.5. For every ε > 0 there exists δ > 0 with the following property. Assume v is a partial isometry and ka − vk < δ. Then there is a partial isometry w ∈ C ∗ (a) such that kv − wk < ε. Moreover, if p and q are projections such that kvv ∗ − pk < δ and kv ∗ v − qk < δ then we can choose partial isometry w ∈ C ∗ (a, p, q) so that ww∗ = p and w∗ w = q. Proof. Fix δ < min(e2 /2, 1/12). If ka − vk < δ then kaa∗ − vv ∗ k ≤ ka(a∗ − v ∗ )k + k(a − v)v ∗ k ≤ δkak + δ ≤ 3δ since kak ≤ 1 + δ. By Lemma 2.8 this implies sp(aa∗ ) ⊆3δ {0, 1} and therefore Exercise 2.4.4 implies sp((aa∗ )1/2 ) ⊆√3δ {0, 1}. Thus sp((aa∗ )1/2 ) is included in two short

page 24

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

25

intervals centered at 0 and 1. Let f ∈ C((sp(aa∗ )1/2 ) be the function that maps the first interval to 0 the second to 1. Then f ((aa∗ )1/2 ) is a selfadjoint element whose spectrum is {0, 1}, and it is therefore a projection. √ It is also within 3δ of (aa∗ )1/2 and it therefore satisfies the assumptions of Lemma 3.2. Hence there is a partial isometry w ∈ C ∗ (a) such that a = (aa∗ )1/2 w. For the moreover part, pick δ small enough so that pvq satisfies the assumptions of the first part of the lemma. Applying it to pvw we obtain w ∈ C ∗ (a, p, q) such that pwq = w, kww∗ − pk < 1 and kw∗ w − qk < 1. By Exercise 1.6.3, we have ww∗ = p and w∗ w = q. The last few lemmas are concerned with stability of classes of operators in C*-algebras. For more information on this exciting subject the reader can consult the excellent [56]. The following lemma will be important in the analysis of the structure of UHF algebras. We write A1 = {a ∈ A : kak ≤ 1}. Lemma 3.6. For every n ∈ N and ε > 0 there exists δ > 0 with the following property. If A and B are unital subalgebras of C such that A is isomorphic to Mn (C) and A1 ⊆δ B, then there exists a unitary u ∈ C such that uAu∗ ⊆ B and k1 − uk ≤ ε. Proof. I shall try to avoid the computation of δ (it is given in [12] and in [39]). Assume δ1 > 0 is very small. Consider a ∈ A defined by a = P diag(1, 2, 3, . . . , n). Then a = j
are partial isometries, too.

page 25

March 11, 2014

10:36

26

WSPC 9 x 6

singapore-final-r

I. Farah

Lemma 3.7. If a is self-adjoint and there exist x and a self-adjoint b such that xa = bx then x∗ x commutes with a. If x is moreover invertible then b is unitarily equivalent to a, i.e., there exists unitary u ∈ A such that b = uau∗ . Proof. By the self-adjointness of a and b we have ax∗ = x∗ b. Therefore x∗ xa = x∗ bx = ax∗ x as required. Now assume x is invertible. By Lemma 3.1 we have x = u|x| for some u ∈ A. By the first part, x∗ x commutes with a. Since |x| = (x∗ x)1/2 belongs to the C*-algebra generated by x∗ x, it also commutes with a. We therefore have b = xax−1 = u|x|a|x|−1 u∗ = uau∗ . We state an immediate consequence of the above lemma for future reference. Lemma 3.8. If a and b are self-adjoint and b = xax−1 for an invertible x then b = uau∗ for a unitary u. 3.3. Exercises 3.3.1. Assume a is normal and f and g are continuous functions on sp(a). Then kf (a) − g(a)k = kf − gk∞ . 3.3.2. Find an operator a such that C ∗ (a) does not contain partial isometry v such that a = |a|v. (Hint: Choose a to be compact but of infinite rank.) 3.3.3. Show that invertible elements form an open set in a unital C*algebra. (Hint: Use the proof of Lemma 1.4 to show that b invertible and kb − ck < kbk implies c is invertible.) 3.3.4. Prove that a is invertible if and only if a∗ is invertible if and only if aa∗ invertible if and only if |a| is invertible. 3.3.5. Prove Lemma 3.4 and express δ in terms of ε and F . 3.4. Murray–von Neumann equivalence of projections Murray–von Neumann equivalence of projections is a noncommutative analogue of equinumerosity relation of sets and also a continuous variant of the dimension of a closed subspace of the Hilbert space (see Example 3.9 (1)). It was introduced by Murray and von Neumann in their seminal series of papers ‘Rings of operators’ (an old name for von Neumann algebras)

page 26

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

27

where it played a fundamental role in type classification (see e.g., [46]). While C*-algebras are not nearly as well-behaved as von Neumann algebras Murray–von Neumann equivalence of projections is a useful tool in classification problem for some well-behaved classes of C*-algebras. Assume A is a C*-algebra. Two projections p and q in A are Murray-von Neumann equivalent if there exists v ∈ A such that vv ∗ = p and v ∗ v = q.

(MvN)

In this case we write p ∼ q and keep in mind that the relation depends on the ambient algebra A. Note that a witness v of Murray-von Neumann equivalence is necessarily a partial isometry. In some C*-algebras p ∼ q is strictly weaker than the requirement that p and q are conjugate by a unitary (see Example 3.9 (1)). Example 3.9. (1) If A = B(H) then p ∼ q if and only if the range of p and the range of q have the same dimension, where the dimension of a closed subspace of the Hilbert space is the minimal cardinality of an orthonormal basis. This is an immediate consequence of the fact that two complex Hilbert spaces with the same dimension are linearly isometric. (2) A special case of (1) is A = Mn (C), where two projections are Murray-von Neumann equivalent if and only if they have the same rank. This extends to the algebra K of compact operators. (3) If A is abelian then p ∼ q if and only if p = q. The following example requires some minimal knowledge of vector bundles; see e.g., [41]. Example 3.10. Projections of C(X, Mn (C)) are maps f : X → Mn (C) such that f (x) is a projection for all x ∈ X. By identifying a projection in Mn (C) with a subspace of Cn one sees that projections of C(X, K) are vector bundles over X. Murray-von Neuman equivalence of these projections is the usual equivalence of vector bundles. Lemma 3.11. Assume p and q are projections in A such that kp−qk < 1/2. Then p ∼ q. If A is unital then there is moreover a unitary u such that u∗ pu = q. Proof. We first prove the case when A is unital. Let a = pq + (1 − p)(1 − q). Since 1 − p and 1 − q are at a distance kp − qk < 1/2, the distance from a to 1 = p2 + (1 − p)2 is < 1. By Lemma 1.4 a is invertible. By Lemma 3.1 we have a = |a|u for a unitary u.

page 27

March 11, 2014

10:36

28

WSPC 9 x 6

singapore-final-r

I. Farah

One easily checks that p1 = aqa−1 satisfies p21 = p1 = p∗1 , and is therefore a projection. Similarly, p2 = a(1 − p)a−1 is a projection and p1 + p2 = 1. By inspecting the definition of a one sees that pp1 = p1 and (1 − p)p2 = p2 , By Exercise 1.6.4 we conclude that p = aqa−1 . By Lemma 3.8 we conclude p = uqu∗ . Then v = uq is a partial isometry such that vv ∗ = p and v ∗ v = q. Now assume A is not unital. By the above, in the unitization of A there exists a unitary u such that uqu∗ = p. The partial isometry v = uq as above belongs to A and witnesses p ∼ q. Lemma 3.11 can be improved; see Exercise 3.5.7. 3.5. Exercises 3.5.1. Prove that p ∼ q is equivalent to the existence of v such that v ∗ pv = q and vqv ∗ = p. Also prove that such v is necessarily a partial isometry. 3.5.2. If F : A → B is a *-homomorphism and p and q are projections in A, show that p ∼ q implies F (p) ∼ F (q). Give an example showing that the converse may fail. 3.5.3. Let Φ : A → B be a unital *-homomorphism between C*-algebras and let a and b be such that b = Φ(a). (1) Prove that if a is normal (self-adjoint, positive, unitary, projection, partial isometry) then b is normal (self-adjoint, positive, unitary, projection, partial isometry). (2) Assume b is self-adjoint (positive) Prove that we can choose a′ such that Φ(a′ ) = b and a′ is self-adjoint (positive). (3) Provide examples showing that b can be normal (unitary, projection, partial isometry, respectively) while no a′ satisfying Φ(a′ ) = b is normal (unitary, projection, partial isometry, respectively). (Hint: For projections and partial isometries consider abelian algebras and use Exercise 2.2.6.) 3.5.4. Find a C*-algebra A and two Murray-von Neumann equivalent projections that are not conjugate. (Hint: Try B(H).) Let P(A) denote the set of all projections of a C*-algebra A. 3.5.5. Two projections are homotopic (in a C*-algebra A) if they belong to the same path-connected component of P(A). Prove that being homotopic implies being conjugate by a unitary.

page 28

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

29

(Hint: Lemma 3.11.) 3.5.6. Prove that kp − qk < 1 implies p and q are homotopic. (Hint: The path tp + (1 − t)q, for 0 ≤ t ≤ 1, consists of nonzero positive elements. Use the continuous functional calculus to morph it into a path consisting of projections.) 3.5.7. Prove that kp−qk < 1 implies p and q are conjugate and in particular p ∼ q. (Hint: Combine Exercise 3.5.5 and Exercise 3.5.6.) 3.5.8. Prove that there exists ε > 0 such that for all A and projections p and q in A we have p ∼ q if and only if there exists a ∈ A such that kaa∗ − pk < ε and ka∗ a − qk < ε. (Hint: Lemma 3.5 and Lemma 3.11.) 3.5.9. Prove that the following two properties of a C*-algebra A are equivalent. (1) the set of invertible self-adjoint elements in the unitization of A is dense in the set of all self-adjoint elements in the unitization of A. (2) Linear combinations of projections are dense in A. (Hint: Every element of A is a linear combination of two self-adjoint operators. Use continuous functional calculus.) C*-algebras A satisfying either of the statements from Exercise 3.5.9 have real rank zero. Recall that on projections we define a relation p ≤ q if and only if pq = p (Exercise 1.6.3). A nonzero projection p in a C*-algebra is minimal if the only projections q ≤ p are 0 and p. 3.5.10. Prove that a projection p in a real rank zero algebra A is minimal if and only if pAp is isomorphic to C. Then prove that all minimal projections in a simple real rank zero algebra are Murray-von Neumann equivalent. (Hint: If p and q are minimal projections in a real rank zero algebra prove that the vector space pAq = {paq : a ∈ A} is one-dimensional.) 3.6. Traces A trace of a C*-algebra A is a state τ such that τ (ab) = τ (ba) for all a and b. We record an immediate consequence of the definition of ∼. Lemma 3.12. If τ is a trace on A and p ∼ q then τ (p) = τ (q).

page 29

March 31, 2014

18:14

WSPC 9 x 6

singapore-final-r

I. Farah

30

Let T (A) = {τ ∈ A∗ : τ is a trace}. If τ and σ are traces and 0 < t < 1 then tτ + (1 − t)σ is a trace. Therefore T (A) is a convex subset of the unit sphere of A∗ . Also, since being a trace is a closed condition, by Birkhoff–Alaoglu theorem T (A) is compact in the weak*-topology. Being a compact and convex set, by the Krein–Milman theorem T (A) is the closure of the convex hull of its extreme points. On Mn (C) define the normalized trace via (below a stands for the matrix (aij )i≤n,j≤n ) n

tr(a) =

1 ajj . n j=1

Lemma 3.13. If p and q are projections in Mn (C) then p ∼ q if and only if tr(p) = tr(q), and tr(p) = k/n where 0 ≤ k ≤ n is the dimension of the range of p. Proof. The direct implication is Lemma 3.12. The converse implication is an exercise in linear algebra. Lemma 3.14. Functional tr is a unique trace on Mn (C). Proof. This is of course a standard linear algebra fact. Fix a and b and note  that the i-th diagonal entry of ab is equal to j
page 30

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

31

a *-homomorphism sends projections to projections, by Lemma 3.13 we conclude that 1/n = m/k for some m, concluding the proof. In order to prove the second part, assume Φ and Ψ are unital *-homomorphisms of Mn (C) into Mk (C). Fix a minimal projection p ∈ Mn (C), for example p = diag(1, 0, 0, . . . , 0). Then Φ(p) and Φ(q) are projections in Mk (C) each with trace 1/n Therefore A = Φ(p)Mk (C)Φ(p) and B = Ψ(p)Mk (C)Ψ(p) are both isomorphic to Mk/n (C). By Exercise 2.6.12 we have that Mk (C) ∼ = Mn (A) ∼ = Mn (B). Therefore an isomorphism α : A → B extends to an automorphism α′ of Mk (C) such that Ψ = α′ ◦ Φ. By the easy finite-dimensional case of Exercise 1.2.7 we have that α is inner and therefore for some unitary u ∈ Mk (C) we have Ψ = Ad u◦Φ, as required. Note that if F : B → C is a unital *-homomorphism and τ is a trace of C then τ ◦ F is a trace of B. The map T (C) ∋ τ 7→ τ ◦ F ∈ T (B) is continuous and affine. Lemma 3.16. Assume A = limn An is unital. If each An has a unique trace then A has a unique trace. More generally, T (A) = lim T (An ). ←− Proof. We prove only the first assertion. Since A is unital, all but finitely many *-homomorphisms from An to A are unital. Let τn be the unique trace of An . Then τn+1 ↾ An = τn . Therefore τ ′ = limn τn is a well-defined trace on a dense subset of A. Since trace is norm-continuous, τ ′ has a unique extension to a trace of A. Assume σ is a trace of A. Then σ ↾ An = τn for all n and therefore σ and τ agree on a dense subset of A. Since σ is a continuous functional, σ = τ. In order to prove the second assertion use Exercise 2.2.10 in addition to the above and observe that the functor is contravariant. Example 3.17. (1) Assume A is unital and abelian. By the Gelfand– Naimark theorem A = C(X) for a compact Hausdorff space X. By the Riesz Representation theorem, every continuous functional φ of A is of the R form φ(f ) = f dµ for a finite Radon measure µ on X. If φ is a state then µφ is a probability measure. Since A is abelian the condition φ(ab) = φ(ba) is automatic and therefore all states are traces. Therefore T (A) is affinely homeomorphic to P (X), the space of Radon probability measures on X. (2) By Lemma 3.14 T (Mn (C)) is a singleton for every n. (3) Furthermore, Lemma 3.16 implies that every UHF algebra carries a unique trace.

page 31

March 31, 2014

18:14

32

WSPC 9 x 6

singapore-final-r

I. Farah

(4) Let A = B ⊕ C. Then clearly T (A) = {λτ + (1 − λ)σ : τ ∈ T (B), σ ∈ T (C), 0 < λ < 1}. Therefore if A is a direct sum of n matrix algebras by (2) we have that T (A) is affinely homeomorphic to the n-simplex, Δn .  (5) By (4) every trace τ of i
page 32

March 11, 2014

10:36

WSPC 9 x 6

singapore-final-r

Logic for C*-Algebras

33

for some l, or ∞ if the set of such k is unbounded. One can show that the generalized integer uniquely determines a separable UHF algebra up to the isomorphism. Let me again emphasize that UHF algebras are by definition unital. Nonunital algebras that are direct limits of full matrix algebras are called matroid algebras, approximately matricial (AM) algebras, or stabilized UHF algebras (note that the latter terminology is somewhat misleading, since they are not necessarily stable; see Exercise 4.3.6). Example 4.1. (1) The CAR (Canonical Anticommutation Relation) algebra is the UHF algebra which is a direct limit of M2n (C) for n ∈ N. It is often denoted by M2∞ . (2) One can similarly define M3∞ as the direct limit of M3n (C) for n ∈ N. (3) The universal UHF algebra is the UHF algebra corresponding to the Q generalized integer j p∞ j .

Let Dn denote the subalgebra consisting of all diagonal matrices in Mn (C). Then Dn is a maximal abelian subalgebra of Mn (C) isomorphic to Cn . If A = limj Mn(j) (C) is a UHF algebra then algebras Dn(j) form a directed system and their limit D is the diagonal masa in A (cf. Exercise 4.3.2). If A is unital and infinite-dimensional then by Exercise 4.3.3 its diagonal masa is isomorphic to C(2N ). Therefore the CAR algebra can be considered as a noncommutative version of the Cantor space. (It is customary to identify compact Hausdorff space X and the C*-algebra C(X), since compact Hausdorff spaces and unital abelian C*-algebras form equivalent categories.) Lemma 4.2. Every UHF algebra A has a unique trace τ . The values of τ on projections of A are all numbers of the form k/n, where k ∈ N and n is a natural number that divides nA . Proof. Each Mn (C) has a unique trace (Lemma 3.14) and the conclusion follows by Lemma 3.16.

We note that Glimm’s result applies to an apparently larger class of algebras (see Theorem 4.4). Theorem 4.3 (Glimm). Separable UHF algebras A and B are isomorphic if and only if they have the same generalized integer. Proof. If kA 6= kB then by Lemma 4.2 A and B are not isomorphic.

page 33

March 11, 2014

10:36

WSPC 9 x 6

34

singapore-final-r

I. Farah

Assume kA = kB . Write A = limj Mn(j) (C) and B = limj Mm(j) (C). By going to subsequences of n(j) and of m(j) we may assume that for all j we have that n(j) divides m(j) and m(j) divides n(j + 1). By Lemma 3.15 we can fix a *-homomorphism φj : Mn(j) (C) → Mm(j) (C) and a *-homomorphism ψj : Mm(j) (C) → Mn(j+1) (C) for every j. By the second part of the same lemma we may choose these maps so that all triangles in Figure 1 commute. Mn(1) (C) φ1 Mm(1) (C)

Mn(2) (C) ψ1

φ2

ψ2

Mm(2) (C)

Fig. 1.

Mn(3) (C)

...

A

...

B

φ3 Mm(3) (C)

Proof of Glimm’s theorem.

S Therefore Φ0 = j φj is a well-defined *-homomorphism from a dense subalgebra of A into B. It is an isometry by Lemma 2.7 and it therefore extends to a *-homomorphism Φ : A → B. By the same argument we have S a *-homomorphism Ψ : B → A extending j ψj . We claim that Ψ ◦ Φ is the S identity on A. It suffices to check this for the dense subalgebra j Mn(j) (C). Indeed, for any j it is the identity on Mn(j) (C) by the commutativity of the above diagram. Similarly Φ ◦ Ψ is the identity on B, and therefore Φ : A → B is a *-isomorphism. 4.2. Another look at the UHF algebras A C*-algebra A is locally matricial (or LM) if for every ε > 0 and every finite F ⊆ A there exist n and a *-homomorphism Φ : Mn (C) → A such that F ⊆ε Φ(Mn (C)). (Recall that K ⊆ε L means that inf y∈L kx − yk ≤ ε for all x ∈ K.) In other words, for every finite subset of A there exists a full matrix subalgebra B of A such that each element of F is within ε of B. Infinite tensor products of unital C*-algebras were defined in §2.5.7. Theorem 4.4 (Glimm). For a separable unital C*-algebra the following are equivalent. (1) A is a tensor product of full matrix algebras. (2) A is a direct limit of full matrix algebras (i.e., it is UHF), and (3) A is LM.

page 34

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

35

An algebra as in (1) is necessarily unital, and (2) and (3) remain equivalent in the case when A is not necessarily unital. Proof. We prove only the equivalence of (2) and (3) (but see the hint to Exercise 4.3.1). By the definition of direct limit (2) implies (3), even without the separability requirement. The implication from (3) to (2) is a consequence of Lemma 3.6. As a corollary to Theorem 4.4 and Theorem 4.3, to each unital separable LM algebra one can associate a generalized integer and that this generalized integer is a complete isomorphism invariant for unital separable LM algebras. The following theorem taken from [30] shows that the situation in nonseparable case is quite different. Recall that a density character of a C*algebra is the minimal cardinality of a dense subset. Theorem 4.5. (1) There exist a unital C*-algebra of density character ℵ1 that is a direct limit of full matrix algebras but not a tensor product of full matrix algebras. (2) Every LM algebra of density character ≤ ℵ1 is a direct limit of full matrix algebras. (3) There exists a unital LM algebra of density character ℵ2 that is not a direct limit of full matrix algebras. All algebras constructed in [30] are indistinguishable from the CAR algebra by their Elliott invariant, Cuntz semigroup, or any other known C*-algebraic invariant (see [31]). 4.3. Exercises 4.3.1. Prove the equivalence of (1) and (2) in Theorem 4.4: a unital separable C*-algebra is a tensor product of full matrix algebras if and only if it is a unital direct limit of full matrix algebras. (Hint: Exercise 2.6.12.) 4.3.2. Prove that the diagonal masa (see the paragraph before Lemma 4.2) is a masa (i.e., a maximal abelian C*-subalgebra). 4.3.3. Prove that the diagonal masa of a separable infinite-dimensional UHF algebra is isomorphic to C(2N ), where 2N denotes the Cantor space. (Hint: It is a direct limit of finite-dimensional abelian C*-algebras. Prove that it is isomorphic to C(X) for X a compact metrizable zero-dimensional space without isolated points.)

page 35

March 11, 2014

36

10:36

WSPC 9 x 6

singapore-final-r

I. Farah

4.3.4. Let A be a UHF algebra. Show that its generalized integer kA is uniquely defined as the number whose finite divisors are those n such that Mn (C) has a unital *-homomorphism into A. 4.3.5. Let A and B be unital separable UHF algebras. Prove that A is elementarily equivalent to B (in the logic of metric structures, [29]) if and only A is isomorphic to B. (Hint: [7].) 4.3.6. A C*-algebra is stable if it is isomorphic to its stabilization (see §2.5.6). Construct a direct limit of full matrix algebras that is neither unital nor stable. (Hint: First prove that a stable algebra cannot have a finite trace and then construct a nonunital direct limit of full matrix algebras with a finite trace.) 4.3.7 (Dixmier). Classify separable non-unital direct limits of full matrix algebras. (Hint: First classify pairs (A, p) where A is a non-unital direct limit of full matrix algebras and p ∈ A is a projection. Being familiar with classification of rank one torsion-free abelian groups may help, but beware of Exercise 4.3.6.) 4.3.8. Characterize when two separable UHF algebras have isomorphic corners (see §2.5.9). 4.3.9. Let A and B be separable UHF algebras. Prove that A is isomorphic to a unital subalgebra of B if and only if kA divides kB . 4.4. Bratteli diagrams The following lemma is a consequence of the Artin–Wedderburn theorem but we sketch a direct proof in Exercise 4.7.6 below. Lemma 4.6. Every finite-dimensional C*-algebra is *-isomorphic to a direct sum of finitely many full matrix algebras that each of these full matrix algebras is a minimal (nontrivial) ideal of the algebra. We shall introduce a tool for describing *-homomorphisms Φ : A → B between finite-dimensional C*-algebras. By Lemma 4.6 every such algebra is a direct sum of its minimal ideals each of which is isomorphic to a full matrix algebras. Recall that there exists a unital *-homomorphism from Mn (C) into Mk (C) if and only if n divides k (Lemma 3.15). If n ≤ k and Φ : Mn (C) → Mk (C) is a *-homomorphism (not necessarily unital)

page 36

March 31, 2014

18:14

WSPC 9 x 6

singapore-final-r

Logic for C*-Algebras

37

then its multiplicity is the rank of Φ-image of any minimal projection in Mn (C) (since Φ preserves Murray–von Neumann equivalence, the rank is well-defined). In other words, if Φ sends a to diag(a, a, . . . , a, 0, . . . , 0) then its multiplicity is the number of the occurrences of a on the right-hand side. (If you have not tackled Exercise 3.7.1 yet, now is a good time.) Bratteli diagram of Φ : A → B is a bipartite graph whose vertices on the left correspond to the minimal ideals of A and vertices on the right correspond to the minimal ideals of B. These vertices may be labelled by numbers indicating the dimension of the corresponding algebra. Two vertices are connected by k edges if and only if the multiplicity of the map between them is k. An example is in order. Consider a unital *-homomorphism between M2 (C) ⊕ M3 (C) and M6 (C) ⊕ M5 (C) ⊕ M6 (C) defined by (a, b) → (diag(a, a, a), diag(a, b), diag(b, b)). The Bratteli diagram describing this map is given in Figure 2.

2

6

3

5 6

Fig. 2.

A Bratteli diagram.

By Lemma 4.7, *-homomorphism described by a Bratteli diagram is unique up to unitary conjugacy. When describing a unital AF algebra A = limn An by a Bratteli diagram we put together diagrams of each Φn : An → An+1 . For convenience we also let A1 = C and assume that all Φn are unital. Under these conventions the labels of vertices can be omitted since the dimension of any of the full matrix algebras can be determined by adding the multiplicities of vertices from the earlier levels. Some examples of C*-algebras defined via Bratteli diagrams are given in Figures 3–7. Lemma 4.7. Every unital *-homomorphism between finite direct sums of matrix algebras corresponds to a Bratteli diagram. Moreover, any two unital *-homomorphisms with the same Bratteli diagram are conjugate.

page 37

March 11, 2014

10:36

WSPC 9 x 6

38

singapore-final-r

I. Farah

•

•

•

•

...

Fig. 3. This diagram describes the unital directed system M2 (C) → M4 (C) → M8 (C) . . . and its limit, the CAR algebra M2∞ .

•

• Fig. 4.

•

•

•

...

This diagram represents M3∞ .

•

...

•

•

...

•

•

...

•

...

Fig. 5. In this diagram every node splits into two nodes. Note that all nodes in the diagram correspond to the abelian algebra C. Therefore the n-th level corresponds to n the algebra C2 and it is not difficult to prove that the direct limit is the algebra C(2N ).

Proof. This is an almost immediate consequence of Lemma 3.15. First note that if F : Mn (C) → Mk (C) is a non-unital *-homomorphism then p = F (1) is a projection. Also, pMk (C)p is isomorphic to Mm (C) where m is the rank of p. By Lemma 3.15 m is a multiple of n. In order to prove the second part of the lemma, fix a unital *L L homomorphism F : the ideni
page 38

March 11, 2014

10:36

WSPC 9 x 6

singapore-final-r

Logic for C*-Algebras

•

39

•

...

•

•

...

•

•

...

•

...

Fig. 6. In this diagram the n-th level has 2n nodes, each one corresponding to C. Hence n the algebra is a direct limit of C2 , just like the algebra C(2N ) from the previous example. However, in the present case only one of the nodes on the n-th level splits, and it splits into 2n other nodes. One can prove that the direct limit is the algebra C(ω + 1), where ω + 1 is a converging sequence together with its limit.

1

1

2

3

5

...

1

1

2

3

...

Fig. 7. In this example nodes are marked by numbers for readability. The n-th level of this diagram corresponds to algebra MF (n) (C) ⊕ MF (n+1) (C), where F (n) is the nth Fibonacci number. It can be shown that this algebra, called Fibonacci algebra, is a simple, unital AF algebra with a unique trace that is not a UHF algebra.

(Hint: Given a Bratteli diagram of A, identify its subsets whose direct limits are ideals of A.) 4.5.3. Which algebra corresponds to the Bratteli diagram given in Figure 8? • •

• •

•

•

• Fig. 8.

...

• •

Diagram for Exercise 4.5.3.

4.5.4 (Bratteli). By Exercise 4.5.1, to each Bratteli diagram D one can associate the unique AF algebra A(D). Describe the equivalence relation

page 39

March 11, 2014

10:36

WSPC 9 x 6

40

singapore-final-r

I. Farah

on Bratteli diagrams defined by D1 E D2 iff A(D1 ) ∼ = A(D2 ). 4.5.5. Construct a Bratteli diagram such that the corresponding AF algebra A is simple and T (A) is affinely homeomorphic to [0, 1]. Hint: See Figure 9. •

•

•

•

...

•

•

•

•

...

Fig. 9.

Hint for Exercise 4.5.5.

4.6. AF algebras A C*-algebra is AF (approximately finite) if it is a direct limit of finitedimensional C*-algebras. If A is a subalgebra of B, X is a subset of B, and ε > 0, we write X ⊆ε A if every element of X is within ε of an element of A. Definition 4.8. A C*-algebra B is an LF-algebra (locally finite) if for every finite F ⊆ B there are a finite-dimensional subalgebra C of A such that X ⊆ε C. Clearly every AF algebra is an LF algebra. Analogously to the case of UHF algebras (Theorem 4.4), the converse is true for separable AF algebras (Exercise 4.7.10). However, this fails for separable AH algebras: direct limits of AH algebras need not be AH ([11]; see §6 for the definition of AH algebras). Analogous statements also fail for nonseparable AF, and even AM, algebras ([30]). 4.7. Exercises 4.7.1. Prove that for n ∈ N every n-dimensional abelian C*-algebra is isomorphic to Cn (with the max norm). A projection p in A is minimal if it is nonzero and the only projections ≤ p are p and 0. 4.7.2. Prove that UHF algebras have no minimal projections.

page 40

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

41

4.7.3. Assume D is a masa (maximal abelian C*-subalgebra) in A. Prove that every minimal projection of D is a minimal projection of A. 4.7.4. Prove that if A is infinite dimensional then every masa in A is infinite-dimensional. (Hint: Use Exercise 4.7.3. It works even if A has no nontrivial projections.) In the following two exercises we sketch a direct proof of the instance of Artin–Wederburn theorem for C*-algebras (Lemma 4.6). 4.7.5. Assume A is a C*-algebra that is both simple and finite-dimensional. Prove that A ∼ = Mn (C) for some n. (Hint: First let D be a maximal abelian subalgebra of A and apply Exercise 4.7.1. Then apply Exercise 3.5.10.) 4.7.6. Assume A is a finite-dimensional C*-algebra. Prove that it is a direct sum of full matrix algebras. (Hint: Apply Exercise 4.7.5 to each minimal ideal of A.) Recall that p ∼ q implies τ (p) = τ (q) for all traces τ . While the converse in general fails, it holds in some well-behaved classes of C*-algebras. The case of UHF algebras was already used earlier. 4.7.7. Assume p and q are projections in an AF algebra A. Prove that if τ (p) = τ (q) for all traces τ then p ∼ q. 4.7.8. Prove that a Bratteli diagram corresponds to an abelian AF algebra if and only if it is a tree. In this case, the corresponding AF algebra is isomorphic to C(X) where X is the set of all branches through this tree. 4.7.9. Prove that a unital abelian algebra is AF if and only if it is of the form C(X) for a zero-dimensional space X. 4.7.10. Prove that every separable LF algebra is AF. (Hint: This is similar to the proof in the case of UHF algebras, i.e., that LM implies AM. See Theorem 4.4. All computations are given in detail in [12].) 4.7.11. Check that if A is a direct sum of n full matrix algebras then T (A) is affinely homeomorphic to the n − 1-dimensional simplex. Given a unital *-homomorphism Φ : A → B between finite-dimensional C*-algebras, use the above to describe which traces in T (A) are Φ∗ -images of traces in B (see Exercise 2.2.10).

page 41

March 11, 2014

10:36

42

WSPC 9 x 6

singapore-final-r

I. Farah

4.7.12. Prove that the Fibonacci algebra has a unique trace. (Hint: Exercise 4.7.11.) 5. The Functor K0 We are ready to define the classifying invariant for AF-algebras, K0 (A). We shall define it only in the unital case since the general case involves some additional technicalities (see e.g., [68]). We first define the Murrayvon Neumann semigroup of a C*-algebra A, denoted by V (A). The underlying set of V (A) is P(M∞ (A))/ ∼. (See §2.5.6 for its definition.) One could consider K ⊗ A instead (see §2.5.6). Since K ⊗ A is the completion of M∞ (A), Lemma 3.3 implies that every projection in K ⊗ A is equivalent to a projection in M∞ (A). Note that a projection p ∈ Mn (A) is Murray-von Neumann equivalent to diag(0, p) ∈ M2n (A) (here 0 is the zero matrix in Mn (A)). This device of ‘moving projections away down the diagonal’ is used to define addition in V (A). For projections p and q in M∞ (A) we can find n such that both p and q belong to Mn (A)  and  define [p] ⊕ [q] in V (A) to be the equivalence   p0 01 class of the projection in M2n (A). By conjugating with we 0q 10   q0 see that this projection is equivalent to . The associativity of ⊕ is 0p also easy to check and therefore ⊕ defines an operation on V (A) that turns it into an abelian semigroup. Note that p + q = r implies [p] ⊕ [q] = [r], but certainly not vice versa; in particular p + q need not be a projection. Recall that a Grothendieck group of a semigroup (V, +) is defined as follows. On V 2 define equivalence relation ≈ via (f, g) ≈ (f ′ , g ′ ) if f + g ′ = f ′ + g. The addition of equivalence classes is defined coordinatewise, by [(f1 , g1 )] + [(f2 , g2 )] = [(f1 + f2 , g1 + g2 )]. This construction results in an abelian group. We define K0 (A) to be the Grothendieck group of (V (A), ⊕). Also, by letting K0+ (A) be the image of V (A) we provide an ordered group structure. Finally, in the unital case K0 (A) is the ordered abelian group with order unit, (K0 (A), K0+ (A), [1A ]), where [1A ] denotes the equivalence class of the identity. A word of caution is due at this point. The ordering on K0 (A) can behave in a very unusual way. In some cases it contains elements p and q

page 42

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

43

such that np > (n + 1)q for some n, while p 6> q. This behaviour can be exhibited by analyzing nontrivial vector bundles (cf. Example 3.10) and it is a key ingredient in the construction of many pathological C*-algebras (see §6.0.1). We shall use an abbreviation and an abuse of notation and write K0 (A) instead of (K0 (A), K0 (A)+ , [1A ]) whenever there is no danger of confusion. 5.1. Computation of K0 in some simple cases 5.1.1. K0 of Mn (C) Since Mn (C) ⊗ K is isomorphic to K, we have that V (Mn (C)) is isomorphic to (N, +) (with 0 ∈ N). Note that V (Mn (C)) and V (Mk (C)) can be distinguished if one keeps track of the ∼-equivalence class of the identity of the algebra, [1A ]. After rescaling, K0 (Mn (C)) becomes (Z[1/n], Z+ [1/n], 1). 5.1.2. K0 of B(H) We are assuming H is infinite-dimensional. Then we have [p] + [1] = [p] for all p, and therefore K0 (B(H)) = {0}. 5.1.3. K0 of the Calkin algebra Calkin algebra, denoted C(H), is the quotient B(H)/K(H). Gelfand–Naimark–Segal theorem implies that it is a C*algebra. All nonzero projections in the Calkin algebra are Murray–von Neumann equivalent. This extends to K ⊗ C(H) and therefore K0 (C(H)) = {0}. 5.1.4. K0 of the CAR algebra By §5.1.1, for every UHF algebra A we have that K0 (A) is a direct limit of copies of Z, with the positive part being exactly the positive integers. For the CAR algebra, the unit of the n-th copy of Z is 2n . Therefore K0 (M2∞ ) is isomorphic to the group of dyadic rationals, {k/2n : k ∈ Z, n ∈ N}, with [1A ] = 1 and the positive part being exactly the positive dyadic rationals. 5.1.5. K0 of other UHF algebras Let A be a UHF algebra corresponding to the generalized integer k. The above argument shows that K0 (A) = {m/k : m ∈ Z and k divides k} with [1A ] = 1 and the natural positive part. Note that for projections p ∈ A the equivalence class [p] exactly corresponds to the normalized trace tr(p).

page 43

March 11, 2014

10:36

WSPC 9 x 6

44

singapore-final-r

I. Farah

5.1.6. K0 of a *-homomorphism If Φ : A → B is a *-homomorphism then by Exercise 3.5.2 it extends to a semigroup homomorphism from V (A) to V (B). If A, B and Φ are unital then Φ can be canonically extended to a *-homomorphism from M∞ (A) to M∞ (B) and we therefore have a homomorphism K0 (Φ) : (K0 (A), K0 (A)+ , [1A ]) → (K0 (B), K0 (B)+ , [1B ]). If Φ is an isomorphism then so is K0 (Φ), but the converse may fail. The following lemma shows that K0 is continuous with respect to inductive limits. Lemma 5.1. If Ai , for i ∈ I and Fij , for i < j is a unital directed system and A = limi Ai then K0 (Ai ), for i ∈ I and K0 (Fij ) for i < j in I is a directed system and K0 (A) = limi K0 (Ai ). Proof. The first claim is an immediate consequence of the above discussion. By Lemma 3.3 every projection in A⊗K is Murray-von Neumann equivalent to an image of a projection in some Ai ⊗K and therefore V (A) = limi V (Ai ) and K0 (A) = limi K0 (Ai ). 5.2. Exercises 5.2.1. Describe K0 (A ⊕ B) in terms of A and B. 5.2.2. Use Exercise 5.2.1 and Lemma 5.1 to show that K0 of an AF algebra is a direct limit of groups of the form Zn(i) , for n(i) ∈ N, with their natural ordering. 5.3. Cancellation property An abelian semigroup (S, +) has the cancellation property if x+y = z+y implies x = z. This is equivalent to stating that in the Grothendieck group of (S, +) no two distinct elements of S belong to the same equivalence class. A C*-algebra A has the cancellation property if its Murray-von Neumann semigroup has cancellation property. Lemma 5.2. A direct limit of algebras with cancellation property has cancellation property. Proof. Assume A = limn An does not have cancellation property. Not having cancellation property is witnessed by the following objects in K ⊗ A: three projections, p, q and r, one partial isometry, v, such that vv ∗ = p + r and v ∗ v = q + r and the absence of a partial isometry w such that ww∗ = p and w∗ w = q. By Lemma 3.3, we may assume p, q and r all belong to

page 44

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

45

Mn (An ) for a large enough n. By Lemma 3.5, we may assume that v also belongs to Mn (An ). Therefore An does not have cancellation property. Lemma 5.3. Every AF algebra has cancellation property. Proof. By Lemma 5.2 it suffices to show that every finite-dimensional C*algebra A has cancellation property. This can be proved in a variety of ways. For example, V (A) is isomorphic to the free abelian semigroup with k generators where k is the number of full matrix direct summands of A, and this semigroup does not have cancellation property. In the category of K0 -groups homomorphisms are group homomorphisms that preserve both positivity and the unit. Note that if p, q and r are projections in M∞ (A) then p + q = r implies [p] + [q] = [r]. The converse is false since p + q need not be a projection. The following weak converse indicates why K0 is defined in P(M∞ (A)) instead of P(A). Lemma 5.4. If p and q are projections in M∞ (A) then there exists q ′ ∈ M∞ (A) such that q ∼ q ′ and p + q is a projection. Proof. Let  n besuch  that p and q both belong to Mn (A) andidentify  p0 q0 01 them with and , respectively, in M2n (A). Let v be (a 00 00 10   00 block-matrix in M2n (A)). Then q ′ = (Ad v)q = is as required. 0q Lemma 5.5. Assume A is finite-dimensional and B has cancellation property. Then every homomorphism of preordered abelian groups with order unit φ : (K0 (A), K0 (A)+ , [1A ]) → (K0 (B), K0 (B)+ , [1B ]) is of the form K0 (Φ) for a unital *-homomorphism Φ : A → B. Proof. Choose n such that there are minimal projections pi , for i < n, P in A satisfying i
page 45

March 11, 2014

10:36

WSPC 9 x 6

46

singapore-final-r

I. Farah

Now we can shed the scaffolding provided by M∞ (B) and work in B. Also, ri ∼ rj if and only if pi ∼ pj . Let ≈ be an equivalence relation on n defined by i ≈ j if and only if pi ∼ pj in A. For each such pair choose a partial isometry wij ∈ A such that ∗ ∗ (1) wij wij = pi and wij wij = pj , and (2) wij wkl = wil δjk

for all i, jk, l. We can recursively choose partial isometries vij for all i ≈ j that, together with ri for i < n, generate a finite-dimensional unital subalgebra of B isomorphic to A and satisfy equalities corresponding to the above. Now define Φ : A → B by Φ(pi ) = ri , Φ(wij ) = vij for i ≈ j and extend it linearly. Then Φ : A → B is a unital *-homomorphism and K0 (Φ) = φ. 5.4. Classification of AF algebras A dimension group is a direct limit of ordered groups with order unit of the form (Zn , (Z+ )n , [e]) where e ∈ (Z= )n . It is an easy consequence of Lemma 5.1 that K0 of a separable unital AF algebra is a dimension group. The converse is also not difficult to prove: a countable ordered group is a dimension group if and only if it is equal to K0 of a separable unital C*-algebra (actually one still has the equivalence if both separability and countability are dropped). A first-order characterization of dimension groups was given by Effros–Handelman–Shen (see [16] or [12]). Lemma 5.6. Assume A = limn An and B = limn Bn . Also assume φn : An → Bn and ψn : Bn → An+1 are *-homomorphisms such that the diagram in Figure 10 commutes. A1 φ1 B1

A2 ψ1

φ2 B2

A3 ψ2

φ3

A4 ψ3

B3

...

A

...

B

φ4 B4

Fig. 10.

Then A is isomorphic to B. Proof. Let Fn : An → A denote the canonical *-homomorphism for each n ∈

page 46

March 11, 2014

10:36

WSPC 9 x 6

singapore-final-r

Logic for C*-Algebras

47

N. Since the diagram commutes, φ(a) = φn (Fn a) if a ∈ An is a well-defined *-homomorphism from a dense subset of A into B. Since each φn is a contraction, so is φ and therefore φ extends to a *homomorphism from A into B. One analogously defines ψ : B → A as a direct limit of ψn for n ∈ N. Since ψ ◦ φ is idA and φ ◦ ψ is idB , we conclude that φ and ψ are *-isomorphisms. Two remarks regarding Lemma 5.6 are in order. First, maps φn and ψn are not required to be isomorphisms or even injections. All we need is commutation of the diagram. Second, this lemma is about direct limits of arbitrary structures. Proof of the following lemma is very similar to the proof of Lemma 5.6. Lemma 5.7. Assume A = limn An and B = limn Bn . Also assume φn : An → Bn are *-homomorphisms such that the diagram in Figure 11 commutes. A1 φ1 B1

A2 φ2 B2

A3 φ3

A4

...

A

...

B

φ4

B3

B4

Fig. 11.

Then there is a *-homomorphism from A to B that extends

S

n

φn .

The assumption that the algebra be unital is not needed in Theorem 5.8 or in Theorem 5.9. However, the definition of K0 of a nonunital algebra is not given in these notes. Theorem 5.8 (Elliott, [18]). Two separable unital AF algebras are isomorphic if and only if their ordered K0 groups are isomorphic. Proof. Write A = limn An and B = limn Bn , where all An and all Bn are finite dimensional and all connecting maps are unital. Then K0 (A) = limn K0 (An ) and K0 (B) = limn K0 (Bn ). Let φ : K0 (A) → K0 (B) be an ordered unit group isomorphism. Each K0 (An ) is finitely generated, and

page 47

March 11, 2014

10:36

WSPC 9 x 6

48

singapore-final-r

I. Farah

therefore we can go to a subsequence of An and Bn so that, after reenumerating, we have that φ sends K0 (An ) into K0 (Bn ) and φ−1 sends K0 (Bn ) into K0 (Bn+1 ) for all n. The plan is to apply Lemma 5.6 with appropriately chosen *homomorphisms Φn and Ψn , for n ∈ N. By applying Lemma 5.5 for each n in both directions we can recursively choose a *-homomorphism Φn : An → Bn and a *-homomorphism Ψ′n : Bn → An which lift the corresponding maps between K0 groups. Assume Φn and Ψ′n were chosen. We shall modify Ψ′n to make the triangle between An , Bn and An+1 in the diagram from Lemma 5.6 commute. We have two *-homomorphisms from An to An+1 , namely Fn,n+1 : An → An+1 given by the directed system and Ψ′n ◦Φn . Lemma 4.7 applies to this pair and gives a unitary un in An+1 such that Ad un ◦ Ψ′n ◦ Φn = Fn,n+1 . Then Ψn = Ad u ◦ Ψ′n is as required. Once all Φn and Ψn are chosen to make the whole diagram commute, Lemma 5.6 implies A ∼ = B. The above proof shows a bit more. Let us say that two *homomorphisms Φj : A → B, for j = 0, 1 are approximately unitarily equivalent if there is a sequence of unitaries un , for n ∈ N, in B such that Φ0 ◦ Ad un converges to Φ1 pointwise. Theorem 5.9 (Elliott, [18]). If A and B are separable unital C*-algebras then for every positive unital group homomorphism Φ : (K0 (A), K0 (A)+ , [1A ]) → (K0 (B), K0 (B)+ , [1B ]) there exists a unital *-homomorphism Φ : A → B such that φ = K0 (Φ). Moreover, K0 (Φ) = K0 (Ψ) if and only if Φ and Ψ are approximately unitarily equivalent. Moreover, if φ is an isomorphism then so is Φ. Proof. The proof of the first statement is similar to the above and uses Lemma 5.7. The approximate unitary equivalence of *-homomorphisms whose K0 coincide is a consequence of Lemma 4.7 applied along the finite stages of the diagram. The last sentence is Theorem 5.8. 5.5. Exercises 5.5.1. Prove that K0 (A) is countable if A is separable. (Hint: Lemma 3.11.) 5.5.2. Prove that if A has cancellation property and p, q are projections in A such that p ∼ q, then 1 − p ∼ 1 − q. Find a C*-algebra in which this is not true, and which therefore does not have the cancellation property.

page 48

March 11, 2014

10:36

WSPC 9 x 6

singapore-final-r

Logic for C*-Algebras

49

(Hint: For the second part use Example 3.9 (1).) 5.5.3. Prove that we could have defined V (A) and K0 (A) in P(A ⊗ K) (see §2.5.6) instead of P(M∞ (A)). (Hint: Lemma 3.11.) 5.5.4. Compute K0 of the UHF algebra corresponding to the generalized Q k(j) integer j pj . √ 5.5.5. Prove that K0 of the Fibonacci algebra is (with s = (1 + 5)/2) (Z2 , {(m, n) : sm + n ≥ 0}, (1, 0)). 5.5.6. Prove the assertions made in §5.1.6. 5.5.7. A state on a preordered abelian group with order unit (G, G+ , [1]) is a homomorphism χ : G → R such that χ[G+ ] ⊆ R+ and χ(1) = 1. Prove that a trace τ ∈ T (A) defines a state of (K0 (A), K0 (A)+ , [1A ]). 5.5.8 (Approximate intertwining, aka Elliott intertwining). Assume A = limn An and B = limn Bn . Furthermore assume Fn ⊆ An , for n ∈ N, is an increasing sequence of finite subsets of A with dense union, and Gn ⊆ Bn , for n ∈ N is an increasing sequence of finite subsets of B with dense union. Finally, assume φn : An → Bn and ψn : Bn → An+1 are *homomorphisms such that the n-th triangle of the diagram in Figure 12 commutes up to 2−n on both Fn and Gn . A1 φ1 B1

A2 ψ1

φ2 B2 Fig. 12.

A3 ψ2

φ3 B3

A4 ψ3

...

A

...

B

φ4 B4

An approximate intertwining argument.

Prove that A ∼ = B. 5.5.9. Prove that K0 (C([0, 1], A) is isomorphic to K0 (A) for all A. 6. Elliott’s Program Around 1990, George Elliott conjectured that separable, unital, nuclear, simple C*-algebras can be classified by K-theoretic invariants. Nuclear C*algebras can be defined in several equivalent but apparently rather different

page 49

March 11, 2014

50

10:36

WSPC 9 x 6

singapore-final-r

I. Farah

ways (e.g., Definition 2.9), but for our purposes the actual definition is irrelevant. See e.g., [4] or [70]. The Elliott invariant of an algebra A is the sixtuple Ell(A) = (K0 (A), K0 (A)+ , [1A ]0 , K1 (A), T (A), rA : T (A) → S(K0 (A)). We have already defined K0 group and the tracial space T (A). Since p ∼ q implies τ (p) = τ (vv ∗ ) = τ (v ∗ v) = τ (q), every trace τ on A defines a state (unital, positive, additive map into R) on K0 (A), and rA is the coupling map that associates states to traces (see Exercise 5.5.7). K1 (A) is an another countable (in case when A is separable) abelian group (see [70]). Fix a family of C*-algebras B. We would like to consider it as the family of building blocks of a larger family of C*-algebras. This can be formalized in (at least) two ways. Definition 6.1. A C*-algebra is an AB-algebra if it is a direct limit of a directed system of algebras in B. Example 6.2. (1) If B consists of all full matrix algebras Mn (C) for n ∈ N, then AB-algebras are UHF algebras. (2) if B consists of all finite-dimensional C*-algebras, then AB-algebras are the approximately finite, or AF, algebras. (3) Recall that T = {z ∈ C : |z| = 1}. Let B be the class of all direct sums of algebras of the form C(T, Mn (C)). Then we arrive at the class of AT algebras (see [70]; T stands for T). (4) If B consists of all algebras of the form C([0, 1], Mn (C)) and their direct sums, then we have the class of AI algebras (see [70]). (5) Take the class of all algebras of the form C(X, Mn (C)) where X is a compact metric space and close it under direct sums and corners (§2.5.9) to obtain B. Then AB is the class of AH algebras (H stands for ‘homogeneous’) (see [70]). After several spectacular successes ([65], [54], [22], [55], [21], [70]), counterexamples to Elliott’s conjecture were found by Rørdam and Toms ([71] and [80]). While these examples rule out functorial classification as conjectured by Elliott, they do not give information about the descriptive complexity of the isomorphism relation of separable nuclear simple unital C*-algebras. In recent years Elliott’s program has been revitalized by influx of new ideas, including invariants such as the Cuntz semigroup (see [6], [9]), regularity properties such as Z-stability ([87]), and a technical tour de force

page 50

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

51

([23]), among other developments. See [20] for a survey and the upcoming [88] for the current state of the art. 6.0.1. Failure of cancellation The following example was first used by Villadsen ([83]) and it appears in one form or another in all known counterexamples to Elliott’s program. Recall that S 2 is the unit sphere in three-dimensional (real) Euclidean space and consider the C*-algebra A = C(S 2 , M2 (C)) (see §2.5.8). By Exercise 2.6.9, projections in A are continuous maps from S 2 into P(M2 (C)), the space of projections in M2 (C). Apart from the ‘trivial’ projections corresponding to constant maps, this algebra has nontrivial projections. Recall that CP 1 is the complex projective space, the space of all lines in C2 . It is homeomorphic to the space P(M2 (N)) of all 1-dimensional projections in M2 (C). It is also homeomorphic to S 2 . The Bott projection p in A corresponds to a natural homeomorphism of S 2 onto P(M2 (C)) (i.e., to the Hopff vector bundle). It is not Murray-von Neumann equivalent to the trivial rank one projection, but there is a partial isometry v ∈ A such that p + vpv ∗ = 1. Villadsen used Bott projection and a clever direct limit construction to construct an AH algebra A such that K0 (A) contains a non-positive element x such that nx is positive for some n ([83]). 6.1. Exercises Exercises given below illustrate basic constructions of AH algebras. See [82] for analysis of the classification problem for such algebras. In the following exercises X and Y are compact metric spaces, and the property of being metric is mostly unnecessary. 6.1.1. Prove that T (C(X, Mn (C)) is affinely homeomorphic to P(X), the space of Borel probability measures on X (cf. Exercise 3.7.3), where to a measure ν one associates trace Z τν (f ) = tr(f (x))dν(x). 6.1.2. Let An = C(X, M2n (C)). If xn,j , for j < n are points in X define a unital *-homomorphism Φn : An → An+1 by the block matrix Φn (f ) = diag(f, f (xn,0 ), f (xn,1 ), . . . , f (xn,n−1 )). (i) Prove that the direct limit of this system A is simple if for every m the S set n≥m {xn,j j < n} is dense in X.

page 51

March 11, 2014

10:36

52

WSPC 9 x 6

singapore-final-r

I. Farah

(ii) With Φ∗ as in Exercise 2.2.10 prove that for a trace τ ∈ T (An+1 ) τ = Φ∗ (τ ) satisfies 1 X 1 tr(f (xn,j ). τ ′ (f ) = τ (f ) + 2 2n j
Assume m = nk and φj : Y → X, for j < k, are continuous maps. Then a unital *-homomorphism Φ : C(X, Mn (C)) → C(Y, Mm (C)) is defined by Φ(f ) = diag(f ◦ φ0 , f ◦ φ1 , . . . , f ◦ φk−1 ). A *-homomorphism of this form is a standard *-homomorphism with characteristic functions φj , for j < n. 6.1.3. Given a standard *-homomorphism Φ : A → B as above, describe the map Φ∗ : T (B) → T (A). 6.1.4. Assume An = C([0, 1], Mk(n) (C)) and Φn : An → An+1 is a standard unital *-homomorphism and let A = limn An . Prove that K0 (A) is isomorphic (as a preordered abelian group with order unit) with K0 of the UHF algebra limn Mk(n) (C). (Hint: Recall that the projections in C([0, 1], Mn (C)) correspond to continuous functions from [0, 1] into P(Mn (C)), and therefore have well-defined rank. In order to prove that p ∼ q if and only if their ranks coincide, one uses Lemma 3.5 together with the compactness and homotopic triviality of [0, 1].) 6.1.5. Let An = C([0, 1], M2n (C)) and let Φn : An → An+1 be the standard *-homomorphism with characteristic functions φ0 (x) = x/2 and φ1 (x) = (1 + x)/2. Let A = limn An . (i) Prove that A is simple. (ii) Compute K0 (A). (iii) Prove that A has a unique trace. By the last exercise, Exercise 5.5.7, and the fact that K1 of both A and the CAR algebra is trivial, one has that Ell(A) ∼ = Ell(M2∞ ). It is then a consequence of Elliott’s classification of simple AI algebras ([21]) that A is isomorphic to the CAR algebra. 6.1.6 (Dimension drop algebras). Fix m and n in N and identify Mmn (C) with Mm (C) ⊗ Mn (C) (Exercise 2.6.11). Consider the following subalgebra of C([0, 1], Mmn (C)). Zm,n = {f : f (0) ∈ Mm (C) ⊗ 1n and f (1) ∈ 1m ⊗ Mn (C)}

page 52

March 11, 2014

10:36

WSPC 9 x 6

singapore-final-r

Logic for C*-Algebras

53

Prove that the only projections in Zm,n are 0 and 1 (i.e., “Zm,n is projectionless”) if and only if m and n are relatively prime. 6.1.7. Prove that T (Zm,n ) is affinely homeomorphic to P([0, 1]), the space of probability measures on [0, 1]. (Hint: See Exercise 6.1.1.) 6.1.8. Construct a direct limit of projectionless dimension drop algebras that is simple and has a unique trace. (Hint: For the trace part keep in mind Exercise 2.2.10(2).) The solution to Exercise 6.1.8 is uniquely defined up to the isomorphism. It is the notorious Jiang–Su algebra Z ([45]). Its Elliott invariant is equal to the Elliott invariant of C, and moreover Ell(Z ⊗ A) = Ell(A) for all well-behaved (as well as many misbehaved) C*-algebras A. A C*-algebra A is Z-stable if A ⊗ Z ∼ = A, and by the above one can only hope to classify Zstable algebras by their Elliott invariants. By remarkable results of Wilhelm Winter, this is true in a number of cases ([82], [86], [87]). 7. Abstract Classification Main references for the remaining sections are [33] and [32]. Recall that a topological space X is Polish if it is separable and completely metrizable. A subset of X is analytic if it is a continuous image of a Borel set in some Polish space. An equivalence relation E on X is analytic if it is an analytic subset of X 2 . The theory of abstract classification can be traced back to the work of Mackey on classification of representations of locally compact metrizable groups, and in particular to the following. Definition 7.1 (Mackey). An equivalence relation E on X is smooth if there is a Borel-measurable f : X → R such that x E y iff f (x) = f (y). Example 7.2. Similarity of n × n complex Hermitian matrices is smooth. This is because one can associate to M the list of its eigenvalues (in the increasing order, with multiplicities). Smooth equivalence relations are effectively classifiable. Example 7.3. Classification of rank 1 torsion-free abelian groups.

page 53

March 11, 2014

10:36

54

WSPC 9 x 6

singapore-final-r

I. Farah

These are exactly the subgroups of Q. To every such group Γ we can Q k(j) associate a generalized integer j pj (see §4.1) as follows. Choose a ∈ Γ and consider {n : (∃b ∈ Γ)nb = a}. This set is the set of all divisors of a generalized integer k(Γ, a). Note that k depends on the choice of a, but only up to a finite factor. A straightforward recursive construction shows that Γ1 and Γ2 are isomorphic if and only if for some (equivalently, all) a1 ∈ Γ1 and a2 ∈ Γ2 the corresponding generalized integers coincide up to a finite factor, i.e., there exist nonzero m1 and m1 in N such that m1 k(Γ1 , a1 ) = m2 k(Γ2 , a2 ).d One could prove that the equivalence relation in Example 7.3 is not smooth. However, it is generally accepted as a simple and natural classification result. In order to compare the complexity of equivalence relations on Polish spaces (and therefore of different classification problems), [36] and [40] have independently introduced the following definition Definition 7.4. Assume E, F are equivalence relations on Polish spaces X, Y , respectively. Then E is Borel reducible to F , or E ≤B F , if there is a Borel-measurable map f : X → Y such that x E y ⇔ f (x) F f (y). Note that E ≤B = R if and only if E is smooth. The relation E ≤B F can be interpreted in the following ways. (1) Borel cardinality of X/E is ≤ than the Borel cardinality of Y /F . (2) Classification problem for E is simpler than the classification problem for F . (3) F -equivalence classes are complete invariants for E-equivalence classes. Thesis 7.5. Almost all classical classification problems deal with analytic equivalence relations on Polish spaces. Thesis 7.6. In almost all cases, the space of invariants has a Polish topology and the computation of invariants is given by a Borel-measurable function. In order to verify a particular instance of these two theses, one needs to (i) Define a Polish space X whose elements correspond to objects that need to be classified, d We

consider generalized integers as formal infinite products.

page 54

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

55

(ii) Define a Polish space Y whose elements correspond to the intended invariants, (iii) Check that the map that associates invariant I(x) to each object x ∈ X is Borel. Once all three steps are verified, the particular classification problem is subject to the well-developed abstract theory of classification (e.g., [44] and [15]), and one can proceed to estimate the complexity of this classification problem. 7.1. Effros Borel space A standard Borel space is a pair (X, Σ) where X is a set and Σ is a σalgebra of subsets of X with the property that X carries a Polish topology τ such that Σ is the corresponding σ-algebra of Borel sets. The actual topology on X is irrelevant for our purposes and by a classical result of Kuratowski all uncountable Polish spaces are Borel-isomorphic. Therefore, considering a standard Borel space instead of a Polish space as the ambient space for our classification problems makes our setting more canonical. However, if the topology on X is particularly natural we shall use it. 7.1.1. Spaces of countable structures Fix any countable signature σ. The set Mσ of all countable models of σ can naturally be identified with a subset of P(N), and the latter carries the compact metric topology. For details see [3]. A case of particular interest for us is when σ is the signature of ordered groups with order unit. This is worked out in [32]. The isomorphism relation on this space is analytic. To see this, note that the space of all triples (A, B, f ) where A and B are in Mσ and f : N → N is an isomorphism between A and B is closed. The set {(A, B) ∈ Mσ : A ∼ = B} is the projection of this set of triples and therefore analytic. For example, the space G of preordered countable groups with order unit looks as follows. The space G consists of all triples (F, G, H) which code (K, K + , [1]) as follows. We may assume that the underlying set of K is N, and F ⊆ N3 is the set {(m, n, k) : m +K n = k}. Also G = K and H = 1K . 7.1.2. Compact metric spaces Every compact metric space is homeomorphic with a subspace of the Hilbert cube [0, 1]N . Thus the space K([0, 1]N ) of all closed subsets of [0, 1[N is the space of all compact metric spaces. The Hausdorff metric on this space, dH (K, L) = inf (K ⊆ε L and L ⊆ε L) ε≥0

page 55

March 11, 2014

56

10:36

WSPC 9 x 6

singapore-final-r

I. Farah

turns it into a compact metric space. The homeomorphism relation on this space is analytic. To see this, identify a homeomorphism f between K and L with its graph. This graph is a closed subset of the square and the set of all triples (K, L, f ) such that K and L are compact subsets of [0, 1]N and f : K → L is (with the above identification) a closed subset of the compact metric space K([0, 1]N )2 × K(([0, 1]N )2 ). Like in the previous example, its continuous image is analytic. 7.1.3. Separable Banach spaces In order to define a Borel space of all separable Banach spaces we shall need some lemmas. All Banach spaces are real, although the complex case neither involves any additional complexity nor brings additional simplicity (as in the case of C*-algebras). Lemma 7.7. Every separable Banach space is isometric with a subspace of C({0, 1}N ) of all continuous functions from the Cantor space into R. Proof. This is related to Exercise 2.2.10. A Banach space X is isometric to a subspace of a Banach space Y if and only if there is an affine continuous surjection f from the unit ball of Y ∗ onto the dual ball of X ∗ . By the Birkhoff–Alaoglu theorem the unit ball of a separable Banach space is compact in the weak*-topology. We can therefore fix a continuous surjection of {0, 1}N onto the unit ball of X ∗ and extend it affinely to the unit ball of C({0, 1}N ). Let Z = C({0, 1}N) and let F (Z) denote the space of all closed subsets of Z. If Y is not locally compact, the Hausdorff metric used in §7.1.2 will not be separable, In order to define a Borel structure on F (Z) we use the following result. Theorem 7.8 (Effros). If Y is a Polish space then the space F (Y ) of closed subsets of Y with respect to the σ-algebra generated by the sets {K ∈ F (Y ) : K ∩ U 6= ∅} for U ⊆ Y open is a standard Borel space. Proof. The idea is to fix a metric compactification Y of X and take advantage of the fact that a subspace of a Polish space is Polish if and only if it is Gδ , in both directions. For details see [48, §12.C]. Since it is not difficult to check that for a Banach space Z the set {Y ∈ F (Z) : Y is a linear subspace} is Effros-Borel, we have a standard Borel space of all separable Banach spaces. There are (at least) three natural equivalence relations on this space:

page 56

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

57

(i) (linear) isometry, (ii) isomorphism (i.e., the existence of a linear homeomorphism) and (iii) bi-embeddability (the existence of linear isometric embedding of X into Y and a linear isometric embedding of Y into X). It is again not difficult to check that all three relations are analytic. Complexities of these equivalence relations were computed in [57] and [35]. 7.1.4. von Neumann algebras with a separable predual (A reader not familiar with these may want to skip this paragraph.) Every such von Neumann algebra is isomorphic to a weakly closed subalgebra of B(H) for a separable complex Hilbert space X. Since B(H) is weakly separable, one can consider this space with respect to the Effros Borel structure. However, unlike in the case of Banach spaces, this space carries a natural Polish topology called Effros–Mar´echal topology. See [74] and [85] for more. 7.1.5. Separable C*-algebras In both cases of Banach spaces and von Neumann algebras there exists a universal separable object (C({0, 1}N ) and B(H), respectively hence the Effros Borel structure provides the setting for analysis of classification problems. This is not the case with C*-algebras. By a result of Junge and Pisier ([47]) there is no universal separable C*algebra. However, there are at least two different Borel spaces of separable C*-algebras. The following space was defined by Kechris ([49]). It takes advantage of a slight refinement of the GNS theorem (Theorem 2.3), to the effect that every separable C*-algebra is isomorphic to a subalgebra of B(H) for the separable Hilbert space H. The space B(H) becomes a standard Borel space when equipped with the Borel structure generated by the weakly open subsets. Let Γ = B(H)N , and equip this with the product Borel structure. For each γ ∈ Γ(H) we let C ∗ (γ) be the C*-algebra generated by γ. If we identify each γ ∈ Γ(H) with C ∗ (γ), then Γ(H) parameterizes all separable C*-algebras acting on H. This gives us a standard Borel parameterization of the category of all separable C*-algebras. The relation γ1 ∼ γ2 is an analytic equivalence relation (see [32] or Exercise 7.4.2). The above can be considered as the space of concrete separable C*algebras. One can also define the space of abstract separable C*-algebras and prove that there is a Borel isomorphism between these spaces that respects the corresponding isomorphism relations (see [32]).

page 57

March 11, 2014

58

10:36

WSPC 9 x 6

singapore-final-r

I. Farah

7.2. Computation of the Elliott invariant is Borel By §7.1.5 and §7.1.1 we have standard Borel spaces Γ of separable C*algebras and G countable ordered groups with order unit, respectively. The following lemma (and corresponding facts for other C*-algebraic invariants) taken from [32] largely justifies taking the descriptive set theoretic view of Elliott’s program. Let Γ1 = {γ ∈ Γ : γ(0) = 1}—the (clearly Borel) space of unital separable C*-algebras. Lemma 7.9. There is a Borel map K : Γ1 → G such that K(γ) is isomorphic to (K0 (C ∗ (γ)), K0 (C ∗ (γ)+ , [1A ]). Sketch of the proof. The details can be found in [32]. First we need a Borel map P that sends Γ to Γ so that P(γ) enumerates a countable dense set of projections in P(M∞ (C ∗ (γ))). For simplicity, we shall instead only construct P1 such that P1 (γ) enumerates a countable dense set of P(C ∗ (γ)). Let pn , for n ∈ N, enumerate all *-polynomials over Q + iQ in variables xj , for j ∈ N, with the property that pn (x) = pn (x)∗ . Then {pj (γ) : j ∈ N} enumerates a countable dense subset of the set of self-adjoint operators in C ∗ (γ). Let f : R → R be any function whose iterates, f n for n ∈ N, uniformly converge to some function g such that g(x) = 0 if x ≤ 1/4 and g(x) = 1 if x ≥ 3/4. If sp(pj (γ)) ∩ [1/4, 3/4] = ∅ then f n (pj (γ)) converges to a projection in the norm topology. A verification that P1 (γ) = (qj : j ∈ N) with qj = limn f n (pj (γ)) if this sequence is norm-convergent and qj = 0 otherwise is a Borel map is straightforward. Clearly, P(γ) is an enumeration of a dense set of projections in C ∗ (γ). By Lemma 3.11, the range of P(γ) intersects all Murray–von Neumann equivalence classes in P(M∞ (C ∗ (γ)). We need to check that the map γ 7→∼γ that associates a binary relation on N to γ such that m ∼γ n if and only if projections P(γ)(m) and P(γ)(n) are Murray-von Neumann equivalent in M∞ (C ∗ (γ)). But this is a consequence of the fact that we have an effective enumeration of a dense subset of C ∗ (γ) akin to pj (γ), for j ∈ N, and Exercise 3.5.8. Similar arguments show that the operation ⊕ and the Groethendieck construction can be effectively defined on P(γ). This gives a Borel map that sends γ to an element of the Borel space G of preordered countable groups with order unit that codes (K0 (C ∗ (γ)), K0 (C ∗ (γ))+ , [1C ∗ (γ) ]).

page 58

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

59

The following was proved in [32]. Theorem 7.10. There is a standard Borel space of Elliott invariants Ell and there is a Borel map Φ : Γ → Ell such that Φ(γ) represents Ell(C ∗ (γ)). 7.3. Comparing complexities of analytic equivalence relations In this section we introduce some important classes of analytic equivalence relations. 7.3.1. Relation E0 On {0, 1} define x E0 y if (∀∞ n)x(n) = y(n). This equivalence relation is Fσ and a simple Baire category argument shows that it is not smooth. By [40], E0 is the minimal non-smooth Borel equivalence relation. (It is, however, not the minimal non-smooth analytic equivalence relation; see Exercise 7.4.3.) It should be noted that the combinatorial essence of this result, called Glimm–Effros dichotomy, first appeared in Glimm’s [38] as a device for embedding M2∞ into C*-algebras with non-smooth dual (non-type I C*-algebras) as a subquotient. 7.3.2. Essentially countable equivalence relations A Borel equivalence relation all of whose classes are countable is said to be countable. A Borel equivalence relation which is Borel-reducible to a countable equivalence relation is essentially countable. By the above, E0 is the ≤B -minimal essentially countable equivalence relation. Orbit equivalence relation of the shift action of the free group on infinitely many generators on its power-set the ≤B -maximal essentially countable equivalence relation and by a result of Adams and Kechris the ≤B ordering on essentially countable equivalence relations has a very rich structure ([1]). 7.3.3. Countable structures An equivalence relation E is classifiable by countable structures if E is Borel-reducible to the isomorphism relation of countable structures in some countable signature (see §7.1.1). By a result of Hjorth–Kechris–Louveau, this is equivalent to E being Borel-reducible to an orbit equivalence relation of a continuous action of a closed subgroup of S∞ on P(N). The ≤B -maximal analytic equivalence relation in this class is the graph isomorphism relation. 7.3.4. Orbit equivalence relations If G is a Polish group that acts continuX ously on a Polish space X, then EG is the orbit equivalence relation, x EX G y iff (∃g ∈ G)g.x = y.

page 59

March 11, 2014

10:36

60

WSPC 9 x 6

singapore-final-r

I. Farah

Being a projection of the closed set {(x, y, g) : g.x = y}, it is clearly analytic. By a result of Becker and Kechris ([3]), considering Borel actions instead of continuous ones does not result in a larger class of equivalence relations. This class also has the ≤B -maximal equivalence relation. It is the shift action of the isometry group G of the Urysohn space on the Effros Borel X∞ space of its closed subsets, denoted EG . ∞ 7.3.5. Turbulence Being classifiable by countable structures is strictly weaker than being below a Polish group action. This is a result of Hjorth ([43]) and it utilizes the notion of turbulence, a dynamical property of group action. This is the main tool used in §8.1. 7.3.6. The dark side On [0, 1]N define x E1 y if (∀∞ n)x(n) = y(n). This equivalence relation is Fσ . By [50], E1 is an immediate ≤B successor of E0 and it is not Borel-reducible to any orbit equivalence relation of a Polish group action. Therefore being ‘above E1 ’ is a measure of an equivalence relation not being simply classifiable. In [72] Rosendal isolated the ≤B -maximal Kσ equivalence relation, EKσ . Its underlying space is the space of nondecreasing functions in NN and we let f EKσ g if and only if there is n such that f (j) ≤ g(j)+n and g(j) ≤ f (j)+n for all n. There exists a maximal analytic equivalence relation EΣ11 , isolated in [57]. Both bi-embeddability of separable Banach spaces ([57]) and the isomorphism of separable Banach spaces ([35]) are bireducible with the maximal analytic equivalence relation. 7.4. Exercises 7.4.1. Compare classification of rank 1 torsion-free abelian groups (Example 7.3) with the classification of UHF algebras (§4.1) and nonunital separable direct limits of full matrix algebras (Exercise 4.3.7). 7.4.2. Prove that both the isomorphism and bi-embeddability of C*algebras are analytic subsets of the square of Kechris’s space Γ (§7.1.5). 7.4.3. On the space of compact subspaces of the Hilbert cube defined in §7.1.2 consider the equivalence relation x E y iff either both x and y are uncountable or both x and y are countable and homeomorphic. Then show that this relation is neither smooth nor ≥B E0 . (Hint: Count the equivalence classes.) The following exercise gives a Borel space of UHF algebras.

page 60

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

61

7.4.4. Prove that there exists a universal separable UHF algebra Q. Also prove that not all unital infinite-dimensional subalgebras of this algebra are UHF, but its UHF subalgebras form a Borel subset of F (Q). 7.4.5. Do the nonunital case of Exercise 7.4.4. 8. Estimating the Complexity of the Isomorphism of C*-Algebras Upper and lower bounds for the complexity of the isomorphism relation for C*-algebras were proved in [33]. 8.1. Turbulence: A lower bound for complexity Dynamical properties of Polish group actions affect the complexity of the orbit equivalence relation. We shall use the following classical result as a warmup. Proposition 8.1. Assume G y X is a continuous action of a Polish group on a Polish space such that (1) every orbit is dense, and (2) every orbit is meager. Then the orbit equivalence relation is not smooth. Proof. Assume f : X → R is a Borel map such that x EX G y implies f (x) = f (y). Let Y be a dense Gδ subset of X such that the restriction of f to Y is continuous. Since x 7→ g.x is a homeomorphism for all g ∈ G, the set {(g, x) : g.x ∈ Y } is by Kuratowski–Ulam theorem comeager in G × X. Therefore there exists x ∈ X such that g.x ∈ Y for comeager many g ∈ G. But this implies that the intersection of the orbit of x with Y is dense in Y . Since f is constant on this set, it is by the continuity constant on Y . X Since all orbits are meager, f cannot be a reduction of EG to =R . Let G y X be a continuous action of a Polish group on a Polish space. Fix x ∈ X, a symmetric open neighborhood U of eG and an open neighbourhood V of x. On V define a graph by letting {x, y} be an edge if x 6= y and there exists g ∈ U such that g.x = y. Since U = U −1 , this relation is symmetric. Let O(x, U, V ) be the connected component of x in this graph. Then O(x, U, V ) is the set of points that can be reached from x by taking small steps (smallness being measured by U ) while staying inside V . Definition 8.2 (Hjorth, [43]). An action G y X as above is turbulent if

page 61

March 11, 2014

10:36

62

WSPC 9 x 6

singapore-final-r

I. Farah

(1) every orbit is dense, (2) every orbit is meager, and (3) the closure of every local orbit of every x ∈ X has a nonempty interior. In the presence of (1) and (2), (3) above is equivalent to the assertion that the closure of every local orbit intersects every G-orbit (exercise!). There are some other equivalent reformulations of the definition, and for all purposes it suffices to assume that the set of points x satisfying (3) is comeager (such actions are generically turbulent ). X Theorem 8.3 (Hjorth, [43]). If the action G y X is turbulent then EG is not classifiable by countable structures.

Example 8.4. (1) Consider c0 as an additive group. It is a Polish group with respect to its norm topology. Then the shift action of ℓ2 on RN is turbulent. In order to prove this, fix x = (xn ), U, V as above and fix y ∈ RN . We shall prove that the orbit of y intersects the closure of O(x, U, V ). Then there exist ε > 0 and k such that U ⊇ {(gn ) ∈ c0 : |gn | < ε for all n} and V ⊇ {(zn ) ∈ RN : |xn − zn | < ε for all n < k}. Let z = (zn ) ∈ RN be such that zn = xn for n < k and zn = yn for n ≥ k. Then z ∈ V ∩ [y]. We shall prove that z ∈ O(x, U, V ). Fix m ∈ N and let K = maxn≤m |xn − zn |. If j > K/ε, then we can find a sequence x = z 0 , z 1 , . . . , z j such that the first m coordinates of z j coincide with the first m coordinates of z and z i − z i+1 ∈ U for all i < j. Therefore all z i belong to O(x, U, V ) and since m was arbitrary z is an accumulation point of O(x, U, V ). (2) The above proof shows that the action of any classical Banach space ℓp , for p ≥ 1 on RN is turbulent, (3) The following example will be used later. Consider the Polish group G = ZN and let G0 = {g ∈ G : lim

n→∞

g(n) = 0}. n

Then G0 is a Polish subgroup of G, and the coset action is turbulent. Here is our main application of turbulence.

page 62

March 11, 2014

10:36

WSPC 9 x 6

singapore-final-r

Logic for C*-Algebras

63

Theorem 8.5 (Farah–Toms–T¨ornquist, [33]). The isomorphism relation ∼ =AI of simple, separable, unital AI algebras has the following two properties: (1) it is not classifiable by countable structures. (2) for any countable signature L the isomorphism relation of countable L-models is Borel-reducible to it. The proof of this theorem proceeds in two steps, via using H2 , the homeomorphism relation of compact subsets of [0, 1]2 : (i) showing that H2 has these properties (Lemma 8.6) and (ii) H2 ≤B ∼ =AI (Lemma 8.9). A curious feature of (ii) is that we shall use classification result to prove a non-classification result. The following is a slight improvement of a result of Hjorth used in [43, 4.21]. Lemma 8.6. (1) The homeomorphism relation of compact subspaces of [0, 1]2 is not classifiable by countable structures. (2) If L is any countable signature then the isomorphism relation of countable L-models is Borel-reducible to the homeomorphism of compact subsets of [0, 1]2 . Proof. (1) It is notationally convenient to work with [−1, 1]2 instead of [0, 1]2 . Let G = ZN . Note that G is a Polish group when given the product group structure and product topology. We let G0 = {g ∈ G : lim

n→∞

g(n) = 0}. n

By [43, 4.16] G0 acts turbulently on G by translation. Let T1 = {(x, y) ∈ [−1, 1] × [0, 1] : |x| ≤ y} and for n ∈ N, let in general Tn =

n [

k=1

[(2k − 2, 0) + T1 ].

Then the Tn are compact and connected and for m 6= n we have Tm 6≃hom Tn . Fix an order-preserving homeomorphism f : R → (−1, 1) such that f (0) = 0. For each m ∈ Z and n, k ∈ N and let     1 1 |m| 2m − 1 2m Jm,n,k ⊆ [ , − ] × [f ,f ] 2n − 1 2n 2n(2n − 1)(|m| + 1) n n

page 63

March 11, 2014

10:36

WSPC 9 x 6

64

singapore-final-r

I. Farah

1 1 , 2n ] × [−1, 1] be a closed set homeomorphic to Tk . Note that Jm,n,k ⊆ [ 2n−1 and that for all n,

lim sup{diam(Jm,n,k ) : k ∈ N} → 0.

m→±∞

Fix a bijection ϕ : Z×N → N and define for each g ∈ G a set K(g) ⊆ [−1, 1]2 by ! G X(g) = Jm,n,ϕ(m+g(n),n) ⊔ (∂([0, 1] × [−1, 1]) ∪ [−1, 0] × {0}) . (∗) m,n

It is easy to see that X(g) is compact for all g ∈ G, and that the map G → K([−1, 1]2 ) : g 7→ X(g) is Borel, where K([−1, 1]2 ) denotes the compact hyperspace of [−1, 1]2 . Note that (∗) provides a decomposition of each X(g) into mutually nonhomeomorphic connected components. By the (m, n) component of X(g) we mean the set Jm,n,ϕ(m+g(n),n) , and by the n-th column of X(g) we mean the set G X(g)n = Jm,n,ϕ(m+g(n),n) . m

It is clear that the set of components of the n-th column {Jm,n,ϕ(m+g(n),n) : m ∈ N} does not depend on g ∈ G. On the other hand, if n0 6= n1 , then the components of X(g)n0 are not homeomorphic to any of the components of X(g)n1 . Therefore any homeomorphism from X(g) to X(h) must map X(g)n to X(h)n , for all n ∈ N. Moreover, any homeomorphism between X(g) and X(h) must fix the point (0, 0). F Claim 8.7. For all g ∈ G, if (xn , yn ) ∈ m,n Jm,n,ϕ(m+g(n),n) is a sequence 1 1 , 2n ] for all n, then the function such that (xn , yn ) → (0, 0) and xn ∈ [ 2n−1 θ : N → Z defined by θ(n) = k ⇐⇒ the connected component of (xn , yn ) isomorphic to Tϕ(k,n) satisfies θ − g ∈ G0 .

page 64

March 31, 2014

18:14

WSPC 9 x 6

singapore-final-r

Logic for C*-Algebras

65

Proof. For each n ∈ N, let mn ∈ Z be such that   2mn 2mn − 1 yn ∈ [f ,f ]. n n mn n Since yn → 0 it follows that f ( 2m n ) → 0, and so n → 0. By definition of X(g) the connected component in which (xn , yn ) lies in is isomorphic to Tϕ(mn +g(n),n) , and so

θ(n) = mn + g(n). Thus θ − g ∈ G0 . Claim 8.8. For all g, h ∈ G, X(g) is homeomorphic X(h) if and only if g − h ∈ G0 . Proof. Suppose first that X(g) and X(h) are homeomorphic, and let π ˆ : X(g) → X(h) witness this. Fix a sequence (x n , yn ) ∈ X(g) such that (x n , yn ) ∈ J0,n,ϕ(g(n),n) for all n ∈ N. Then (x n , yn ) → (0, 0), and so since π ˆ (0, 0) = (0, 0) we have (xn , yn ) = π ˆ (x n , yn ) → 0. Since X(g)n is mapped 1 1 , 2n ]. Moreover, to X(h)n if holds for all n that xn ∈ [ 2n−1 g(n) = k ⇐⇒ the connected component of (xn , yn ) isomorphic to Tϕ(k,n) and so g − h ∈ G0 by Claim 8.7. Suppose conversely that z = g − h ∈ G0 . Then define π : X(g) → X(h) by letting π  ∂([0, 1] × [−1, 1]) ∪ [−1, 0] × {0} = id, and for each n ∈ N and m ∈ Z letting π  Jm,n,ϕ(m+g(n),n) be a homeomorphism Jm,n,ϕ(m+g(n),n) → Jm+z(n),n,ϕ(m+z(n)+h(n),n) . To see that π is a homeomorphism it suffices to see that π is continuous, since it is clearly 1-1 and onto. To see this, for each x ∈  m,n Jm,n,ϕ(m+g(n),n) let mx ∈ Z and nx ∈ N be such that x∈[

1 1 ] × [f , 2nx − 1 2nx



2mx − 1 nx



 ,f

2mx ]. nx

page 65

March 31, 2014

66

18:14

WSPC 9 x 6

singapore-final-r

I. Farah

Then by the definition of π we have    2(mx + z(nx )) 2mx ,f d(x, π(x)) ≤ d f nx nx    2mx − 1 2mx + d f ,f nx nx  2mx − 1 2mx |mx | + d , − nx nx 2nx (2nx − 1)(|mx | + 1)  2(mx + z(nx )) − 1 2(mx + z(nx )) , + d nx nx |mx + z(nx )| − 2nx (2nx − 1)(|mx + z(nx )| + 1) which shows that d(x, π(x)) → 0 as x → ∂([0, 1] × [−1, 1]). Thus π is continuous. (2) follows from the argument suggested in [43, 4.22] but I shall provide a slightly different proof. By [36], it suffices to reduce the isomorphism of countable graphs. With [Z]2 = {(m, n) : m < n} fix a bijection φ : [Z]2 → N. Let Em,n be the union of straight lines in R3 connecting (m, 0, 0) with (0, φ(m, n), φ(m, n)) and (0, φ(m, n), φ(m, n)) with (n, 0, 0). These lines are  pairwise disjoint and m,n Em,n is closed. Let Ln be union of three straight lines connecting (m, 0, 0) and (m, 0, −1), (m, 0, 0) and (m, −1, −1) and (m, 0, 0) and (m, −1, 0) for all n. To a countable graph G = (N, E) associate the set   X(G) = n∈Z Ln ∪ {m,n}∈E Lm,n. This map is clearly continuous. If graphs G and H are isomorphic, then X(G) and X(H) are clearly isomorphic. Now assume X(G) and X(H) are homeomorphic. Homeomorphism has to send each endpoint of the form (m, 0, −1), (m, −1, −1), or (m, −1, 0) to a point of the same form. Therefore each (m, 0, 0) goes to some (f (m), 0, 0) and It is then easy to show that f is an isomorphism between G and H. The following lemma completes the proof of Theorem 8.5. Lemma 8.9. There is a Borel map Φ from K([0, 1]N ), the space of compact subspaces of [0, 1]N , to Γ such that each A(K) = C ∗ (Φ(K)) is a simple, unital AI algebra and A(K) ∼ = A(L) if and only if K and L are homeomorphic. Proof. In [78] Thomsen constructed (among other things) for every compact metric K algebra A(K) such that

page 66

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

(1) (2) (3) (4)

singapore-final-r

67

A(K) is a simple, unital AI algebra, K0 (A(K)) = (Q, Q+ , [1]), K1 (A(K)) = {0}, T (A(K)) is affinely homeomorphic to P(K), the space of Borel probability measures on K.

In [33] it was demonstrated that Thomsen’s construction can be performed effectively. Clearly, if K is not homeomorphic to L then A(K) and A(L) are not isomorphic. For the converse we use a result of Elliott who proved that simple, separable AI algebras are classified by their Elliott invariant (see [70]). (Since (Q, Q+ , [1]) has the unique state, the pairing function ρ is uniquely determined in each A(K).) 8.2. Below a group action: An upper bound for complexity While the isomorphism of separable Banach spaces is the ≤B maximal analytic equivalence relation ([35]), the isomorphism of von Neumann factors is reducible to an orbit equivalence relation of a Polish group action (classical, see e.g., [74]). This is not surprising since Banach spaces are much wilder objects than von Neumann algebras (note, however, that the isometry relation of separable Banach spaces is reducible to an orbit equivalence relation of a Polish group action, by [57]). C*-algebras are not as wild as arbitrary Banach spaces and not as wellbehaved as von Neumann algebras, and one can ask what is the complexity of the isomorphism of separable C*-algebras. While [33] left the general problem open, it did show that the isomorphism of algebras relevant to Elliott’s program is below a group action. The proof, however, took a detour and used some of the deepest results on the structure of separable nuclear C*-algebras. This detour gives us an excuse to introduce a fascinating object. 8.3. Cuntz algebra O2 Let H be a separable complex Hilbert space with orthonormal basis en , for n ∈ N. The equations s(en ) = e2n

t(en ) = e2n+1 for all n

uniquely define linear operators s and t. We have that s∗ s = 1,

t∗ t = 1, and ss∗ + tt∗ = 1

(8.1)

since ss∗ and tt∗ are mutually orthogonal projections. Consider the C*-algebra generated by s and t. The unit in this algebra satisfies the following definition (note that the property of p is computed

page 67

March 11, 2014

68

10:36

WSPC 9 x 6

singapore-final-r

I. Farah

relative to the ambient C*-algebra A, and recall that for projections p and q we write p ≤ q if pq = p). Definition 8.10. A projection p in A is properly infinite if there are projections q ∼ p and r ∼ p such that q + r ≤ p. The first separable simple C*-algebra in which the unit is properly infinite was constructed by Dixmier. He considered C ∗ (s, t)/J with the above s and t, where J is the maximal ideal of C ∗ (s, t). However, moding out by J was not necessary. Theorem 8.11 (Cuntz, [10]). If s and t are isometries satisfying (8.1) then the algebra C ∗ (s, t) is simple. Moreover, any two algebras generated in this way are isomorphic. This algebra generated by s and t is denoted by O2 and we shall spend some time investigating its properties. Lemma 8.12. If s and t satisfy (8.1) then O2 is the closed linear span of Q Q all monomials of the form i≤m xi j≤n yj∗ , where {xj , yi } ⊆ {s, t}.

Proof. We need to show that any monomial in s, t, s∗ and t∗ is either 0 or Q Q equal to a monomial of the form i≤m xi j≤n yj∗ , where {xj , yi } ⊆ {s, t}. Since s∗ s = t∗ t = 1, it will suffice to prove that s∗ t = t∗ s = 0. But by (8.1) we have ss∗ t + tt∗ t = t, and therefore ss∗ t = 0. By multiplying by s∗ on the left we obtain s∗ t = 0. A proof of t∗ s = 0 is similar. The structure of O2 is discussed below in Exercises 8.4.1 and 8.4.2. Every separable simple C*-algebra, including O2 , has outer automorphisms. The following lemma shows that automorphisms of O2 are coded by its unitaries. However, not only that αu is distinct from Ad u, but moreover u 7→ αu is not an action of the unitary group O2 on O2 . Lemma 8.13. Every automorphism α of O2 is determined by α(s) = us and α(t) = ut for some unitary u. Conversely, every unitary u ∈ O2 uniquely determines an automorphism αu of O2 such that αu (s) = us and αu (t) = ut. Proof. Let u = α(s)s∗ + α(t)t∗ . We claim that u is a unitary. By applying Lemma 8.12 we have uu∗ = α(s)α(s)∗ + α(t)α(t)∗ = α(ss∗ + tt∗ ) = 1. Similarly, u∗ u = 1 and therefore u is a unitary. Using Lemma 8.12 again we get us = α(s) and ut = α(t).

page 68

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

69

More information on this fascinating object, as well as its relatives On for n ∈ {2, 3, . . . , ∞}, including the proofs of the results below, can be found in [10] and in [70]. The definition of approximate unitary equivalence was given in the paragraph before Theorem 5.9. Theorem 8.14. Every endomorphism β : O2 → O2 is approximately unitarily equivalent to the identity map. By the following remarkable result O2 tensorially absorbs exactly the algebras that are subject of the Elliott’s program. Its precursor, the proof that O2 ⊗O2 is isomorphic to O2 , was proved by Elliott using Theorem 8.14 and a clever use of ultrapowers and an approximate intertwining argument (see [70]). The difficult proof of Theorem 8.15 proof can be found in [53] or (with fewer details) in [70]. Theorem 8.15 (Kirchberg). Every separable nuclear unital C*-algebra is isomorphic to a unital subalgebra of O2 . Also, A is separable, nuclear, simple and unital C*-algebra if and only if A ⊗ O2 ∼ = O2 . Unlike the case of Thomsen’s theorem (see the proof of Theorem 8.5), we were unable to find a direct Borel proof of Kirchberg’s theorem. Its known proofs are not taking place in the ‘Borel world’ but are instead considering embeddings into the corona of the stabilization of O2 . Nevertheless, Kirchberg’s theorem can be used to prove its Borel version (cf. Exercise 8.4.6). I can now give a rough sketch of the proof of an upper bound for the complexity of the isomorphism relation. By SA(O2 ) we denote the space of subalgebras of O2 , with respect to the Effros Borel structure. Theorem 8.16 (Farah–Toms–T¨ornquist, [33]). The isomorphism relation of separable, simple, unital and nuclear C*-algebras is Borel-reducible to an orbit equivalence relation of a Polish group action. Sketch of a proof. More precisely, the set ΓN = {γ ∈ Γ : C ∗ (γ) is simple, unital and nuclear} is Borel. There is a Borel reduction Φ : N → SA(O2 ) SA(O ) such that C ∗ (γ1 ) ∼ = C ∗ (γ2 ) if and only if Φ(γ1 ) EAut(O22 ) Φ(γ2 ). By a Borel version of Kirchberg’s Theorem 8.15 proved in [33] there is a Borel map Φ from N into SA(O2 ) such that A = Φ(γ) is a unital subalgebra of O2 isomorphic to C ∗ (γ) with the property that the relative commutant of A in O2 , B = {b ∈ O2 : ab = ba for all a ∈ A}

page 69

March 11, 2014

10:36

70

WSPC 9 x 6

singapore-final-r

I. Farah

is isomorphic to O2 and C ∗ (A, B) is both equal to O2 and is naturally isomorphic to A ⊗ O2 . Clearly we only need to check that if C ∗ (γ1 ) ∼ = C ∗ (γ2 ) then there is α ∈ Aut(O2 ) that sends Φ(γ1 ) onto Φ(γ2 ). Fix an isomorphism α0 between Φ(γ1 ) and Φ(γ2 ). Let B1 and B2 be the relative commutants of these algebras. Since they are both isomorphic to O2 , we can fix an isomorphism α2 : B1 → B2 . Then α = α1 ⊗ α2 is as required. 8.4. Exercises 8.4.1. Let s and t be the generators of O2 . For n ∈ N let Fn be the Q Q linear span of j≤n xj j≤n yj∗ , where {xj , yj } ⊆ {s, t}. Prove that Fn is isomorphic to M2n (C). (Hint: First consider n = 1 and check that ss∗ , tt∗ , st∗ and ts∗ are the matrix units in M2 (C) (cf. Exercise 2.6.7). Then go up.) 8.4.2. Using notation of the previous exercise, show that the ‘balanced’ Q Q products j≤n xj j≤n yj∗ , for n ∈ N and {xj , yj } ⊆ {s, t} generate a subalgebra isomorphic to the CAR algebra. Then prove that O2 is generated by the CAR algebra A and a partial isometry s such that a 7→ sas∗ is an endomorphism sending A onto a corner pAp where p ∈ A is a projection of trace 1/2. 8.4.3. Show that all nonzero projections in O2 are Murray-von Neumann equivalent and that K0 (O2 ) is trivial. 8.4.4. Prove that O2 has no normalized traces. (Hint: 1O2 is properly infinite.) One can prove that K1 (O2 ) is also trivial, and therefore by the previous two exercises the Elliott invariant of O2 is equal to the Elliott invariant of the C*-algebra {0} (assuming that we accept the latter as a C*-algebra). Compare this with Theorem 8.15 and think 0 · A = 0. 8.4.5. Let An , Fmn , for m ≤ n, m, n ∈ N, be a directed unital system of C*-algebras all of them isomorphic to O2 . Show that the direct limit is isomorphic to O2 . (Hint: Theorem 8.14 and approximate intertwining.) I do not know what happens if one considers a transfinite direct limit of an ℵ1 -directed sequence of copies of O2 , or whether such algebra of

page 70

March 31, 2014

18:14

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

71

density character ℵ1 is uniquely determined.e I also do not know whether the analogue of Kirchberg’s embedding theorem (see Theorem 8.15) holds for nuclear, or exact, C*-algebras of density character ℵ1 . Note that the conclusion of the following exercise is strictly weaker than what is required in the proof of Theorem 8.16. 8.4.6. Using notation from the proof of Theorem 8.16, prove that there exists function Φ : N → SA(O2 ) that is C-measurable. (Hint: Use Jankov, von Neumann selection theorem.)

9. Concluding Remarks 9.1. The Borel-reducibility diagram Figure 13 summarizes some of the known Borel reductions, concentrating on results relevant to operator algebras.f All classes of C*-algebras occurring in the diagram are separable and unital (unless otherwise specified). Unless otherwise specified, the equivalence relation on a given class is the isomorphism relation. The bi-reducibility between the isomorphism for UHF algebras and biembeddability of UHF algebras is an immediate consequence of Exercise 4.3.9, or rather of its (straightforward) Borel version. Classification of compact metric spaces up to isometry is due to Gromov. Borel bireducibility between abelian C*-algebras and compact metric spaces was proved in [32]. Borel reductions from compact metric spaces to Choquet simplexes to simple AI algebras (as well as the definition of the latter) are given in [33]. A Borel-reduction of Choquet simplexes to the isometry of Banach spaces is given by sending a Choquet simplex K to the Banach space of affine functions on K. A Borel version of Elliott’s reduction of simple AI algebras to Elliott invariant follows from Elliott’s classification result and the fact that the computation of the Elliott invariant is Borel was proved in [32] (see Lemma 8.9). Equireducibility of the maximal orbit equivalence relation of a Polish group action with the isometry of Polish spaces and the isometry of Banach spaces was proved in [8] and [61], respectively. e Added

on April 20, 2013: Now I do know. It is not. The tensor product of the algebra  constructed in [26] with O2 is not isomorphic to ℵ1 O 2 . f Added in July 2013. Recently Marcin Sabok proved that the affine homeomorphism of metrizable Choquet simplexes is the maximal orbit equivalence relation ([73]).

page 71

March 11, 2014

10:36

WSPC 9 x 6

72

singapore-final-r

I. Farah

maximal analytic

iso of Banach spaces

Borel relations are encircled

bi-emb. of Banach spaces

bi-emb.

E Kσ

Cuntz semigroups

orbit equivalence relations

of AF

maximal orbit

isometry of Polish spaces

Elliott invariants

C*-algebras

simple AI

isometry of Banach spaces

Irr(A), A non-type I

conjugacy of unitary operators

RN /ℓq p ≤ q

Choquet simplexes

E1

homeomorphism relation of cpct metric spaces

von Neumann factors

abelian C*-algebras

RN /c0

RN /ℓp

Louveau –Velickovic

RN /ℓ1

countable structures ctble graphs

dimension groups

AF

ctble linear orders

minimal Cantor systems

maximal S∞ orbit

Kirchberg algebras

E0

smooth

isometry of cpct metric spaces

Fig. 13.

UHF

bi-emb. of UHF

Borel reducibility diagram.

non-unital AM

page 72

March 11, 2014

10:36

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

73

The reduction of the Elliott invariant to the maximal orbit equivalence relation, as well as the facts about the Cuntz semigroup, are proved in [32]. Dougherty–Hjorth ([14]) proved that RN /ℓp ≤B RN /ℓq if and only if p ≤ q, and Hjorth proved ([42]) that the translation action of ℓ1 on RN is a minimal turbulent action. On the other hand, I proved ([25]) that there is no minimal turbulent action Borel-reducible to RN /c0 and that the complicated structure of turbulent equivalence relations constructed in [58] occurs below orbit equivalence relation of any turbulent action Borel-reducible to RN /c0 . Also, RN /c0 and RN /ℓ1 are ≤B -incomparable (see [25]). A class of 2ℵ0 many Fσ equivalence relations not classifiable by countable structures that are incompatible, in the sense that any relation Borel-reducible to any two of them is classifiable by countable structures, was constructed in [24], but I could not fit this family into the picture. The fact that RN /ℓ1 is Borel-reducible to the conjugacy of unitary operators on B(H) was proved in [51]. As Todor Tsankov and Alain Louveau pointed out, Dellacherie proved that the conjugacy relation of unitary operators in B(H) is Borel. Characterization of ≤B -maximal isomorphism relation of countable structures as the graph isomorphism or isomorphism of countable linear orderings is in [36] (and so is the original definition of ≤B !). AF algebras are classifiable by countable structures by Elliott’s Theorem 5.8 and the fact that the computation of K0 is Borel (Lemma 7.9). The fact that the topological orbit equivalence of minimal Cantor systems is equireducible with AF algebras was proved in [37], and that these are equivalent with the isomorphism of dimension groups is in [16]. Classification of Kirchberg algebras by countable structures is a consequence of Kirchberg–Phillips theorem ([65], [54]) and Borel-computability of the Elliott invariant. Classification of separable, but not necessarily unital direct limits of full matrix algebras (i.e., non-unital AM algebras) by countable structures is given in [13]; see also Exercise 4.3.7 . Non-classifiability of the isomorphism relation in every major class of von Neumann factors that was not already classified is given in [74] and [75]; see also [60]. Since the maximal isomorphism relation for countable structures is not Borel ([43]), it is not Borel-reducible to any Borel equivalence relation. This in particular implies that the conjugacy of unitary operators is strictly ≤B below the isometry of Polish space. By Irr(A) we denote the space of irreducible representations of a C*algebra A, or equivalently, the space of pure states of A up to the unitary conjugacy. This relation satisfies a dichotomy result: it is either smooth

page 73

March 11, 2014

74

10:36

WSPC 9 x 6

singapore-final-r

I. Farah

(when A is a type I C*-algebra, by the definition of type I algebra) or turbulent was proved independently in [52] and [27], and the latter proof gives that this relation Borel-reduces RN /ℓ1 (and not RN /ℓ2 , as stated in this paper). The fact that this relation is Borel (actually Fσ ) follows from a result of Glimm and Kadison (see [27]). Bi-embeddability of AF algebras is proved to be above EKσ in [33, §8]. The isomorphism of separable Banach spaces is the complete analytic equivalence relation by [35]. Several results on non-classifiability of automorphisms of C*-algebras with respect to unitary equivalence were obtained in [59]. Last, but not least, the isomorphism relation of separable C*-algebras is reducible to an orbit equivalence relation of a Polish group action [19]. This result (stated as an open problem in my last lecture) was obtained in email correspondence between the authors of [19] in the wake of the June 2012 BIRS meeting on applications of descriptive set theory to functional analysis. Some of the results presented in Figure 13 may not be optimal. For example, it is not known whether the homeomorphism of compact metric spaces is equireducible with the maximal orbit equivalence relation. See also [73]. 9.2. Selected open problems The following problem was suggested by N. Christopher Phillips. Question 9.1. Not necessarily self-adjoint, norm-closed algebras are wellstudied (see e.g., [66]). Finite-dimensional algebras in this class are not necessarily direct sums of matrix algebras, and there is no simple classification of such algebras. Is there a descriptive set-theoretic explanation for this? For example, are these algebras classifiable by countable structures? Let H be the class of homogeneous algebras of the form C(X, Mn (C)) for X compact metric and form the class AH using H as the building blocks (Definition 6.1). By [11], not every direct limit of AH algebras is AH. Question 9.2. Let α be the minimal ordinal such that the class of algebras obtained by closing H as above under direct limits α times is closed under direct limits. By the above, α ≥ 2. What is the value of α? (Warning: I do not have an impression that the C*-algebraists are too keen to find an answer to Question 9.2.)

page 74

March 31, 2014

18:14

WSPC 9 x 6

singapore-final-r

Logic for C*-Algebras

75

Question 9.3. For 2 ≤ m ≤ ∞ let Em be the homeomorphism relation of compact subspaces of [0, 1]m . Clearly En ≤B En+1 ≤B E∞ for all n X∞ . Does any of the converses hold? In particular, is E2 the and E∞ ≤ EG ∞ ≤B -maximal orbit equivalence relation of a Polish group action? The following important question is vague since it is really a large number of questions in one. Question 9.4. g How does the complexity of the isomorphism relation of a class of C*-algebras change as the class increases? In particular, is the isomorphism relation of all C*-algebras Borel-reducible to the isomorphism relation of nuclear C*-algebras, or to the isomorphism relation of simple C*-algebras, or to the isomorphism relation of simple nuclear C*-algebras? Question 9.5. What is the complexity of the isomorphism relation of (nuclear, simple) C*-algebras with the same fixed value of the Elliott invariant? In [79] and [81] Toms constructed infinitely, and then continuum many, nonisomorphic nuclear simple C*-algebras with the same Elliott invariant. The present paragraph contains a vague suggestion instead of a question or a problem. By the basic theory of vector bundles ([41]), if 2 dim(X) < n then C(X, Mn (C)) contains only trivial projections (cf. §6.0.1). Therefore counterexamples to Elliott’s conjecture (e.g., [80]) are AH algebras that are direct limits of direct sums of algebras C(X, Mn (C)) where dim(X) > n. On the other hand, in a technical tour de force the AH algebras with ‘slow dimension growth’ were classified by Elliott’s invariant ([23]). One could therefore speculate that the complexity of the isomorphism relation for AH algebras can be tied with very fast growing functions, especially because the latter were a source of interesting logical result for decades. George Elliott suggested that the substantial paper of Rieffel ([69]) could be a useful source for this project. Many classification results have matching range of invariant results and it would be good to have their Borel versions. We have already mentioned [78] whose partial Borel version was used in Lemma 8.9 and the Effros– Handelman–Shen result that every dimension group corresponds to an AF algebra ([16]). A Borel version of this result is not difficult to prove and it was used implicitly above. In [77] it was proved that a pair of countable abelian groups appears as the Elliott invariant of a Kirchberg algebra if and only if the second group is free. It would also be good to have a Borel version g By

Sabok’s results ([73]), all of these classes have the same Borel complexity.

page 75

March 31, 2014

18:14

76

WSPC 9 x 6

singapore-final-r

I. Farah

of the result of [6], that the Cuntz semigroup is recovered functorially from the Elliott invariant for a large class of C*-algebras. Question 9.6. Do the above range of invariant results have Borel versions? It would be nice to have a general selection theorem that provides such results automatically. However, the obvious strategy falls short of obtaining Borel maps (see Exercise 8.4.6). See, however, [19, Theorem A] and [32, Lemma 6.2] for some partial results. As pointed out before, the proof of Kirchberg’s Theorem makes a detour into the nonseparable world. The following question may have an interesting answer. Problem 9.7. What is the strength of Kirchberg’s Theorem 8.15 in the sense of reverse mathematics? It is not known whether nuclear simple separable C*-algebras of real rank zero (see Exercise 3.5.9) are classifiable by K0 and K1 . An obvious set-theoretic take on this problem is the following. Problem 9.8. Can separable nuclear C*-algebras of real rank zero be classified by countable structures? What about separable nuclear simple C*-algebras or real rank zero? It is known that K0 and K1 do not classify non-nuclear separable C*algebras of real rank zero (a L¨ owenheim–Skolem type argument is given in [65]). A distinguishing feature of Elliott’s view of the classification program of C*-algebras is its functoriality. This does not seem to be captured by the present theory of Borel equivalence relations and the following problem is an attempt to remedy this situation. Polish groupoids were defined in [67] and one can similarly define a ‘Polish category’ whose objects are metric structures based on Polish spaces. Some preliminary results on the following problem were obtained by the author in a joint work with S. Coskey, G. Elliott, and M. Lupini. Problem 9.9. Develop Borel reduction theory for Polish groupoids, and more generally for Polish categories. References 1. S. Adams and A.S. Kechris, Linear algebraic groups and countable Borel equivalence relations, J. Amer. Math. Soc. 13 (2000), no. 4, 909–943. 2. W. Arveson, A short course on spectral theory, Graduate Texts in Mathematics, vol. 209, Springer-Verlag, New York, 2002.

page 76

March 31, 2014

18:14

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

77

3. H. Becker and A.S. Kechris, The descriptive set theory of Polish group actions, Cambridge University Press, 1996. 4. B. Blackadar, Operator algebras, Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin, 2006, Theory of C ∗ -algebras and von Neumann algebras, Operator Algebras and Non-commutative Geometry, III. 5. N. Brown and N. Ozawa, C ∗ -algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. 6. N.P. Brown, F. Perera, and A.S. Toms, The Cuntz semigroup, the Elliott conjecture, and dimension functions on C ∗ -algebras, J. Reine Angew. Math. 621 (2008), 191–211. 7. K. Carlson, E. Cheung, I. Farah, A. Gerhardt-Bourke, B. Hart, L. Mezuman, N. Sequeira, and A. Sherman, Omitting types and AF algebras, Arch. Math. Logic 53 (2014), no. 1–2, 157–169. 8. J.D. Clemens, S. Gao, and A.S. Kechris, Polish metric spaces: their classification and isometry groups, Bull. Symbolic Logic 7 (2001), no. 3, 361–375. 9. K.T. Coward, G.A. Elliott, and C. Ivanescu, The Cuntz semigroup as an invariant for C ∗ -algebras, J. Reine Angew. Math. 623 (2008), 161–193. 10. J. Cuntz, Simple C ∗ -algebras generated by isometries, Comm. Math. Phys. 57 (1977), no. 2, 173–185. 11. M. D˘ ad˘ arlat and S. Eilers, Approximate homogeneity is not a local property, J. Reine Angew. Math. 507 (1999), 1–13. 12. K.R. Davidson, C ∗ -algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. 13. J. Dixmier, On some C ∗ -algebras considered by Glimm, J. Functional Analysis 1 (1967), 182–203. 14. R. Dougherty and G. Hjorth, Reducibility and nonreducibility between lp equivalence relations, Trans. Amer. Math. Soc. 351 (1999), no. 5, 1835–1844. 15. E.G. Effros, Classifying the unclassifiables, Group representations, ergodic theory, and mathematical physics: a tribute to George W. Mackey, Contemp. Math., vol. 449, Amer. Math. Soc., Providence, RI, 2008, pp. 137–147. 16. E.G. Effros, D.E. Handelman, and C.L. Shen, Dimension groups and their affine representations, Amer. J. Math. 102 (1980), no. 2, 385–407. 17. E.G. Effros and Z.-J. Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press Oxford University Press, New York, 2000. 18. G.A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976), no. 1, 29–44. 19. G.A. Elliott, I. Farah, V. Paulsen, C. Rosendal, A.S. Toms, and A. T¨ ornquist, The isomorphism relation of separable C*-algebras, Math. Res. Letters (to appear), preprint, arXiv 1301.7108. 20. G.A. Elliott and A.S. Toms, Regularity properties in the classification program for separable amenable C ∗ -algebras, Bull. Amer. Math. Soc. 45 (2008), no. 2, 229–245. 21. G.A. Elliott, A classification of certain simple C ∗ -algebras, Quantum and non-commutative analysis (Kyoto, 1992), Math. Phys. Stud., vol. 16, Kluwer Acad. Publ., Dordrecht, 1993, pp. 373–385.

page 77

March 31, 2014

78

22.

23.

24.

25. 26. 27.

28. 29. 30. 31. 32.

33. 34.

35.

36. 37. 38. 39. 40.

41.

18:14

WSPC 9 x 6

singapore-final-r

I. Farah

, The classification problem for amenable C ∗ -algebras, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z¨ urich, 1994) (Basel), Birkh¨ auser, 1995, pp. 922–932. G.A. Elliott, G. Gong, and L. Li, On the classification of simple inductive limit C ∗ -algebras. II. The isomorphism theorem, Invent. Math. 168 (2007), no. 2, 249–320. I. Farah, Basis problem for turbulent actions I: Tsirelson submeasures, Proceedings of XI Latin American Symposium in Mathematical Logic, Merida, July 1998, Annals of Pure and Applied Logic, vol. 108, 2001, pp. 189–203. , Basis problem for turbulent actions II: c0 -equalities, Proceedings of the London Mathematical Society 82 (2001), 1–30. , Graphs and CCR algebras, Indiana Univ. Math. Journal 59 (2010), 1041–1056. , A dichotomy for the Mackey Borel structure, Proceedings of the 11th Asian Logic Conference In Honor of Professor Chong Chitat on His 60th Birthday (Yang Yue et al., eds.), 2011, pp. 86–93. I. Farah, B. Hart, and D. Sherman, Model theory of operator algebras I: Stability, Bull. London Math. Soc. 45 (2013), 825–838. , Model theory of operator algebras II: Model theory, Israel J. Math. (to appear), arXiv:1004.0741. I. Farah and T. Katsura, Nonseparable UHF algebras I: Dixmier’s problem, Adv. Math. 225 (2010), no. 3, 1399–1430. , Nonseparable UHF algebras II: Classification, Math. Scand. (to appear), preprint arXiv:1301.6152. I. Farah, A.S. Toms, and A. T¨ ornquist, The descriptive set theory of C*algebra invariants, Int. Math. Res. Notices 22 (2013), 5196–5226, Appendix with Caleb Eckhardt. , Turbulence, orbit equivalence, and the classification of nuclear C*algebras, J. Reine Angew. Math. 688 (2014), 101–146. I. Farah and E. Wofsey, Set theory and operator algebras, Appalachian set theory 2006-2010 (J. Cummings and E. Schimmerling, eds.), Cambridge University Press, 2013, pp. 63–120. V. Ferenczi, A. Louveau, and C. Rosendal, The complexity of classifying separable Banach spaces up to isomorphism, J. Lond. Math. Soc. (2) 79 (2009), no. 2, 323–345. H. Friedman and L. Stanley, A Borel reducibility theory for classes of countable structures, The Journal of Symbolic Logic 54 (1989), 894–914. T. Giordano, I.F. Putnam, and C.F. Skau, Topological orbit equivalence and C ∗ -crossed products, J. Reine Angew. Math. 469 (1995), 51–111. J. Glimm, Type I C ∗ -algebras, Ann. of Math. (2) 73 (1961), 572–612. J.G. Glimm, On a certain class of operator algebras, Trans. Amer. Math. Soc. 95 (1960), 318–340. L.A. Harrington, A.S. Kechris, and A. Louveau, A Glimm–Effros dichotomy for Borel equivalence relations, Journal of the American Mathematical Society 4 (1990), 903–927. A. Hatcher, Vector bundles and K-theory, 2003.

page 78

April 22, 2014

16:39

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

79

42. G. Hjorth, Actions by classical Banach spaces, The Journal of Symbolic Logic 65 (2000), 392–420. 43. , Classification and orbit equivalence relations, Mathematical Surveys and Monographs, vol. 75, American Mathematical Society, 2000. , Borel equivalence relations, Handbook of set theory (M. Foreman 44. and A. Kanamori, eds.), Springer, 2010, pp. 297–332. 45. X. Jiang and H. Su, On a simple unital projectionless C∗ -algebra, Amer. J. Math 121 (1999), 359–413. 46. V.F.R. Jones, Von Neumann algebras, 2010, lecture notes, http://math.berkeley.edu/∼vfr/. 47. M. Junge and G. Pisier, Bilinear forms on exact operator spaces and B(H) ⊗ B(H), Geom. Funct. Anal. 5 (1995), no. 2, 329–363. 48. A.S. Kechris, Classical descriptive set theory, Graduate texts in mathematics, vol. 156, Springer, 1995. , The descriptive classification of some classes of C ∗ -algebras, Pro49. ceedings of the Sixth Asian Logic Conference (Beijing, 1996), World Sci. Publ., River Edge, NJ, 1998, pp. 121–149. 50. A.S. Kechris and A. Louveau, The structure of hypersmooth Borel equivalence relations, Journal of the American Mathematical Society 10 (1997), 215–242. 51. A.S. Kechris and N.E. Sofronidis, A strong generic ergodicity property of unitary and self-adjoint operators, Ergodic Theory Dynam. Systems 21 (2001), no. 5, 1459–1479. 52. D. Kerr, H. Li, and M. Pichot, Turbulence, representations, and tracepreserving actions, Proc. Lond. Math. Soc. (3) 100 (2010), no. 2, 459–484. 53. E. Kirchberg and N.C. Phillips, Embedding of exact C*-algebras in the Cuntz algebra O2 , J. reine angew. Math. 525 (2000), 17–53. 54. E. Kirchberg, Exact C∗ -algebras, tensor products, and the classification of purely infinite algebras, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z¨ urich, 1994) (Basel), Birkh¨ auser, 1995, pp. 943–954. 55. H. Lin, Classification of simple C ∗ -algebras of tracial topological rank zero, Duke Math. J. 125 (2004), no. 1, 91–119. 56. T.A. Loring, Lifting solutions to perturbing problems in C ∗ -algebras, Fields Institute Monographs, vol. 8, American Mathematical Society, Providence, RI, 1997. 57. A. Louveau and C. Rosendal, Complete analytic equivalence relations, Trans. Amer. Math. Soc. 357 (2005), no. 12, 4839–4866. 58. A. Louveau and B. Velickovic, A note on Borel equivalence relations, Proceedings of the American Mathematical Society 120 (1994), 255–259. 59. M. Lupini, Unitary equivalence of automorphisms of separable C*-algebras, arXiv preprint arXiv:1304.3502 (2013). 60. M. Lupini and A. T¨ ornquist, Set theory and von Neumann algebras, Appalachian set theory 2006-2010 (J. Cummings and E. Schimmerling, eds.), Cambridge University Press, to appear. 61. J. Melleray, Computing the complexity of the relation of isometry between separable Banach spaces, MLQ Math. Log. Q. 53 (2007), no. 2, 128–131. 62. G.J. Murphy, C ∗ -algebras and operator theory, Academic Press Inc., Boston, MA, 1990.

page 79

March 31, 2014

80

18:14

WSPC 9 x 6

singapore-final-r

I. Farah

63. V. Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. 64. G.K. Pedersen, Analysis now, Graduate Texts in Mathematics, vol. 118, Springer-Verlag, New York, 1989. 65. N.C. Phillips, A classification theorem for nuclear purely infinite simple C ∗ algebras, Doc. Math. 5 (2000), 49–114 (electronic). 66. S.C. Power, Limit algebras: an introduction to subalgebras of C ∗ -algebras, Pitman Research Notes in Mathematics Series, vol. 278, Longman Scientific & Technical, Harlow, 1992. 67. A. Ramsay, The Mackey-Glimm dichotomy for foliations and other Polish groupoids, Journal of Functional Analysis 94 (1990), no. 2, 358–374. 68. M. Rørdam, F. Larsen, and N.J. Laustsen, An introduction to K-theory for C∗ algebras, London Mathematical Society Student Texts, no. 49, Cambridge University Press, 2000. 69. M.A. Rieffel, Projective modules over higher-dimensional noncommutative tori, Canad. J. Math. 40 (1988), no. 2, 257–338. 70. M. Rørdam, Classification of nuclear C ∗ -algebras, Encyclopaedia of Math. Sciences, vol. 126, Springer-Verlag, Berlin, 2002. , A simple C∗ -algebra with a finite and an infinite projection, Acta 71. Math. 191 (2003), 109–142. 72. C. Rosendal, Cofinal families of Borel equivalence relations and quasiorders, Journal of Symbolic Logic 70 (2005), 1325–1340. 73. M. Sabok, Completeness of the isomorphism problem for separable C*algebras, arXiv preprint arXiv:1306.1049 (2013). 74. R. Sasyk and A. T¨ ornquist, Borel reducibility and classification of von Neumann algebras, Bulletin of Symbolic Logic 15 (2009), no. 2, 169–183. , Turbulence and Araki-Woods factors, J. Funct. Anal. 259 (2010), 75. no. 9, 2238–2252. 76. P. Skoufranis, Operator theory notes, notes available at http://www.math.ucla.edu/%7Epskoufra/OperatorAlgebras.html, 2012. 77. W. Szyma´ nski, The range of K-invariants for C ∗ -algebras of infinite graphs, Indiana Univ. Math. J. 51 (2002), no. 1, 239–249. 78. K. Thomsen, Inductive limits of interval algebras: the tracial state space, Amer. J. Math. 116 (1994), no. 3, 605–620. 79. A.S. Toms, An infinite family of non-isomorphic C ∗ -algebras with identical K-theory, Trans. Amer. Math. Soc. 360 (2008), no. 10, 5343–5354. 80. , On the classification problem for nuclear C ∗ -algebras, Ann. of Math. (2) 167 (2008), no. 3, 1029–1044. , Comparison theory and smooth minimal C ∗ -dynamics, Comm. 81. Math. Phys. 289 (2009), no. 2, 401–433. 82. A.S. Toms and W. Winter, The Elliott conjecture for Villadsen algebras of the first type, J. Funct. Anal. 256 (2009), no. 5, 1311–1340. 83. J. Villadsen, Simple C ∗ -algebras with perforation, J. Funct. Anal. 154 (1998), no. 1, 110–116. 84. N. Weaver, Set theory and C ∗ -algebras, Bull. Symb. Logic 13 (2007), 1–20.

page 80

April 22, 2014

16:39

WSPC 9 x 6

Logic for C*-Algebras

singapore-final-r

81

85. C. Winslow and U. Haagerup, The Effros–Mar´echal topology in the space of von Neumann algebras, American Journal of Mathematics 120 (1998), 567–617. 86. W. Winter, Decomposition rank and Z-stability, Invent. Math. 179 (2010), 229–301. 87. , Nuclear dimension and Z-stability of pure C*-algebras, Invent. Math. 187 (2012), 259–342. , Regularity of nuclear C*-algebras, CBMS conference notes, to ap88. pear.

page 81

May 2, 2013

14:6

BC: 8831 - Probability and Statistical Theory

This page intentionally left blank

PST˙ws

March 14, 2014

16:18

WSPC 9 x 6: BC9154

SUBCOMPLETE FORCING AND L-FORCING

Ronald Jensen Humboldt-Universit¨ at zu Berlin, Institut f¨ ur Mathematik Sitz: Rudower Chaussee 25, D-10099 Berlin, Germany [email protected] In his book Proper Forcing (1982) Shelah introduced three classes of forcings (complete, proper, and semi-proper) and proved a strong iteration theorem for each of them: The first two are closed under countable support iterations. The latter is closed under revised countable support iterations subject to certain standard restraints. These theorems have been heavily used in modern set theory. For instance using them, one can formulate “forcing axioms” and prove them consistent relative to a supercompact cardinal. Examples are PFA, which says that Martin’s axiom holds for proper forcings, and MM, which says the same for semiproper forcings. Both these axioms imply the negation of CH. This is due to the fact that some proper forcings add new reals. Complete forcings, on the other hand, not only add no reals, but also no countable sets of ordinals. Hence they cannot change a cofinality to ω. Thus none of these theories enable us e.g. to show, assuming CH, that Namba forcing can be iterated without adding new reals. More recently we discovered that the three forcing classes mentioned above have natural generalizations which we call “subcomplete”, “subproper” and “semi-subproper”. It turns out that each of these is closed under Revised Countable Support (RCS) iterations subject to the usual restraints. The first part of our lecture deals with subcomplete forcings. These forcings do not add reals. Included among them, however, are Namba forcing, Prikry forcing, and many other forcings which change cofinalities. This gives a positive solution to the above mentioned iteration problem for Namba forcing. Using the iteration theorem one can also show that the Subcomplete Forcing Axiom (SCFA) is consistent relative to a supercompact cardinal. It has some of the more striking consequences of MM but is compatible with CH (and in fact with ♦). (Note: Shelah was able to solve the above mentioned iteration problem for Namba forcing by using his ingenious and complex theory of 83

jensen

page 83

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

84

“I-condition forcing”. The relationship of I-condition forcing to subcomplete forcing remains a mystery. There are, however, many applications of subcomplete forcing which have not been replicated by I-condition forcing.) In the second part of the lecture, we give an introduction to the theory of “L-forcings”. We initially developed this theory more than twenty years ago in order to force the existence of new reals. More recently, we discovered that there is an interesting theory of L-forcings which do not add reals. (In fact, if we assume CH +2ω1 = ω2 , then Namba forcing is among them.) Increasingly we came to feel that there should be a “natural” iteration theorem which would apply to a large class of these forcings. This led to the iteration theorem for subcomplete forcing. Combining all our methods, we were then able to prove: (1) Let κ be a strongly inaccessible cardinal. Assume CH. There is a subcomplete forcing extension in which κ becomes ω2 and every regular cardinal τ ∈ (ω1 , κ) acquires cofinality ω. (2) Let κ be as above, where GCH holds below κ. Let A ⊂ κ. There is a subcomplete forcing extension in which: – κ becomes ω2 ; – If τ ∈ (ω1 , κ) ∩ A is regular, then it acquires cofinality ω; – If τ ∈ (ω1 , κ)\A is regular, then it acquires cofinality ω1 . We will not be able to fully prove these theorems in our lectures, but we hope to develop some of the basic methods involved. Contents 0 Preliminaries 1 Admissible Sets 1.1 Introduction 1.2 Ill founded ZF− models 1.3 Primitive recursive set functions 2 Barwise Theory 3 Subcomplete Forcing 3.1 Introduction 3.2 Liftups 3.3 Examples 4 Iterating Subcomplete Forcing 5 L-Forcing 6 Examples 6.1 Example 1 6.2 Example 2 6.3 Example 3 6.4 The extended Namba problem References

85 94 94 100 101 102 112 112 116 127 136 155 167 167 171 174 175 181

page 84

April 22, 2014

16:42

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

85

0. Preliminaries ZF− (“ZF without power set”) consists of the axioms of extensionality and foundation together with:  (1) ∅, {x, y}, x are sets. (2) (Axiom of Subsets or “Aussonderungsaxiom”) x ∩ {z | ϕ(z)} is a set. (3) (Axiom of Collection)       x y ϕ(x, y) → u v x ∈ u y ∈ v ϕ(x, y). (4) (Axiom of Infinity) ω is a set. Note (3) implies the usual replacement axiom, but cannot be derived from it without the power set axiom. ZFC− is ZF− together with the strong form of the axiom of choice: (5) Every set is enumerable by an ordinal. Note The power set axiom is required to derive (5) from the weaker forms of choice. The Levy hierarchy of formulae is defined in the usual way: Σ0 formulae are the formulae containing only bounded quantification – i.e. Σ0 = the smallest set of formulae containing the primitive formulae and closed under sentential operations and bounded quantification:   x ∈ y ϕ, x∈yϕ     (where x ∈ y ϕ = x(x ∈ y → ϕ) and x ∈ y ϕ = x(x ∈ y ∧ ϕ)). (In some contexts it is useful to introduce bounded quantifiers as primitive signs rather than defined operations.) We set: Π0 = Σ0 . Σn+1 formulae are then the formulae of the form   x ϕ, where ϕ is Πn . Similarly Πn+1 formulae have the form x ϕ, where ϕ is Σn . A relation R on the model A is called Σn (A) (Πn (A)) iff it is definable over A by a Σn (Πn ) formula. R is Σn (A) (Πn (A)) in the parameters p1 , . . . , pm iff it is Σn (Πn ) definable in the parameters p1 , . . . , pn ∈ A. It is Σn (A) (Πn (A)) iff it is Σn (Πn ) definable in some parameters. It is Δn (A) iff it is Σn (A) and Πn (A).      x or card(x) denotes the cardinality of x. (We reserve the notation |x| for other uses.)

page 85

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

86

If r is a well ordering or a set of ordinals, then otp(r) denotes its order type. crit(f ) is the critical point of the function f (i.e. α = crit(f ) ↔ (f  α = id ∧ f (α) > α). F  A is the image of A under the function (or relation) F . rng(R) is the range of the relation R. dom(R) is the domain of the relation R. TC(x) is the transitive closure of x, Hα = {x | TC(x) < α}. Boolean Algebras and Forcing The theory of forcing can be developed using ”sets of conditions“ or complete Boolean algebras. The former is most useful when we attempt to devise a forcing for a specific end. The latter is more useful when we deal with the general theory of forcing, as in the theory of iterated forcing. We adopt here an integrated approach which begins with Boolean algebras. By a Boolean algebra we mean a partial ordering B = |B|, cB with maximal and minimal elements 0, 1, lattice operations ∩, ∪ defined by: a ⊂ (b ∩ c) ←→ (a ⊂ b ∧ a ⊂ c) (b ∪ c) ⊂ a ←→ (b ⊂ a ∧ c ⊂ a) and a complement operation ¬ defined by: a ⊂ ¬b ←→ a ∩ b = 0, satisfying the usual Boolean equalities. We call B a complete Boolean algebra   if, in addition, for each X ⊂ B there are operations B X, B X defined by: B  a⊂ X ←→ b ∈ X a ⊂ b,  B X ⊂ a ←→ b ∈ X b ⊂ a,

s.t.

a∩



b∈I

b=



b∈I

(a ∩ b),

a∪



b∈I

=



(a ∩ b).

b∈I

We shall generally write ’BA’ for ’Boolean algebra’. We write A ⊆ B to mean that A, B are BA’s, A is complete, and A is completely contained in B – i.e.  B  B X = A X, X = A X for X ⊂ A.  If A ⊆ B and b ∈ B, we define h(b) = hA,B (b) by: h(b) = {a ∈ A | b ⊂ a}. Thus:

page 86

April 22, 2014

16:42

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

87

  • h( bi ) = h(bi ) if bi ∈ B for i ∈ I. i

i

• h(a ∩ b) = a ∩ h(b) if a ∈ A. • b = 0 ↔ h(b) = 0 for b ∈ B.

     If B is a complete BA, we can form the canonical maximal B-valued model VB . The elements of VB are called names and there is a valuation function ϕ → [[ϕ]]B attaching to each statement ϕ = ϕ (t1 , . . . , tn ) a truth value in B. (Here ϕ is a ZFC formula and t1 , . . . , tn are names.) All axioms of ZFC have truth value 1 (assuming ZFC). The sentential connectives are interpreted by: [[ϕ ∧ ψ]] = [[ϕ]] ∩ [[ψ]];

[[ϕ ∨ ψ]] = [[ϕ]] ∪ [[ψ]];

[[ϕ → ψ]] = [[ϕ]] ⇒ [[ψ]], where (a ⇒ b) =Df ¬a ∪ b; [[¬ϕ]] = ¬[[ϕ]]. The quantifiers are interpreted by:   [[ϕ(x)]], [[ v ϕ(v)]] =

  [[ v ϕ(v)]] = [[ϕ(x)]].

x∈VB

x∈VB

If u ⊂ VB is a set and f : u → B, then there is a name x ∈ VB s.t.  [[y ∈ x]] = [[y = z]] ∩ f (x) z∈u

B

for all y ∈ V . Conversely, for each x ∈ VB there is a set ux ⊂ VB s.t.  [[y = z]] ∩ [[z ∈ x]]. [[y ∈ x]] = z∈ux

We can, in fact, arrange things s.t. {z, x | z ∈ ux } is a well founded relation. If U ⊂ VB is a class j and A : U → B, we may add to the ◦ ◦  [[x = z]] ∩ A(z). We language a predicate A interpreted by: [[Ax]] = inductively define for each x ∈ V a name xˇ by:  [[y ∈ x ˇ]] = [[y = zˇ]],

z∈u

z∈x

ˇ by: and a predicate V

ˇ = [[y ∈ V]]



[[y = zˇ]].

τ ∈V

∼ If σ : A ↔ B is an isomorphism, then we can define an injection σ∗ : VA → VB as follows: Let R = {z, x | z ∈ ux } be the above mentioned well

page 87

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

88

founded relation for VA . By R-induction we define σ ∗ (x), picking σ∗ (x) to be a w ∈ VB s.t.  [[y = σ ∗ (z)]]B ∩ σ([[z ∈ x]]A ). [[y ∈ w]]B = z∈ux

Then: (1)

σ([[ϕ(x )]]A ) = [[ϕ(σ ∗ (x ))]]B

for all ZFC formulae and all x1 , . . . , xn ∈ VA . If σ : A → B is a complete ∼ embedding (i.e. σ : A ↔ A ⊆ B for some A ), then σ∗ can be defined the same way, but (1) then holds only for Σ0 formulae. In such contexts it is often useful to take VB as a B-valued identity model, meaning that [[x = y]] = 1 −→ x = y

for x, y ∈ VB .

(If VB does not already have this property, we can attain it by factoring.) ∼ If σ : A ↔ B and VA , VB are identity models, then σ∗ is bijective (and is, in fact, an isomorphism of VA , I A , E A onto VB , I B , E B , where I = (x, y) = [[x = y]], E(x, y) = [[x ∈ y]]). Another advantage of identity models in that {z | [[z ∈ x]] = 1} is then a set, rather than a proper class. There are many ways to construct a maximal B-valued model VB and we can take its elements as being anything we want. Noting that A ⊆ B means that id  A is a complete embedding, it is useful, when dealing with such a pair A, B, to arrange that VA ⊂ VB and (id  A)∗ = id  VA . (We express this by: VA ⊆ VB .) The forcing relation B is defined by: b  ϕ ←→Df (b = 0 ∧ b ⊂ [[ϕ]]). We also set:  ϕ ↔Df [[ϕ]] = 1. Now suppose that W is an inner model of ZF and B ∈ W is complete in the sense of W . We can form W B internally in W , and it turns out that all ZF axioms are true in W B . (If W satisfies ZFC, then ZFC holds in W B .) W could also be a set rather than a class. If W is only a model of ZF− , we can still form W B , which will then model ZF− (or ZFC− if W models ZFC− ). (In this case, however, we may not be able – internally in W – to factor W B to an identity model.) We say that G ⊂ B is B-generic over W iff G is an ultrafilter on B which respects all intersections and unions of X ⊂ B s.t. X ∈ W – i.e.     x ∈ G ←→ b ∈ x b ∈ G, x ∈ G ←→ b ∈ X b ∈ G. If G is generic, we can form the generic extension W [G] of W by:

W [G] = {xG | x ∈ W G }, where xG = {z G | z ∈ ux ∧ [[z ∈ x]] ∈ G}.

page 88

March 14, 2014

16:18

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

89

Then W ⊂ W [G], since x ˇG = x (by G-induction on x ∈ W ). Then:  G b ∈ G b  ϕ(x1 , . . . , xn ). W [G]  ϕ(xG 1 , . . . , xn ) ←→

If we suppose, moreover, that for every b ∈ B \ {0} there is a generic G  b (e.g. if ϕ(B) ∩ W is countable), then: G b  ϕ(x1 , . . . , xn ) ←→ (W [G]  ϕ(xG 1 , . . . , xn ) for all generic G  b).

If B is complete in V we shall often find it useful to work in a mythical universe in which: (∗) V is an inner model and for every b ∈ B \ {0} there is a G  b which is B-generic over V. ˇ B ˇ in VC . This is harmless, since if C collapsed 2B to ω, then (∗) holds of V, ◦ ◦ ◦ ◦ ˇ and [[ˇb ∈ G]] = b for b ∈ B. (G We note that there is a G ∈ VB s.t.  G ⊂ B

is in fact unique if VB is an identity model.) If then G is B-generic over V, ◦ ◦ ◦ ˇ ˇ We call G the canonical over V. we have GG = G. Thus  G is B-generic B-generic name.

◦

If our language contains predicates A other than 0, ∈, we set: ◦

◦

AG = {xG | [[x ∈ A]] ∈ G}. Since [[x ∈ Vˇ ]] =



[[x = zˇ]], we get:

z∈V

ˇ G = {ˇ V z G | z ∈ V} = V.

Sets of Conditions By a set of conditions we mean P = |P|, ≤P s.t. ≤=≤p is a transitive relation on |P|. (Notationally we shall not distinguish between P and |P|.)  We say that two conditions p, q are compatible (pq) if r r ≤ p, q. Otherwise they are incompatible (p⊥q). For each set of conditions P there is a canonical complete BA over P (BA(P)) defined as follows: For X ⊂ P set:  ¬X = {q | p ∈ X p⊥q}. Then X ⊂ ¬¬X and ¬¬¬X = ¬X. Hence ¬¬ is a hull operator on P(P). Set |B| = {X ⊂ P | X = ¬¬X}. Then BA(P) = |B|, c , where c is the ordinary inclusion relation on |B|. B = BA(P) is then a complete BA with the complement operation ¬ and intersection and union operations given by: B  B  X = X, X = ¬¬ X.

page 89

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

90

  We say that Δ ⊂ P is dense in P iff p ∈ P q ⊆ p, q ∈ Δ. Δ is predense   in P iff p ∈ P q (qp and q ∈ Δ). (In other words, the closure of Δ under ≤ is dense in P.)  Set: [p] = ¬¬{p} for p ∈ P (i.e. [p] = {a ∈ BA(P) | a ⊃ b}). The forcing relation for P is defined by: p  ϕ ←→Df [p] ⊂ [[ϕ]].

If P ∈ W and W is a transitive model of ZF, we say that G is P-generic over W iff the following hold: • If p, q ∈ G, then pq. • If p ∈ G and p ≤ q, then q ∈ G. • If Δ ∈ W is dense in P, then G ∩ Δ = ∅. If B = BA(P)W is the complete BA over P (as defined in W ), then it follows that G is P-generic over W iff F = FG = {b ∈ B | b ∩ G = ∅} is B-generic over W . Conversely, if F is B-generic, thus G = GF = {p | [p] ∈ F } is P-generic. We also note that if B is a complete BA, then B \ {0}, ⊂ is a set of ∼ conditions, and there is an isomorphism σ : B ↔ BA(B \ {0}) defined by: σ(b) = {a | a ⊃ b}. Moreover, G is a B-generic filter iff it is a B \ {0}-generic set. When dealing with Boolean algebras, we shall often write: ”Δ is dense in B“ to mean ”Δ is dense in B \ {0}“. The Two Step Iteration Let A ⊆ B, where A, B are both complete. If (in some larger universe) G  is A-generic over V, then G = {b ∈ B | a ∈ G a ⊂ b} is a complete filter on B and we can form the factor algebra B/G (which we shall normally denote by B/G). It is not hard to see that B/G is then complete in V[G]. By the definition of the factor algebra there is a canonical homomorphism σ : B → B/G s.t. σ(b) ⊂ σ(c) ↔ ¬b ∪ c ∈ G . When the context permits we shall write b/G for σ(b). We now list some basic facts about this situation. Fact 1 Let B0 ⊆ B1 , B0 and B1 being complete. Let G0 be B0 -generic over ˜ =Df {b ∈ ˜ be B ˜ = B1 /G-generic over V[G]. Set G1 = G0 ∗ G V and let G ˜ Then G1 is B1 -generic over V and V[G1 ] = V[G0 ][G]. ˜ B1 | b/G0 ∈ G}. Conversely we have: ˜ = {b/G0 | Fact 2 If G1 is B1 -generic over V and we set: G0 = B0 ∩ G1 , G ˜ is B1 /G0 -generic over V[G0 ] and b ∈ G1 }. Then G0 is B0 -generic over V, G ˜ G1 = G0 ∗ G.

page 90

April 22, 2014

17:9

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

91

Fact 3 Let A ⊆ B and let h = hA,B as defined above. Then ◦

h(b) = [[ˇb/G = 0]]A , ◦

G being the A-generic name. Proof. h(b) =

◦   ([[ˇ a ⊃ ˇb]] ⇒ a) where a = [[ˇ a ∈ G]] = {a ∈ A | a ⊃ b} = a∈A



a∈A

◦ ◦ ◦  ˇ ⊃ ˇb → a ∈ G)]] = [[ˇb/G = 0]] [[ˇ a ⊃ ˇb → a ˇ ∈ G]] = [[ a ∈ A(a

QED(Fact 3)

◦

ˇ G, where ˆb ∈ VA . There is a unique Let A ⊆ B and A ˇb ∈ B/ ◦ ◦ b ∈ B s.t. A b = ˇb/G.

Fact 4

◦

◦

◦

Proof. To see uniqueness, let  ˇb/G = ˇb /G. Then  ˇb \ ˇb /G = 0. Hence ◦ h(b \ b ) = [[ˇb \ ˇb /G = 0]] = 0. Hence b \ b = 0. Hence b ⊂ b . Similarly b ⊂ b. ◦ ◦  To see the existence, note that Δ = {a ∈ A | b a  b = ˇb/G} is dense in ◦ ◦ A. Let X be a maximal antichain in Δ. Let a  b = ˇba /G for a ∈ X. Set: ◦ ◦  a ∩ ba . Then  b = ˇb/G, since if G is A-generic there is a ∈ X ∩ G b= a∈X

◦

by genericity. Hence b G = ba /G = b/G.

QED(Fact 4)

Fact 2 shows that, if B0 ⊆ B1 , then forcing with B1 is equivalent to a two ˜ ∈ V[G0 ]. step iteration: Forcing first by B0 to get V[G0 ] and then by a B ˜ is equivWe now show the converse: Forcing by B0 and then by some B alent to forcing by a single B1 : ◦

Fact 5 Let B0 be complete and let B0 B is complete. There is B1 ⊇ B0 ◦ ◦ ˇ 1 /G). (Hence, whenever G0 is B0 -generic, we s.t. B0 ( B is isomorphic to B ◦ ˜ =Df B G0 .) have B1 /G0  B In order to prove this we first define: ◦

◦

Definition Let A B is complete. B = A ∗ B is the BA defined as follows: Assume VA to be an identity model and set: ◦

|B| =Df {b ∈ VA | A b ∈ B},

b ⊂ c in B ←→Df A b ⊂ c.

page 91

April 22, 2014

17:9

WSPC 9 x 6: BC9154

jensen

R. Jensen

92

This defines B = |B|, ⊂ . B is easily seen to be a BA with the operations: a ∩ b = that c s.t. A c = a ∩ b, a ∪ b = that c s.t. A c = a ∪ b, ¬b = that c s.t. A c = ¬b. ◦

Similarly, if bi | i ∈ I is any sequence of elements of B, there is a B ∈ VA defined by: ◦

◦

A B : Iˇ −→ B;

◦

A B(ˇi) = bi for i ∈ I.

We then have: 

bi = that c s.t. A c =



bi = that c s.t. A c =

i∈I

◦



B(i),



B(i),

i∈Iˇ

i∈I

◦

i∈Iˇ

showing that B is complete. Now define σ : A → B by: ◦

◦

σ(a) = that c s.t. A (a ∈ G ∧ c = 1) ∨ (a ∈ / G ∧ c = 0). σ is easily shown to be a complete embedding. Clearly, if G is A-generic, then σ  G is σ A-generic, and V[G] = V[σ  G]. ˜ = B/G. ˜ = σ  G, B ˜ We then have for b, c ∈ B: Set G  ˜ ⊂ c/G ˜ ←→ a ∈ G σ(a) ⊂ (¬b ∪ c) b/G ←→ (¬bG ∪ cG ) = 1 ←− bG ⊂ cG (since σ(a)G = 1 for a ∈ G).

∼ ◦G ˜↔ ˜ = bG . Hence: Hence there is k : B B defined by: k(b/G) ◦

ˇ A ( B is isomorphic to B/G). ∼ If A = B0 and we pick B1 ,π : B ↔ B1 with πσ = id, then B1 satisfies Fact 5. QED ◦

The algebra A ∗ B constructed above is often useful. General Iterations It is clear from the foregoing that an n-step iteration – i.e. the result of n successive generic extensions of V – can be adequately described by a sequence Bi | i < n s.t. Bi ⊆ Bj for i ≤ j < n. The final model is the result of forcing with Bn−1 . What about transfinite iterations? At first glance it might seem that there is no such notion, but in fact we can define

page 92

April 22, 2014

17:9

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

93

the notion by turning the previous analysis on its head. We define: Definition By an iteration of length α > 0 we mean a sequence Bi | i < α of complete BA’s s.t. • Bi ⊆ Bj for i ≤ j < α.  Bi , i.e. there is • If λ < α is a limit ordinal, then Bλ is generated by i<λ    no proper B ⊂ Bλ s.t. Bi ⊂ B and X, X ∈ B for all X ⊂ B. i<λ

˜ i ] where If Gi is Bi+1 -generic and Gi = G ∩ Bi , then V[G] = V[Gi ][G ˜ ˜ Gi = {b/Gi | b ∈ G} is Bi = Bi+1 /Gi -generic. If G is λ-generic for a ˜ i -generic limit λ, then V[G] can be regarded as a ”limit“ of successive B ˜ i = Bi+1 /Gi for i < λ. extensions, where Gi = G ∩ Bi , B ◦

◦

In practice, we usually at the i-th stage pick a B i s.t. Bi ( B i is a complete BA), and arrange that: ◦

◦

ˇ i /G is isomorphic to B). Bi (B If the construction of the Bi ’s is sufficiently canonical, then the iteration is ◦

completely characterized by the sequence of B i ’s. However, our definition of ”iteration“ gives us great leeway in choosing Bλ for limit λ < α. We shall make use of that freedom in these notes. Traditionally, however, a handful of standard limiting procedures has been used. The direct limit takes Bλ as  Bi . It is characterized the minimal completion of the Boolean algebra i<λ  up to isomorphism by the property that Bi \ {0} lies dense in Bλ . (If i<λ  B∗ = BA( Bi \ {0}), there is then a unique isomorphism of Bλ onto B∗ i<λ  taking b to [b] for b ∈ Bi \ {0}.) Another frequently used variant is the i<λ

inverse limit , which can be defined as follows: By a thread in Bi | i < λ we mean a b = bi | i < λ s.t. bj ∈ Bj \{0} and hBi Bj (bj ) = bi for i ≤ j < λ. We call Bλ an inverse limit of Bi | i < λ iff  • If b is a thread, then b∗ = bi = 0 in Bλ . i<λ

• The set of such b∗ is dense in Bλ .

Bλ is then characterized up to isomorphism by these conditions. (If T is the set of all threads, we can define a partial ordering of T by: b ≤ c iff  i < λ bi ⊂ ci .) If we then set: B∗ = BA(T ), there is a unique isomorphism of Bλ onto B∗ taking b∗ to [b] for each thread b.)

page 93

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

94

By the support of a thread we mean the set of j < λ s.t. bi = bj for all i < j. The countable support (CS) limit is defined like the inverse limit using only those threads which have a countable support. A CS iteration is one in which Bλ is a CS limit for all limit λ < α. (This is equivalent to taking the inverse limit at λ of cofinality ω and otherwise the direct limit.) Countable support iterations tend to work well if no cardinal has its cofinality changed to ω in the course of the iteration. Otherwise – e.g. if we are trying to iterate Namba forcing – we can use the revised countable support (RCS) iteration, which was invented by Shelah. The present definition is due to Donder: By an RCS thread we mean a thread b s.t. either there is i < λ s.t. ˇ = ω or the support of b is bounded in λ. The RCS limit is then bi Bi cf(λ) defined like the inverse limit, using only RCS threads. An RCS iteration is one which uses the RCS limit at all limit points. Note Almost all iterations which have been employed to date make use of sublimits of the inverse limit – i.e. {b∗ | b is a thread ∧ b∗ = 0} is dense  in Bλ for all limit λ. This means that ( Bi )+ remains regular. In these i<λ

notes, however, we shall see that it is sometimes necessary to employ larger limits which do not have this consequence. In dealing with iterations we shall employ the following conventions: If B = Bi | i < α is an iteration we assume the VBi to be so constructed that VBi ⊆ VBj (in the sense of our earlier definition). In particular [[ϕ(x )]]Bi = [[ϕ( x ]]Bj for x1 , . . . , xn ∈ VBi , i ≤ j, when ϕ is a Σ0 formula. We shall also often simplify the notation by using the indices i < α as in: hij for hBi Bj , i for Bi , [[ϕ]]i for [[ϕ]]Bj . If i0 < α and G is Bi0 -generic, we set: B/G = Bi0 +j /G | j < α − i0 . We can assume the factor algebras to be so ˜ =  Bi is a BA. defined that Bi0 +h /G ⊆ Bi0 +j /G for h ≤ j < α − i0 . (B i<α

˜ Hence we can form B/G and identify Bi0 +j /G with {b/G | b ∈ Bi0 +j }.) It then follows easily that B/G is an iteration in V[G]. 1. Admissible Sets 1.1. Introduction

Let H = Hω be the collection of hereditarily finite sets. We use the usual Levy hierarchy of set theoretic formulae: Π0 = Σ0 = the set of all formulae in which all quantifiers are bounded.  Σn+1 = the set of all formulae x ϕ where ϕ is Πn .  Πn+1 = the set of all formulae x ϕ where ϕ is Σn .

page 94

March 14, 2014

16:18

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

95

The use of H offers an elegant way to develop ordinary recursion theory. Call a relation R ⊂ H n r.e. (or ”H-r.e.“) iff R is Σ1 -definable over H. We call R recursive (or H-recursive) iff it is Δ1 -definable (i.e. R and its complement ¬R are Σ1 -definable). Then R ⊂ ω n is rec (r.e.) in the usual sense iff it is the restriction of an H-rec. (H-r.e.) relation to ω. Moreover, there is an H-recursive function π : ω ↔ H s.t. R ⊂ H n is H-recursive iff {x1 , . . . , xn | R(π(x1 ), . . . , π(xn ))} is recursive. (Hence {x, y | π(x) ∈ π(y)} is recursive.)      This suggests a way of relativizing the concepts of recursion theory to transfinite domains: Let N = |N |, ∈, A1 , A2 , . . . be a transitive structure (with finitely or infinitely many predicates). We define: R ⊂ N n is N -r.e. (N -rec.) iff R is Σ1 (Δ1 ) definable over N. Since N may contain infinite sets, we must also relativize the notion ”finite“: u is N -finite iff u ∈ N. There are, however, certain basic properties which we expect any recursion theory to possess. In particular: • If A is recursive and u finite, then A ∩ u is finite. • If u is finite and F : u → N is recursive, then F  u is finite. The transitive structures N = |N |, ∈, A1 , A2 , . . . which yield a satisfactory recursion theory are called admissible. They were characterized by Kripke and Platek as those which satisfy the following axioms:  (1) ∅, {x, y}, x are sets. (2) The Σ0 -axiom of subsets (Aussonderung) x ∩ {z | ϕ(z)} is a set, where ϕ is any Σ0 formula. (3) The Σ0 -axiom of collection       x y ϕ(x, y) → u v x ∈ u y ∈ v ϕ(x, y) where ϕ is any Σ0 formula. Note Applying (3) to: x ∈ u → ϕ(x, y), we get:      x ∈ u y ϕ(x, y) −→ v x ∈ u y ∈ v ϕ(x, y).

Note Kripke-Platek set theory (KP) consists of the above axioms together with the axiom of extensionality and the full axiom of foundation (i.e. for all formulae, not just Σ0 ones). These latter axioms of course hold trivially in transitive domains. KPC (KP with choice) is KP augmented by: Every set is enumerable by an ordinal.

page 95

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

96

We now show that admissible structures satisfy the criteria stated above. Lemma 1 Let u ∈ M . Let A be Δ1 (M ). Then A ∩ u ∈ M .   Proof. Let Ax ↔ y A0 yx, ¬Ax ↔ y A1 yx, where A0 , A1 are Σ0 . Then     x y(A0 yx ∨ A1 yx). Hence there is v ∈ M s.t. x ∈ u y ∈ v(A0 yx ∨  QED(Lemma 1) A1 yx). Hence u ∩ A = u ∩ {x | y ∈ v A0 yx} ∈ M . Before verifying the second criterion we prove:

Lemma 2 M satisfies:      x ∈ u y1 . . . yn ϕ(x, y ) −→ v x ∈ u y1 . . . yn ∈ v ϕ(x, y ) for Σ0 formulas ϕ.   Proof. Assume x ∈ u y1 . . . yn ϕ(x, y ). Then



 x w(x ∈ u → y1 . . . yn ∈ w ϕ(x, y )).   Σ0



Hence there is v ∈ M s.t.     x ∈ u w ∈ v  y1 . . . yn ∈ w ϕ(x, y ). Take v = v  .

QED(Lemma 2)

Finally we get:

Lemma 3 Let u ∈ M , u ⊂ dom(F ), where F is Σ1 (M ). Then F  u ∈ M .   x ∈ Proof. Let y = F (x) ↔ z F  zyx, where F  is Σ0 (M ). Since      u y y = F (x), there is v s.t. x ∈ u y, z ∈ v F zyx. Hence F u =   QED(Lemma 3) v ∩ {y | x ∈ u z ∈ v F  zyx}. By similarly straightforward proofs we get:  Lemma 4 If Ryx is Σ1 , so is y Ryx.    Lemma 5 If Ryx is Σ1 , so is y ∈ u Ryx (since y ∈ u z ϕ(y, z) ↔    v y ∈ v z ∈ v ϕ(y, z)).   Σ0

Lemma 6 If R, Q ⊂ M n are Σ1 , then so are R ∪ Q, R ∩ Q.

Lemma 7 If R (y1 , . . . , yn ) is Σ1 and f (x1 , . . . , xm ) is a Σ1 function for i = 1, . . . , n, then R(f1 (x ), . . . , fn (x )) is Σ1 . Proof. R(f (x )) ↔



y1 . . . yn (

n 

i=1

yi = fi (x ) ∧ R(y )).

QED(Lemma 7)

page 96

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

97

Note The boldface versions of Lemmas 4–7 follow immediately. Corollary 8 If the functions f (z1 , . . . , zn), gi (x ) (i = 1, . . . , n) are Σ1 in a parameter p, then so is h(x )  f (g1 (x ), . . . , gn (x )).  Lemma 9 The following functions are Δ1 : x, x ∪ y, x ∩ y, x \ y (set difference), {x1 , . . . , xn }, x1 , . . . , xn , dom(x), rng(x), x y, x  y, x−1 , x × y, (x)ni , where: (z0 , . . . , zn−1 )i = zi ; (u)ni = ∅ otherwise;  x(z) if x is a function and z ∈ dom(x), x[z] = ∅ if not. Note As a corollary of Lemma 3 we have: If f is Σ1 , u ∈ M , u ⊂ dom(f ). Then f  u ∈ M , since f  u = g  u, where g(x)  f (x), x . Lemma 10 If f : M n+1 → M is Σ1 in the parameter p, then so are: F (u, x ) = {f (z, x ) | z ∈ u},

F  (u, x ) = f (z, x ) | z ∈ u .

    Proof. y = F (u, x ) ↔ z ∈ y v ∈ u z = f (y, z ) ∧ v ∈ u z ∈ y z = f (y, x ). But F  (u, x ) = {f  (z, x ) | z ∈ u}, where f  (y, x ) = f (y, x ), x . QED(Lemma 10) (Note The proof of Lemma 10 shows that, even if f is not defined everywhere, F is Σ1 in p, where: F (u, x )  {f (y, x ) | y ∈ u}, where this equation means that F (u, x ) is defined and has the displayed value iff f (y, x ) is defined for all y ∈ u. Similarly for F  .) Lemma 11 (Set Recursion Theorem) Let G be an n + 2-ary Σ1 function in the parameter p. Then there is F which is also Σ1 in p s.t. F (y, x )  G(y, x, F (z, x ) | z ∈ y ) (where this equation means that F is defined with the displayed value iff F (z, x ) is defined for all z ∈ y and G is defined at y, x, F (z, x ) | z ∈ y .) Proof. Set u = F (y, x) ↔

f (ϕ(f, x ) ∧ u, y ∈ f ), where  ϕ(f, x ) ←→ (f is a function ∧ dom(f ) ⊂ dom(f )  ∧ y ∈ dom(f ) f (y) = G(y, x, f  y)). 

The equation is verified by ∈-induction on y.

QED(Lemma 11)

page 97

April 22, 2014

16:42

WSPC 9 x 6: BC9154

jensen

R. Jensen

98

Corollary 12 TC, rn are Δ1 functions, where TC(x) = the transitive closure of x = x ∪ rn(x) = the rank of lub{rn(z) | z ∈ x}.



TC(z),

z∈x

Lemma 13 ω, On ∩ M are Σ0 classes.   Proof. x ∈ On ↔ ( x ⊂ x ∧ z, w ∈ x(z ∈ w ∨ w ∈ z)), x ∈ ω ↔  (x ∈ On ∧ ¬Lim(x) ∧ y ∈ x¬Lim(y)), where Lim(x) ↔ (x = 0 ∧ x ∈  On ∧ x = x). Corollary 14 The ordinal functions α + 1, α + β, α · β, αβ , . . . are Δ1 . An even more useful version of Lemma 11 is Lemma 15 Let G be as in Lemma 11. Let h : M → M be Σ1 in p s.t. {x, z | x ∈ h(z)} is well founded. There is F which is Σ1 in p s.t., F (y, x )  G(y, x, F (z, x ) | z ∈ h(y) ). The proof is just as before. We also note: Lemma 16.1 Let u ∈ Hω . Then the class u and the constant function f (x) = u are Σ0 .   x = z, x = u ↔ ( z ∈ x z ∈ Proof. ∈-induction on u: x ∈ u ↔ z∈u  u∧ z ∈ x). z∈u

QED

Lemma 16.2 If ω ∈ M , then the constant function x = ω is Σ0 .   Proof. x = ω ↔ ( z ∈ x z ∈ ω ∧ ∅ ∈ x ∧ z ∈ x z ∪ {z} ∈ x).

Lemma 16.3 If ω ∈ M , the constant for x = Hω is Σ1 (hence Δ1 ).      Proof. x = Hω ↔ ( z ∈ x u f n ∈ ω( n ⊂ u ∧ x ⊂ u ∧ f : n ↔  x)) ∧ ∅ ∈ x ∧ z, w ∈ x({z, w}, z ∪ w ∈ x).

Lemma 17 Fin, Pω (x) are Δ1 , where Fin = {x ∈ M | x < ω}, Pω (x) = Fin ∩ P(x).   Proof. x ∈ Fin ↔ n ∈ ω f fin ↔ x,     x∈ / Fin ↔ y(y = ω ∧ n ∈ y f n ⊂ x fin ↔ n),    y = Pω (x) ↔ u ∈ y(u ∈ Fin ∧ u ⊂ x) ∧ z ∈ x ({z} ∈ y ∧ u, v ∈ y u ∪ v ∈ y). QED

page 98

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

99

The constructible hierarchy relative to a class A is defined by: L0 [A] = ∅; Lν+1 [A]d = Def(Lν [A], A ∩ Lν [A] )  Lν [A] for limit λ, Lλ [A] = ν<λ

where Def(A) is the set of B ⊂ A which are A-definable in parameters from A. We also define Lν = Lν [∅]. The constructible hierarchy over a set u is defined by: L0 (u) = TC({u}), Lν+1 (u) = Def(Lν (u)),  Lν (u) for limit λ. Lλ (u) = ν<λ

It is easily seen that: Lemma 18 If A ⊂ M is Δ1 (M ) in p, then Lν [A] | ν ∈ M is Δ1 (M ) in p. • If u ∈ M , then Lν (u) | ν ∈ M is Δ1 (M ) in u. By set recursion we can also define a sequence 


ν<α


a Δ1 (M ) well ordering of M . Moreover, there is a Δ1 (M ) map h : M → M s.t. h(x) = {z | z <M x}. Using this, it follows easily that every Σ1 (M ) relation is uniformizable by a Σ1 (M ) function. Thus the KP axioms give us a “reasonable” recursion theory. They do not suffice, however, to get Σ1 -uniformization. In fact, since we have not posited the axiom of choice, we do not even have N -finite uniformization. However, the admissible structures dealt with in these notes will almost always satisfy Σ1 -uniformization. This can happen in different ways. If N =

page 99

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

100

LA τ =Df Lτ [A], A , there is a well ordering < of N s.t. the function h(x) = {z | z < x} is Σ1 . We can then uniformize R(y, x ) as follows: Let R(y, x ) ↔  z R (y, z, x ), where R is Σ0 . R is then uniformized by:   z(R (y, z, x ) ∧ y  , z  ∈ h(y, z )¬R(u , z  x )).

The same holds for N = Lτ (a) where a is a transitive set with a well ordering constructible from a below τ . If N is a ZFC− model with a definable well ordering <, then every definable relation has a definable uniformization. If N ∗ = N, A1 , A2 , . . . is the result of adding all N -definable predicates to N , then the Σ1 (N ∗ ) relations are exactly the N -definable relations, so uniformization holds trivially. 1.2. Ill founded ZF− models We now prove a lemma about arbitrary (possibly ill founded) models of ZF− (where the language of ZF− may contain predicates other than ’∈’). Let A = A, ∈A , B1 , B2 , . . . be such a model. For X ⊂ A we of course write A|X = X, ∈A ∩X 2 , . . . . By the well founded core of A we mean the set of all x ∈ A s.t. ∈A ∩ C(x)2 is well founded, where C(x) is the closure of {x} under ∈A . Let wfc(A) denote the restriction of A to its well founded core. Then wfc(A) is a well founded structure satisfying the axiom of extensionality, and is, therefore, isomorphic to a transitive structure. Hence there is A s.t. A is isomorphic to A and wfc(A ) is transitive. We say that a model A of ZF− is solid iff wfc(A) is transitive and ∈wfc(A) =∈ ∩wfc(A)2 . Thus every consistent set of sentences in ZF− has a solid model. Note that if A is solid, then ω ⊂ wfc(A). By Σ0 -absoluteness we of course have: (1)

wfc(A)  ϕ(x ) ←→ A  ϕ(x )

if x1 , . . . , xn ∈ wfc(A) and ϕ is a Σ0 -formula. By ∈-induction on x ∈ wfc(A) it follows that the rank function is absolute: (2)

rn(x) = rnA (x) for x ∈ wfc(A). Using this we prove:

Lemma 21 Let A be a solid model of ZF− . Then wfc(A) is admissible. Proof. Let ϕ be Σ0 and let   (3) wfc(A))  x y ϕ(x, y, z )

where z1 , . . . , zn ∈ wfc(A). Let u ∈ wfc(A). By (3) and Σ0 absoluteness:   (4) A  x ∈ u y ϕ(x, y, z ).

page 100

March 14, 2014

16:18

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

101

Since A is a ZFC− model, there must then be v ∈ A of minimal A-rank rnA (v) s.t.   (5) A  x ∈ u y ∈ v ϕ(x, y, z ).

It suffices to note that rnA (v) ∈ wfc(A), hence rnA (v) = rn(v) and v ∈ wfc(A). (Otherwise there is r ∈ A s.t. A  r < rn(v) and there is v  ∈ A s.t. A  v  = {x ∈ v | rn(x) < r}. Hence v  satisfies (5) and rnA (v  ) < rnA (v). Contradiction!) By Σ0 absoluteness, then:   (6) wfc(A)  x ∈ u y ∈ v ϕ(x, y, z ). QED (Lemma 21) As immediate corollaries we have:

Corollary 21.1 Let δ = On ∩ wfc(A). Then Lδ (a) is admissible for a ∈ wfc(A). Corollary 21.2 definable.

LA δ = Lδ [A], A ∩ Lδ [A] admissible whenever A is A-

(Proof. We may suppose w.l.o.g. that A is one of the predicates of A.) Note In Lemma 21 we can replace ZF− by KP. In this form it is known as Ville’s Lemma. However, a form of Lemma 21 was first employed in our paper [NA] with Harvey Friedman. If memory serves us, the idea was due to Friedman. 1.3. Primitive recursive set functions A function f : V → V is called primitive recursive (pr) iff it is generated by successive applications of the following schemata: (i) (ii) (iii) (iv) (v)

f (x ) = xi (here x is x1 , . . . , xn ) f (x ) = {xi , xj } f (x ) = xi \ xj f (x ) = g(h1 (x ), . . . , hm (x ))  f (y, x ) = g(z, x ) z∈y

(vi) f (y, x ) = g(y, x, f (z, x ) | z ∈ y ) We call A ⊂ Vn a pr relation iff its characteristic function is a pr function. (However , a function can be a pr relation without being a pr function.) pr functions are ubiquitous. It is easily seen for instance that the functions listed in Lemma 9 are pr. Lemmas 4–7 hold with ’Σ1 ’ replaced by ’pr’. The functions TC(x), rn(x) are easily seen to be pr. We call f :

page 101

March 14, 2014

16:18

102

WSPC 9 x 6: BC9154

jensen

R. Jensen

Onn → V a pr function if it is the restriction of a pr function to On. The functions α + 1, α + β, α · β, αβ , . . . etc. are then pr. Since the pr functions are proper classes, the above discussion is carried out in second order set theory. However, all that needs to be said about pr functions can, in fact, be adequately expressed in ZFC. To do this we talk about pr definitions: By a pr definition we mean a finite list of schemata of the form (i)–(vi) s.t. • the function variable on the left side does not occur in a previous equation in the list. • every function variable on the right side occurs previously on the left side. Clearly, every pr definition s defines a pr function Fs . Moreover, for each s, Fs has a canonical Σ1 definition ϕs (y, x1 , . . . , xn ). (Indeed, the relation {x, s | x ∈ Fs } is Σ1 .) The canonical definition has some remarkable absoluteness properties. If u is transitive, let Fsu be the function obtained by relativizing the canonical definition to u. Hence Fsu ⊂ Fs is a partial map on u. Then: • If u is pr closed, then Fsu = Fs ∩ u. • If α is closed under the functions ν + 1, ν · τ, ν τ , . . . etc., then Lα [A] is pr closed for every A ⊂ V. These facts are provable in ZFC− . The proofs can be found in [AS] or [PR] As corollaries we get: V[G]

(1) Let V[G] be a generic extension of V. Then V ∩ Fs = FsV . (2) Let A be a solid model of ZFC− . Let A = wfc(A). Then FsA ∩ A = FsA = Fs . Proof. We prove (2). Clearly FsA = Fs , since A being admissible, is pr closed. But each x ∈ A is an element of a transitive pr closed u ∈ A, since A is admissible. Hence y = FsA (x) ↔ y = Fsu (x) ↔ y = FsA (x). QED 2. Barwise Theory Jon Barwise worked out the syntax and model theory of certain infinitary (but M -finitary) languages on countable admissible structures M . In so doing, he created a powerful and flexible tool for set theorists, which enables us to construct transitive structures using elementary model theory. In this

page 102

March 14, 2014

16:18

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

103

section we give an introduction to Barwise’ work, whose potential for set theory has, we feel, been unduly neglected. Let M be admissible. Barwise develops a first order theory in which arbitrary M -finite conjunctions and disjunctions are allowed. The predicates, however, have only a (genuinely) finite number of argument places and there are no infinite strings of quantifiers. If we wish to make use of the notion of M -finiteness, we must “arithmetize” the language – i.e. identify its symbols with objects in M . A typical arithmetization is: Predicates: Pxn = 0, n, x (x ∈ M , 1 ≤ n < ω) (Pxn = the x-th n-place predicate) Constants: cx = 1, x (x ∈ M ) Variables: vx = 2, x (x ∈ M ) Note The set of variables must be M -infinite, since otherwise a single formula could exhaust all the variables. We let P02 be the identity predicate ˙ which will be a part of most (=) ˙ and also reserve P12 as the ∈-predicate (∈), interesting languages. By a primitive formula we mean P t1 . . . tn = 3, P, t1 , . . . , tn , where P is an n-place predicate and t1 , . . . , tn are variables and constants. We then define: ¬ϕ = 4, ϕ ,

(ϕ ∨ ψ) = 5, ϕ, ψ ,

(ϕ ∧ ψ) = 6, ϕ, ψ ,  (ϕ → ψ) = 7, ϕ, ψ , (ϕ ↔ ψ) = 8, ϕ, ψ , v ϕ = 9, v, ϕ ,      v ϕ = 10, v, ϕ , and: f = 11, f , f = 12, f .

The set Fml of 1-st order M -formulas is the smallest set X which contains all primitive formulae, is closed under ¬, ∨, ∧, →, ↔, and s.t.   • If v is a variable and ϕ ∈ X, then v ϕ, v ϕ ∈ X.     f and • If f = ϕi | i ∈ I ∈ M and ϕi ∈ X for i ∈ I, then ϕi =Df i∈I     ϕi =Df f are in I. i∈I

Then the usual syntactical notions are Δ1 , including: Fml, Cnst (set of constants), Vbl (set of variables), Sent (set of all sentences), Fr(ϕ) = the set of free variables in ϕ, and: ϕ(v1 , . . . , vn /t1 , . . . , tn )  the result of replacing all free occurences of the vbl vi by ti (where ti ∈ Vbl ∪ Const), as long as this can be done without any new occurence of a variable ti being bound; otherwise undefined.

page 103

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

104

That Vbl, Const are Δ1 (in fact Σ0 ) is immediate. The characteristic function X of Fml is definable by a recursion of the form: X (x) = G(x, X (z) | z ∈ TC(x) ). Similarly for the functions Fr(ϕ) and ϕ(v /). Then Sent = {ϕ | Fr(ϕ) = ∅}. t Note We of course employ the usual notation, writing ϕ(t1 , . . . , tn ) for ϕ(v1 , . . . , vn /t1 , . . . , tn ), where the sequence v1 , . . . , vn is taken as known. M -finite predicate logic has as axioms all instances of the usual predicate logical axiom schemata together with:



ϕi −→ ϕj , ϕj −→ ϕi for j ∈ u ∈ M. i∈u

i∈u

The rules of inference are: ϕ, ϕ → ψ (modus ponens), ψ ψ→ϕ ϕ→ψ  ,  for x ∈ / Fr(ϕ), ϕ→ xψ xψ→ϕ ψi → ϕ (i ∈ u) ϕ → ψi (i ∈ u)     , . ϕ→ ϕi ψi → ϕ i∈u

i∈u

We say that ϕ is provable from a set of statements A if ϕ is in the smallest set which contains A and the axioms and is closed under the rules of inference. We write A  ϕ to mean that ϕ is provable from A. (Note: By the last rule,       ∅ → ϕ for every ϕ, hence  ¬ ∅. Similarly  ∅.) A formula is provable if and only if it has a proof. Because we have not assumed choice to hold in our admissible structure M , we must use a somewhat unorthodox concept of proof, however. Definition By a proof from A we mean a sequence pi | i < α s.t. α ∈ On and for each i < α, if ψ ∈ pi , then either ψ ∈ A or ψ is an axiom or ψ  ph by a single application of one of the rules. follows from h
If A is Σ1 (M ) in a parameter q it follows easily that {p ∈ M | p is a proof from A} is Σ1 (M ) in the same parameter. It is also easily seen that A  ϕ iff there exists a proof of ϕ from A. A more interesting conclusion is:

page 104

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

105

Lemma 1 Let A be Σ1 (M ). Then A  ϕ iff there is an M -finite proof of ϕ from A. Proof . (←) is trivial. We prove (→). Let X = the set of ϕ s.t. there exists a p ∈ M which proves ϕ from A. Claim {ϕ | A  ϕ} ⊂ X. Proof . We know that A ⊂ X and all axioms lie in X. Hence it suffices to show that X is closed under the rules of proof. This must be demonstrated rule by rule. As an example we show:   ψi ∈ X. Claim Let ϕ → ψi ∈ X for i ∈ u, where u ∈ M . Then ϕ → i∈u

Proof . Let P (p, ψ) mean: p is a proof of ψ from A. Then P is Σ1 (M ). By our assumption:   (1) i∈u p P (p, ϕ → ψi ).  Now let P (p, ψ) ↔ z P  (z, p, ψ), where P  is Σ0 . We then have:    (2) i∈u z p P  (z, p, ϕ → ψi )

whence follows easily that there is v ∈ M with:    (3) i∈u z∈v p ∈ v P  (z, p, ϕ → ψi ).   Set w = {p ∈ v | i ∈ u z ∈ v P  (z, p, ψ)}. Then   (4) i∈u p ∈ w P (p, ϕ → ψi ) and w consists of proofs from A.

Let α ∈ M , α ≥ dom(p) for all p ∈ w. Define a proof p∗ of length α + 1 by: ⎧ ⎨ {pi | p ∈ w ∧ i ∈ dom(p)} for i < α,  p∗ (i) = {ϕ →  ψi } for i = α. ⎩ i∈u

Then p∗ ∈ M proves ϕ →

 

i∈u

ψi from A.

QED(Lemma 1)

From this we get the M -finiteness lemma: Lemma 2 Let A be Σ1 (M ). Then A  ϕ iff there is u ∈ M s.t. u ⊂ A and u  ϕ. Proof . (←) is trivial. We prove (→). Let p ∈ M be a proof of ϕ from A. Let u = the set of ψ s.t. for some  ph i ∈ dom(p), ψ ∈ pi , but ψ is not an axiom and does not follow from h
by a single application of a rule. Then u ∈ M , u ⊂ A, and p is a proof from u. Hence u  ϕ. QED(Lemma 2)

page 105

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

106

Another consequence of Lemma 1 is Lemma 3 Let A be Σ1 (M ) in q. Then {ϕ | A  ϕ} is Σ1 (M ) in the same parameter q (uniformly in the Σ1 definition of A from q).  Proof . {ϕ | A  ϕ} = {ϕ | p ∈ M p proves ϕ from A}. QED Corollary 4 Let A be Σ1 (M ) in q. Then “A is consistent” is Π1 (M ) in the same parameter q (uniformly in the Σ1 definition of A from q). Note that, since u ∈ M is uniformly Σ1 (M ) in itself, we have: Corollary 5 {u, ϕ | u ∈ M ∧ u  ϕ} is Σ1 (M ). Similarly: Corollary 6 {u ∈ M | u is consistent} is Π1 (M ). Note Call a proof p strict iff pi = 1 for i ∈ dom(p). This corresponds to the more usual notion of proof. If M satisfies the axiom of choice in the form: Every set is enumerable by an ordinal, then Lemma 1 holds with “strict proof” in place of “proof”. We leave this to the reader. Languages We will normally not employ all of the predicates and constants in our M finitary first order logic, but cut down to a smaller set of symbols which we intend to interpret in a model. Thus we define a language to be a set L of predicates and constants. By a model of L we mean a structure A = |A|, tA | t ∈ L s.t. |A| = ∅, P A ⊂ |A|n whenever P is an n-place predicate, and cA ∈ |A| whenever |A| is a constant. By a variable assignment we mean a map f : Vbl → A (Vbl being the set of all variables). The satisfaction relation A  ϕ[f ] is defined in the usual way, where A  ϕ[f ] means that the formula ϕ becomes true in A if the free variables in ϕ are interpreted by f . We leave the definition to the reader, remarking only that:  ϕi [f ] iff i ∈ u A  ϕi [f ], A i∈u

A



i∈u

ϕi [f ] iff

 i ∈ u A  ϕi [f ].

We adopt the usual conventions of model theory, writing A = |A|, tA 1 , . . . if we think of the predicates and constants of L as being arranged in a fixed

page 106

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

107

sequence t1 , t2 , . . . Similarly, if ϕ = ϕ(v1 , . . . , vn ) is a formula in which at most the variables v1 , . . . , vn occur free, we write: A  ϕ[x1 , . . . , xn ] for: A  ϕ[f ] where f (vi ) = xi (i = 1, . . . , n). If ϕ is a statement, we write: A  ϕ. If A is a set of statements we write: A  A to mean: A  ϕ for all ϕ ∈ A. The correctness theorem says that if A is a set of L-statements and A  A, then A is consistent. (We leave this to the reader.) Barwise’ Completeness Theorem says that the converse holds if our admissible structure M is countable: Theorem 7 Let M be a countable admissible structure. Let A be a set of statements in the M -language L. If A is consistent in M -finite predicate logic, then A has a model A. Proof (sketch). We make use of the following theorem of Rasiowa and Sikorski: Let B be a Boolean algebra. Let Xi ⊂ B (i < ω) s.t. the Boolean union  Xi = bi exists in the sense of B. Then B has an ultrafilter U s.t. bi ∈ U ←→ Xi ∩ U = ∅

for i < ω.

(Proof . Successively choose ci (i < ω) by c0 = 1, ci+1 = ci ∩ b = 0, where b ∈ Xi ∪ {¬bi }. Let U = {a ∈ B | Vi ci ⊂ a}. Then U is a filter and extends to an ultrafilter on B.) Extend the language L by adding an M -infinite set C of new constants. Call the extended language L∗ and set: [ϕ] = {ψ | A  ψ ↔ ϕ} for L∗ -statements ϕ. Then B = {[ϕ] | ϕ ∈ StL∗ } in the Lindenbaum algebra of L∗ with the operations: [ϕ] ∪ [ψ] = [ϕ ∨ ψ], [ϕ] ∩ [ψ] = [ϕ ∧ ψ], ¬[ϕ] = [¬ϕ], 

  

  (u ∈ M ), (u ∈ M ), [ϕi ] = ϕi [ϕi ] = ϕi

i∈u

i∈u



 [ϕ(c)] = [ v ϕ(v)],

c∈C

i∈u



c∈C

i∈u

 [ϕ(c)] = [ v ϕ(v)].

The last two equations hold because the constants in C, which do not occur in the axioms A, behave like free variables. By Rasiowa and Sikorski there is then an ultrafilter U on B which respects the above operations. We define

page 107

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

108

a model A = |A|, tA | t ∈ L as follows: For c ∈ C set [c] = {c ∈ C | [c = c ] ∈ U }. If P ∈ L is an n-place predicate, set: P A ([c1 ], . . . , [cn ]) ←→ [P c1 . . . cn ] ∈ U. If t ∈ L is a constant set: tA = [c], where c ∈ C,

[t = c] ∈ U.

A straightforward induction then shows: A  ϕ[[c1 ], . . . , [cn ]] ←→ [ϕ(c1 , . . . , cn )] ∈ U for formulae ϕ = ϕ(v1 , . . . , vn ) with at most the free variables v1 , . . . , vn . In particular A  ϕ ↔ [ϕ] ∈ U for L∗ -statements ϕ. Hence A  A, since [ϕ] = 1 for all ϕ ∈ A. QED(Theorem 7) Combining the completeness theorem with the M -finiteness lemma, we get the well known Barwise compactness theorem: Corollary 8 Let M be countable. Let L be Δ1 and A be Σ1 . If every M -finite subset of A has a model, then so does A. By a theory or axiomatized language we mean a pair L = L0 , A s.t. L0 is a language and A a set of L0 -statements. We say that A models L iff A is a model of L0 and A  A. We also write: L  ϕ for (A  ϕ ∧ ϕ ∈ FmlL0 ). We say that L = L0 , A is Σ1 (M ) (in the parameter p) iff L0 is Δ1 (M ) (in p) and A is Σ1 (M ) (in p). Similarly for: L is Δ1 (M ) (in p).  We now consider the class of axiomatized languages containing a fixed pred˙ the special constants x (x ∈ M ) (We can set e.g. x = 1, 0, x .) icate ∈, and the basic axioms • Extensionality    v = z) (x ∈ M ) • v(v ∈˙ x ↔ z∈x

(Further predicates, constants, and axioms are allowed, of course.) We call any such theory an “∈ -theory”. Then: Lemma 9 Let A be a solid model of the ∈-theory L. Then xA = x ∈ wfc(A) for x ∈ M . Proof . ∈-induction on x. Definition Let L be an ∈-theory. ZF− L is the set of (really) finite Lstatements which are axioms of L. (Similarly for ZFC− L .)

page 108

March 14, 2014

16:18

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

109

− We write L  ZF− for L  ZF− L . (Similarly for L  ZFC .)

 ∈-theories are a useful tool in set theory. We now bring some typical applications. We recall that an ordinal α is called admissible if Lα is admissible and admissible in a ⊂ α if Laα = Lα [a], a is admissible. Lemma 10 Let α > ω be a countable admissible ordinal. There is a ⊂ ω s.t. α is the least ordinal admissible in a. This follows straightforwardly from: Lemma 11 Let M be a countable admissible structure. Let L be a consistent Σ1 (M ) ∈-theory s.t. L  ZF− . Then L has a solid model A s.t. On ∩ wfc(A) = On ∩ M . We first show that Lemma 11 implies Lemma 10, and then prove Lemma 11. Take M = Lα , where α is as in Lemma 10. Let L be the M -theory with: Predicate: ∈˙ ◦ Constants: x (x ∈ M ), a ◦ Axioms: Basic axioms + ZF− , and β is not admissible in a (β < α). Then L is consistent, since Hω1 , ∈, a is a model, where a is any a ⊂ ω which codes a well ordering of type ≥ α (and x is interpretedly x for x ∈ M ). Now let A be a solid model of L s.t. On ∩ wfc(A) = α. Then wfc(A) is ◦ admissible by Section 1, Lemma 21. Hence so is Laα , where a = a A . But β is not admissible in a for ω < β < α, since “β is admissible in a” is Σ1 (Laα ); hence the same Σ1 statement would hold of β in A. Contradiction! QED(Lemma 10) Note Pursuing this method a bit further we can prove: Let ω < α0 < . . . < αn−1 be a sequence of countable admissible ordinals. There is a ⊂ ω s.t. αi = the i-th α > ω which is admissible in a (i < n). A similar theorem holds for countable infinite sequences, but the proof requires forcing and is much more complex. It is given in §5 and §6 [AS]. We now prove Lemma 11 by modifying the proof of the completeness theorem. Let Γ(v) be the set of formulae v ∈ On, v > β (β ∈ M ). Add an M -infinite (but Δ1 (M )) set E of new constants to L. Let L be L with the new constants and the new axioms Γ(e) (e ∈ E). Then L is consistent, since any M -finite subset of the axioms can be modeled by interpreting the new constants as ordinals in wfc(A), A being any solid model of L. As in

page 109

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

110

the proof of completeness we then add a new class C of constants which is not M -finite. We assume, however, that C is Δ1 (M ). We add no further axioms, so the elements of C behave like free variables. The so extended language L is clearly Σ1 (M ). Now set:   Δ(v) = {v ∈ / On} ∪ {v ≤ β} ∪ {e < v}. β∈M

e∈E

 Claim Let c ∈ C. Then {[ϕ] | ϕ ∈ Δ(c)} = 1 in the Lindenbaum algebra of L .

Proof . Suppose not. Set Δ = {¬ϕ | ϕ ∈ Δ(c}. Then there is an L statement ψ s.t. A∪{ψ} is consistent, where L = L0 , A and A∪{ψ}  Δ . Pick an e ∈ E which does not occur in ψ. Let A∗ be the result of omitting the axioms Γ(e) from A. Then A∗ ∪ {ψ} ∪ Γ(e)  c ≤ e. By the M -finiteness lemma there is β ∈ M s.t. A∗ ∪ {ψ} ∪ {β ≤ e}  c ≤ e. But e behaves here like a free variable, so A∗ ∪ {ψ}  c ≤ β. But A ⊃ A∗ and A ∪ {ψ}  β < c. Thus A ∪ {ψ} is inconsistent. Contradiction! QED(Claim)

Now let U be an ultrafilter on the Lindenbaum algebra of L which respects both the operations listed in the proof of the completeness theorem and the  unions {[ϕ] | ϕ ∈ Δ(c)} for c ∈ C. Let X = {ϕ | [ϕ] ∈ U }. Then as before, L has a model A, all of whose elements have the form cA for a c ∈ C and such that A  ϕ ↔ ϕ ∈ X for L -statements ϕ. We assume w.l.o.g. that A is solid. It suffices to show that Y = {x ∈ A | x > ν in A for all v ∈ m} has no minimal element in A. Let x ∈ Y , x = cA . Then A  e < c for some QED(Lemma 11) e ∈ E. But eA ∈ Y . Another – very typical – application is: Lemma 12 Let W be an inner model of ZFC. Suppose that, in W , U is a normal measure on κ. Let τ > κ be regular in W and set M = HτW , U . Assume that M is countable in V. Then for any α ≤ κ there is M = H, U s.t. U is a normal measure in M and M iterates to M in exactly α many steps. (Hence M is iterable, since M is.) Proof . The case α = 0 is trivial, so assume α > 0. Let δ be least s.t. Lδ (M ) is admissible. Then N = Lδ (M ) is countable. Let L be the ∈-theory on N with: Predicate: ∈˙ ◦ Constants: x (x ∈ N ), M Axioms:

◦

◦

◦

◦

The basic axioms; ZFC− ; M = H, U is a transitive ZFC−

model; M iterates to M in α many steps.

page 110

April 22, 2014

16:42

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

111

It suffices to prove: Claim L is consistent. We first show that the claim implies the theorem. Let A be a solid model ◦

of L. Then N ⊂ wfc(A). Hence M, M ∈ wfc(A), where M = M A . There is M i | i < α which, in A, is an iteration from M to M . But then M i | i < α ∈ wfc(A) really is an iteration by absoluteness. QED We now prove the claim. Case 1 α < κ. Iterate W, U α many times, getting Wi , Ui (i ≤ α) with iteration maps πij : Wi , Ui ≺ Wj , Uj . Set Mi = π0i (M ). Then Mi , Ui (i ≤ α) is  = πij  Mi . It suffices to show that the iteration of M, U with maps πij Lα = π0,α (L) is consistent. This is clear, however, since Hτ + , M models ◦

◦

Lα (with M interpreting the constant M α = π0,α (M )).

QED(Case 1)

Case 2 α = κ. This time we iterate W, U β many times where π0β (κ) = β and β ≤ κ+ . Hτ + , M again models Lβ . QED(Lemma 12) Barwise theory is useful in situations where one is given a transitive structure Q and wishes to find a transitive structure Q with similar properties inside an inner model. Another tool used in such situations is Schoenfield’s lemma, which, however requires coding Q by a real. Unsurprisingly, Schoenfield’s lemma can itself be derived from Barwise theory. We first note the well known fact that every Σ12 condition on a real is equivalent to a Σ1 (Hω1 ) condition, and conversely. Thus it suffices to show: Lemma 13 Let Hω1  ϕ[a], a ⊂ ω, where ϕ is Σ1 . Then HωL[a]  ϕ[a]. 1  Proof . Let ϕ = z ψ, where ψ is Σ0 . Let Hω1  ψ[z, a], where rn(z) < α and α is admissible in a. Let L be the language on Lα (a) with: Predicate: ∈˙ ◦ Constants: z , x (x ∈ Lα (a)) ◦ Axioms: Basic axioms, ZFC− , ψ( z , a). Then L is consistent since Hω1 , z is a model. Applying L¨owenheimSkolem in L(a), we find a countable α and a map π : Lα (a) ≺ Lα (a). Let ◦ ◦ w.l.o.g. π( z ) = z and let L be defined over Lα (a) like L over Lα (a). Then

page 111

April 22, 2014

16:42

WSPC 9 x 6: BC9154

jensen

R. Jensen

112

◦

L is consistent and has a solid model A in L(a). Let z = z A . Then z ∈ L(a) and Hω1  ψ[z, a] in L(a). QED(Lemma 13) 3. Subcomplete Forcing 3.1. Introduction In §10 of [PF] Shelah defines the notion of complete forcing: Definition Let B be a complete BA. B is a complete forcing iff for sufficiently large θ we have: Let B ∈ Hθ . Let σ : H ≺ H, where H is countable and transitive. Let σ(B) = B. If G is B-generic over H, then there is b ∈ B which forces that, that whenever G  b is B-generic, then σ  G ⊂ G. Note If G, G, H, H, σ are as above, then σ extends uniquely to a σ ∗ s.t. σ∗ : H[G] ≺ H[G] and σ ∗ (G) = G. Proof . To see uniqueness, note that each x ∈ H[G] has the form x = tG where t ∈ H is a B-name. Thus σ ∗ (x) = σ(t)G . To see existence, note that:  G H[G]  ϕ(tG b ∈ G b H ϕ(t1 , . . . , tn ) 1 , . . . , tn ) ←→ B  H −→ b ∈ G B ϕ(σ(t1 ), . . . , σ(tn )) −→ H[G]  ϕ(σ(t1 )G , . . . , σ(tn )G ).

Hence there is σ∗ : H[G] ≺ H[G] defined by: σ ∗ (tG ) = σ(t)G . But then σ∗ ⊃ σ since ˇ G = σ(x) σ ∗ (x) = σ∗ (ˇ xG ) = σ(x)

˙ be the B-generic name and G˙ the B-generic name we for x ∈ H. Letting G then have: ˙ G ) = G˙ G = G. σ ∗ (G) = σ∗ (G

QED

Lemma 1.1 Let B be a complete forcing. Let G be B-generic. Then V[G] has no new countable sets of ordinals. Proof . Let  f : ω ˇ → On. Claim f G ∈ V. ˇ for some b. Let θ be big enough and let Suppose not. Then b  f ∈ / V σ : H ≺ Hθ s.t. σ(f , b, B) = f, b, B, where H is countable and transitive. Let G  b be B-generic over H. Let G be B-generic s.t. σ G ⊂ G. Let σ∗ be the above mentioned extension of σ. Then σ∗ (f G ) = f G . But clearly σ ∗ (f G ) = σ  f G ∈ V, where b = σ(b) ∈ G. Contradiction! QED(Lemma 1)

page 112

March 14, 2014

16:18

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

113

We note without proof that Lemma 1.2 If B is the result of a countable support iteration of complete forcings, then B is complete. Remark In fact, the notion of complete forcing reduces to that of an ω-closed set of conditions. (P is called ω-closed iff whenever pi | i < ω is a sequence with pi ≤ pj for all j ≤ i, then there is q with q ≤ pi for all i.) It is shown in [FA] that: Lemma 1.3 B is a complete forcing iff it is isomorphic to BA(P) for some ω-closed set of conditions P. The properties of ω-closed forcing are well known and Lemmas 1.1, 1.2 follow easily from Lemma 1.3. The knowledgable reader will recognize the complete forcings as being a subclass of Shelah’s proper forcings. Proper forcings satisfy Lemma 1.2 but not Lemma 1.1. In fact, many proper forcings add new reals. However, a proper forcing can never change the cofinality of an uncountable regular cardinal to ω. Thus, the notion is useless in dealing e.g. with Namba forcing. What we want is a class of forcings which do not add new reals but do permit new sets of ordinals – even to the extent of changing cofinalities. We of course want these forcings to be iterable – i.e. some reasonable analogue of Lemma 1.2 should hold. The proof of Lemma 1.1 gives us a clue as to how such a class might be defined: The proof depends strongly on the fact that σ  G ⊂ G for a σ ∈ V. Instead, we might require that, if H, σ, θ, B, G are as in the definition of “completed forcing”, then there is b ∈ B which forces that, if G  b is B-generic, there is σ ∈ V[G] s.t. σ : H ≺ Hθ , σ (B) = B and σ   G ⊆ G. We can even require b to force σ  (s) = σ(s) for an arbitrarily chosen s ∈ H. If we now try to carry out the proof of Lemma 1 with a σ : H ≺ Hθ s.t. σ (f , b, B) = f, b, B, in place of σ, we can conclude only that f G = σ  f G . Since we do not know that σ ∈ V, we cannot conclude that f G ∈ V. However, if we assume  f : ω → ω, then f G = σ   f G , where f G ∈ V and f G ⊂ ω 2 . Since σ   ω = id, we can then conclude that f G ∈ V. Thus such forcings will add no reals, but may permit us to add new countable sets of ordinals. In order to carry out this program we must address several difficulties, the first being this: Suppose that Hθ has definable Skolem functions. (This is certainly the case if V = L.) We could then form σ : H ≺ Hθ s.t. σ(b, f , B) = b, f, B simply by transitivizing the Skolem closure of {b, f, B}.

page 113

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

114

But then σ is the only possible elementary map to Hθ with σ(b, f , B) = b, f, B. Thus we perforce have: σ = σ. In order to avoid this we must place a stronger condition on H which implies the possibility of many maps to the top. We shall define such a condition for the case that H is a ZFC− -model. Definition Let N be transitive. N is full iff ω ∈ N and there is γ s.t. Lγ (N ) models ZFC− and N is regular in Lγ (N ) – i.e. if f : x → N , x ∈ N , f ∈ Lγ (N ), then rng(f ) ∈ N . It follows that N itself is a ZFC− model. In fact, regularity in Lγ (N ) is equivalent to saying that N models 2nd order ZFC− in Lγ (N ). If N is full and σ : N ≺ N , then there will, indeed, be many different maps σ : N ≺ N . Since fullness is a property of ZFC− models, however, we shall have to reformulate Shelah’s definition so that we do not work directly with Hθ but rather with ZFC− models containing Hθ . It also turns out that, in order to prove iterability, we must apparently impose a stronger similarity between σ  and σ than we have hitherto stated. In order to formulate this we define: Definition Let B be a complete BA. δ(B) = the smallest cardinality of a set which lies dense in B \ {0}. Note If we were working with sets P of conditions rather than complete BA’s, we would normally choose P to have cardinality δ(BA(P)). Hence the above definition would be superfluous and we would work with P instead. − Definition Let N = LA τ =Df Lτ [A], ∈, A ∩ Lτ [A] be a ZFC model. Let X ∪ {δ} ⊂ N .

CδN (X) =Df the smallest Y ≺ N s.t. X ∪ {δ} ⊂ Y. We are now ready to define: Definition Let B be a complete BA. B is a subcomplete forcing iff for sufficiently large cardinals θ we have: B ∈ Hθ and for any ZFC− model N = LA τ s.t. θ < τ and Hθ ⊂ N we have: Let δ : N ≺ N where N is countable and full. Let σ(θ, s, B) = θ, s, B where s ∈ N . Let G be B-generic over N . Then there is b ∈ B \ {0} s.t. whenever G  b is B-generic over V, there is σ ∈ V[G] s.t. (a) σ : N ≺ N , (b) σ (θ, s, B) = θ, s, B, (c) CδN (rng(σ )) = CδN (rng(σ)) where δ = δ(B),

page 114

March 14, 2014

16:18

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

115

(d) σ   G ⊂ G. (Hence σ  extends uniquely to a σ ∗ : N [G] ≺ N [G] s.t. σ∗ (G) = G.) Note We define N [G] in such a way that A is still a predicate. Thus N = LA τ is N [G]-definable. Note This is expressible in V, since the last part can be expressed as: ◦  ˇ,N ˇ N ˇ , σ◦, G, G), b ∈ B b  ϕ(B, ◦

G being the generic name.

Note If we omitted (c) from the definition of subcompleteness, the resulting class of forcings would still satisfy Lemma 1.2 for countable support iterations of length ≤ ω2 . Since such forcings might change the cofinality of ω2 to ω, we would thereafter have to use the revised countable support (RCS) iteration. (We will also have to make some further assumptions on the component forcings Bi of the iteration Bi | i < α .) (c) appears to be needed to get past regular limits points λ of the iteration s.t. λ > δ(Bi ) for i < λ. Definition θ verifies the subcompleteness of B iff θ is as in the definition of subcompleteness. In the following discussion write ’ver(B, θ)’ to mean ’θ verifies the subcompleteness of B’. Now let B ∈ Hθ and let θ > H θ be a cardinal. A L¨owenheim-Skolem argument that, in order to determine whether ver(B, θ), we need only consider N = LA τ s.t. N ∈ Hθ  . By the well known fact: V[G] Hθ  [G] = (Hθ  ) for B-generic G, where B ∈ Hθ  , we see that, in fact, the definition of ver(B, θ) relativizes to Hθ  – i.e. Lemma 2.1 Let B ∈ Hθ . Let θ > H θ be a cardinal. The statement ver(B, θ) is absolute in Hθ . This holds in particular for θ = (H θ )+ . But then the elements of Hθ  can be coded by subsets of Hθ and we get: Lemma 2.2 Let θ > ω1 be a cardinal. {B | ver(B, θ)} is uniformly 2nd order definable over Hθ . Hence: Corollary 2.3 Let W be an inner model s.t. P(Hθ ) ⊂ W . Then ver(B, θ) is absolute in W .

page 115

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

116

Finally, we note: Lemma 2.4 Let θ verify the subcompleteness of B. Then B is subcomplete. (Thus “sufficiently large θ” can be replaced by “some θ” in the definition of ’subcomplete’.) Proof of Lemma 2.4. It suffices to show: Claim Let B ∈ Hθ . Let θ > H θ be a cardinal. Then ver(B, θ ). Proof . We can assume w.l.o.g. that θ is least with ver(B, θ). Then by Lemma 2.1: (1)

Hθ   θ is least s.t. ver(B, θ).

 Now let N = LA τ s.t. θ < τ and Hθ  ⊂ N . Then θ < τ and Hθ ⊂ N . Let σ : N ≺ N where N is countable and full. Let σ(B, θ , s) = B, θ , s. By (1) there is θ s.t. σ(θ) = θ. By ver(B, θ) there is then a b ∈ B with the desired property. QED(Lemma 2.4)

When actually verifying the subcompleteness of a specific B we often find it convenient to employ an additional parameter. Thus we define: Definition θ, p verifies the subcompleteness of B (ver(B, θ, p)) iff p, B ∈ Hθ and for any ZFC− model N = LA τ with θ < τ and Hθ ⊂ N we have: Let σ : N ≺ N where N is countable and full. Let σ(p, θ, s, B) = p, θ, s, B. Let G be B-generic over N . Then the previous conclusion holds. The natural analogues of Lemma 2.1 – Corollary 2.3 follow as before. But then we can repeat the proof of Lemma 2.4 to get: Lemma 2.5 Let θ, p verify the subcompleteness of B. Then B is subcomplete. This will often be tacitly used in verifications of subcompleteness. 3.2. Liftups In order to better elucidate the concept of fullness, we make a digression on the topic of cofinal embeddings. Definition Let A, A be models which satisfy the extensionality axiom. Let π : A → A be a structure preserving map. We call π cofinal (in symbols: π : A → A cofinally) iff for all x ∈ A there is u ∈ A s.t. x ∈A π(u).

page 116

March 14, 2014

16:18

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

117

Note In this definition we did not require A, A to be transitive or even well founded. Most of our applications will be to transitive models, but we must occasionally deal with ill founded structures. We shall, however, normally assume such structures to be solid in the sense of Section 1. (I.e. the well founded core of A (wfc(A)) is transitive and ∈A ∩wfc(A)2 =∈ ∩wfc(A)2 .) Definition Let τ be a cardinal in A. HτA = the set of x s.t. A  x ∈ Hτ . Note Even if A were a transitive ZFC− model, we would not necessarily have: HτA ∈ A. Definition Let τ ∈ A be a cardinal in A. We call π : A → A τ -cofinal iff for all x ∈ A there is u ∈ A s.t. u < τ in A and x ∈A π(u). We shall generally work with elementary embeddings but must sometimes consider a finer degree of preservation: Definition π : A → A is Σn -preserving (π : A →Σn A) iff for all Σn formulae ϕ and all x1 , . . . , xn ∈ A: A  ϕ[x1 , . . . , xn ] ←→ A  ϕ[π(x1 ), . . . , π(xn )]. Definition Let A be a solid model of ZFC− . Let τ ∈ wfc(A) be an uncountable cardinal in A. Set H = HτA . (Hence H ⊂ wfc(A).) Let π : H →Σ0 H cofinally, where H is transitive. Then by a liftup of A, π we mean a pair A, π s.t π ⊃ π, H ⊂ wfc(A), and π : A →Σ0 A τ -cofinally, where A is solid. (We also say: π : A → A is a liftup of A by π : H → H.) Lemma 3.1 Let A, τ , H, H, π be as in the above definition. The liftup A, π of A, π (if it exists) is determined up to isomorphism (i.e. if A , π ∼ is another liftup, there is σ : A ↔ A with σπ = π ). Proof . Set Δ = the set of f ∈ A s.t. A  (f is a function ∧ dom(f ) ∈ Hτ ). For each f ∈ Δ let d(f ) = that u ∈ H s.t. u = dom(f ) in A. Set: Γ = {f, x | f ∈ Δ ∧ x ∈ π(d(f ))}. It is easily seen by τ -cofinality that each a ∈ A has the form: a = π(f )(x) in A, where f, x ∈ Γ. The same holds for A , π if A , π is another liftup. But: π(f )(x) ∈ π(g)(y) in A ←→ x, y ∈ π({z, w | f (z) ∈ g(w) in A}) ←→ π (f )(x) ∈ π (g)(y) in A !

page 117

April 22, 2014

16:42

WSPC 9 x 6: BC9154

jensen

R. Jensen

118

Similarly: π(f )(x) = π(g)(y) in A ←→ π  (f )(x) = π  (g)(y) in A . ∼ Hence there is σ : A ↔ A defined by σ(π(f )(x)A ) = π (f )(x)A for f, x ∈ Γ. But for any a ∈ A, we have: A  a = ka (0), where ka = {a, 0 } in A. Thus π(a) = π(ka )(0) in A, where ka , 0 ∈ Γ. Hence σ(π(a)) = π (ka )(0) = π  (a). QED(Lemma 3.1) Since the identity is the only isomorphism of a transitive structure onto a transitive structure, we have: Corollary 3.2 Let A, π be the liftup A, π , where A, A are transitive. Then A, π is the unique liftup. ∼ Proof . Let A , π  be a liftup. Let σ : A ↔ A s.t. π  = σπ. Then A is well founded, hence transitive, by solidity. Hence σ = id and π = π, A = A. QED(Corollary 3.2) A transitive liftup does not always exist, even when A is transitive. However, a straightforward modification of the ultrapower construction does give us: Lemma 3.3 Let A be a solid model of ZFC− . Let τ > ω, τ ∈ wfc(A) be a cardinal in A and set: H = HτA . Let π : H →Σ0 H cofinally, where H is transitive. Then A, π has a liftup A, π . A Proof . Define Δ, Γ as above. Let A = |A|, ∈A , AA 1 , . . . , An . Define an ∗ ∗ ∗ ∗ ∗ equality model Γ = Γ, = , ∈ , A1 , . . . , An by:

f, x =∗ g, y ←→ x, y ∈ π({z, w | f (z) ∈ g(w) in A}) f, x ∈∗ g, y ←→ x, y ∈ π({z, w | f (z) ∈ g(w) in A}) f, x ∈ A∗i ←→ x ∈ π({z | f (z) ∈ Ai in A}). A straightforward modification of the usual proof gives us L  os’ Theorem for Γ∗ : (1) Γ∗  ϕ[f1 , x1 , . . . , fn , xn ] ←→ x1 , . . . , xn ∈ π({z | A  ϕ[f1 (z1 ), . . . , fn (zn )]}). This is proven by induction on ϕ. The case that ϕ is a primitive formula is immediate. We display the induction step for ϕ = ϕ(v1 . . . . , vn ) =  v0 ψ(v0 , . . . , vn ). (→) Let Γ∗  ϕ[f1 , x1 , . . . , fn , xn ]. Then Γ∗  ψ[f0 , x0 , . . . , fn , xn ]

page 118

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

119

for some f0 , x0 ∈ Γ. Hence x0 , . . . , xn ∈ π({z | A  ψ[f0 (z0 ), . . . , fn (zn )}) 

π(d(f0 ) × {z | A  ϕ[f1 (z1 ), . . . , fn (zn )]})

−→ x1 , . . . , xn ∈ π({(z ) | A  ϕ[f1 (z1 ), . . . , fn (zn )]}). (←) Set u = {z | A  ϕ[f1 (z1 ), . . . , fn (zn )]}. Then u ∈ H and x ∈   π(u). In A we have z y(y, f1 (z1 ), . . . , fn (zn )). Hence, by ZFC− , there is f0 ∈ A s.t.  z ψ(f0 (z ), f1 (z1 ), . . . , fn (zn )) in A. But then f0 , z ∈ Γ and

x , x1 , . . . , xn ∈ π({z | A  ψ[z0 , . . . , zn ]}). ∗

Hence Γ  ψ[f0 , x , f1 , x1 , . . . , fn , xn ].

QED(1)

Now let Γ = |Γ |, ∈ , A1 , . . . , An be the result of factoring Γ∗ by =∗ , the elements being the =∗ -equivalence classes x of x ∈ Γ. Since Γ satisfies ∼ extensionality, there is an isomorphism σ : Γ ↔ A, where A is solid. Set: [f, x] = σ(f, x  ), where f, x ∈ Γ. Then A  ZFC− by (1). We now define π : A ≺ A by: Definition For a ∈ A let k = {a, 0 } in A. Set: π(a) =Df [k, 0]. Then: π : A ≺ A.

(2) Proof .

A  ϕ[a1 , . . . , an ] ←→ 0 − 0 ∈ {z | A  ϕ[ka1 (z1 ), . . . , kan (zn )]} ←→ 0 − 0 ∈ π({z | A  ϕ[ka1 (z1 ), . . . , kan (zn )]}) ←→ A  ϕ[π(a1 ), . . . , π(an )] by (1).

QED(2)

Now set: Definition Δ0 = the set of functions f ∈ H. Γ0 = the set of f, x s.t. f ∈ Δ0 and x ∈ π(dom(f )). Since π : H → H cofinally, H is the set of π(f )(x) s.t. f, x ∈ Γ0 . Now set: ˜ = {[f, x] | f, x ∈ Γ0 }. Definition H

page 119

April 22, 2014

16:42

WSPC 9 x 6: BC9154

R. Jensen

120

(3)

jensen

˜ is “A-transitive” – i.e if a ∈A b ∈ H, ˜ then a ∈ H. ˜ H

Proof . Let a = [f, x], b = [g, y], where g, y ∈ Γ0 and f, x ∈ Γ. Set: u = {z ∈ d(f ) | f (z) ∈ H}: Then f, x ∈∗ g, y implies f, x =∗ f  u, x , where f  u, x ∈ Γ0 . QED(3) But for f, x , g, y ∈ Γ0 we have: [f, x] ∈ [g, y] in A ←→ x, y ∈ π({z, w | f (z) ∈ g(w)}) ←→ π(f )(x) ∈ π(g)(y). Similarly: [f, x] = [g, y] ↔ π(f )(x) = π(g)(y). Hence there is an isomor∼ ˜ ∈A ↔H, phism σ : H, ∈ defined by: σ([f, x]) = π(f )(x) for f, x ∈ Γ0 . ˜ is A-transitive it follows that ˜ Hence H, ∈A is well founded. Since H 2 2 ˜ ⊂ wfc(A); hence ∈A ∩H ˜ =∈ ∧H by solidity. Hence H ˜ is transitive. H Thus σ = id and (4)

˜ = H ⊂ wfc(A) and [f, x] = π(f )(x) for f, x ∈ Γ0 . H

But then: (5)

[f, x] = π(f )(x) in A for all f, x ∈ Γ.

Proof . x ∈ π(d(f )), where d(f ) = {x | f (x) = f (x)} = {x | f (x) = (kf (0))(id  d(f ))(x) in A} where kf = {f, 0 } in A. Hence x, 0, x ∈ π({z, y, w | f (z) = kf (y)(id  d(f ))(z) in A}. Thus [f, x] = [kf , 0]([(id  d(f )), x]) in A, where: [kf , 0] = π(f ) and [id  d(f ), x] = π(id  d(f ))(x) = x by (4). QED(5) (6) π  H = π, since for a ∈ H we have π(a) = [ka , 0] = π(ka )(0) = kπ(a) (0) = π(a) by (4). Finally, since every a ∈ A has the form π(f )(x) for an x ∈ H, it follows that a ∈ π(rng(f )) in A, where rng(f ) < τ in A. Thus (7)

π : A ≺ A τ -cofinally.

QED(Lemma 3.3)

The above proof yields more than we have stated. For instance: Lemma 3.4 Let π : N →Σ0 N confinally, where N is a ZFC− model and N is transitive. Then π : N ≺ N . (Hence N is a ZFC− model.)

page 120

April 22, 2014

16:42

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

121

Proof . Repeat the above proof with τ = On ∩ N (hence H = N ). All steps ˜ = N. go through and we get A = H QED(Lemma 3.4) Lemma 3.5 Let A, A, H, H, τ , π be as in Lemma 3.3. Set τ˜ = On ∩ H. Then τ˜ ∈ wfc(A) and H = Hτ˜A . Proof . By the definition of wfc(A) we have: (∗)

If x ∈ A and y ∈ wfc(A) whenever y ∈A x, then x ∈ wfc(A).

We consider two cases: Case 1 τ is regular in A. A Claim H = Hπ(τ ˜ ∈ wfc(A)). ) (hence π(τ ) = τ

Proof . (⊂) is trivial. We prove (⊃). Let x ∈ Hπ(τ ) in A. We claim that x ∈ H. Let x ∈ π(u) in A, where u ∈ A, u < τ in A. Let v = u ∩ Hτ in A. Then v ∈ H = HτA by regularity of τ . But then x ∈ π(v) ∈ H. Hence x ∈ H. QED(Case 1) Case 2 Case 1 fails. Let κ = cf(τ ) in A. Then κ ∈ H. Let f : κ → τ in A be normal and cofinal ˜ = sup π  κ. Then κ ˜ ≤ π(κ) ∈ H. Hence in τ . Then f ∈ wfc(A) by (∗). Let κ κ ˜ ∈ H. Let g = π(f )  κ ˜ in A. It follows easily by (∗) that g ∈ wfc(A). Thus ˜ ∈ wfc(A). τ˜ = sup g  κ Claim H = Hτ˜A (⊂) Let x ∈ H. Then x ∈ π(u) where u ∈ H. Hence x ∈ π(u) ∈ Hτ˜ . Hence x ∈ Hτ˜ . A for a ν < τ which is regular in A, since (⊃) Let x ∈ Hτ˜A . Then x ∈ Hπ(ν)  τ˜ = sup π τ and τ˜ is a limit cardinal in A. Let x ∈ π(u) in A, where u ∈ A, u < τ in A. We can choose ν large enough that u < ν in A. Let v = u ∩ Hν in A. Then v ∈ Hν ⊂ H and x ∈ π(v) ∈ H. QED(Lemma 3.5) An immediate corollary of the proof is: Corollary 3.6 A H = Hπ(τ ).

If τ is regular or cf(τ ) = ω in A. Then τ˜ = π(τ ) and

Note that if N , N are transitive ZFC− models, τ ∈ N is a cardinal in N and π : N ≺ N τ -cofinally, then π is κ cofinal for every κ ≥ τ which is a cardinal in N . Hence, by Corollary 3.6 we conclude: Corollary 3.7 Let π : N →Σ0 N τ -cofinally, where N , N are transitive, τ ∈ N is a cardinal in N , and N  ZFC− . Let κ ≥ τ be regular in N or

page 121

March 14, 2014

16:18

122

WSPC 9 x 6: BC9154

jensen

R. Jensen

N cf(κ) = ω in N . Then π(κ) = sup π  κ and Hπ(κ) =





π(u).

N u∈Hκ

We are now ready to develop the concept of fullness further. We first generalize it as follows: Definition Let N be a transitive ZFC− model. N is almost full iff ω ∈ N and there is a solid A s.t. • A  ZFC− , • N ∈ wfc(A), • N is regular in A – i.e. if f : x ∈ N , x ∈ N , and f ∈ A, then rng(f ) ∈ N . The last condition can be alternatively expressed by: |N | = HτA , where τ = On ∩ N . Definition A verifies the almost fullness of N iff the above holds. Clearly every full structure is almost full. By Lemmas 3.3 and 3.5 we then have: Lemma 4.1 Let N be almost full. Let π : N →Σ0 N cofinally, where N is transitive. Then N is almost full. (In fact, if A verifies the almost fullness of N and A, π is a liftup of A, π , then A verifies the almost fullness of N.) Definition Let N be a transitive ZFC− model. δN = the least δ s.t. Lδ (N ) is admissible. By Section 1 Corollary 21.1 we then have: Lemma 4.2 If A verifies the almost fullness of N , then LδN (N ) ⊂ wfc(A). Combining this with Lemma 4.1 we get a conclusion that is rich in consequences: Lemma 4.3 Let π : N →Σ0 N cofinally where N is almost full and N is transitive. Let ϕ be a Π1 condition. Let a1 , . . . , an ∈ N . Then LδN (N )  ϕ[N , a ] −→ LδN (N )  ϕ[N, π(a )]. ˜ be a liftup of Proof . Let A verify the almost fullness of N and let A, π A, π . We assume: LδN (N )  ψ[N, π(a )], where ψ is a Σ1 condition, and prove:

page 122

March 14, 2014

16:18

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

123

Claim LδN (N )  ψ[N , a ]. Set: ν = the least ordinal s.t. Lν (N )  ψ[N, π(a )]. Then ν < δN . Noting that A  ψ[N, π(a )], we see that ν is A-definable, hence has a preimage ν under π ˜ AO

π ˜

/A O

? wfc(A) O

? wfc(A) O

? Lδ (N ) O ν ? N

? Lδ (N ) O ν=π ˜ (ν) ? /N

π

Since ν ∈ wfc(A), we conclude that ν ∈ wfc(A). Hence Lν (N )  ψ[N , a ]. But Lη (N ) is not admissible for any η ≤ ν. Hence Lη (N ) is not admissible for any η ≤ ν. Hence ν < δN and the conclusion follows. QED(Lemma 4.3) We now combine this with Barwise’ theory. Recall that by a theory or axiomatized language on an admissible structure M we mean a pair L0 , A where L0 is a language (i.e. a set of predicates and constants) in M -finitary predicate logic, and A is a set of axioms in L0 . We defined L = L0 , A to be Σ1 (M ) in parameters p1 , . . . , pn ∈ M iff L0 is Δ1 (M ) in p and A is Σ1 (M ) in p. By Section 2 Corollary 4 we get: Lemma 4.4 Let M be admissible. Let L = L0 , A be a theory on M which is Σ1 (M ) in parameters p1 , . . . , pn ∈ M . The statement: ’L is consistent’ is then Π1 (M ) in p (uniformly in the Σ1 definition of A from p ). Hence Lemma 4.5 Let π : N →Σ0 N cofinally, where N is almost full. Let L be an infinitary theory on LδN (N ) which is Σ1 in parameters N , p1 , . . . , pn ∈ N . Let the theory L on LδN (N ) be Σ1 in N , π( p ) by the same definition. If L is consistent, so is L. A typical application is: Corollary 4.6 Let π : N →Σ0 N cofinally, where N is almost full. Let ϕ(v1 , . . . , vn ) be a first order (finite) formula in the N -language with one

page 123

April 22, 2014

16:42

WSPC 9 x 6: BC9154

jensen

R. Jensen

124 ◦

additional predicate A. Let card(N ) = τ , card(N ) = τ . Let x1 , . . . , xn ∈ N .   If coll(ω, τ ) forces AN , A  ϕ[x ]. Then coll(ω, τ ) forces AN, A  ϕ[π(x )], (coll(ω, τ ) being the usual conditions for collapsing τ to ω). Proof . Let L be the language on LδN (N ) with the basic axioms. The addi◦

tional constant a , and the additional axiom: ◦

N , a  ϕ[x1 , . . . , xn ]. Let L have the same definition over LδN (N ) in the parameters π(x1 ), . . . , π(xn ). By Barwise’ completeness theorem, L is consistent iff  coll(ω, τ ) forces AN , A  ϕ[x ]. Similarly for L, N , π(x ). The conclusion then follows by Lemma 4.5. QED(Corollary 4.6) The theory of liftups also reveals the import of condition (c) in the definition of “subcomplete”. To this end we prove the interpolation lemma: Lemma 5.1 Let π : N ≺ N where N is a transitive ZFC− model and N  ˜ = {π(u) | is transitive. Let τ be a cardinal in N . Set: H = HτN and H u ∈ N and u < τ in N }. Then: ˜, π (a) The transitive liftup N ˜ of N , π  H exists. ˜ ˜ = id. (b) There is σ : N ≺ N s.t. σ˜ π = π and σ  H   ˜ ˜ = π and σ   π ˜ = id, where (c) σ is the unique σ : N →Σ0 N s.t. σ π ˜ τ˜ = On ∩ H. Proof . Let A, π ˜ be a liftup of N , π  H . Letting Γ be as in the proof of Lemma 3.3 we see that each y ∈ A has the form π ˜ (f )(x) in A for some f, x ∈ Γ. Moreover: ˜ (fn )(xn )] A  ϕ[˜ π (f1 )(x1 ), . . . , π ←→ x1 , . . . , xn ∈ π({z | N  ϕ[f1 (z1 ), . . . , fn (zn )]}) ←→ N  ϕ[π(f1 )(x1 ), . . . , π(fn )(xn )]. Hence there is σ : A ≺ N defined by: σ(˜ π (f )(x)) = π(f )(x) for f, x ∈ Γ. Thus A is well founded, hence transitive by solidity. This proves (a), (b). We ˜ τ -cofinally, it follows now prove (c). Let σ  be as in (c). Since π : N ≺ N ˜ that any y ∈ N has the form π(f )(ν) for an f, ν ∈ Γ s.t. dom(f ) ⊂ τ . Hence σ (y) = π(f )(ν) = σ(y). QED(Lemma 5.1)

page 124

March 14, 2014

16:18

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

125

Just as in the proof of Lemma 3.4 we can repeat this using τ = On ∩ N , getting: Lemma 5.2 Let π : N ≺ N where N , N are transitive ZFC− models. Set: ˜ cofinally.) Then N ˜ ≺ N. ˜ =  π(u). (Hence π : N ≺ N N u∈N

We now utilize this to examine the meaning of (c) in the definition of “subcomplete”.

− Lemma 5.3 Let σ : N ≺ N where N = LA model and N is α is a ZFC transitive. Let σ(δ) = δ, where δ is a cardinal in N . Set C = CδN (rng(σ)), ˜ =  σ(u). Let N ˜, σ H = (Hδ+ )N , H ˜ be the liftup of N , σ  H and let u∈H

˜ ≺ N s.t. k˜ ˜ = id. Then C = rng(k). k=N σ = σ and k  H

Proof . (⊂) rng(σ) ⊂ rng(k) and σ ⊂ rng(k). (⊃) Let x ∈ rng(k), x = k(˜ x) where x ˜∈σ ˜ (u), u ∈ N , u < δ + in N . Let onto f ∈ N , f : δ −→ u. Then x = k˜ σ (f )(ν) = σ(f )(ν) for a ν < δ. Hence x ∈ C. QED(Lemma 5.3) ˜ , k from C by the definition: k : Stating this differently, we can recover N ∼ ˜ ˜ N ↔ C, where N is transitive. We can then recover σ ˜ from C by σ ˜ = k −1 ·σ.   N If we now have another σ : N ≺ N s.t. σ (δ) = δ and C = Cδ (rng(σ )), ˜, σ ˜  = k −1 σ  . Thus σ = k˜ σ, then N ˜  is the liftup of N , σ   H , where σ   σ = kσ where σ ˜, σ ˜ are determined entirely by σ  H, σ  H, respectively. Hence Corollary 5.4 Let σ, σ  be as above. Let τ ∈ N be regular in N s.t. τ > δ and σ(τ ) = σ  (τ ). Then sup σ  τ = sup σ   τ . ˜  τ = sup σ ˜   τ , since σ ˜, σ ˜ Proof . Let k(˜ τ ) = σ(τ ) = σ  (τ ). Then τ˜ = sup σ are τ -cofinal and τ is regular in N . But then: sup σ τ = sup σ   τ = sup k  τ˜. QED(Corollary 5.4) A similar argument yields: Corollary 5.5 Let τ = On ∩ N , where σ, σ are as above. Then sup σ  τ = ˜. sup σ  τ = sup k  τ˜, where τ˜ = On ∩ N Our original version of (c) was weaker, and can be stated as: (c ) Let s = s0 , λ1 , . . . , λn and s = s0 , λ1 , . . . , λn where λi > δ is regular in N . Let λ0 = On ∩ N . Then sup σ λi = sup σ  λi for i = 0, . . . , n.

page 125

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

126

This is, of course, an immediate consequence of the above two corollaries. The weaker definition of ’subcomplete’ should not be forgotten, since we might someday encounter a forcing which satisfies the weaker version but not the stronger one. That has not happened to date, however, and in fact our original verifications of (c ) turned essentially on first verifying (c). Before leaving the topic of τ -cofinal embeddings, we mention that these concepts can be applied to structures that are not ZFC− models. For our purposes it will suffice to deal with the class of smooth models: Definition Let N be a transitive model. N is smooth iff either N  ZFC−  or else there is a sequence Ni , αi | i < λ of limit length s.t. N = Ni i

and Nj  ZFC− , Nj is transitive, and Ni ∈ Nj s.t. αi is regular in Nj and N Ni = Hαij for i < j < λ. Then: Lemma 5.6 If N is smooth, N transitive, and π : N →Σ0 N cofinally, then N is smooth. Proof . If N  ZFC− , this is immediate from the foregoing. Otherwise there is a sequence N i , αi which verifies the smoothness of N . Set Ni = π(N i ), αi = π(αi ). Then Ni , αi | i < λ verifies the smoothness of N. QED(Lemma 5.6) Note It does not follow that π : N ≺ N . The concepts “τ -cofinal” and “liftup” are defined as before, and it follows as before that if N is smooth, τ is a cardinal in N and π : HτN →Σ0 H cofinally, then N , π has at most one transitive liftup. Lemma 5.7 Let π : N →Σ0 N τ -cofinally, where N is smooth. Let N . Then κ ∈ N be regular in N , where κ > τ . Let H = HκN , H = Hπ(κ) π  H : H →Σ0 H τ -cofinally. Proof . Exactly as in Case 1 of Lemma 3.5. Lemma 5.8 Let N i , αi | i < λ verify the smoothness of N . Let τ ∈ N be a cardinal. Let π : HτN →Σ0 H cofinally. The transitive liftup of N , π exists iff for each i s.t. τ < αi the transitive liftup of N i , π exists. Proof . (→) Let N, π be the liftup of N , π . Set: αi = π(αi ), Ni = π(N i ). Then Ni , π  N i is the liftup of N i , π . (←) Let Ni , πi be the liftup of N i , π for τ < αi . By Lemma 5.7 we

page 126

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

127

have: πj  N i : N i → Ni τ -cofinally. Hence πj  N i = πi and we can set:  π = πi . π : N →Σ0 N is then τ -cofinal and π = π  H.QED(Lemma 5.8) i

Lemma 5.9 Let N be smooth and π : N →Σ0 N , where N is transitive. Let τ be a cardinal in N . Set H = HτN . Then: ˜, π (a) The transitive liftup N ˜ of N , π  H exists. ˜ ˜ = id, where H ˜ = (b) There is σ : N →Σ0 N s.t. σ˜ π = π and σ  H  π(u). u∈H

˜ →Σ0 N s.t. σ˜ (c) σ is the unique σ : N π = π and σ  τ˜ = id, where ˜ τ˜ = On ∩ H. Proof . Case 1 N  ZFC− .  Set: N  = π(u). Then π : N →Σ0 N  cofinally. Hence π : N ≺ N  ⊂ N u∈N

and we apply our previous lemmas.

Case 2 Case 1 fails. Let N i , αi | i < λ verify the smoothness of N . Assume w.l.o.g. that τ ∈ N 0 . (Hence H = HτN i for all i < λ.) (a) follows by Lemma 5.8. Moreover ˜i is the liftup of N , π  H by Lemma 5.7, where N ˜i = π(N i ). Let N i , N  ˜ ˜ σi : Ni →Σ0 π(N i ) be defined by σi π ˜i = π  N i , σi  H = id. Set σ = σi . i

˜ = id. This proves (b). But σi ˜ →Σ0 N and σ˜ π = π, σ  H Then σ : N ˜ is unique s.t. σi : Ni →Σ0 π(N i ), σi π ˜ = π  N i and σi  τ˜ = id. Hence ˜i = σi for i < λ if σ is as in (c). This proves (c). QED(Lemma 5.9) σN 3.3. Examples We are now ready to prove that some specific forcings are subcomplete. Since these forcings will be presented as sets of conditions rather than Boolean algebras, we set: Definition Let P be a set of conditions. VP =Df VBA(P) ,

δ(P) =Df δ(BA(P))

where BA(P) is the canonical Boolean algebra over P as defined in Section 0. We may refer to the elements of VP as ’P-names’. We note: − model. Let BA(P) ∈ Hθ ⊂ N . Let Fact 1 Let N = LA τ be a ZFC δ ⊂ C ≺ N , where BA(P) ∈ C and δ = δ(P). Then for each p ∈ P there is

page 127

March 14, 2014

16:18

128

WSPC 9 x 6: BC9154

jensen

R. Jensen

q ∈ C ∩ P s.t. [q] ⊂ [p]. (Hence every set predense in C ∩ P is predense in P.) Proof . Let B = BA(P). By definition there are f, Δ ∈ Hθ s.t. Δ is dense in B and f : δ ↔ Δ. Hence there are such f, Δ ∈ C. But Δ ⊂ C, since δ ⊂ C. Let p ∈ P. There is b ∈ Δ s.t. b ⊂ [p]. Hence there is q ∈ C ∩ P s.t. [q] ⊂ b, since C ≺ N . Hence [q] ⊂ [p]. QED(Fact 1) Our first example is Prikry forcing. Lemma 6.1 Prikry forcing is subcomplete. Proof . Let U be a normal ultrafilter on a measurable cardinal κ. We define the Prikry forcing determined by U to be the set P = PU consisting of all pairs s, X s.t. X ∈ U and s ⊂ κ is finite. The extension relation ≤P is defined by: s, X ≤ t, Y iff X ⊂ Y, s ⊃ t, and t = lub(t) ∩ s. P does not collapse cardinals or add new bounded subsets of κ If G is P-generic, the P-sequence added by G is   S = SG = {s | Xs, X ∈ G}. Then S is unbounded in κ and has order type ω. G is, in turn, definable from S by: G = GS = {s, X ∈ P | s = S ∩ lub(s) ∧ S \ s ⊂ X}. Definition We call S ⊂ κ a P-sequence (or Prikry sequence) iff S = SG for some P-generic G. The following characterization of Prikry sequences is well known: Fact 2 S is a Prikry sequence iff S ⊂ κ has order type ω and is almost  contained in every X ∈ U (i.e. ν < κ S \ ν ⊂ X). κ

We now prove that P is subcomplete. To this end we let θ > 22 and let − N = LA τ be a ZFC model s.t. τ > θ and Hθ ⊂ N . Furthermore we assume that σ : N ≺ N where N is countable and full. We also suppose that σ(θ, U , P, s) = θ, U, P, s. Hence σ(B) = B, where B = BA(P) and B = BA(P) in N . We must show:

Main Claim There is p ∈ P s.t. whenever G  p is P-generic. Then there is σ ∈ V[G] s.t.

page 128

April 22, 2014

17:9

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

(a) (b) (c) (d)

129

σ : N ≺ N , σ (θ, U , P, s) = θ, U, P, s, CδN (rng σ ) = CδN (rng σ), where δ = δ(P). σ  G ⊂ G.

Note that if we set: S = SG and S = SG in N [G], then (d) becomes equivalent to: (d ) σ   S = S. Let C = CδN (rng σ). Using Fact 1 we get: (1) Let X ∈ U . Then there is Y ∈ C ∩ U s.t. Y ⊂ X. Proof . Suppose not. Then for each ν < κ the set Δν is dense in P ∩ C where Δν = {s, Y ∈ P ∩ C | s \ ν ⊂ X}. Hence Δν is predense in P by Fact 1. Let G be P-generic. Then G ∩ Δν = ∅ for ν < κ. Hence SG is not almost contained in X. Contradiction! by Fact 2. QED(1) Hence: (2) S is a P-generic sequence iff S has order type ω and is almost contained in every X ∈ C ∩ U . (3) δ ≥ κ, since otherwise C < κ and C ∩ U would have a minimal element Y =  (C ∩ U ). ∼ Definition We define N0 , k0 , σ0 by: k0 : N0 ↔ C, where N0 is transitive σ0 = k0−1 ◦ σ. We also set: Θ0 , P0 , U0 , s0 = σ0 (θ, P, U , s). By Section 3.2, however, we have an alternative characterization: (4) Let σ0 (δ) = δ, ν = δ +N , H = HνN . Then N0 , σ0 is the liftup of N , σ  H . Moreover k0 is defined by the condition: k0 : N0 ≺ N,

k0 σ0 = σ,

k0  ν0 = id,

where ν0 = sup σ  ν. Since ν is regular in N , we conclude: (5) σ0 is a ν-cofinal map and σ0 (ν) = ν0 . Definition α0 = δN0 = the least α s.t. Lα (N0 ) is admissible.

page 129

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

130

Our Main Claim will reduce to the assertion that a certain language L0 on Lα0 (N0 ) is consistent. We define: Definition L0 is the language on Lα0 (N0 ) with: Predicate: ∈ ◦ ◦ Constants: S , σ , x (x ∈ Lα0 (N0 )) Axioms: • Basic axioms and ZFC− ◦ • S is P0 -sequence over N0 ◦ • σ : N ≺ N 0 κ-cofinally, where σ(κ) = κ ◦ • σ (θ, P, U , s) = θ 0 , P0 , U 0 , s0 •

◦

◦

σ  S = S .

We first show that L0 is consistent. To this end we define: Definition N1 , σ1 = the liftup of N , σ  HκN . k1 = the unique k : N1 ≺ N0 s.t. kσ1 = σ0 and k  κ1 = id, where κ1 = sup σ κ. θ1 , P1 , U1 , s1 = σ1 (θ, P, U , s), S1 = σ1  S. Note that κ1 = σ1 (κ), since σ1 is κ-cofinal into N1 and κ is regular in N . Then: (6) (a) (b) (c) (d)

S1 is a P1 -sequence over N1 , σ1 : N ≺ N1 κ-cofinally, σ1 (θ, P, U , s) = θ1 , P1 , U1 , s1 , σ1  S = S1 .

Proof . (b)–(d) are immediate. (a) follows by: Claim Let X ∈ U1 . Then S1 is almost contained in X.  Proof . Let X ∈ σ1 (w), where w < κ in N . Then Y = (U ∩ w) is almost contained in every z ∈ U ∩ w and Y ∈ U . Hence Y = σ1 (Y ) is almost contained in every Y ∈ U1 ∩ σ1 (w). In particular, Y is almost contained in X. But S1 is almost contained in Y . QED(6) Now let: α1 = δN1 = the least α s.t. Lα (N1 ) is admissible. Let L1 be the language Lα1 (N1 ) which is defined as L0 was defined on Lα0 (N0 ), substituting θ1 , P1 , U1 , s1 , κ1 for θ0 , P0 , U0 , s0 , κ0 . Then (7) L1 is consistent. Proof . Hκ , S1 , σ1 models L1 by (6).

QED(7)

page 130

April 22, 2014

16:42

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

131

Note, however, that: NO 1 σ1 N

k1 / > N0 || | | || σ || 0

where all maps are cofinal and all models are almost full. Then L0 is Σ1 (Lα0 [N0 ]) in N0 and the parameters: κ, P0 , κ, N , θ, P, U , s, θ0 , P0 , U0 , s0 . But L1 is Σ1 (Lα1 [N1 ]) in N1 and the k1 -preimages of these parameters by the same Σ1 -formula. Since N1 is almost full and k1 : N1 ≺ N0 cofinally, we conclude by Lemma 4.5: (8) L0 is consistent. From this we now derive the Main Claim: Work in a generic extension V[F ] of V in which Lα0 [N ] is countable. Then L0 has a solid model ◦

◦

A = |A|, S A , σ A . ◦

◦

Set S = S A , σ  = k0 ◦ σ A . Then (8) (a) (b) (c) (d) (e)

σ : N ≺ N , S is P-generic over V, σ (θ, P, U , s) = θ, P, U, s, CδN (rng σ ) = C, S = σ   S. ◦

Proof . (a), (c) are immediate. To see (d) note that N0 = CδN0 (rng σ A ), ◦ ◦ since σ A is κ-cofinal and δ ≥ κ = σ A (κ). Hence: ◦

C = k0  N0 = CδN (rng k0 ◦ σ A ). Since k0  (κ + 1) = id we have U ∩ N0 = U ∩ C. Hence (b) follows by (2). ◦ (e) follows by σ  κ = σ A  κ. QED(9) We have almost proven the Main Claim, the only problem being that σ is not necessarily an element of V[S]. We now show: (10)

There is σ  ∈ V[S] satisfying (9).

Proof . Work in V[S]. Let μ be regular in V[S] s.t. N ∈ Hμ . Set: M = Hμ , N, S, θ, P, U, s, σ . We define a language L2 on the admissible structure M as follows:

page 131

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

132

Definition L2 is the language on M with Predicate: ∈ ◦ Constants: σ , x, (x ∈ M ) Axioms: • ZFC− and basic axioms ◦ • σ :N ≺ N ◦ • σ (θ, P, U , s) = θ, P, U , s ◦ N • Cδ (rng σ ) = C •

◦

S = σ  S

L2 is clearly consistent, since M, σ  is a model of L2 in V[F ], where σ  is defined as above. ˜ ≺ M , where M ˜ is countable and transitive. Let L˜2 be the Now let π : M ˜ with the same Σ1 definition, replacing all parameters by language on M ˜ ∈ HωV[S] their preimages under π. Then L˜2 is consistent. Since M = HωV1 , 1 ◦A ˜  ˜ in V. Let σ ˜ = σ and set: σ = π ◦ σ ˜. it follows that L˜2 has a solid model A The verification of (9) is then straightforward.

QED(10)

But, since S is a Prikry sequence, there must be p ∈ GS which forces the QED(Lemma 6.1) existence of such a σ . This proves the Main Claim. Lemma 6.2 Assume CH. Then Namba forcing is subcomplete. Proof . We first define Namba forcing. The set ω2<ω of monotone finite sequences in ω2 is a tree ordered by inclusion. The set N of Namba conditions is the collection of all subtrees T = ∅ of ω2<ω s.t. T is downward closed in ω2<ω and for each s ∈ T the set {t | s ≤T t} has cardinality ω2 . The extension relation ≤N is defined by: T ≤ T  ←→Df T ⊂ T  .  If G is N-generic, then S = G is a cofinal map of ω into ω2V . We rite S = SG and call any such S a Namba sequence. G is then recoverable from S by:  G = GS = {T ∈ N | n < ω S  n ∈ T }. It is known that, if CH holds, then Namba forcing adds no reals. We shall also make use of the following fact, which is proven in the Appendix to [DSF]:

Fact Let S be a Namba sequence. Let S  ∈ V[S] be a cofinal ω-sequence in ω2V . Then S  is a Namba sequence and V[S  ] = V[S]. Note that δ(N) ≥ ω2 , since otherwise ω2 would not be collapsed.

page 132

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

133

ω2

− We now turn to the proof. Let θ > 22 . Let N = LA model τ be a ZFC s.t. τ > θ and Hθ ⊂ N . Let σ : N ≺ N where N is countable and full. Let σ(θ, N, s) = θ, N, s. Let G be N-generic over N . It suffices to show:

Main Claim There is p ∈ N s.t. whenever G  p is N-generic, then there is σ  ∈ V[G] with: (a) (b) (c) (d)

σ : N ≺ N , σ (θ, N, s) = θ, N, s, CδN (rng σ ) = CδN (rng σ) where δ = δ(N), σ  G ⊂ G.

Note We shall actually prove a stronger form of (c): CωN2 (rng σ ) = CωN2 (rng σ). Note (d) can equivalently be replaced by: σ   S = S, where S = SG , Definition

S = SG .

Set C = CωN2 (rng σ). Define k0 by:

∼ k0 : N0 ↔ C, where N0 is transitive, σ0 = k0−1 ◦ σ,

θ0 , N0 , s0 = σ0 (θN, s).

Just as before we get: (1) N0 , σ0 is the liftup of N , σ  HωN3 , k0 is the unique k : N0 ≺ N s.t. k0 σ0 = σ and k0  ω3N0 = id, (where ω3N0 = sup σ0  ω3N ). Now let α0 be the least α s.t. Lα (N0 ) is admissible. We define a language L0 on Lα0 (N0 ) as follows: Definition L0 is the language on Lα0 (N0 ) with: Predicate: ∈ ◦ Constants: σ , x (x ∈ Lα0 (N0 )) Axioms: • Basic axioms and ZFC− ◦ N • σ : N ≺ N 0 ω 2 -cofinally ◦ • σ (θ, N , s) = θ0 , N0 , s0 . (2) L0 is consistent. Proof . Let N1 , σ1 be the liftup of N , σ  HωN2 . Define k1 : N1 ≺ N0 by: k1 σ1 = σ0 ,

k1  γ1 = id, where γ1 = sup σ ω2N = σ1 (ω2N ).

page 133

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

134

Let L1 be the corresponding language on Lα1 (N1 ), where α1 = δN1 . Just as before it suffices to show that L1 is consistent. This clear, however, since QED(2) Hω2 , σ1 is a model. Now let S  be a Namba sequence. Work in V[S  ]. Let μ be a regular cardinal in V[S  ] with N ∈ Hμ . Set: M = Hμ , N, σ, N, s . ˜ ≺ M , where M ˜ is transitive and countable. Then M ˜ ∈ Hω1 ⊂ V Let π : M  in V [S ]. Let ˜ C) ˜ σ ˜ , L, ˜ k, ˜ = N, σ, N, L0 , k0 , C0 . π(N, ˜, N ◦ ˜ Set σ ˜ . It follows Let A ∈ V be a solid model of L. ˜ = k˜ ◦ σ A ; σ  = π ◦ σ easily that:

(3) (a) σ : N ≺ N (b) σ  (θ, N, s) = θ, N, s (c) CωN1 (rng σ ) = C Now let S = SG and set: S = σ   S. Then S ∈ V[S  ] is a cofinal ω-sequence in ω2V ; hence: (4) S is a Namba sequence and V[S] = V[S  ]. (Hence σ  ∈ V[S].) But we know: (5) S = σ   S. Let G = GS . There is then a p ∈ G which forces the existence of a σ ∈ V[S] satisfying (3), (5). This proves the Main Claim. QED(Lemma 6.2) Now let κ > ω1 be a regular cardinal. Let A ⊂ κ be a stationary set of ω-cofinal ordinals. Our final example is the forcing PA which is designed to shoot a cofinal normal sequence of order type ω1 through A: Definition PA is the set of normal functions p : ν + 1 → A, where ν < ω1 . The extension relation is defined by: in PA ←→Df q ⊂ p.  Clearly, if G is PA -generic, then G : ω1 → A is normal and cofinal in κ. PA adds no new countable subsets of the ground model. If, however, {λ < κ | cf(λ) = ω ∧ λ ∈ / A} is stationary, then PA will not be a complete forcing. p≤q

Lemma 6.3 PA is subcomplete.

page 134

March 14, 2014

16:18

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

135

Proof . Clearly δ(PA ) ≥ κ, since otherwise κ would remain regular. Now κ − let θ > 22 . Let N = LB model s.t. τ > θ and Hθ ⊂ N . Let τ be a ZFC σ : N ≺ N where N is countable and full. Let σ(θ, P, A, κ, s) = θ, PA , A, κ, s. Let G be P-generic over N . It suffices to show: Main Claim There is p ∈ P s.t. whenever G  p is PA -generic, there is σ ∈ V[G] s.t. (a) (b) (c) (d)

σ : N ≺ N , σ (θ, P, κ, A, s) = θ, P, κ, A, s, CκN (rng σ ) = CκN (rng σ), σ   G ⊂ G.

(1) Let σ  ∈ V satisfy (a), (b), (c) and (e) sup σ   κ ∈ A. Then the Main Claim holds.  Proof . Let F = G. Then F is a cofinal normal map of ω1N into A, where σ(A) = A. Define p ∈ PA by:  σ F (ξ) for ξ < ω1N , p(ξ) =   for ξ = ω1N . sup σ κ Clearly p ≤ σ  (q) for q ∈ G. Hence if G  p is generic, then σ   G ⊂ G. QED(1) We must produce a σ satisfying (a), (b), (c) and (e). For ξ < κ set: Cξ = CξN (rng σ). Set: D = {τ < κ | τ = κ ∩ Cτ }. Then D is club in κ. Hence then is κ0 ∈ D ∩ A. Set: ∼ Definition k0 : N0 ↔ Cκ0 , where N0 is transitive; σ0 = k0−1 ◦ σ; θ0 , P0 , s0 , A0 = σ0 (θ, P, s, A). (Hence κ0 = σ0 (κ).) We again let α0 = δN0 be least s.t. Lα0 (N0 ) is admissible and define: Definition L0 is the language on Lα0 (N0 ) with: Predicate: ∈ ◦ Constants: σ , x (x ∈ Lα0 (N0 )) Axioms: • Basic axioms and ZFC− ◦ • σ : N ≺ N 0 κ-cofinally ◦ • σ (θ, P, s, κ, A) = θ 0 , P0 , s0 , κ0 , A0 .

page 135

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

136

(2) L0 is consistent. Proof . Let N1 , σ1 be the liftup of N , σ  HκN . Note that σ  HκN = σ0  HκN . Hence there is k1 : N1 ≺ N0 defined by: k1 σ1 = σ0 ,

k1  κ1 = id,

where κ1 = σ1 (κ) = sup σ  κ. Let L1 be the corresponding language on Lα1 (N1 ), where α1 = δN1 . By the usual argument it suffices to show that L1 is consistent: Since N0 = CκN00 (rngσ0 ), we can conclude that σ0 : N ≺ N0 is κ+N -cofinal. Hence: NO 1 σ1 N

k0 / > N0 || | | || σ || 0

where all maps are cofinal and all structures are almost full. L1 is trivially consistent, however, since Hκ , σ1 models L1 . QED(2) ˜ ≺ M s.t. M ˜ is countable and Now let M = Hκ , N0 , κ0 , A0 , σ0 . Let π : M ˜ = L0 . Then L˜ is a consistent language on Lα˜ (N ˜) = transitive. Let π(L) −1 ˜ π (Lα0 (N0 )). Hence L has a solid model A. Set: ◦

σ  = k0 ◦ π ◦ σ A . Then σ  satisfies (a), (b), (c), (e) of (1).

QED(Lemma 6.3)

4. Iterating Subcomplete Forcing The two step iteration theorem for subcomplete forcing says that if A is subcomplete and ◦

A B

is subcomplete,

◦

then A ∗ B is subcomplete. Equivalently: Theorem 1 Let A ⊆ B where A is subcomplete and ◦

ˇ G A B/

is subcomplete.

Then B is subcomplete. ◦

◦

ˇ G and other Boolean conventions em(Note The definitions of A ∗ B, B/ ◦

ployed here can be found in Section 0. G is the canonical generic name – ◦ ◦ ˇ ˇ and [[ˇ i.e. A G is A-generic over V, a ∈ G]] = a for a ∈ A.)

page 136

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

137

Proof of Theorem 1. Let θ be big enough that θ verifies the subcompleteness of A and: ◦

ˇ G. A θˇ verifies the subcompleteness of B/ − model s.t. Hθ ⊂ N and τ > θ. Let σ : N ≺ N Let N = LA τ be a ZFC where N is countable and sound. Let:

σ(θ, A, B, s) = θ, A, B, s where s ∈ N . Let G be B-generic over N . We must find b ∈ B \ {0} s.t. whenever G  b is B-generic, there is σ ∈ V[G] satisfying (a)–(d) in the definition of subcompleteness. Let G0 = G ∩ A. Then G0 is A-generic over N . Since θ verifies the sub◦ completeness of A, there exist a ∈ A \ {0}, σ 0 ∈ VA s.t. whenever G0  a ◦ 0 is A-generic and σ0 = σ G 0 , then (a)–(d) hold with A, G0 , A, G0 , σ0 in  place of B, G, B, G, σ . Let B∗ = B/G0 . Let G0  a be A-generic. Set: B∗ = B/G0 . Clearly, σ0 extends to σ0∗ s.t σ0∗ : N [G0 ] ≺ N [G0 ]

and

σ0∗ (G0 ) = G0 .

A,G0 ∗ , N = LA In other words, σ0∗ : N ∗ ≺ N ∗ where: N = LA τ, τ , N = Lτ V[G0 ] ∗ A,G0 V ∗ ∗ N = Lτ . Note that Hθ = Hθ [G0 ] ⊂ N . Moreover G is B∗ ∗ generic over N where:

G∗ = G/G0 = {b/G0 | b ∈ G}. Clearly σ0∗ (θ, A, B, B∗ , s) = θ, A, B, B∗ , s. Since θ verifies the subcompleteness of B∗ in V[G0 ], we conclude that there is b∗ ∈ B∗ s.t. whenever G∗  b∗ is B∗ -generic over V[G0 ], then there is σ ∗ ∈ V[G0 ][G∗ ] with: (a∗ ) (b∗ ) (c∗ ) (d∗ )

σ∗ : N ∗ ≺ N ∗ , σ∗ (θ, A, B, B∗ , s) = θ, A, B, B∗ , s ∗ ∗ CδN∗ (rng(σ∗ )) = CδN∗ (rng(σ0∗ )), where δ ∗ = δ(B∗ ). σ∗  G∗ ⊂ G∗ .

Note that G = G0 ∗ G∗ =Df {b ∈ B | b/G0 ∈ G∗ }. Set G = G0 ∗ G∗ . Set σ = σ∗  N . Then σ  ∈ V[G] = V[G0 ][G∗ ]. We show: Claim σ satisfies: (a) σ : N ≺ N (b) σ (θ, A, B, s) = θ, A, B, s

page 137

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

138

(c) CδN (rng(σ )) = CδN (rng(σ)), where δ = δ(B). (d) σ  G ⊂ G. We note first that the claim proves the theorem, since G is B-generic and there must, therefore, be a b ∈ G which forces the existence of such a σ  . We now prove the claim. (a), (b), (d) are immediate. We prove (c). Note that δ ≥ δ∗ , we have: ∗

∗

CδN (rng(σ∗ )) = CδN (rng(σ0∗ )). Since CδN (rng(σ0 )) = CδN (rng(σ)), it suffices to show: ∗

(1) CδN (rng(σ  )) = N ∩ CδN (rng(σ ∗ )) ∗

(2) CδN (rng(σ0 )) = N ∩ CδN (rng(σ0∗ )). We proof (1), the proof of (2) being virtually identical. (⊂) is trivial. We prove (⊃). ∗ Let x ∈ N ∩CδN (rng(σ ∗ )). Then x is N [G0 ]-definable in ξ, σ ∗ (z), G0 , where ξ < δ, z ∈ N . But, letting t ∈ N A s.t. z, G0 = tG0 , we have: σ∗ (z), G0 = σ∗ (z, G0 ) = σ∗ (tG0 ) = σ  (t)G0 . Hence: x = that x s.t. N [G0 ]  ϕ[x, ξ, σ  (t)G0 ]. But since σ  (B) = B, we have: σ  (δ) = δ, where δ = δ(B). Since δ ≥ δ(A), there is f ∈ N mapping δ onto a dense subset of A. Hence σ  (f ) maps δ onto a dense subset of A. Hence there is ν < δ s.t. σ  (f )(ν) ∈ G0 and σ  (f )(ν) ˇ σ (t)). Hence ˇ σ (t)). Thus: x = that x s.t. σ (f )(ν) N ϕ(ˇ x, ξ, forces ϕ(ˇ x, ξ, A N  x ∈ Cδ (rng(σ )). QED(Theorem 1) The proof of Theorem 1 shows more than we have stated. We can omit the assumption that A is subcomplete and omit the map σ, assuming, ◦ ◦ ˇ G0 is subcomplete, where G0 is the canonical A-generic however, that A B/ name. Let θ be big enough that ◦

ˇ G0 . A θˇ verifies the subcompleteness of B/ Let N be as before and let N be countable and full. Suppose that a ∈ A\{0} ◦ ◦ 0 and σ ∈ VA are given s.t. whenever G0  a is A-generic and σ0 = σ G 0 , then • σ0 : N ≺ N • σ0 (θ, A, B, s) = θ, A, B, s • σ0  G0 ⊂ G0 .

page 138

April 22, 2014

16:42

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

139

Our proof then yields a b∗ ∈ B∗ = (B/G0 ) \ {0} s.t. if G ⊃ G0 is B-generic and b∗ ∈ G∗ = G/G0 = {c/G0 | c ∈ G}, then there is σ  ∈ V[G] s.t. (a), (b), (d) and: (c ) CδN (rng(σ  )) = CδN (rng(σ0 )) hold, where δ = δ(B). We can improve on this still further. Suppose that ˇ . This means that tG0 ∈ N whenever G  a is t ∈ VA s.t. and a  t ∈ N 0 A-generic. We can then select our b∗ so as to force: (e*) σ∗ (tG0 ) = σ0 (tG0 ) in addition to (a*)–(d*). It then follows that: (e ) σ (tG0 ) = σ0 (tG0 ). Since, whenever G0  a is A-generic, we can find a b∗ ∈ B/G0 forcing (a), ◦

(b), (d), (c ), (e ), we conclude that there is b0 ∈ VA s.t. a forces b∗ = b G0 ◦ ◦ ˇ G0 and to have these properties. We may assume w.l.o.g. that A b ∈ B/ ◦

 0]]A = a. By Section 0, Fact 4 there is then a unique b ∈ B s.t. [[ b = ◦ ◦  ˇ A b/G0 = b . Letting h = hA,B be defined as in Section 0 by h(c) = {a ∈ A | c ⊂ a} for a ∈ B, we conclude by Section 0, Fact 3 that: ◦

◦

h(b) = [[ˇb/G0 = 0]] = [[ b = 0]] = a. Clearly, if G  b is B-generic, then G0  a is A-generic, where G0 = G ∩ A. ◦

Thus b/G0 = b G0 = b∗ has the above properties and (a), (b), (d), (c ),(e ) hold. Putting all of this together, we get a very useful technical lemma: ◦

ˇ G is subcomplete. Let θ be big Let A ⊆ B and let: A B/ ◦ ˇ G. Let enough that B ∈ Hθ and: A θˇ verifies the subcompleteness of B/ − N = LA model s.t. Hθ ⊂ N and θ < τ . Let N be countable τ be a ZFC and full. Let A ⊆ B in N , where G is B-generic over N . Set: G0 = G ∩ A. ◦ Suppose that a ∈ A \ {0}, σ 0 ∈ VA s.t. whenever G0  a is A-generic and ◦ 0 σ0 = σ G 0 , then: Lemma 1.1

(i) (ii) (iii) (iv)

σ0 : N ≺ N σ0 (θ, A, B, s) = θ, A, B, s σ0  G0 ⊂ G0 t G0 ∈ N .

page 139

April 22, 2014

16:42

140

WSPC 9 x 6: BC9154

jensen

R. Jensen ◦

Let h = hA,B . Then there are b ∈ B \ {0}, σ ∈ VB s.t. a = h(b) and ◦ whenever G  b is B-generic, σ = σ G , and G0 = G ∩ A, then (a) (b) (c) (d) (e)

σ:N ≺N σ(θ, A, B, s) = θ, A, B, s CδN (rng(σ)) = CδN (rng(σ0 )) (δ = δ(B)) σ  G ⊂ G σ(tG0 ) = σ0 (tG0 ). 

We now prove a theorem about iterations of length ω. ◦

ˇ i+1 /G Theorem 2 Let Bi | i < ω be s.t. B0 = 2, Bi ⊆ Bi+1 and Bi (B is subcomplete) for i < ω. Let Bω be the inverse limit of Bi | i < ω . Then Bω is subcomplete. Proof . Let θ be big enough that Bi θˇ verifies the subcompleteness of ◦ ˇ i+1 /G for i < ω. Let N = LA s.t. Hθ ⊂ N , θ < τ , and N is a ZFC− B τ model. Let σ : N ≺ N s.t. σ(θ, Bi | i ≤ ω , s) = θ, Bi | i ≤ ω , s, and N is countable and full. Let G = Gω be Bω -generic over N and set Gi = G ∩ Bi . Then Gi is Bi -generic over N . We claim that there is b ∈ Bω \ {0} s.t. whenever G  b is Bω -generic, there is σ  ∈ V[G] s.t. (a) (b) (c) (d)

σ : N ≺ N σ (θ, Bi | i < ω , s) = θ, Bi | i < ω , s CδN (rng(σ )) = CδN (rng(σ)), where δ = δ(Bω ). σ   G ⊂ G. ◦

We first construct a sequence bi , σ i (i < ω) s.t. bi ∈ Bi , hi (bi+1 ) = bi (where hi = hBi ,Bi+1 ) and whenever Gi  bi is Bi -generic, then, letting σi = σiGi , we have: (a ) (b ) (c ) (d )

σi : N ≺ N σi (θ, Bi | i ≤ ω , s) = θ, Bi | i ≤ ω , s CδN (rng(σi )) = CδN (rng(σ)) σi G ⊂ Gi .

Now let xi | i < ω enumerate N . Set: ui = the N -least u s.t. σ(xi ) ∈ σi (u) and u ≤ δ =Df δ(B) in N . (This exists, since rng(σ) ⊂ CδN (rng(σi )) =  {σi (u) | u ≤ δ in N } by Section 3, Lemma 5.5.) σi will satisfy the additional requirements:

page 140

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

141

(e ) σ0 = σ ◦ (f ) σi (xh ) = σh (xh ) for h < i, where σh =Df σ h (Gi ∩ Bh ). ◦

Bh i (Note Then σh = σ G ⊆ VBi (i.e. the identity is h , since we assume: V Bh Bi Gi ∩Bh the natural injection of V into V ). Thus t = tGi for t ∈ VBh , h < i.)

(g ) σi (uh ) = σh (uh ) for h < i. ◦

◦

◦

i Note that ui = u G for a u i ∈ VBi . We set: b0 = 1, σ 0 = σ ˇ . Given i ◦ ◦ bi , σ i , Lemma 1.1 then gives us bi+1 , σ i+1 . (Take σi+1 (tGi ) = σi (tGi ) ◦ ◦ where Bi t = ˇ x0 , . . . , x ˇi , u 0 , . . . , u i .) Since hi (bi+1 ) = bi , the sequence b = bi | i < ω is a thread in Bi | i < ω . Hence b =  bi = 0 in Bω ,

i

since Bω is the inverse limit. Now let G  b be Bω -generic. Set Gi = G ∩ Bi , ◦ ◦ Gi    σi = σ G i = σ i . Then (a )–(g ) hold for i < ω. By (f ) we can define   σ : N ≺ N by: σ (x) = σi (x) for i s.t. σi (x) = σj (x) for i ≤ j. (a), (b) are then trivial. We prove: (c) CδN (rng(σ )) = CδN (rng(σ)). Proof . Set Ci = CδN (rng(σi )) for i < ω. (Hence C0 = CδN (rng(σ)).) (⊂) It suffices to show rng(σ  ) ⊂ C0 . But σ (xi ) = σi (xi ) ∈ Ci = C0 . (⊃) We show rng(σ) ⊂ CδN (rng(σ  )).  σ(xi ) ∈ σi (ui ) = σ  (ui ) ⊂ {σ  (u) | u ≤ δ in N } = CδN (rng(σ  )). QED(c) Finally we show: (d) σ   G ⊂ G. Proof . We first note that σ   Gi ⊂ G for i < ω, since if a ∈ Gi , then σ (a) = σj (a) ∈ Gj ⊂ G for some j ≥ i. Now let a ∈ G. Since Bω is the inverse limit  of Bi | i < ω , we may assume w.l.o.g. that a = ai where a | i < ω i<ω

is a thread in Bi | i < ω . Let σ  (ai | i < ω ) = ai | i < ω . Then   ai | i < ω is a thread in Bi | i < ω and σ  (a) = σ  ( ai ) = ai ∈ G by the completeness of G wrt. V, since ai ∈ G for i < ω.

i

i

QED(Theorem 2)

Note Theorem 2 can be generalized to countable support iterations of length < ω2 . At ω2 it can fail, however, since in a countable support iteration we are required to take a direct limit at ω2 . If some earlier stage changed the cofinality of ω2 to ω (e.g. if B1 were Namba forcing), then the direct limit would not be subcomplete. Hence for longer iterations we must employ revised countable support iterations, which we discuss in the next section.

page 141

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

142

Revised Countable Support Iterations Definition By an iteration we mean a sequence Bi | i < α s.t. • B0 = 2 • Bi ⊆ Bj for i ≤ j < α  Bi generates Bλ . • If λ < α is a limit ordinal, then i<λ

In dealing with an iteration we shall employ obvious notational simplifications, writing e.g. i for Bi , [[ϕ]]i for [[ϕ]]Bi etc. We also write:   hi (b) = hBi (b) =Df {a ∈ Bi | b ⊂ a} in Bi , for b ∈ Bj . Recall that: j<α

• hi (b) = 0 ↔ b = 0   hi (bj ) • hi ( bj ) = j∈I

i∈I

• a ∩ hi (b) = hi (a ∩ b) for a ∈ Bi ◦ • hi (b) = [[ˇb/G = 0]]i . Our definition of “iteration” permits great leeway in defining Bλ at limit λ. In practice people usually employ one of a number of standard limiting procedures, such as finite support (FS), countable support (CS) or revised countable support (RCS) iterations. RCS iterations are particulary suited to subcomplete forcing. The definition of RCS iteration is given in Section 0. For present purposes all we need to know is: Fact Let B = Bi | i < α be an RSC iteration. Then: (a) If λ < α and ξi | i < ω is monotone and cofinal in λ, then:  bi = ∅ in (i) If bi | i < ω is a thread through Bξi | i < ω , then Bi .  (ii) The set of such bi is dense in Bλ .

i<ω

i

ˇ > ω for i < λ, then (b) If λ < α and i cf(λ)



i<λ

Bi is dense in Bλ .

(c) If i < λ and G is Bi -generic, then the iteration Bi+j /G | j < α − i satisfies (a), (b) in V[G]. (Note By a “thread” through Bi | i < ω we mean a sequence bi | i < ω wrt. b0 = 0, bi ∈ Bi , and hi (bj ) = bi for i ≤ j < ω.) Theorem 3 Let B = Bi | i < α be an RCS-iteration s.t. for all i + 1 < α: (a) Bi = Bi+1

page 142

March 14, 2014

16:18

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

143

◦

ˇ i+1 /G is subcomplete) (b) i (B ˇ i ) has cardinality ≤ ω1 ). (c) i+1 (δ(B Then every Bi is subcomplete. Proof . Set: δi = δ(Bi ). Then (1) δi ≤ δj for i ≤ j < α, since if X is dense in Bj , then {hi (a) | a ∈ X} is dense in Bi . (2) ν ≤ δν for ν < α. Proof . Suppose not. Let ν be the least counterexample. Then ν > 0 is a cardinal. If ν < ω, then δν < ω and hence Bν is atomic with δν the number of atoms. Let ν = n + 1. Then δn < δν < n + 1 by (a). Hence δn < n. Contradiction! Hence ν ≥ ω is a cardinal. If ν is a limit cardinal, then δν ≥ sup δν ≥ ν. Contradiction! Thus ν is a successor cardinal. Let X ⊂ Bν i<ν

be dense in Bν with X = δν < ν. Then X ⊂ Bη for an η < ν by the regularity of ν. Hence Bη = Bν , contradicting (a). QED(2) By induction on i we prove: Claim Let G be Bh -generic, h ≤ i. Then Bi /G is subcomplete in V[G]. (Hence Bi  Bi /{1} is subcomplete in V, taking h = 0.) The case h = i is trivial, since then Bi /G  2. Hence i = 0 is trivial. Now let i = j + 1. ˜ be Bj /G-generic over V[G]. Then G = G ∗ Then Bj /G ⊂ Bi /G. Let G ˜ ˜  ˜ G = {b ∈ Bj | b/G ∈ G} is Bj -generic over V. But then (Bi /G)/G   ˜ by (b). Hence we have shown: Bi /G is subcomplete in V[G ] = V[G][G] ◦  ((Biˇ/G)/G is subcomplete). But Bj /G is subcomplete in V[G] by Bj /G

the induction hypothesis, so it follows by the two step theorem that Bi /G is subcomplete in V[G]. There remains the case that i = λ is a limit ordinal. By our induction hypothesis Bj /Gh is subcomplete in V[Gh ] for h ≤ j < λ. But then Bh+i /Gh | i < λ − h satisfies the same induction hypothesis, ˜ ˆ is Bh+i /Gh -generic over V[Gh ], then G = Gh ∗G since if i ≤ k < λ−h and G ˜ is Bh+i -generic over V and (Bh+k /Gh )/G  Bh+k /G is subcomplete in V. Case 1 cf(λ) ≤ δi for an i < λ. Then cf(λ) ≤ ω1 in V[Gj ] for i < j < λ whenever Gj is Bj -generic. It suffices to prove the claim for such j, since if h < j and Gh is Bh -generic, we can then use the two step theorem to show – exactly as in the successor case – that Bλ /Gh is subcomplete in V[Gh ]. Hence it will suffice to prove:

page 143

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

144

Claim Assume cf(λ) ≤ ω1 in V. Then Bλ is subcomplete, since the same proof can then be carried out in V[Gj ] to show that Bλ /Gj is subcomplete. Fix f : ω1 → λ s.t. sup f  ω1 = λ. Let θ > λ be a cardinal s.t. B < θ and θ is big enough that: ◦

ˇ j /G) i (θˇ witnesses the subcompleteness of B − for i ≤ j < λ. Let N = LA model s.t. Hθ ⊂ N , θ < τ . Let τ be a ZFC σ : N ≺ N s.t. N is countable and full. Suppose also that: σ(θ, B, λ, f , s) = θ, B, λ, f, s.

Claim There is b ∈ Bλ \ {0} s.t. whenever G  b is Bλ -generic, there is σ ∈ V[G] s.t. (a) (b) (c) (d)

σ : N ≺ N σ (θ, B, λ, f , s) = θ, B, λ, f, s CδN (rng(σ )) = CδN (rng(σ)), where σ = sup{δi | i < λ}. σ  G ⊂ G.

˜ = sup σ  λ. It is easily verified that there is a sequence νi | i < ω Set: λ in ω1N s.t., setting ξ i = f (νi ), we have: ξ 0 = 0, and ξ | i < ω is monotone and cofinal in λ. (We can assume w.l.o.g. that f (0) = 0.) Set ξi = f (νi ). ˜ Moreover: Then ξi = σ(ξ i ) and ξi | i < ω is monotone and cofinal in λ. (3) σ (ξ i ) = ξi whenever σ  : N ≺ N s.t. σ  (f ) = f . We now closely imitate the proof of Theorem 2, constructing a sequence bi , ◦ σ i (i < ω) s.t. bi ∈ Bξi , hξi (bi+1 ) = bi , and whenever Gi  bi is Bξi -generic, ◦ i then, letting σi = σ G i , we have: (a ) (b ) (c ) (d ) (e ) (f )

σi : N ≺ N σi (θ, B, f , s) = θ, B, f, s CδNi (rng(σi )) = CδN (rng(σ)) σi  Gi ⊂ Gi σ0 = σ ◦ i σi (xh ) = σh (xh ) (h ≤ i) where σh = σ G h (x |  < ω being an arbitrarily chosen enumeration of N .) (g ) σi (uh ) = σh (uh ) (h ≤ i), where ui = the N -least u s.t. σ(xi ) ∈ σi (u) and u < δ =Df σ −1 (δ) in N . The construction is exactly as before using that σi (Bξj ) = Bξj for all j and ◦

ˇ ξj+1 /G is subcomplete). As before set: σ  (x) = σi (x), where i that Bξj (B

page 144

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

145

is big enough that σi (x) = σj (x) for i ≤ j. The verification of (a)–(c) is exactly as before. To verify (d), we first note that, as before, (4) σ   Gi ⊂ G for i < ω. We then consider two cases: If cf(λ) = ω in N , then cf(λ) = ω in N and ˜ = λ. B is then the inverse limit of B | i < ω and Bλ is the inverse λ λ ξi limit of Bξi | i < ω . We then proceed exactly as before. If cf(λ) = ω1 , Bλ  is the direct limit – i.e. Bi is dense in Bλ . The conclusion then follows i<ω

by (4).

QED(Case 1)

Case 2 Case 1 fails. Then λ is regular and δi < λ for all i < λ. Hence λ = sup δi . Let N , N , i<λ

θ, σ, G be as before with σ(θ, B, s, λ) = θ, B, s, λ. (However, there is now ˜ = sup σ  λ. It nothing corresponding to the function f .) As before set: λ suffices to show: Claim There is c ∈ Bλ s.t. whenever G  c is Bλ -generic, there is σ  ∈ V[G] with: (a) (b) (c) (d)

σ : N ≺ N σ (θ, B, λ, s) = θ, B, λ, s CλN (rng(σ )) = CλN (rng(σ)) σ   G ⊂ G.

Choose a sequence ξ i | i < ω which is monotone and cofinal in λ with ◦ ξ 0 = 0. Set: ξi = σ(ξ i ). As before, our strategy is to construct ci , σ i (i < ω) ◦ ˇ , ci | i < ω is a thread in Bξi | i < ω , and ci forces s.t. c0 = 1, σ 0 = σ  ◦ ˇ ˇ σ i : N ≺ N . The intention is, again, that if c = ci ∈ G and G is Bλ i

generic, then we can define the embedding σ  ∈ V[G] from the sequence ◦ σi = σ G i (i < ω). However, since we no longer have the function f available in defining ξi | i < ω , we shall not be able to enforce: σi (ξ j ) = ξj for j < ω. ˜ and shall have to make do with Nonetheless we can enforce: sup σi  λ = λ, ◦ that. We inductively construct ci ∈ Bξi , σ i ∈ VBξi with the properties: (I) (a) c0 = 1 (b) hξh (ci ) = ch for i = h + 1. (II) Let G  ci be Bξi -generic. Set: Gη = G ∩ Bη (η ≤ ξi ), Gη = G ∩ Bη ◦ ◦ Gξ h (η ≤ ξ i ), σh = σ G for h ≤ i. Then: h = σh (a) σi : N ≺ N (b) σi (θ, B, λ, s) = θ, B, λ, s

page 145

April 22, 2014

16:42

WSPC 9 x 6: BC9154

jensen

R. Jensen

146

(c) CλN (rng(σi )) = CλN (rng(σ)) (d) Let σi (ξ m ) ≤ ξi < σi (ξ m+1 ). Then σi  Gξm ⊂ G. (e) σi (xh ) = σh (xh ) for h < i, x |  < ω being a fixed enumeration of N . (f) σi (uh ) = σh (uh ) for h < i, where uh = the N -least u s.t. σ(xh ) ∈ σh (u). (g) σi = σh if σh (ξ m ) ≤ ξh < ξi < σh (ξ m+1 )  (I), (II) are easily seen to imply the claim. Set c = ci . Then c = 0, since i

◦

c is a thread in Bξi | i < ω . Let G  c be Bλ -generic. Define σi = σ G i (i < ω) and define σ (x) = σj (x) where σj (x) = σk (x) for all k ≥ j. (a)– (c) follow exactly as before. We prove (d). Since Bλ is the direct limit of Bξ | i < ω , it suffices to show: i

(d’) σ   G ⊂ G for i < ω, where Gη = G ∩ B η . Proof . Let a ∈ Gξi . We first note that for j ≥ i sufficiently large we have: σj (ξ m ) ≤ ξj < σj (ξ m+1 ) for an m ≥ i, since otherwise ξj < σj (ξ i ) for arbitrarily large j. But σj (ξ i ) = σ  (ξ i ) for sufficient large j. Hence σ (ξ i ) ≥ sup ξj = λ. Contradiction! If we also pick j large enough that j

σj (a) = σ (a), then σ  (a) = σj (a) ∈ G, since a ∈ Gξm .

QED(d)

◦

It remains only to construct ci , σ i and verify (I), (II). This will be somewhat trickier than the construction in Theorem 2. We shall also have to add further induction hypotheses to (I), (II). Before defining ci we define a bi ∈ Bξi s.t. ◦

(III) (a) b0 = 1, σ 0 = σ ˇ (b) hξj (bi ) = cj if i = j + 1 (c) (II)(a)–(g) hold whenever bi ∈ G. ◦

σ i will be defined simultaneously with bi , before defining ci . Our next induction hypothesis states an important property of bi : ˜ s.t. ξh < ν for h < i, Definition Let ν ≤ ξi < μ < λ ◦ ◦ ˇ]]ξi . ajνμ =Df bi ∩ [[ σ i (ˇξ j ) = νˇ ∧ σ i (ˇξ j+1 ) = μ

It follows easily that: (5) ajνμ ∩ aj



ν  μ

=0

if

j, ν, μ = j  , ν  , μ .

page 146

April 22, 2014

16:42

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing 



147



Proof . Suppose ajνμ ∩ aj ν μ ∈ G where G is Bξi -generic. Then j = j  , since if e.g. j < j  , then μ ≤ σi (ξ j+1 ) ≤ σi (ξ j  ) = ν  ≤ ξi . Contradiction! QED(1) But then ν = σi (ξj ) = ν  , μ = σi (ξ j+1 ) = μ . Contradiction! Our final induction hypothesis reads: ◦

x) = yˇ]]ξi ∈ Bν if sup ξh < ν ≤ ξi < μ. (IV) ajνμ ∩ [[ σ i (ˇ h
◦

Hence ajνμ = ajνμ ∩ [[ σ i (ˇ 0) = ˇ0]] ∈ Bν . Definition A = Ai = the set of all ajνμ = 0 s.t. sup ξh < ν ≤ ξi < μ. h
By (IV) we see that for each a = a

jνμ

◦

∈ A there is σ a ∈ VBν s.t.

◦

G ν (6) σ G a = σi for Bξi -generic G  a.

But: (7) If G  a is Bν -generic, then G extends to a Bξi -generic G s.t. G = ◦ ◦ G ◦G G ∩ Bν . Hence: σ G a = σa = σi . Thus we have: (8) Let G  a be Bν -generic, where a = ajνμ ∈ Ai . Then (II) holds with ◦ ◦G ◦Gξh for h < i, where Gη =Df G ∩ Bη σa = σ G a in place of σi , σh = σ h = σ h (η ≤ ν). Note Since a ∈ G, (d) then reduces to: σa  Gξ ⊂ G. j

Note We then have: σa (xh ) = σh (xh ), σa (uh ) = σh (uh ) for h < i. Whenever ν < μ < λ and G is Bν -generic, we know that Bμ /G is subcomplete in V[G]. Then, using (8), Lemma 1.1, and repeating the construction ◦ ◦ of bi+1 , σ i+1 from bi , σ i in the proof of Theorem 2, we get: (9) ◦ ˜ ∈ Bμ , σ a ∈ VBμ s.t. hν (˜ a) = a and Let a ∈ Ai , a = ajνμ . There are a ◦ ◦ whenever G  a ˜ is Bμ -generic, σa = σ G , and σa = σ G , then we have: a (a) (b) (c) (d)

σa : N ≺ N τa (θ, B, λ, s) = θ, B, λ, s CλN (rng(σa )) = CλN (rng(σa )) σa  Gξj+1 ⊂ G

page 147

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

148

(e) Let r be least s.t. μ ≤ ξr . Then σa (xh ) = σa (xh ) for h < r. (f) Let r be as above. Then σa (uah ) = σa (uah ) for h < r, where uah = the N -least u ∈ N s.t. u ≤ λ in N and σ(xh ) ∈ σa (uh ). (g) σa (ξ  ) = σa (ξ  ) for  ≤ j + 1. ◦

˜, σ a , which can be regarded as an inFor each a ∈ Ai fix such a pair a struction to be used later in forming br , where r is least s.t. μ ≤ ξr . If G ◦ ˜ ∈ G and σr = σ G is Bξr -generic and a ∩ br ∈ G, we shall want: a a (where ◦ σr = σ G ). In particular, we want: a ∩ b = a ˜ . But we shall also require: r r hξi (br ) = ci . Hence we need: a ∩ ci = hξ (a ∩ br ) = hξ (˜ a). This is why bi must be “shrunk” to ci . Accordingly we define ci as follows:  Definition Let bi be given. Set b = bi \ Ai . Then:  hξi (˜ a). ci =Df b ∪ a∈Ai

We are working by induction on i. We assume (I)–(IV) to hold below i and (III), (IV) to hold at i. We must now verify (I), (II) at i. (II) is immediate by a) = hξj (˜ a) = hξj hν (˜ a) = (III)(c), since ci ⊂ bi . (I)(b) holds, since hξj hξi (˜ hξj (a) for i = j + 1 and a = aμν ∈ Ai . Hence:   hξj (ci ) = hξj (b) ∪ hξj (˜ a) = hξj (b ∪ hξi (˜ a)) = hξj (bi ) = cj . a∈A

a∈A

For (I)(a) note that A0 = {a} where a = a0,0,ξ1 = 1, since σ0 = σ by a) = 1. This completes the verification of (I)–(IV) (III)(a). Hence c0 = h0 (˜ at i, given (III), (IV) at i and (I)–(IV) below i. Now let (I)–(IV) hold below ◦ i. We must define bi , σ i and verify (III), (IV) at i. For i = 0 set: b0 = 1, ◦ σ0 = σ ˇ . The verifications are trivial. a | a ∈ A , σa | a ∈ A have been Now let i = j + 1. Note that A , ˜ defined for  ≤ j. Set:  A s.t. ξj < μ. Definition Aˆj = the set of a = ahνμ ∈ ≤j



hνμ

(10) Let a, a ∈ Aˆj , a = a h , ν  , μ .



,a =a

h ν  μ

. Then a ∩ a = 0 if h, ν, μ =

Proof . Suppose not. Let a ∈ A , a ∈ A . Then  =  by (4). Let e.g.  <  ◦ Let a ∩ a ∈ G, where G is Bj -generic. Set σ = σ G  for  ≤ j. Then: σ (ξ h ) = ν ≤ ξ < ν  ≤ ξ ≤ ξj < μ = σ (ξ h+1 ). Hence σ = σ by (II)(g). Hence h < h , since σ (ξ h ) = ν  > σ (ξ h ). Hence σ (ξ h+1 ) ≤ ν  < μ. Contradiction! QED(10)

page 148

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

149

We now define: Definition bi =



{hξi (˜ a) | a ∈ Aˆj } for i = j + 1.

◦ To define σi we set: A˜ = the set of aiνμ ∈ Aˆj s.t. μ ≤ ξi . σ i ∈ VBi is then ◦ ◦ ◦ ◦ ˜ ˜ [[ σ i = σ j ]] ∩ bi = bi \  A. a name s.t. [[ σ i = σ a ]] = a ˜ if a ∈ A,

Then:

(11) (III)(c) holds at i. Proof . Let G  bi be Bξi -generic. ˜ Case 1 a ˜ ∈ G for an a ∈ A.

Let a = ahνμ ∈ A , μ ≤ ξi (hence ξj < μ ≤ ξi ). Thus σi = σa . (II)(a)–(d) hold by (9)(a)–(d). Note that the r in (9)(e), (f) is r = i. But, if a ∈ A ,  ≤ j, then σ = σa . Hence σ (ξ h ) = ν ≤ ξ ≤ ξ < σ (ξ h+1 ) = μ for  ≤  ≤ j. Hence: σa = σ for  ≤  ≤ j. But then (II)(e), (f) hold by (9)(e), (f). Finally (II)(g) holds vacuously, since ξj < μ = σi (ξ h+1 ) ≤ ξi , hence ξj < σi (ξ m ) where σi (ξ m ) ≤ ξi < σi (ξ m+1 ). Case 2 Case 1 fails. Then σi = σj . (II)(a)–(g) then follow trivially.

QED(11)

(III)(a) holds vacuously at i = j + 1. We prove: (12) (III)(b) holds at i. Proof . Clearly hξj (bi ) = Claim cj =





ˆj a∈A

hξj (˜ a).

hξj (˜ a). Hence we need:

ˆj a∈A

 hξj (˜ a), For j = 0 this is trivial, so let j =  + 1. Recall that cj = b ∪ a∈Aj  where b = bj \ a, so it suffices to show: a∈Aj

Claim b =



a∈A 

hξj (˜ a) where A = Aˆj \ Aj .

(⊃) Let a ∈ A . Then a ∈ Aˆ . Hence hξj (˜ a ) ⊂



ˆ a∈A

hj (˜ a) = bj . But for all

a) ∩ hξj (˜ a ) = a ∩ hξj (˜ a ) = a ∈ Aj we have a ∩ a = 0 by (11). Hence hξj (˜     hξj (a ∩ a ˜ ) = 0, since a ∩ a ˜ ⊂ a ∩ a = 0. Hence hξj (˜ a ) ⊂ b. (⊂) Suppose not. Then there is a ∈ Aˆj \ A s.t. b ∩ hξj (˜ a) = 0. But then a) = a. Hence a ∩ b = 0 by the definition of b. QED(12) a ∈ Aj and hξj (˜ It remains only to show: (13) (IV) holds at i.

page 149

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

150

◦

Proof . Let a = ah,ν,μ ∈ Ai . Then ξj < ν ≤ ξi . a ∩ [[ σ i (ˇ x) = yˇ]] = bi ∩ d, ◦ ◦ ◦ ◦ where d = [[ σ i (ˇξ h ) = νˇ ∧ σ i (ξ h+1 ) = μ ˇ ∧ σ i (ˇ x) = yˇ]]Bξi = [[ϕ( σ i )]], where , νˇ, μ ˇ, xˇ, yˇ, all of which the formula ϕ(v) is Σ in the parameters ˇξ , ˇξ h

0

h+1

lie in V2 . Recall that we are assuming VBη ⊆ VBτ for η ≤ τ (i.e. Bη is completely contained in Bτ and the identity is the natural embedding of VBη in VBτ ). As mentioned in Section 0, this has the consequence that if ψ is a Σ0 formula and t1 , . . . , tm ∈ VBη , then: a Bτ ψ(t ) ←→ a Bη ψ(t )

for a ∈ Bη ,

or in other words: [[ψ(t )]]Bτ = [[ψ(t )]]Bη ∈ Bη . We shall make strong use of this. We know: bi =



hξi (˜ e). Hence it

ˆj e∈A

suffices to assign to each e ∈ Aˆj an e∗ ∈ Bν s.t. e) ∩ d = e ∗ , hξi (˜ since then we have: bi ∩ d =



e∗ ∈ Bν .

ˆj e∈A

e) ∩ d = 0 we, of course, set e∗ = 0. Now let hξi (˜ e) ∩ d = 0. Let For hξi (˜ ◦ ◦ ◦ e = ah,ν,μ ∈ Aˆj . Let G  hξi (˜ e) ∩ d be Bξi -generic. Set: σi = σ G , σ j = σ G j . Case 1 μ ≤ ξi . ◦ ◦ e) ∈ Bμ ∧ G. Hence σi = σe =Df σ e G . Hence [[ϕ( σ e )]] ∈ G. Then e˜ = hξi (˜ ◦ Conversely, if e˜ ∩ [[ϕ( σ e )]] ∈ G, then σi = σe and hence e˜ ∩ d ∈ G. Since this holds for all G, we conclude: ◦

e˜ ∩ d = e˜ ∩ [[ϕ( σ e )]] ∈ Bμ . However, μ ≤ ν, since otherwise we would have σi (ξ  ) = σj (ξ  ) for  ≤ h+1 and σi (ξ h ) = ν < μ = σi (ξ h+1 ). Hence h ≤ h and σi (ξ h ) ≤ σj (ξ h ) = ν ≤ QED(Case 1) ξj < ν. Contradiction! Case 2 μ > ξi . We show that this cannot occur. Clearly, if G  hξi (˜ e) ∩ d is Bξi -generic,   then σi = σj = σe by the definition of σe . But then e˜ ∩ d = 0, since if G  e˜ ∩ d were Bμ -generic, then σi (ξ h ) = ν ≤ ξj < ν ≤ ξi < μ = σi (ξ h+1 ). Hence ν = σi (ξ h ) is impossible. Contradiction! e) ∩ d = hξi (˜ e ∩ d) = 0. Contradiction. Since d ∈ Bξi , we conclude: hξi (˜ QED(13) This completes the proof of Theorem 3.

page 150

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

151

The above theorem can be adapted to iterations which allow more freedom in the formation of limit algebras. Definition An iteration B = Bi | i < α is nicely subcomplete iff the following hold: (a) For all i + 1 < α: ◦

ˇ i+1 /G is subcomplete, (i) i B ˇ i )) ≤ ω1 . (ii) i+1 card(δ(B (b) If λ < α and ξn | n < ω is monotone and cofinal in λ, then  bn = 0 in Bλ whenever b = bn | n < ω is a thread in Bξn | (i) n

n < ω , (ii) Bλ is subcomplete if Bi is subcomplete for i < λ. ˇ > ω for all i < λ, then  Bi is dense in Bλ . (c) If λ < α and i cf(λ) i<λ

(d) If i < α and G is Bi -generic, then (a)–(c) hold for Bi+j /G | j < α − i in V[G]. (This allows greater freedom in forming limit algebras at points which acquire cofinality ω, but requires us to take direct limits at other points.) Theorem 4 Let B = Bi | i < α be nicely subcomplete. Then every Bi is subcomplete. Proof . (sketch) By induction on i we again prove: Claim Let h ≤ i. Let G be Bh -generic. Then Bi /G is subcomplete in V[G]. The cases h = i, i = j + 1 are again trivial, so assume that i = λ is a limit ordinal. We again have the two cases: Case 1 cf(λ) ≤ δ(Bh ) for an h < λ. Case 2 Case 1 fails. In Case 1 it again suffices to prove the claim for sufficiently large h < λ, so we assume cf(λ) ≤ ω1 in V[G] whenever G is Bh -generic. But then we can assume cf(λ) ≤ ω1 in V, since the same proof can be carried out in V[G] for Bh+j /G | j < α − h . This splits into two subcases: Case 1.1 cf(λ) = ω. Then Bλ is subcomplete by (b)(ii). Case 1.2 cf(λ) = ω1 . We then literally repeat the argument in the proof of Theorem 3, using that Bλ is the direct limit of Bi | i < λ .

page 151

April 22, 2014

16:42

WSPC 9 x 6: BC9154

jensen

R. Jensen

152

(Note If we instead assumed cf(λ) = ω, the proof in Theorem 3 would  no longer work, since the set of b s.t. b = bn | n < ω is a thread in n

Bξn | n < ω may not be dense in Bλ .)

QED(Case 1) ˜ = In Case 2 we literally repeat the proof in Theorem 3, using that if λ   sup σ λ, then by (b)(i), if c = cn is a thread in Bξn | n < ω (ξn = σ(ξ n ), n

where ξ n | n < ω is monotone and cofinal in λ), then c ∈ Bλ˜ . Just as before we utilize the fact that we can ensure that sup σn  λ = λ, even though we cannot fix the values of σn (ξ i ) (i < ω). QED(Theorem 4) Forcing Axioms We say that a complete BA B satisfies Martin’s Axiom iff whenever Δi | i < ω1 is a sequence of dense sets in B, there is a filter G on B s.t. G∩Δi = ∅ for i < ω. The original Martin’s Axiom said that this holds for all B satisfying the countable chain condition. This axiom is consistent relative to ZFC. It was later discovered that very strong versions of Martin’s Axiom can be proven consistent relative to a supercompact cardinal. The best known of these are the proper forcing axiom (PFA), which posits Martin’s Axiom for proper forcings and Martin’s Maximum (MM) which is equivalent to Martin’s Axiom for semiproper forcings. Both of these strengthen the original Martin’s Axiom, hence imply the negation of CH. Here we shall consider the subcomplete forcing axiom (SCFA), which says that Martin’s Axiom holds for subcomplete forcings. This, it turns out, is compatible with CH, hence cannot be a strengthening of the original Martin’s Axiom (though it is, of course, a strengthening of Martin’s Axiom for complete forcings). Nonetheless it turns out that SCFA has some of the more striking consequences of MM. A fuller account of this can be found in [FA]. We recall from Section 3.1 that the notion of subcompleteness is “locally based” in the sense that, if θ, θ  are cardinals with H θ < θ , then we  need only consider N = LA τ of size less than θ , in order to determine whether θ verifies the subcompleteness of a given B. In other words, P(Hθ ) contains all the information needed to determine this. As a consequence we get Section 3, Corollary 2.3, which says that, if W is an inner model and P(Hθ ) ⊂ W , then the question, whether θ verifies the subcompleteness of B, is absolute in W . Using this we prove: Theorem 5 Let κ be supercompact. There is a subcomplete B ⊂ Vκ s.t. whenever G is B-generic, then:

page 152

April 22, 2014

16:42

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

153

(a) κ = ω2 , (b) CH holds, (c) SCFA holds. Proof . Let f be a Laver function (i.e. for each x and each cardinal β there is a supercompact embedding π : V → W s.t. x = π(f )(κ) and W β ⊂ W ). We define and RCS iteration Bi | i ≤ κ by: • B0 = 2.

◦

ˇ i+1 /G  f (i) ∗ • If i f (i) is a subcomplete forcing, then i B coll(w1 , f (i)).

◦

ˇ i+1 /G  coll(w1 , w2 ). • If i f (i) is a subcomplete forcing, then i B Let G be B = Bκ -generic. Then CH holds in V[G], since there will be a stage Bi+1 which makes CH true by collapsing. But then it remains true at later stages, since no reals are added. We now show that SCFA holds in ◦

V[G]. Let A ∈ V[G] be subcomplete. Let A = A G . Let U ∈ V, U ⊂ VBκ s.t. ◦  [[x = z]] for x ∈ VBκ . (1) [[x ∈ A]] ⊂ z∈U

◦

We may also assume w.l.o.g. that A is forced to be subcomplete and in fact: ◦

(2) κ θˇ verifies the subcompleteness of A. ◦

Let β = Vβ where A, U, θ ∈ Vβ . Let π : V → W be a supercompact ◦

embedding s.t. A = π(f )(κ) and W β ⊂ W . (Hence Vβ+1 ⊂ W .) Then: (3) θ verifies the subcompleteness of A in W [G], since this depends only on P(Hθ ) ⊂ W . Now let: Bi | i ≤ κ = π(Bi | i ≤ κ ). Then Bκ = Bκ and G is Bκ -generic over W . Hence we can form G ⊃ G which is Bκ -generic over W . Since Bκ+1  A ∗ coll(ω1 , A), there is A ∈ W [G ] which is A-generic over W [G]. Now let π ∗ be the unique π∗ ⊃ π s.t. (4) π ∗ : V[G] ≺ W [G ] ∧ π ∗ (G) = G . Then ◦



(5) π∗ (A) = A , where A = π(A)Bκ .

page 153

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

154

But (6) ˜ π ∗  A ∈ W [G ], since π  U ∈ W and π∗  A is definable as that π  s.t. π ˜ (tG ) = π(t)G whenever t ∈ U and tG ∈ A. Let A be the filter on A generated by π∗  A. Let Δi | i < ω1 be a sequence of dense sets in A in V[G]. Let Δi | i < ω1 = π∗ (Δi | i < ω1 ). Obviously A ∩ Δi = ∅ for i < ω1 . Since π : V[G] ≺ W [G], we conclude that there is a filter A˜ on A in V[G] s.t. A˜ ∩ Δi = ∅ for all i < ω2 . QED(Theorem 5) In [FA] we show that subcomplete forcings are ♦-preserving – i.e. if ♦ holds in V, it continues to hold in V[G]. It follows easily from this that ♦ holds in the model V[G] just constructed. If we do a prior application of Silver forcing to make GCH true, then the ultimate model will also satisfy GCH. Hence we have, in fact, shown the consistency of SCFA + ♦ + GCH relative to a supercompact cardinal. SCFA has two of the more striking consequences of MM: Friedman’s principle and the singular cardinal hypothesis at singular strong limit cardinals. Friedman’s principle at a regular cardinal τ > ω1 says that if A ⊂ τ is any stationary set of ω-cofinal ordinals, then there is a normal function f : ω1 → A (i.e. f is monotone and continuous at limits). It is easily seen that Friedman’s principle at β + implies the negation of β . Lemma 6 Assume SCFA. Let κ > ω1 be regular. Then Friedman’s principle holds at κ. Proof . Let PA be as in the final example of Section 3.3, where A ⊂ κ is a stationary set of ω-cofinal ordinals. Let Δi = the set of p ∈ PA s.t. i + 1 ⊂ dom(p) for i < ω1 . Then Δi is dense in PA . By SCFA there is a set G of mutually compatible  conditions s.t. G ∩ Δi = ∅ for i < ω1 . But then the function f = G has the desired property. QED(Lemma 6) By essentially the same proof we get. Lemma 7.1 Assume SCFA. Let τ > ω1 be regular. Let Ai ⊂ τ be a stationary set of ω-cofinal points for i < ω1 . Let Di | i < ω1 be a partition of ω1 into disjoint stationary sets. Then there is a normal function f : ω1 → τ s.t. f (j) ∈ Ai for j ∈ Di .

page 154

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

155

Proof . We need only to show that the appropriate forcing P is subcomplete The proof is exactly like Section 3.3, Lemma 6.3. QED(Lemma 7.1) The singular cardinal hypothesis for strong limit cardinals then follows by a well known argument of Solovay: Corollary 7.2 Assume SCFA. Let τ be as above. Then τ ω1 = τ . Proof . Let Aξ | ξ < τ partition {λ < τ | cf(λ) = ω} into disjoint stationary sets. For each a ∈ [τ ]ω1 let ξi | i < ω1 enumerate a. Let  f : ω1 → Aξi be normal s.t. f (j) ∈ Aξi if j ∈ Di , where Di | i < ω1 i<ω1

partitions ω1 into stationary sets. Let λ = sup f  ω1 . Then

a = Bλ =Df {ξ | Aξ ∩ λ is stationary in λ}. Hence [τ ]ω1 ⊂ {Bλ | λ < τ }.

QED(Corollary 7.2) β

Corollary 7.3 Assume SCFA. If cf(β) ≤ ω1 < β and 2  ≤ β + , then 2β = β + . β

Proof . 2β = (2  )cf(β) ≤ (β + )ω1 = β + . Using Silver’s Theorem we conclude: Corollary 7.4 then 2β = β + .

Assume SCFA. If β is a singular strong limit cardinal,

5. L-Forcing β

In the following, assume CH. Let β > ω1 be a cardinal and assume: 2  = β (i.e. 2α ≤ β for α < β). Let M = LA β =Df Lβ [A], ∈, A ∩ Lβ [A] s.t. Lβ [A] = Hβ and A ⊂ Hβ . Suppose we have forcing conditions which do not collapse ω1 , but do add a map collapsing β onto ω1 . The existence of such a map is equivalent to the existence of a commutative “tower” Mi | i < ω1 , πij | i ≤ j < ω1 s.t. each Mi is countable and transitive, πij : Mi → Mj for i ≤ j < ω1 , and the tower converges to M (i.e. there are πi | i < ω1 s.t. (M, πi | i < ω1 ) is the direct limit of Mi | i < ω1 , πij | i ≤ j < ω1 ). In L-forcing we attempt to collapse β onto ω1 by conditions which directly describe such a tower (or at least a commutative directed system converging to M ). The “L” in “L-forcing” refers to an infinitary language on a structure of the form: N = Hβ + , ∈, M, . . .

page 155

April 22, 2014

17:9

156

WSPC 9 x 6: BC9154

jensen

R. Jensen

in the ground model V. L then determines a set of conditions PL . L-forcing has been used to add new reals with interesting properties. In these notes, however, we shall concentrate wholly on a form of L-forcing which does not add new reals. This means, of course, that Hω1 is absolute. Hence all countable initial segments of our “tower” will lie in V. The theory of L-forcing is developed in [LF]. In that paper, however, we dealt only with forcings which literally added a tower converging to M in the aforementioned sense. In later applications we found it better to replace the tower by other sorts of convergence systems. We therefore adopt a more general approach here. The proofs in [LF] can be readily adapted to this approach. Recall that we are working in first order set theory, so we cannot literally quantify over arbitrary classes. Instead we work with “virtual classes”, which are expressions of the form {x | ϕ(x)} where ϕ = ϕ(x) is a formula of ZF. Normally we suppose x to be the only variable occurring free in ϕ. We define: Definition An approximation system is a pair Γ, Π of virtual classes s.t. (I)–(VII) below are provable in ZFC− . (I) Γ is a class of pairs M, C s.t. 1 ,...,An (a) M = LA for some A1 , . . . , An , τ . τ (b) C ⊂ M .

(Definition For u ∈ Γ set: u = Mu , Cu .) (II) Π is a class of triples π, u, v s.t. u, v ∈ Γ, π : Mu ≺ Mv , Cu = π −1  Cv .  (Definition π : u  v ↔Df π, u, v ∈ Π, u  v ↔Df π π : u  v.) (III) There is at most one π s.t. π : u  v. (Definition πuv Df that π s.t. π : u  v.) (IV) (a) u  u ∧ πuu = id for u ∈ Γ. (b) u  v  w → (u  w ∧ πuw = πvw ◦ πuv ). (c) If u, v  w and rng(πuw ) ⊂ rng(πvw ), then u  v and πuv = −1 ◦ πuw . πvw We say that a set X ⊂ Γ is -directed iff for all u, v ∈ X there is w s.t. u, v  w. In this case we can form a direct limit v, πu | u ∈ X of u | u ∈ X , πuu | u  u ∧ u, u ∈ X . Then v = A, C , where A is a (possibly ill founded) ZFC− model. If A is well founded, we can take it as transitive. Clearly the transitivized direct limit of X, if it exists, is uniquely determined by X.

page 156

April 22, 2014

17:9

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

157

(V) Let X ⊂ Γ be -directed. Let v, πu | u ∈ X be the transitivized direct limit. Then v ∈ Γ and πu = πuv for u ∈ X. Moreover, if u  w for all u ∈ X, then v w. (Hence πvw is uniquely determined by: πvw πu = πuw for u ∈ X.) If t = {x | ϕ(x)} is a virtual class and W is any set or class, we can form tW (the interpretation of t in W ) by relativizing all quantifiers in ϕ to W . (VI) If M is an admissible set, then Γ ∩ M = ΓM and Π ∩ M = ΠM . If A = |A|, ∈A is any binary structure we can form the relativization tA by relativizing quantifiers to |A| and simultaneously replacing ∈ by ∈A in ϕ. (VIII) If A is a solid model of ZFC− and A = wfc(A), then Γ ∩ A = ΓA ∩ A, and Π ∩ A = ΠA ∩ A. Hence: (1) If V[G] is a generic extension of V, then ΓV[G] ∩ V = ΓV , ΠV[G] ∩ V = ΠV . Proof . Let x ∈ V. Then x ∈ M ∈ V, where M is admissible. Hence x ∈ ΓV[G] ↔ x ∈ ΓM ↔ x ∈ ΓV , applying (VI) first in V[G], then in V. QED(1) Remark In practice (I)–(VII) will follow readily from the definitions given for Γ, Π, so we shall not bother to verify them in detail. In all cases Γ and Π will also be provably primitive recursive in ZFC− , so the absoluteness properties (VI), (VII) will follow by Section 1.3. However, it will also be easy to verify these properties directly without going through the theory of pr functions.  A simple example of an approximation system is: Γ is the set of all M, C s.t. • M = LA τ for some A, τ . • M models ZFC− and ω1 exists and CH. • C maps ω1M onto M . Π is then the set of all π, u, v s.t. u, v ∈ Γ, π : Mu ≺ Mv and π ◦ Cu ⊂ Cv . (Note that in this example we have Γ, Π ⊂ Hω2 .) The absoluteness properties are straightforward, since if M, N, π ∈ A and A is admissible,

page 157

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

158

then π : M ≺ N is uniformly expressible by a Σ1 formula in any solid A extending A. Now let an approximation system Γ, Π be given. Let M = LA β be as described at the outset with β > ω1 and Hβ = Lβ [A]. Our aim in L-forcing is to generically add C ⊂ M in the extension V[C] s.t. M, C ∈ ΓV[C] and M, C is the limit of a directed X ⊂ Γ ∩ Hω1 . At the same time we want to add no reals, so that Γ ∩ Hω1 remains absolute. Since we are assuming CH it follows easily that card(M ) = ω1 in V[G]. (In the above example we would accomplish this explicitly, since C would map ω1 onto M .) L is a language on N = Hβ + , ∈, M, <, . . . , where < is a well ordering of Hβ + . (Note N remains a ZFC− model, hence admissible, no matter which predicates and constants we adjoin to it.) The only nonlogical predicate of L is ∈. In addition to the constants ◦

x (x ∈ N ) there will be one further constant C. We always suppose L to contain the following core axioms: • ZFC− (here the usual finite axioms are meant, so we could write them as a single M -finite conjunction).    v = z) for x ∈ N . • v(v ∈ x ↔ z∈x

• Hω1 = Hω1 (or equivalently P(ω) = P(ω)). ◦

• M , C ∈ Γ. • For all countable X ⊂ M there is u ∈ Γ ∩ Hω1 s.t. X ⊂ rng(π

◦

u,M , C

).

(L might, of course, contain further axioms as well.) Assume that L is consistent. Then it is forced to be consistent by the forcing collapsing the cardinality of N to ω with finite conditions. Hence, in such generic extensions, L has solid models. In what follows, without confusion, when we claim any solid model of L, it’s understood that we work in such generic extensions. A (u A v) are Definition Let A be a solid model of L. ΓA , ΠA , A , πuv defined in the obvious way. Set: ◦

˜=Γ ˜ A =Df {e ∈ Γ ∩ Hω1 | e  M, C A in A}. Γ ˜ set: πeA =Df π A For e ∈ Γ

◦

e,M, C A

.

˜ is a -directed system with limit Lemma 1.1 Let A be as above. Then Γ ◦ ˜ M, C A , πeA | e ∈ Γ .

page 158

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

Proof . M =

159

˜ is directed. Let e0 , e1 ∈ rng(πeA ) is trivial. We show that Γ



˜ e∈Γ

˜ Let u ∈ Γ ˜ s.t. rng(π A ) ∪ rng(π A ) ⊂ rng(πA ). Then e0 , e1  u and Γ. e0 e1 u A −1 πeh u = (πu ) ◦ πeAh . QED(Lemma 1.1) ˜ s.t. Lemma 1.2 Let A be as above. Let A ∈ A s.t. A ⊂ M . There is e ∈ Γ A rng(πe ) ≺ M, A .  rng(πeAh ) ⊂ Xi ⊂ Proof . In A construct ei | i < ω , Xi | i < ω s.t. h
rng(πeAi ) and Xi ≺ M, A . It follows easily that eh  ei for h ≤ i < ω and ◦

{eh | h < ω} has a direct limit e, πei e | i < ω . But then e  M, C A and   rng(πeAi ) = Xi ≺ M, A . QED(Lemma 1.2) rng(πeA ) = i<ω

i<ω

Corollary 1.3 Let A be as above. Let U ⊂ P(M ) s.t. U ∈ A is countable ˜ s.t. rng(π A ) ≺ M, A for all A ∈ U . in A. There is e ∈ Γ e

Proof . Let Ai | i < ω ∈ A enumerate U and apply Lemma 1.2 to A = QED(Corollary 1.3) {x, i | x ∈ Ai }. Corollary 1.4 Let A be as above. Let U , V be countable in A s.t. U ⊂ M , ˜ s.t. U ⊂ rng(π A ) ≺ M, A for A ∈ V . V ⊂ P(M ). There is e ∈ Γ e Proof . Apply Corollary 1.3 to U ∪ V .

QED(Corollary 1.4)

If L is consistent, we can define a set P = PL of conditions as follows: ˜ be the set of p = p0 , p1 s.t. p0 ∈ Γ ∩ Hω and Definition Let P 1 p1 ⊂ P(M ) × P(Mp0 )

is countable.

˜ let ϕp be the conjunction of the L statements: For p ∈ P ◦

• p0  M , C • If π = π

◦

p0 ,M , C

, then π : M p0 , a ≺ M , a for all a, a ∈ p1 .

˜ | con(L(p))}, where con(L(p)) is Set L(p) = L + ϕp . We set: P = {p ∈ P the statement that “L(p) is consistent”. The extension relation on P is then defined by: Definition Let p, q ∈ P p ≤ q ←→Df (q0  p0 ∧ rng(q1 ) ⊂ rng(p1 ) ∧ πq0 p0 : Mq0 , a ≺ Mp0 , a whenever a, a ∈ q1 , a, a ∈ p1 ).

page 159

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

160

Lemma 2.1 ≤ is a partial ordering. Proof . Transitivity is immediate. Now let p ≤ q, q ≤ p. We claim that p = q, p0 = q0 is immediate. But if a, a ∈ q1 , a, a ∈ p1 , then a = a , QED(Lemma 2.1) since πq0 p0 = id. Hence q1 = p1 . Definition Let p ∈ P. Mp = Mp0 , Cp = Cp0 , F p = p1 , Rp = rng(p1 ), Dp = dom(p1 ). Lemma 2.2 Let p ∈ P. Then (a) F p is a function. (b) If Rp is closed under set difference, then F p : Dp ↔ Rp . (c) F p  Mp injects Mp into M . Proof . Let A be a solid model of L(p). Let π = πpA =Df (π

◦

p0 ,M, C A

(a) Let a, a , a, a ∈ F p . Then a = a = π−1  a. (b) Let a, a , b, b ∈ F p . It suffices to show:

)A .

Claim: a ⊂ b → a ⊂ b. Set c = a \ b, c = a \ b = ∅. Then (F p )−1 (c) = π −1  c = π −1  a \ π −1  b = b \ a = ∅. Hence c = ∅, since π : Mp0 , c ≺ M, c . QED(b) (c) Let x ∈ Mp0 , x, x ∈ F p . Then π(x) = x ∈ M since π : Mp , x ≺ M, x . QED(Lemma 2.2) We define: Definition π p = F p  Mp . Note By the proof of (c) we have: L(p)  π p ⊂ π

◦

p,M, C

.

We now prove the main lemma on extendability of conditions. Lemma 3.1 P = ∅. Moreover, if p, q ∈ P and L(p) ∪ L(q) is consistent, there is r s.t. r ≤ p, q. Moreover, if X ⊂ P(M ) is any countable set, we may choose r s.t. X ⊂ Rr . ◦

Proof . To see P = ∅ let A be any solid model of L. Let e  M, C A in A where e ∈ Γ ∩ Hω1 . Then A  L(p) where p = e, ∅ . Hence p ∈ P. Now let A  L(p) ∪ L(q). Let X ⊂ P(M ) be countable in V. Let Y = ◦

X ∪ Rp ∪ Rq . There is e ∈ Hω1 ∩ Γ s.t. e  M, C A in A and πeA ≺ M, A for all A ∈ Y . For A ∈ Y set A = (πeA )−1  A. Letting Ai | i < ω be an enumeration of Y in V, we see that Ai | i < ω ∈ Hω1 . Hence F ∈ V

page 160

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

161

where F = {A, A | A ∈ Y } = {Ai , Ai | i < ω}. Set r = e, F . Then A  L(r) and r ≤ p, q. QED(Lemma 3.1) Corollary 3.2 p, q are compatible in P iff L(p) ∪ L(q) is consistent. Proof . (←) Lemma 3.1. (→) If r ≤ p, q, then L(r)  L(p) ∪ L(q).

QED(Corollary 3.2)

Corollary 3.3 Let p ∈ P, X ⊂ P(M ) where X is countable. There is r ≤ p with X ⊂ Rr . Corollary 3.4 Let p ∈ P, u ⊂ M , u is countable. There is r ≤ p with u ⊂ rng(π r ). Lemma 3.5 Let p ∈ P, u ⊂ Mp , u finite. There is r ≤ p s.t. r0 = p0 and u ⊂ dom(π r ). Proof . Let A be a solid model of L(p). Set: r0 = p0 , F r = F p ∪ (πpA  u). Then A  L(r). QED(Lemma 3.5) Using these extension lemmas we get: Lemma 3.6 Let G be P-generic. For p ∈ G set: πpG = q0 }. Then:

 q {π | q ∈ G ∧ p0 =

(a) {p0 | p ∈ G} is a -directed system with limit M, C G , πpG | p ∈ G ,  where C G = πpG  Cp . p

(b) πpG : Mp , a ≺ M, a for a, a ∈ F p .

Note πpG : p0  M, C G in V[G] by (a). The proof is straightforward. Now let κ > (2β ) be regular in V. Then κ V[G] remains regular in V[G], since P ∈ Hκ . Hκ , C G then models all of the core axioms except possibly the axiom: Hω1 = Hω1 . We now state a condition called revisability which will guarantee that V[G] no reals are added – hence that all core axioms hold in Hκ , C G . We first define: Definition Let N ∗ = Hδ , M, <, . . . be a model of countable or finite type, where δ > 2β is a cardinal and < well orders Hδ . Let p ∈ P. p conforms to N ∗ iff whenever a1 , . . . , an ∈ Rp (n ≥ 0) and b ⊂ M is N ∗ -definable in a1 , . . . , an , then b ∈ Rp . Note If p conforms to N ∗ then Rp = ∅ and F p : Dp ↔ Rp by Lemma 2.2. Note {p | p conforms to N ∗ } is dense in P by the extension lemmas.

page 161

April 22, 2014

16:42

WSPC 9 x 6: BC9154

jensen

R. Jensen

162

Before defining revisability we prove a theorem: Lemma 4 Let p conforms to N ∗ . There is a unique N ∗ = N ∗ (p, N ∗ ) s.t. (i) N ∗ is transitive and of the same type as N ∗ . (ii) If a1 , . . . , an ∈ Rp (n ≥ 0) and b ⊂ M is N ∗ -definable in a1 , . . . , an , then ap1 , . . . , apn ∈ N ∗ (where api = (F p )−1 (ai )) and bp (= (F p )−1 (b)) is N ∗ -definable in ap1 , . . . , apn by the same definition. (iii) Each x ∈ N ∗ is N ∗ -definable from parameters in Mp ∪ Dp . Moreover, if A is a solid model of L(p), then πpA ∪ F p extends uniquely to a π ⊃ πpA ∪ F p s.t. π : N ∗ ≺ N ∗ . Proof . We use the following: Fact For any X ⊂ M the following are equivalent: (a) X ≺ M, a for all a ∈ Rp . (b) Let Y = the smallest Y ≺ N ∗ s.t. X ∪ Rp ⊂ Y . Then Y ∩ M = X. ((b) → (a) is trivial. (a) → (b) follows from the fact that each z ∈ Y is N ∗ -definable from parameters in X ∪ Rp .) Let Y˜ = the smallest Y˜ ≺ N ∗ s.t. M ∪ Rp ⊂ Y˜ . Then Y˜ has cardinality β in V. Hence, if – in some extension V[G] – A is a solid model of L(p), ∼ ˜ ˜∗ ↔ ˜ ∗ ∈ N ⊂ A, where π Y is the transitivation of Y˜ . Working ˜:N then N ˜ ∗ s.t. X ∪ Rp ⊂ Z, where in A, we now form Z = the smallest Z ≺ N A ∗ ∼ X = rng(πp ). Transitivize Z to get π : N ↔ Z. Then N ∗ ∈ HωA1 = HωV1 . Claim 1 N ∗ satisfies (i)–(iii). ∼ Proof . Let π = π ˜ π : N ∗ ↔ Y = the smallest Y ≺ N ∗ s.t. X ∪ Rp ⊂ Y . A Then π  Mp = πp , since X = Y ∩ M by the above Fact. For a ∈ Rp we have π −1 (a) = π−1  (X ∩ a) = (πpA )−1  (X ∩ a) = ap . Thus π ⊃ πpA ∪ F p . Using this, (i)–(iii) follow easily. Claim 2 At most one N ∗ satisfies (i)–(iii). Proof . Let N ∗0 , N ∗1 be two different ones. Then (1) Let x1 , . . . , xn ∈ Mp , b1 , . . . , bm ∈ D p . Then N ∗0  ϕ(x, b ) ↔ N ∗1  ϕ(x, b ). Proof . Let bi = api , ai ∈ Rp . Set: c = {x ∈ M | N ∗  ϕ(x, a )}. Then by (ii): cp = {x ∈ Mp | N ∗h  ϕ(x, b )}. QED(1) But it then follows straightforwardly that id  (Mp ∪ D p ) extends to a ∼ σ : N ∗0 ↔ N ∗1 . Hence σ = id, since the models are transitive. QED(Claim 2)

page 162

March 14, 2014

16:18

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

163

In the proof of Claim 1, we have shown that, if A is a solid model of L(p), then πpA ∪ F p extends to a π : N ∗ ≺ N ∗ . It remains only to note that π is unique, since every z ∈ rng(π) is N ∗ -definable from elements of X ∪ Rp = rng(πpA ∪ F p ). QED(Lemma 4) Note Clearly Mp = M , where N ∗ = H, M , <, . . . . We now define: Definition P = PL is revisable iff for sufficiently large cardinals Ω > 2β : Let N ∗ = HΩ , M, <, P, . . . where < well orders HΩ . Let p conform to N ∗ and set N ∗ = N ∗ (p, N ∗ ). Let G be P-generic over N ∗ , where N ∗ = H, M , <, P, . . . . Then there is q ∈ P s.t. Mq = Mp , Cq = C G , and F q = F p. (In other words q = Mp , C G , F p ∈ P.) Lemma 5.1 Let P be revisable. Then P adds no new reals. ◦

Proof . Let  f : ω → 2. ◦  Claim Δ = {p | f p  f = fˇ} is dense in P.

Let r ∈ P. Pick Ω big enough to verify revisability and set N ∗ = ◦

HΩ , M, <, P, f , r, . . . . Let p conform to N ∗ . Set N ∗ = N ∗ (p, N ∗ ). Let N ∗ = H, M , <, P, f , r, . . . . Let G  r be P-generic over N ∗ . Let f = f G . Let q = M , C G , F p ∈ P. ◦

Claim q ≤ r and q  f = fˇ. Proof . Let A be a solid model of L(q). Let σ ⊃ πqA ∪ F q s.t. σ : N ∗ ≺ N ∗ . (1) q ≤ r. Proof . Let C = C G , r0 = r 0  M , C = q0 and πr0 ,q0 = πrG . But Rr ⊂ Rq , ∗ since r is N -definable. Let a, a ∈ F r , a, a ∈ F q . Claim πr0 ,q0 : Mr , a ≺ Mq , a . This is clear, since a = (F q )−1 (a) = σ−1 (a) and hence a, a ∈ σ−1  F r = F r ). QED(1) (2) Let s ∈ G, s = σ(s). Then A |= L(s). Proof . s0 = r0  q0 = M , C  M, C˙ A , and πsA0 = σ ◦ πs0 ,q0 .

page 163

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

164

Let a, a ∈ F s . Then a = σ(a ), where a , a ∈ F s . Hence, πsA0 : Ms , a ≺ M, a . QED(2) ◦

(3) q  f = fˇ. ◦

ˇ Let q  ≤ q s.t. Suppose not. Then there is i s.t. f (i) = h and q  f (ˇi) = h. ◦ ˇ Let A be a solid model of L(q  ), hence of L(q). Let s ∈ G s.t. q   f (ˇi) = h. ◦

ˇ Hence s P f (ˇi) = ˇ h. Let σ be as above. Let s = σ(s). Then s  f (ˇi) = h.   q , s are incompatible. But A  L(q ) ∪ L(s). Contradiction! by Lemma 3.1. QED(Lemma 5.1) Now let Lc be L with its axioms reduced to the core axioms. (Thus Lc is uniquely determined by Γ, Π.) By Lemma 5.1 we have: Lemma 5.2 Let P be revisable. Let G be P-generic. Let p ∈ G. Set: V[G] A = Hκ , C G , where κ > 2β is regular. Then A models Lc (p). An examination of the proof of Lemma 4 shows, however, the proof of the final clause in that Lemma used only that A models Lc (p). Hence: Corollary 5.3 Let P be revisable. Let G be P-generic. Let p ∈ G where p conforms to N ∗ = HΩ , M, <, . . . . Let N ∗ = N ∗ (p, N ∗ ). There is a unique σ ⊃ πpG ∪ F p s.t. σ : N ∗ ≺ N ∗ . Proof . πpG = πpA where A is as in Lemma 5.2.

QED(Corollary 5.3)

Combining this with the proof of Lemma 5.1 we get: Lemma 5.4 Let P be revisable. Let N ∗ = HΩ , M, <, P, . . . where Ω verifies revisability. Let p conform to N ∗ . Set: N ∗ = N ∗ (p, N ∗ ) = H, M , <, P, . . . . Let G be P-generic over N ∗ and set: q = Mp , C G , F p . Let G  q be P-generic. Let σ ⊃ πpG ∪ F p s.t. σ : N ∗ ≺ N ∗ . Then σ  G ⊂ G. (Hence σ extends uniquely to σ ∗ : N ∗ [G] ≺ N ∗ [G] with σ∗ (G) = G.) Proof . The proof of (2) in Lemma 5.1 made use of a solid model A of L(q). An examination of this proof shows, however, that it is enough that V[G] A models Lc (q). Hence we can take A = A, where A = Hκ , C G is as ∗ A G G q above. Hence πq = πq and if σ ⊃ πq ∪ F is s.t. if σ : N ≺ N ∗ then A |= L(s) whenever s ∈ G and s = σ(s). If s ∈ G, there would be p ∈ G

page 164

March 14, 2014

16:18

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

165

incompatible with s. But A |= L(p) ∪ L(s). Contradiction! QED(Lemma 5.4) We say that L is modest if all of its axioms can be forced by PL -more precisely: Definition Let L satisfy the core axioms. L is modest iff whenever G is V[G] PL -generic there is a regular κ > 2β s.t. A = Hκ , C G satisfies L. Lemma 5.2 says that Lc is modest. Assuming modesty, we have a simple criterion for deciding whether a given condition lies in a generic set G: Lemma 5.5 Let P = PL where L is modest. Let G be P-generic. Let p ∈ P. Then p ∈ G iff the following hold: • p0  M, C G , • πpG0 : Mp , a ≺ M, a whenever a, a ∈ F p . Proof . (→) is trivial. We prove (←). Let κ be regular s.t. κ > 2β and / G there would be a q ∈ P A = Hκ , C G satisfies L. Then A  L(p). Iff p ∈ s.t. p, q are incompatible. But A  L(p) ∪ L(q). QED(Lemma 5.5) Note In [LF], §4 we have shown that the assumption of modesty can be omitted from Lemma 5.5 assuming that P adds no reals. This is because P = PL∗ , where L∗ is the set of L statements forced to hold in V[G] A = Hκ , C G , where κ > 2β is regular. We shall not use that here, however, since our languages will always be modest. (We are unlikely to adopt an axiom without the expectation that it will be forced.) Finally, we note that there is an apparently weaker notion of revisability relative to a parameter: Definition P is weakly revisable iff there exist a cardinal Ω > 2β and an s ∈ HΩ s.t. whenever N ∗ = HΩ , M, <, P, s, . . . and p conforms to N ∗ , then, letting N ∗ = N ∗ (p, N ∗ ) = H, M , <, P, s, . . . , we have: Let G be P-generic over N ∗ . Then q = M , C G , F p ∈ P. It turns out that this is equivalent to full revisability. This fact is useful (and may be used tacitly) in verifying revisability. Lemma 5.6 Let P be weakly revisable. Then it is fully revisable. Proof . Let Ω be the smallest cardinal verifying weak revisability. Let Ω > H Ω be a cardinal. Let N ∗ = HΩ , M, < , P, . . . . Let p conform to N ∗ and let N ∗ = N ∗ (p, N ∗ ) = H  , M  , <, P , . . . . Let G be P -generic over N ∗ .

page 165

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

166

Claim q = M  , C G , F p ∈ P. Note that Ω, s are N ∗ -definable, where s = the < -least s s.t. Ω, s verifies weak revisability. Let A be a solid model of L(p) and let σ  ⊃ πpA ∪ F p s.t. σ : N ∗ ≺ N ∗ . Let σ  (Ω, s) = Ω, s.

(1)

Set: N ∗ = HΩ , M, <, P, s, . . . where <=< ∩HΩ2 . Then p conforms to N ∗ . Set: N ∗ = N ∗ (p, N ∗ ) = H, M , <, P, s, . . . . Let σ ⊃ πpA ∪ F p s.t. σ : N ∗ ≺ N ∗ . Then each x ∈ rng(σ) is N ∗ -definable in parameters from rng(πpA ∪ F p ). Hence it is N ∗ -definable in these parameters. Hence: rng(σ) ⊂ rng(σ ).

(2) But: (3)

M = Mp = M  ; σ  M = πpA = σ   M .

Moreover, each a ∈ P(M ) ∩ N ∗ is M , b -definable from parameters from M , where b ∈ D p . Similarly for P(M ) ∩ N ∗ . Hence: (4)

P(M ) ∩ N ∗ = P(M ) ∩ N ∗ .

Since σ  Dp = F p = σ   D p and σ  M = σ   M  , we conclude (5)

σ  P(M ) = σ   P(M ).

P = |P|, ≤P is canonically codable as a subset of P(M ). Similarly for P . But σ(P) = σ  (P ) = P. It follows easily that. (6)

P = P and σ  P = σ   P . But if Δ ∈ P(P) ∩ N ∗ , then Δ = (σ−1 ) · σ(Δ) ∈ N ∗ . Hence:

(7)

P(P) ∩ N ∗ ⊂ N ∗ .

Hence G is generic over N ∗ and we conclude: (8)

q = M , C G , F p ∈ P.

QED(Lemma 5.6)

In conclusion we say a few words about the difference between the present approach and that taken in [LF]. There too we approximated A M = LA τ s.t. Lτ = Hβ for some β > ω1 . Our intention, however, was simply to make M the limit of a tower of countable models. In place of an approximation system Γ, Π we worked with a collection T of tower segments Mi | i ≤ α , πij | i ≤ j ≤ α satisfying: Mi Mi i • Mi = LA βi , i ≤ ω1 , Mh ∈ Hω1 for h < i.

page 166

April 22, 2014

16:42

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

167

• πij : Mi ≺ Mj (i ≤ j) with πii = id. • πij πhi = πhj .  rng(πiλ ). • If λ ≤ α is a limit ordinal, then Mλ = i<λ

We sometimes imposed further requirements on T , but T was always primitive recursive. For t ∈ T we set: t | i ≤ j ≤ αt . t = Mir | i ≤ αt , πij

Call t a segment of s iff αt ≤ αs and Mit = Mis ,

t s πij = πij

for i ≤ j ≤ αt . ◦

Our language contained a single constant t in addition to x (x ∈ N ) and the core axioms: 



◦   ZFC− , Hω1 = H ω1 , v v∈x↔ w = z , t ∈ T, z∈x

α

◦

t

= ω1 ,

Mωt 1 = M ,

 i < ω 1 Mit ∈ Hω1 .

We now show how to convert this approach into our present one. For each t ∈ T set: et = Mαt , {y, x, i | i < αt ∧ πiαt (x) = y} . Set Γ = {et | t ∈ T }. Note that t is uniquely recoverable from et. We set: et  es iff et is a segment of e1 , π : et  es iff (et  es and π = παs t ,αs ). Then Γ, Π is an approximation system and the above core axioms translate into our usual core axioms. 6. Examples We now display some specific examples of L-forcing. All of them are revisable and will turn out to be subcomplete as well. 6.1. Example 1 Assume CH and 2ω1 = ω2 . Without adding reals we wish to make ω2 become ω-cofinal. We first define our approximation system: Definition Γ = the set of M, C s.t. − • M = LA τ models ZFC and “ω1 is the largest cardinal”. • C is a cofinal subset of OnM of order type ω.

page 167

April 22, 2014

16:42

WSPC 9 x 6: BC9154

jensen

R. Jensen

168

Definition For u ∈ Γ set u = Mu , Cu . Definition Π = the set of π, u, v s.t. u, v ∈ Γ, π : Mu ≺ Mv , π  Cu = Cv . We again write π : u  v for π, u, v ∈ Π and u  v for Definition αu = ω1Mu for u ∈ Γ.



π π : u  v.

We note that: (1) Let v ∈ Γ. Let α ≤ αv . There is at most one u ∈ Γ s.t. uv and α = αu . Thus  is a tree. Now let M = LA ω2 , where Lω2 [A] = Hω2 . Set: N = Hω3 , M, <, . . . where < well orders Hω3 . Let L be the language on N constaining exactly the core axioms (wrt. Γ, Π). Lemma 1 L is consistent. Proof . Let θ > 2β be a regular cardinal. Let H = Hθ and σ : H ≺ H, ˜ = L. Set where H is countable and transitive. Let σ(M , N ) = M, N , σ(L)  ˜ ˜ ˜ M= σ(u). Then σ  M : M ≺ M cofinally. Let H, σ ˜ be the liftup of u∈M

˜ ω2M -cofinally. Let k˜ : H ˜ ≺ H s.t. k˜ M , σ  M . Then σ ˜ :H ≺H σ = σ, ˜ H ˜ ˜ ˜ and it suffices to k  ω2 = id. Let k(L) = L. Then L is a language on N show: Claim L˜ is consistent. Proof . Let C ⊂ M be cofinal in OnM with order type ω. Set C˜ = σ  C. ˜ ˜ ˜ Then HωH2 , C models L. QED(Lemma 1) Now let P = PL . We show that P satisfies a particularly strong form of revisability. Lemma 2 Let p ∈ P. Let C be cofinal in OnMp with order type ω. Then q = Mp , C , F p ∈ P. Proof . Let A be a solid model of L(p). We shall “resection” A to get a solid model A of L(q). Let A = |A|, C A . Set A = |A|, C  where C  = πpA  C. Since C  is defined in A, we have A  (ZFC− ∧ H ω1 = Hω1 ). Since HωA1 = Hω1 it follows easily that whenever X ⊂ M is countable in A, then there is u  M, C  s.t. u ∈ Hω1 and X ⊂ rng(πuA ). Hence all core axioms hold. QED(Lemma 2)

page 168

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

169

An immediate corollary is: Corollary 2.1 P is revisable. Thus, if G is P-generic and κ > 2β is regular, HκG , C c satisfies all core axioms. But these are exactly the axioms of L. Hence L is modest. Making use of Lemma 2 we now prove: Lemma 3 P is subcomplete. ω2

− Proof . Let θ > 22 . Let W = LA τ be a ZFC model s.t. Hθ ⊂ W and θ < τ . Let π : W ≺ W , where W is countable and full. Let π(θ, P, s) = θ, P, s. Since ω2 ≤ δ(P) it suffices to show:

Claim Let G be P-generic over W . There is q ∈ P s.t. whenever G  q is P-generic, then there is σ ∈ V[G] s.t. (a) (b) (c) (d)

σ:W ≺W σ(P, θ, s) = P, θ, s CωW2 (rng(σ)) = CωW2 (rng(π)) σ  G ⊂ G.

∼ ˜ ↔ ˜ is transitive. Set π C, where W ˜ = k −1 ·π. Now let C = CωW2 (rng(π)), k : W W ˜ ˜ = k−1 σ, Then π ˜ : W ≺ W is ω3 -cofinal. If σ satisfies (a)–(d) and we set: σ W W ˜ is also ω3 -cofinal. But since σ then σ ˜ : W ≺ W ˜ takes ω2 cofinally to ˜ W ω2 = ω2 , it follows that σ ˜ is ω2 -cofinal. The following lemma hints at the possibility of such a σ ˜ : Let π ˜ (θ, P, s) = ˜ P, ˜ s˜. θ, ˜ Sublemma 3.1 Let δ = δW ˜ = the least δ s.t. Lδ (W ) is admissible. Then ˜ ˜ the following language L on Lδ (W ) is consistent: Predicate: ∈ ◦ ˜ )) Constants: σ , x (x ∈ Lδ (W    ◦ − ˜ v = z), σ : W ≺ W Axioms: ZFC , v(v ∈ x ↔ z∈x

˜

ωW 2 -cofinally,

˜ s˜. ˜ θ, σ(P, θ, s) = P,

ˆ ,π Proof . Let W ˆ be the liftup of W , π  H , where H = (Hω2 )W . (Hence ˆ ˆπ = π ˆ ≺N ˜ s.t. kˆ ˜  H.) Let kˆ : W ˜ , kˆ  ω2W = id. Then kˆ is cofinal πH =π ˜ . Let δˆ = δ ˆ be least s.t. L ˆ(W ˆ ) is admissible. Let Lˆ be defined on in W W δ ˆ sˆ in place of W ˜ s˜, ˆ ˜ ˜ ˆ , P, ˆ θ, ˜ , P, ˜ θ, Lδˆ(W ) as L was defined on Lδ (W ) with W ˆ sˆ. It suffices to show: ˆ θ, where π ˆ (P, θ, s) = P, Claim Lˆ is consistent. ˆ ,π ˆ This is trivial, however, since W ˆ models L. QED(Sublemma 3.1)

page 169

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

170

Now let Ω > 2β be a cardinal. Set: N ∗ = HΩ , M, W, P, θ, s, π, . . . . Let p conform to N ∗ and set: N ∗ = N ∗ (p, N ∗ ) = H  , M  , W  , P , θ , s , π  , . . . . ˜ ,π ˜ , π ˜  , L˜ be defined in N ∗ as W ˜ , L˜ were defined in N ∗ . Since N ∗ Let W ◦ ˜ Set σ is countable, there is a solid model A of L. ˜ = σ A . Then ˜ σ ˜:W ≺W

ω2W -cofinally.

˜ ˜  = CW σ )). Set: C = C G , C  = π   C. Then C  is cofinal Hence W  (rng(˜ ωW 2



in ω2W and has order type ω. Set: q = M  , C  , F p . Then q ∈ P by the strong revisability lemma. Let G  q be P-generic. Let π ∗ ⊃ πqG ∪ F q s.t. ˜ , σ = π ∗ σ  . Then σ ∈ V[G]. π∗ : N ∗ ≺ N ∗ . Let π∗ (k  ) = k. Set: σ  = k  σ Claim σ satisfies (a)–(d). 

Proof . (a), (b) are trivial. We prove (c). Set ω2 = ω2W . (1)





CωW (rng(σ )) = CωW (rng(π  )), 2

2

˜  = C W  (rng(π  )) by definition and k   W ˜  = C W  (rng(σ )), since k   W ω2 ω2 ˜      ˜  = CW since W (rng(˜ σ )), k rng(˜ σ ) = rng(σ ), and k  ω = id.  2 ω 2

(2)

CωW2 (rng(σ)) ⊂ CωW2 (rng(π)), 



since rng(σ) = π ∗  rng(σ ) ⊂ π ∗  CωW (rng(π  )) ⊂ π∗ (CωW (rng(π  )) = 2 2 CωW2 (rng(π)). (3)

CωW2 (rng(π)) ⊂ CωW2 (rng(σ)), 

since rng(π) = π∗  rng(π  ) ⊂ π ∗  CωW2 (rng(σ  )) ⊂ CωW2 (rng(σ)), since QED(c) π ∗  rng(σ  ) = rng(σ) and π∗  ω2 ⊂ ω2 . We now prove (d). Since L is modest we have: Sublemma 3.2 Let C = C G . Then G = GC = the set of p ∈ P s.t. p0  M, C and π : Mp , a ≺ M, a whenever a, a ∈ F p , when π = πp0 ,M,a . Now let r ∈ G, r = σ0 (r). Then r0 = r 0  M , C and πr 0 ,M,C = πrG , where C = C G . Obviously,

σ  M : M , C  M  , C  and πqG : M  , C   M, C ,

page 170

April 22, 2014

16:42

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

171

where πqG = π∗  M  . Hence σ  M : M , C  M, C . Let r = σ(r). Then r0 = r 0 and F r = {π ∗ (a), a | a, a ∈ F r }. Clearly r0  M, C and: πr0 ,M,C = σ ◦ πrG . Now let a, a = π ∗ (a ), a ∈ F r . Then πrG : Mr , a ≺ M  , a and σ(M  , a ) = M, a . QED(Lemma 3) Note We could in this case have omitted the predicate C and simply taken − and “ω1 is the largest Γ as the set of M, ∅ s.t. M = LA τ models ZFC cardinal”. Π would then be defined as the set of π, u, v s.t. u, v ∈ Γ, π  Mu ≺ Mv cofinally. If we call P the resulting set of conditions, then it is the “same” as P in the sense that BA(P)  BA(P ). Note P is, in fact, equivalent to Namba forcing in the sense that BA(P)  BA(N). This is surprising, since P not only looks different and has a different motivation, but the combinatorics involved in the proofs are quite different. 6.2. Example 2 β 

Now let β > ω2 be a cardinal and assume: 2ω = ω1 , 2ω1 = ω2 , 2 = β. We shall develop a forcing very much like the previous forcing which, however, gives cofinality ω not only to ω2 but to every regular τ ∈ [ω2 , β]. There will be some variation in the definition of the forcing, depending on whether cf(β) = ω1 . Thus, in this example, we assume cf(β) = ω1 . In Example 3 we shall then detail the changes which must be made if cf(β) = ω1 . Let M = LA β where Hω2 = Lω2 [A] and Hβ = Lβ [A]. M is then smooth in the sense defined in Section 3.2.  Definition Relabel the classes Γ,  defined in Example 1 as Γ0 , 0 . Set: Γ = the collection of M, C s.t. • M = LA β is smooth. • γ = ω2M exists and Lγ [A] = Hω2 in M . • C ⊂ γ, sup C = γ, otp C = ω. For u = Mu , Cu ∈ Γ set: αu = αMu = ω1Mu ,

γu = γMu = ω2Mu ,

u Mu0 = LA γu ,

Hence: u0 ∈ Γ0 .

u0 = Mu0 , Cu .

u Mu = LA βu ,

page 171

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

172

Definition Let u, v ∈ Γ. π : u  v iff the following hold: • π 0 : u0 0 v0 where π 0 = π  Mu0 . • π : Mu ≺ M v . • Let π : Mu →Σ0 Muv cofinally. Then Muv , π is the liftup of Mu , π 0 . (In other words π : Mu →Σ0 Muv γu -cofinally.) Γ, Π is easily seen to be an approximation system.  LA β

We return to M = as stated at the outset. Let N = Hβ + , M, <, . . . where < well orders Hβ + . Let L be the language on N containing only the core axioms (wrt. Γ, Π). Lemma 4 L is consistent. Proof . Let θ > 2β be a regular cardinal. Let π : H ≺ Hθ s.t. H is countable ˆ = HωM and set and transitive and π(M , N , L) = M, N, L. Let H 2 ˆ ˜ π H, ˜ = the liftup of H, π  H . ˜ ˜,N ˜ , L˜ = π ˜ ≺ Hθ s.t. k˜ σ = σ. k  ω2H = id. Set M ˜ (M , N , L). Then Let k : H ˜ k(L) = L and it suffices to show: Claim L˜ is consistent.

Let C ⊂ ω0M cofinally s.t. otp(C) = ω. Set C˜ = π  C = π ˜  C. We prove: ˜ models L. ˜ Claim Hω2 , C Proof . All axioms are trivial except for the last one. We show that if ˜ and X ⊂ M is countable, then there is u ∈ Γ ∩ Hω1 s.t. u  M, C ˜ X ⊂ rng(πu,M,C ˜ ). We construct such a u: Let Z ≺ H be countable s.t. ∼ ˜ ), C  = π −1  C, ˜ X ∪ rng(˜ π ) ⊂ Z. Let π : H  ↔ Z. Set: M  = π −1 (M     π = π  M . Then X ⊂ rng(π ) and it suffices to show: ˜ , C . ˜ Claim π  : M  , C   M ˜ 0 , C ˜ is obvious. We therefore need only to show: π   M 0 : M 0 , C  0 M Claim Let π  : M  →Σ0 M ∗ cofinally. Then the map π  is ω2 M  -cofinal into M ∗ . ˜ ω H  -cofinally, since if x ∈ H, ˜ then Proof . First note that π  : H  ≺ H 2  −1   x∈π ˜ (u), where u < ω2 in H. Set u = π π ˜ (u). Then x ∈ π (u ), u < ω2 in H  . Now let x ∈ M ∗ . By cofinality there is v ∈ M  s.t. x ∈ π  (v). Let ˜ s.t. x ∈ π  (u) and u < ω2 in H  . Set: w = u∩v. Then x ∈ π (w) where u∈H ˜ w < ω in M  , since M  = Hβ  in H  , where β  = (π  )−1 (β).QED(Lemma 4) We then define P = PL as before. Exactly as before we get:

page 172

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

173

Lemma 5 Let p ∈ P. Let C ⊂ γp be cofinal in γp with order type ω, where M γp =Df γp0 = ω2 p . Then q = Mp , C , F p ∈ P. Hence: Corollary 5.1 P is revisable. Hence Hκ [G], C G models the core axioms whenever G is P-generic and κ > 2β . But L has only the core axioms and is, therefore, modest. Using this we obtain: Lemma 6 P is subcomplete. The proof is virtually identical to that of Lemma 3. However, in the verification of (d) at the end of the proof we need additional justification for: σ   M : M , C  M  , C  . Letting σ   M map M cofinally to M ∗ , we must show: σ   M : M −→Σ0 M ∗

ω2M -cofinally.

This follows from ˜ σ : N ≺ N

ω2N -cofinally

by the argument used in Lemma 4 to get: π  : M  −→Σ0 M ∗



ω2M -cofinally

˜  ω H  -cofinally. from: π  : H  ≺ H 2

QED(Lemma 6)

P obviously collapses β to ω1 . We now show that its successor is not collapsed: Lemma 7 Let G be P-generic. Then β + is regular in V[G]. This is immediate from: Sublemma 7.1 B = BA(P) has a dense subset of size β. Proof . We defined a collection S of statements in the forcing language s.t. S ≤ β (in V), and for each p ∈ P there is a ψ ∈ S s.t. 0 = [[ψ]] ⊂ [p]. ◦

([p] being the smallest a ∈ B s.t. p ∈ a.) Let C be the canonical term s.t. ◦

C G = C G for P-generic G. For each triple u, a, a s.t. u = Mu , Cu ∈ Γ ∩ Hω1 ,

a : ω → P(Mu ),

a : ω → M,

page 173

April 22, 2014

16:42

WSPC 9 x 6: BC9154

jensen

R. Jensen

174

let ψuaa be the statement: ◦ ˇ , C ∧ i < ω z(z ∈ a ˇ(i) ←→ π◦(z) ∈ a ˇ(i)) u ˇ  M ◦

where π = π

◦

ˇ C u ˇ M,

. All such triples are elements of M , so the set S of

such statements has at most cardinality β. We now show that for each p ∈ P there is ψ ∈ S with 0 = [[ψ]] ⊂ [p]. It suffices to prove this for a dense subset of P, so assume w.l.o.g. that p conforms to N ∗ = H(2β )+ , M, < . Let ˜ = LA , where G  p be P-generic. Let β˜ = sup πpG  βp . Then β˜ < β. Set M β˜ ˜ . Let ai , ai | i < ω enumerate M = LA . For each a ∈ Rp set a ˜ =a∩M β

F p in V. Set a = ai | i < ω , a ˜ = ˜ ai | i < ω . Let ψ = ψp0 ,a,˜a . Then [[ψ]] = 0, since ψ is true in V[G]. We claim that [[ψ]] ⊂ [p], or equivalently: Claim Let G be P-generic. Then G ∩ [[ψ]] = ∅ → p ∈ G. Then p0  M, C , since ψ is true in V[G]. Let a, a ∈ F p . We must show: Claim π : Mp , a ≺ M, a , where π = πp0 ,M,C .

Set: b = {z1 , . . . , zn | M, a  X (z1 , . . . , zn )}. Then b ∈ Rp , since p conforms to N ∗ . Moreover b, b ∈ F p where b has the same definition over Mp , a . Hence: ˜ ∩b Mp , a  X (z1 , . . . , zn ) ←→ z1 , . . . , zn ∈ b ←→ π(z1 , . . . , zn ) ∈ ˜b = M −→ π(z1 , . . . , zn ) ∈ b −→ M, a  X (π(z1 ), . . . , π(zn )). Since this holds for all X we have: π : Mp , a ≺ M, a .

QED(Lemma 7)

6.3. Example 3 β 

We now assume 2ω = ω1 , 2ω1 = ω2 , 2 = β, and cf(β) = ω1 . We again want to give cofinality ω to all regular cardinals τ ∈ [ω2 , β]. It is clear that β will also acquire cofinality ω, since it either already has cofinality ω, or its cofinality lies in [ω2 , β). The simplest way of handling this is to revise the definition of  to: Definition Let u, v ∈ Γ. π : u  v iff the following hold: • π0 : u0 0 v 0 where π 0 = π  Mu0 . • π : Mu ≺ Mv γu -cofinally. Let M = LA β where Lβ [A] = Hβ . As before set N = Hβ + , <, M, . . . . Let L be the language on N with only the core axioms. Exactly as before we prove:

page 174

March 14, 2014

16:18

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

175

Lemma 8 L is consistent. (Note that if N is countable and transitive, π : N ≺ N , π(M ) = M , and ˜, π ˜ M 0 = HωM2 , then if N ˜ is the liftup of N , π  M 0 , then π  M : M ≺ M M ˜ =π ω2 -cofinally, where M ˜ (M ).) We then set P = PL . Exactly as before we get: Lemma 9 P is strongly revisable. Corollary 9.1 P is revisable. Hence L is modest, since it has only the core axioms. Exactly as before we get: Lemma 10 P is subcomplete. Lemma 7 does not go through, however. In fact 2β acquires cardinality ω1 . This follows from the very general theorem: Lemma 11 Let W be an inner model of ZFC and CH. Let Hω1 = HωW1 . β 

Let β > ω1 s.t. 2 = β in W . Suppose that cf(β) = ω and β = ω1 in V. Then card((2β )W ) = ω1 in V. Proof . Let M = LA β where Lβ [A] = Hβ in W . Let f map ω1 onto M in V. Let βi | i < ω ∈ V be cofinal in β. Set: Xα = f  α for α < ω1 . Set: C = {α < ω1 | α = ω1 ∩ Xα ∧ Xα ≺ M ∧ {βi | i < ω} ⊂ Xα }. ∼ For α ∈ C set πα : Mα ↔ Xα , where Mα is transitive. Then Mα ∈ Hω1 . For any B ⊂ β, B ∈ W , there is α ∈ C, s.t. B ∩ βi ∈ Xα for i < ω. Set:  B = {πα−1 (B ∩ βi ) | i < ω}.

Then α, B ∈ Hω1 and B is recoverable from α, B by:  πα (u ∩ B). π ˜ (α, B) = u∈Mα

Thus π ˜ maps a subset of Hω1 onto (P(β))W .

QED(Lemma 11)

6.4. The extended Namba problem Shelah was the first to show that Namba forcing can be iterated without adding reals. If we iterate it out to a strongly inaccessible κ, then κ becomes the new ω2 and arbitrarily large regular cardinals below κ become ω-cofinal. However, many regular cardinals become ω1 -cofinal. The “extended Namba

page 175

March 14, 2014

16:18

176

WSPC 9 x 6: BC9154

jensen

R. Jensen

problem” asks whether, without adding reals, one can make κ become ω2 while giving all of the regular cardinals in the interval (ω1 , κ) cofinality ω. This problem seemed so difficult that at one point we conjectured a provably negative answer in ZFC for all κ. Moti Gitik then disproved this conjecture by constructing a ZFC model in which the extended Namba problem had a positive solution for some κ. His model was a generic extension of a universe containing a supercompact cardinal. Following Gitik’s breakthrough we then obtained a positive solution in ZFC for all κ. It is impossible to give the full proof of that result in these notes, but we shall endeaver to give some account of the methods used. We may assume w.l.o.g. that GCH holds below κ, since we may achieve this by a prior forcing in which all collapsed regular cardinals acquire a cofinality ≥ ω2 . If we then give the surviving regular cardinals in (ω1 , κ) the cofinality ω, the collapsed ones will also become ω-cofinal. It is natural to try to solve this problem by an iteration Bi | i ≤ κ . We ask now what the initial steps of this iteration should look like. We follow the convention that B0 = 2. Thus B1 is the first stage which “does something”. We certainly expect it to give ω2 the cofinality ω without adding reals. By Lemma 11 it follows that ω3 will be collapsed, so ω3 must acquire cofinality ω. But then ω4 is collapsed etc. Thus every ωn must be collapsed with cofinality ω. By Lemma 11, it then follows that ωω+1 is collapsed etc. This chain of implications does not break down until we reach ωω1 . There, however, it does break down, since we can use the P of Example 2 with β = ωω1 . All regular cardinals in (ω1 , ωω1 ) acquire cofinality ω and ωω1 +1 is not collapsed, thus becoming the new ω2 . We take B1  BA(P). We can then repeat the process, getting B2 ⊇ B1 which collapses ωω1 ·2 to ω1 etc. This gives us the first ω stages Bi | i < ω . Our job now is to find an appropriate limit Bω . Since each Bi is subcomplete, the inverse limit B∗ is also subcomplete. However, a bit of reflection shows that B∗ is too small to do the job: At the limit stage ωω1 ·ω will be collapsed to ω1 . Hence by Lemma 11 ω(ω1 ·ω)+1 will be collapsed and hence must acquire cofinality ω etc. Proceeding in this fashion we see that ωω1 ·(ω+1) must be collapsed. Thus our limit algebra must be large, not containing any dense set of size less than ωω1 (ω+1) . At the same time it should have a dense subset of size ωω1 (ω+1) in order that the successor is preserved. It turns out that a limit with the requisite properties can be obtained by a construction rather like that of Example 2. We shall now sketch that construction, but a full verification of its properties is beyond the purview of these notes.

page 176

April 22, 2014

16:42

WSPC 9 x 6: BC9154

Subcomplete Forcing and L-Forcing

jensen

177

Let M 0 = LA γ where γ = ωω1 ω , Lγ [A] = Hγ , and A canonically codes Bi | i < ω . We define Γ0 , Π0 as follows: Γ0 = the set of u = Mu , Bu s.t. u • M = LA γu where Mu models Zermelo set theory and Au canonically codes a sequence Bui | i < ω of complete Boolean algebras in the sense of M with Bui ⊆ Buj (i ≤ j < ω).  • Bu ⊂ Bui s.t. Bu ∩ Bi is Bi -generic over M for i < ω.

i

• sup{δ(Bi ) | i < ω} = β and Bi collapses δ(Bi ) to ω1M for i < ω.

Π0 = the set of π, u, v s.t. u, v ∈ Γ0 , π : Mu ≺ Mv and π  Bu ⊂ Bv . We write π : u 0 v for π, u, v ∈ Π0 . Setting Bui = Bu ∩ Bui , we see that π has a unique extension πi s.t. π i : Mu [Bui ] ≺ Mv [Bvi ] and πi (Bui ) = Bvi .   Set: Mu∗ = Mu [Bui ] and π ∗ = π i . Then π∗ : Mu∗ → Mv∗ cofinally. i

i

u , we see that π is Letting fui be the canonical map of ω1 onto LA δ(Biu ) i i uniquely characterized by: π ◦ fu = fv for i < ω. It follows easily that π = πuv is the unique π : u 0 v and that Γ0 , Π0 is an approximation 0 A system. Now let M = LA β where β = ωω1 (ω+1) , Lβ [A] = Hβ and M = Lγ (γ = ωω1 ·ω ). Set:

Γ = the set of u = Mu , Bu s.t. Mu is smooth and there is γ = γu ∈ Mu u u s.t. u0 = Mu0 , Bu ∈ Γ0 , where Mu = LA and Mu0 = LA γ . β

We then set: Π = the set of π, u, v s.t. u, v ∈ Γ and: • π0 : u0  v 0 where π 0 = π  M 0 . • π : M u ≺ Mv . • Let π : Mu → Mu,v cofinally. Then Mu,v , π is the liftup of Mu , π0 . We again set: π :uv

iff

π, u, v ∈ Π.

Thus Γ, Π is an approximation system which is related to Γ0 , Π0 exactly as in Section 6.2. Again, letting M = LA β be as above, and N = Hβ + , <, M, . . . , we form the language L on N containing only the core axioms. Lemma 12 L is consistent. Proof . Let B∗ = the inverse limit of Bi | i < ω . Then B∗ is subcomplete. Let B ∗ be B∗ -generic. We prove the consistency of L in V[B ∗ ]. Let Bi =

page 177

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

178

B ∗ ∩ Bi , B =



i<ω

Bi . Let H = H(2β )+ in V. Let π : H ≺ H in V[B ∗ ] s.t. H

is countable and transitive. Let: π(N , M , M 0 , Bi | i < ω ) = N, M, M 0 , Bi | i < ω . Set B i = π−1  Bi for i < ω. Since we are working in V[B ∗ ] we may assume that B i is Bi -generic over M for i < ω. Clearly π takes M 0 to M 0 cofinally. Moreover: π  M 0 : M 0 , B 0 M 0 , B . ˜ π ˜,N ˜ , L, ˜ Now let H, ˜ be the liftup of H, π  M 0 . Let: π ˜ (M , N , L) = M ˜ ≺ H with k(L) ˜ = L, it suffices to where π(L) = L. Since there is k : H prove that L˜ is consistent. We claim: ˜ where κ > 2β is regular in V. Claim Hκ , B models L, ˜ be countable. There Proof . The only problematical case is: Let X ⊂ M ˜ ˜ be is u ∈ Γ ∩ Hω1 s.t. u  M , B and X ⊂ rng(π ˜ ). Let Y ≺ H u,M ,B

countable s.t. rng(˜ κ) ∪ X ⊂ Y and whenever Δ ∈ Y is dense in Bi (i < ω), then Δ ∩ B = ∅. Let:  ∼ ˜ , Bi | i < ω . π  : H  ↔ Y, π  (M 0 , M  , Bi | i < ω ) = M 0 , M Set: B  = π −1  Bi , π  = π   M  . ˜ , B . Claim π  : M  , B   M

˜ , it suffices Clearly: π  M 0 : M 0 , B  0 M 0 , B . Since π  : M  ≺ M to show that: If π : M  → M ∗ cofinally, then M ∗ , π  is the liftup of M  , π   M 0 – i.e. that π  takes M  γ  -cofinally to M ∗ wheres γ  =  QED(Lemma 12) (ω1 · ω)M . This follows by the usual argument. The strong revisability lemma reads: Lemma 13 For sufficiently large θ > 2β we have: Let N ∗ = Hθ , M, P, < , . . . . Let p conform to N ∗ and set: N ∗ = N ∗ (N ∗ , p) = H, M , P, <, . . . .  M Let B ⊂ Bi s.t. B ∩ BM is BM i i -generic over M for i < ω. Then i<ω

q = M , B , F p ∈ P.

We must forego the proof of Lemma 13, since it is very long and involves properties of the algebras Bi which we have not developed here. An immediate corollary is: Corollary 13.1 P is revisable, since revisability says that the above holds ∗ when B = B G for a G which is P-generic over N .

page 178

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

179

Since L has only the core axioms, it is then modest. But then we get: Lemma 14 P is subcomplete. We sketch briefly the proof of Lemma 14, which is largely the same as before. Let θ be big enough to verify the subcompleteness of Bi for i < ω. − Let W = LA τ be a ZFC model with Hθ ⊂ W and θ < τ . Let π : W ≺ W where W is countable and full. Let π(θ, P, s) = θ, P, s. Claim There is q ∈ P s.t. if G  q is P-generic, there is σ ∈ V[G] with: (a) σ : W ≺ W (b) σ(θ, P, s) = θ, P, s (c) CγW (rng(σ)) = CγW (rng(π)), where γ = On ∩ M 0 = sup δ(Bi ). i<ω

(d) σ  G ⊂ G.

(Note γ ≤ δ(P), since otherwise γ would not be collapsed.) Let Ω > θ be big enough to verify the strong revisability of P. Set: N ∗ = HΩ , <, M, N, P, W, π, . . . . Let p conform to N ∗ . Set: N ∗ = N ∗ (N ∗ , p) = H  , M  , N  , P , W  , π  , . . . .  (i < ω). Set θ , P , s = π  (θ, P, s). Set γ  = π (γ), where Set: Bi = BM i π(γ) = γ = On ∩ M 0 . Noting that W  is countable and imitating the proof of Section 4, Theorem 2 we get:   Sublemma 14.1 There are σ and B  ⊂ Bi s.t. Bi = B  ∩ Bi is i<ω

Bi -generic over W  for i < ω and:

(a) σ  : W ≺ W  (b) σ  (θ, P, s) = θ , P , s   (c) CγW (rng(σ )) = CγW (rng(π  )) (d) σ   B ⊂ B  , where B = B G . ◦

To get this we successively define σ i , bi ∈ Bi s.t. whenever Bi  bi is P ◦  generic over W  and σi = σ i Bi , then σi satisfies (a)–(c) and: σi  B i ⊂ Bi (where B i = B ∩ Bi ). We ensure hi (bi+1 ) = bi for i < ω. We then successively choose Bi  bi  with: Bi is Bi -generic over W  and Bi ⊃ B for  < i. We set: B  = Bi ◦



i

and let σ  be the ’limit’ of σi = σ i Bi (i < ω) exactly as in the proof of Section 4, Theorem 2. QED(Sublemma 14.1)

page 179

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

R. Jensen

180

By the strong revisability lemma we have: q = M  , B  , F p ∈ P. Let G  q be P-generic. Then πqG ∪ F q extends uniquely to: σ ∗ : N ∗ ≺ N ∗ . Set σ = σ ∗ · σ . It follows by a virtual repetition of previous proofs that σ has the desired properties. QED(Lemma 14) Now let B = BA(P). We define a map μ : ◦

μ(b) = [[ˇb ∈ B]] where

◦

BG = BG



i<ω

Bi → B by:

for all generic G.

Then: (1)

μ is injective. ◦

Proof . It suffices to show: μ(b) = 0 → b = 0. Let b = 0. Then L + b ∈ B is consistent by the proof that L is consistent. Hence there is p ∈ P, b ∈ Bp ◦ s.t. π p (b) = b. Hence p  ˇb ∈ B – i.e. p ∈ μ(b). QED(1) μ  Bi is a complete embedding.      ◦ ˇ bi ∈ B . bi = Proof . μ (2)

i∈X

QED(2)

i∈X

Hence we can take B ⊃



i<ω

∼ Bi s.t. for some k, k : B ↔ B and kμ = id.

B is then a limit of Bi | i < ω which collapses  = ωω1 (ω1 +1) to ω1 while making all regular τ ∈ (ω1 , ) become ω-cofinal. A proof like that of Lemma 7 shows that + is not collapsed, becoming the new ω2 . Hence we apply Example 2 at the next stage to collapse (ω1 ) = the ω1 -th successor of  to ω1 . We continue in this fashion. We define an iteration Bi | i ≤ κ and a sequence i | i ≤ κ as follows: 0 = ω1 , B0 = 2. (ω )

i+1 = i 1 and Bi+1 is constructed using Example 2 so as to collapse all regular τ ∈ (ω1 , i+1 ) without collapsing + i+1 . For limit λ we proceed as follows: Case 1 λ has cofinality ω or has acquired cofinality ω at an earlier stage (i.e. cf(λ) < λ ∧ cf(λ) = ω1 in V). By essentially the above construction we form a limit Bλ which collapses (ω1 )  λ = sup i without collapsing + . i<λ

page 180

March 14, 2014

16:18

WSPC 9 x 6: BC9154

jensen

Subcomplete Forcing and L-Forcing

181

Case 2 Case 1 fails. We set λ = sup i and let Bλ be the direct limit of Bi | i < λ . If cf(λ) = ω1 i<λ

in V, then + λ becomes the new ω2 . Otherwise λ = λ is inaccessible. Using the fact that we took the direct limit stationarily often below λ it follows that Bλ satisfies the λ-chain condition. Hence λ is the new ω2 .  By induction on i we verify that Bi is subcomplete for i ≤ κ, using Section 4, Theorem 4 for Case 2 above. We stress, however, that in order to carry out the induction we must also verify many other properties of the Bi which have not been dealt with here. These include some strong symmetry properties. Given that GCH holds below κ, we can modify the above construction by making selective regular τ ∈ (ω1 , κ) ω1 -cofinal. The set of such τ can be chosen arbitrarily in advance. Hence: Theorem Let κ be inaccessible. Let GCH hold below κ. Let A ⊂ κ. There is a set of conditions P ⊂ Vκ s.t. whenever G is P-generic, then in V[G] we have: • κ is ω2 . • If τ ∈ (ω1 , κ) is regular in V, then cf(τ ) =



ω1 if τ ∈ A, ω if not.

References [ASS] J. Barwise. Admissible Sets and Structures, Perspectives in Math. Logic Vol. 7, Springer Verlag, 1976. [NA] H. Friedman, R. Jensen. A Note on Admissible Ordinals, in: The Syntax and Semantics of Infinitary Languages. Springer Lecture Notes in Math. Vol. 72, 1968. [PR] R. Jensen, C. Karp. Primitive Recursive Set Functions, in Axiomatic Set Theory, AMS Proceedings of Symposia in Pure Math. Vol. XIII, Part 1, 1971. [PF] S. Shelah. Proper and Improper Forcing, Perspectives in Math. Logic, Springer Verlag, 1998. [AS] R. Jensen. Admissible Sets*. [LF] R. Jensen. L-Forcing* [SPSC] R. Jensen. Subproper and Subcomplete Forcing*

page 181

April 22, 2014

16:42

182

WSPC 9 x 6: BC9154

jensen

R. Jensen

[ENP] R. Jensen. The Extended Namba Problem* [ITSC] R. Jensen. Iteration Theorems for Subcomplete and Related Forcings* [DSP] R. Jensen. Dee-Subproper Forcing* [FA] R. Jensen. Forcing Axioms Compatible with CH *

* These handwritten notes can be downloaded from http://www.mathematik.hu-berlin/de/∼raesch/org/jensen.html (or enter ‘Ronald B. Jensen’ in Google).

page 182

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

E-RECURSION 2012∗

Gerald E. Sacks Department of Mathematics, Harvard University, Cambridge, MA 02138, USA [email protected] math.harvard.edu/˜sacks In Memory of S. C. Kleene Premier Recursion Theorist E-recursive versus Σ1 admissible, two competing intuitions of recursion theory. Partial E-recursive functions, E-recursive enumerability and computation trees. Divergence witnesses and reflecting ordinals. Priority arguments and forcing constructions for E-closed structures. Logic on E-closed structures.

Contents 1

2

Introduction 1.1 New intuitions for old 1.2 Failure of the least number operator 1.3 History 1.3.1 Sources 1.4 Intuitions behind E-recursion 1.5 The Normann schemes 1.6 Classical recursion theory Σ1 Admissibility versus E-Recursion definable versus E-recursive 2.1 ΣZF 1 2.2 Equality is E-recursive

185 185 185 185 186 186 187 187 188 188 188

∗ This paper is a recreation of a course given by the author at the Asian Initiative for Infinity (AII) Graduate Summer School held at the National University of Singapore between 20 June 2012 and 17 July 2012. The author thanks Professors C. T. Chong, T. Slaman and H. Woodin for their kind invitation.

183

page 183

April 22, 2014

16:44

ΔZF predicates are E-recursive 0 E-recursive evaluations The natural enumeration of E 2.5.1 Transfinite E-recursion 2.6 E-recursive enumerability 2.7 G¨ odel’s O is not E-recursive Computations 3.1 Computation instructions 3.2 Convergent equals wellfounded 3.3 Length of computations 3.4 Divergence witnesses Gandy Selection 4.1 Existential number quantifiers 4.2 E-recursive Skolem functions E-Recursion in L 5.1 E-closed sets 5.2 Reflection 5.3 Kechris’s basis theorem 5.4 From Gandy to Kechris 5.5 Divergence witness definability sketch 5.5.1 Scheme T 5.5.2 Definition of w by recursion on t 5.6 Divergence witness details 5.7 Σ1 admissibility and divergence 5.8 The divergence-admissibility split 5.9 Relativization and reducibility 5.10 Transitivity of E-reducibility 5.11 Regularity of E-re degrees 5.12 Projecta 5.13 Post’s problem 5.14 Splitting and density Forcing Computations to Converge 6.1 Genericity versus E-closure 6.2 The forcing language 6.3 Forcing relations 6.4 Definability of forcing The Tree of Possibilities 7.1 Effective bounding 7.2 Genericity 7.3 Enumerable forcing relations 7.3.1 An Annoying Assumption (AAF) needed for forcing Countably Closed Forcing 8.0.1 Enumerability 8.1 A general forcing fact Countable Chain Condition Forcing 9.1 Less-than-ω1 selection 2.3 2.4 2.5

4

5

6

7

8

9

New˙Erecursion

G. E. Sacks

184

3

WSPC 9 x 6: BC9154

189 189 190 190 191 192 192 192 193 193 193 194 195 195 196 196 197 197 198 198 199 199 199 200 200 201 201 202 203 204 205 205 205 205 206 207 207 207 208 208 209 209 210 210 211 211

page 184

April 22, 2014

16:44

WSPC 9 x 6: BC9154

E-Recursion 2012

9.2 Effectiveness of c.c.c. forcing 10 Selection 10.1 Grilliot selection 10.2 Harrington-MacQueen selection 10.3 Selection and admissibility 10.3.1 Moschovakis selection 11 van de Wiel’s Theorem 12 Logic on E-Closed Sets 12.1 Deductions 12.2 Completeness References

New˙Erecursion

185

211 212 212 212 213 213 214 215 215 215 216

1. Introduction 1.1. New intuitions for old The phrase heard most often in classical recursion theory is {e}(n), where e is a code number for a finite instruction derived from schemes and n is a nonnegative integer. The corresponding phrase in E-recursion theory is {e}(x), where e is again a code number for a finite instruction, but x is an arbitrary set. To put it bluntly, x ∈ V . The transition from classical recursion to E-recursion is more than a long jump from HF , the set of hereditarily finite sets, to V , the class of all sets. The move leaves behind some comfortable intuitive notions of computability. For example there exists a function that fails to be partial E-recursive despite the E-recursiveness of its graph. The theorems of E-recursion foster the development of new intuitions. By contrast Σ1 admissible recursion theory, in particular α-recursion theory, greatly extends classical notions of computability without losing sight of them. 1.2. Failure of the least number operator In classical recursion theory the proof that a function is partial recursive, if its graph is recursively enumerable, is an application of the least number operator. Suppose f (n) is defined. To evaluate f (n) enumerate the graph of f until an ordered pair of the form < n, y > is found. In E-recursion theory the idea of enumerating all computations still makes sense but unbounded search is no longer an effective procedure. 1.3. History Objects of type 0 are non-negative integers. An object of type n + 1 is a set of objects of type n. Kleene [9], [10] defined {e}(x), the e-th partial

page 185

April 22, 2014

16:44

186

WSPC 9 x 6: BC9154

New˙Erecursion

G. E. Sacks

recursive function for x an object of finite type. He defined of all non-empty objects of type n, and proved:

n+1

E, the set

∀x ⊆ ω[x is recursive in 2 E ←→ x ∈ HY P ]. The type n + 1 object, n+1 E, is equivalent to the equality predicate for objects of type n. In Kleene’s theory equality is not recursive when n > 0. But he did investigate what was recursive in 2 E and 3 E. E-recursion extended his ideas from objects of finite type to arbitrary sets with one major change: equality is E-recursive. The initial ground breaking results were due to Gandy, Moschovakis [12] and Normann. The schemes defining E-recursion were introduced by Normann, but discovered independently by Moschovakis who drew attention to the concept of divergence witness, whose great importance in the study of computation higher up was not anticipated in classical recursion theory or in Σ1 admissible recursion theory. Further results were obtained by Fenstad, Griffor, Grilliot, Harrington, Kechris, MacQueen, Moldstad, Shoenfield and Slaman. Back in the 70’s there was a golden age of E-recursion with centers in Cambridge Mass, Oslo and UCLA.

1.3.1. Sources The most complete source for E-recursion is Sacks [21]. The most polished sources are Moschovakis [12] and Slaman [22], [23]. 1.4. Intuitions behind E-recursion Let x and y be arbitrary sets. The predicate, x = y, is E-recursive. Hence infinitely long computations are necessary. Let {e}(x) be the e-th partial E-recursive function; e is a code number, a non-negative integer that encodes one of finitely many E-recursion schemes; the result (if there is one) of applying that scheme to x is the value of {e}(x). The function f (e, x) = {e}(x) is partial E-recursive and its existence is required by the only non-trivial E-recursion scheme.

page 186

April 22, 2014

16:44

WSPC 9 x 6: BC9154

E-Recursion 2012

New˙Erecursion

187

1.5. The Normann schemes → Let − x denote x1 , . . . , xn . < 1, n, i > is a positive integer that effectively encodes 1, n, i. There are seven Normann schemes. The first four are needed to show fundamental set theoretic notions are E-recursive. → (1) projection {e}(− x ) = xi if e =< 1, n, i >. → − if e =< 2, n, i, j >. (2) difference {e}( x ) = xi − xj → (3) pairing {e}(− x ) = {xi , xj } if e =< 3, n, i, j >. → (4) union {e}(− x ) = ∪{y | y ∈ x } if e =< 4, n, i, j >. 1

Scheme (5) is the source of infinitely long computations. If x1 is infinite, → then the computation of {e}(− x ) leads immediately to infinitely many subcomputations. (5) E-recursive bounding → {e}(− x ) = {{c}(y, x2 , . . . , xn ) | y ∈ x1 } if e =< 5, n, c >. (6) composition → → → x ), . . . , {dm )(− x )) if e =< 6, n, m, c, d1 , . . . , dm >. {e}(− x ) = {c}({d1 }(− (7) enumeration − → → {e}(c, → x ,− y ) = {c}(− x ) if e =< 7, n, m >. The enumeration scheme, it will be seen, does all the work. 1.6. Classical recursion theory A set y is transitive iff (∀x ∈ y)(x ⊆ y). Let tc(z) denote the transitive closure of z, the least transitive w ⊇ z. Intuitively, tc(z) is z ∪ (∪z) ∪ (∪(∪z)) . . . ; tc is an E-recursive function by Proposition 9. HF is the set of hereditarily finite sets. x ∈ HF ←→ tc({x}) is finite. Proposition 1. Let f be a function whose domain and range are subsets of ω. Then (i) ←→ (ii) ←→ (iii). (i) f is partial recursive. (ii) f is partial E-recursive. (iii) f is ΣZF definable over < HF, ∈ >. 1 The two major extensions of classical recursion theory, Σ1 admissibility and E-recursion, differ in spirit and on many initial segments of G¨odel’s L, but agree on ω. The fact that two intuitively different formulations of recursion coincide on ω adds to the evidence for Church’s Thesis.

page 187

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

G. E. Sacks

188

The classical partial recursive functions can be derived from trivial finitary schemes and one non-trivial scheme, enumeration. The derivation begins with the enumeration scheme, proceeds to the fixed point theorem, then to definition by recursion, and finally to the least number operator. 2. Σ1 Admissibility versus E-Recursion 2.1. ΣZF definable versus E-recursive 1 iff all of its quantifiers are A formula in the language of set theory is ΔZF 0 bounded; a formula is ΣZF iff it is of the form ∃xG, where G is ΔZF 1 0 . Let A be a transitive set. A is Σ1 admissible iff A is closed under the operations of pairing and union; A satisfies ΔZF comprehension, for every ΔZF F (w), 0 0 ∀x∃z∀w[w ∈ z ←→ w ∈ x ∧ F (w)] holds in A; bounding, for every ΔZF G(x, y), A satisfies ΔZF 0 0 [∨x ∈ z∃yG(x, y) −→ ∃v∀x ∈ z∃y ∈ vG(x, y)] holds in A. The formulas F (w) and G(x, y) are boldface; that is, they may contain parameters from A. From now on all formulas of ZF are boldface. comprehension and ΣZF Note that a Σ1 admissible A satisfies ΔZF 1 1 bounding. A set x is: A-finite iff x ∈ A; A-recursive iff x is a ΔZF 1 definable subset of A; A-recursively enumerable iff x is a ΣZF definable 1 subset of A. A function is partial A-recursive iff its graph is a ΣZF 1 definable subset of A. The range of an A-recursive function defined on and restricted to an Afinite set is A-finite. The intersection of an A-recursive set and an A-finite set is A-finite, but the intersection of an A-re set and an A-finite set need not be A-finite. A set is A-recursive iff it and its complement relative to A are A-re. According to Lemma 1 every partial E-rec function is ΣZF 1 . The converse is false by Proposition 7. 2.2. Equality is E-recursive The representing function f (x) of a predicate P (x) is defined by: ∀x[f (x) = 1 ←→ P (x)] ∀x[f (x) = 1 ∨ f (x) = 0]. A predicate is E-recursive iff its representing function is E-recursive.

page 188

April 22, 2014

16:44

WSPC 9 x 6: BC9154

E-Recursion 2012

New˙Erecursion

189

Proposition 2. Equality is E-recursive. Proof. Let 0 denote x − x, and 1 denote {0}. t(y) = 0 f (z) = {t(y) | y ∈ z} z = 0 ←→ f (z) = 0 z = 0 ←→ f (z) = 1 g(z) = 1 − f (z) represents z = 0. g((x − y) ∪ (y − x)) represents x = y. 2.3. ΔZF predicates are E-recursive 0 Proposition 3. If G(x, y) is E-recursive, then (∃y ∈ x)G(x, y) is Erecursive. Proof. Let t(x, y) = 0 if G(x, y) and = 1 otherwise. Define f (x) = {t(x, y) | y ∈ x ∧ G(x, y)]. f represents (∃y ∈ x)G(x, y). Corollary 1. If P (x) is ΔZF 0 , then P (x) is E-recursive. Proof. By induction on the logical complexity of P (x). It follows from Corollary 1 that a ΣZF formula Q(x) can be put in the 1 form ∃yR(x, y), where R is E-recursive. It follows from Proposition 6 there exists an E-recursive R(x, y) such that ∃yR(x, y) is not E-re. 2.4. E-recursive evaluations → E is the class of E-recursive evaluations. A tuple < e, − x , y > is put in E → iff y is a value for {e}(− x ) determined by the Normann schemes. E is the range of a Σ1 transfinite recursion on the ordinals. → {e}(− x ) ↓ y (converges to y) iff → < e, − x , y >∈ E. f is a partial E-recursive function iff → → → ∃e∀− x f (− x ) = {e}(− x ).

page 189

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

G. E. Sacks

190

2.5. The natural enumeration of E → (− x is x , . . . , x ) 1

n

Stage σ = 0: → < e, < − x , xi >> is put in E0 if e =< 1, n, i >. Schemes (2), (3) and (4) are treated similarly. Stage σ > 0: E<σ = ∪{Eγ | γ < σ}. − < e, → x , z > is put in Eσ if e =< 5, n, c >, ∀y ∈ x1 ∃w[< c, < y, x2 , . . . , xn >, w >∈ E<σ ], z = {w | ∃y ∈ x1 [< c, < y, x2 , . . . , xn >, w >∈ E<σ ]}. Schemes (5) and (6) are treated similarly. E = ∪{Eσ | σ ∈ ORD}. The enumeration of E lend itself to a straightforward induction that shows {e}(x) converges to at most one y: ∃ ≤1 y < e, < x >, y >∈ E. Thus for any e, λx | {e}(x) is a partial E-recursive function. transfinite recursion on the ordinals, so The enumeration of E is a ΣZF 1 the graph of any partial E-recursive function is ΣZF 1 . The converse is false according to Proposition 7. 2.5.1. Transfinite E-recursion Proposition 4. (fixed point theorem) Let f be a partial E-recursive function defined for all n ∈ ω. Then there exists a c ∈ ω such that {f (c)} = {c}. Proof. The enumeration scheme leads to a partial E-recursive function such that ∀e t(e) ↓ ∧{t(e)} = {{e}(e)}. There is a d such that {d} = f ◦ t. Then {t(d)} = {{d}(d)} = {f (t(d))}. Let c be t(d). Then {f (c)} = {c}. Proposition 5. Let I be a total E-recursive function. Then there exists a partial E-recursive function f : ORD −→ V such that

page 190

April 22, 2014

16:44

WSPC 9 x 6: BC9154

E-Recursion 2012

New˙Erecursion

191

∀γ f (γ) ↓ and f (γ) = I(f  γ). [(f  γ) is the graph of f restricted to γ.] Proof. There exists a recursive g such that ∀e [g(e) ↓ ∧ ∀γ {g(e)}(γ) = I({e}  γ)]. By Proposition 5 ∃c g(c) = {c}. By induction on γ, {c}(γ) ↓ and {c}(γ) = I({c}  γ). Then f is {c}. G¨ odel’s L, the class of constructible sets is defined by a ΣZF transfinite 1 definable class. recursion, and consequently is a ΣZF 1 L(0) = ∅. L(γ + 1) = F od((L(γ)). (F od(x) is the set of first order definable subsets of x.) L(λ) = ∪(L(γ) | γ < λ). (λ is a limit.) L = ∪(L(γ) | γ ∈ ORD). It follows from Corollary 1 and Proposition 5 that L(γ) is an E-recursive function of γ. Nonetheless according to Proposition 6, L is not E-recursively enumerable if V = L. 2.6. E-recursive enumerability Definition 1. A is E-recursively enumerable in y (E-re in y) iff ∃e A = {x | {e}{x, y} ↓}. Definition 2. A is E-recursively enumerable (E-re) iff ∃e A = {x | {e}{x} ↓}. According to Gandy selection (Section 4), if A and its complement are E-re, then A is E-recursive. The proof from classical recursion does not work in E-recursion because it depends on the least number operator. Proposition 6. If V = L, then L is not E-re. Proof. Suppose L = {x | {e}(x) ↓}. Let b ∈ V − L. Then {e}(b) ↑. according to Lemma 2. By Levy The predicate, {e}(x) ↑, is lightface ΣZF 1 absoluteness, ∃x ∈ L such that {e}(x} ↑.

page 191

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

G. E. Sacks

192

(Levy absoluteness: assume F is a sentence in the language of ZF whose parameters belong to L(ω1L ); if V |= F , then L |= F .) A stronger result is available. Theorem 1. If V = L, then L is not E-re in any c ∈ L. The proof is a combination of forcing and iterated Σn hull formation. 2.7. G¨ odel’s O is not E-recursive The function O : L −→ Ord is defined by: O(x) = (least γ) x ∈ L(γ). Proposition 7. The function O is not the restriction of a partial Erecursive function to L. Proof. Suppose ∀x x ∈ L −→ [{e}(x) ↓ ∧ {e}(x) = O(x)]. Let α = ω be countable in L and Σ1 admissible. There exists a b ⊆ ω such that b is generic over L(α) and b is a member of L. Thus L(α, b) is Σ1 admissible and b ∈ L(α). But {e}(b) ↓, so {e}(b) ∈ L(α) by Proposition 10 and Remark 1, hence b ∈ L(α). The above proof is easily stretched to show O is not the restriction to L of a function partial E-re in c for any c ∈ L. definable. Note that the graph of O is lightface ΣZF 1 3. Computations 3.1. Computation instructions → → A computation instruction is an (n + 1)-tuple < e, − x >. (− x is x1 , . . . , xn .) → − < e, − x > is the top node of the computation tree T <e,→ x > . Every node − of T<e,→ other than the top node is an immediate subcomputation x> instruction (isi) of the node above it. − (i) If e =< 1, n, i > then < e, → x > has no immediate subcomputations. < 2, n, i, j >, < 3, n, i, j > and < 4, n, c > are treated similarly. − x > for (ii) If e =< 5, n, c >, then < c, y, x2 , . . . , xn > is an isi of < e, → every y ∈ x1 . < 6, n, m, c, d1 , . . . , dm > and < 6, n, m > are treated similarly. (iii) If e is not an index of a Normann scheme or n is not the correct number → − of arguments, then < e, − x > is an isi of < e, → x >.

page 192

April 22, 2014

16:44

WSPC 9 x 6: BC9154

E-Recursion 2012

New˙Erecursion

193

3.2. Convergent equals wellfounded Define b >i a by: a is an isi of b. Let >U be the transitive closure of >i : b >U a iff ∃n, ∃c1 , . . . , cn such that b = c1 >i ... >i cn = a. The relation b >i a is E-re, but not E-recursive. The relation b >U a is ΣZF 1 , but not E-re. → − − T<e,→ x > is an upside down tree consisting of < e, x > and the nodes → − − of >U below < e, x >. To say T<e,→ x > is wellfounded is to say it has no infinite descending branch. → − is wellfounded. Lemma 1. {e}(− x ) ↓←→ T → <e, x >

− Proof. Suppose T<e,→ x > is wellfounded. Then clause (iii) of the definition of isi in Subsection 3.1 implies e is an index. A transfinite induction on → − → − − T<e,→ x > shows: if < c, z > is a node, then {c}( z ) ↓. → Now suppose {e}(− x ) ↓. Then < e, x, {e}(x) > was put in E at stage → σ of the natural enumeration of E. Proceed by induction on E. If {c}(− z) → − → − → − is an immed. subcomp. instruc. of {e}( x ), then < c, z , {c}( z ) > was − − put in E prior to stage σ, and so T is wellfounded. Hence T<e,→ x > is wellfounded.

3.3. Length of computations Proposition 8. There exists a partial E-recursive function g such that for → all e and − x: → → {e}(− x ) ↓ ←→ g(e, − x)↓. → → − {e}(− x ) ↓ ←→ g(e, − x ) = T<e,→ x >.

− − Suppose T<e,→ x > is wellfounded. Then each node b of T<e,→ x > has an ordinal rank r(b) defined by recursion. If b is terminal, then r(b) = 0. Otherwise r(b) is the least ordinal greater than r(c) for every node c immediately → − − below b. Let | T<e,→ x > | be r({e}( x )). → − − | if T → is wellfounded, and = ∞ Definition 3. | {e}(− x ) |=| T → <e, x >

<e, x >

otherwise. 3.4. Divergence witnesses → − Suppose {e}(− x ) ↑. Then T<e,→ x > is not wellfounded. A witness w to the → − − divergence of {e}( x ) is an infinite descending branch of T<e,→ x >.

page 193

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

G. E. Sacks

194

w is a function with domain ω. → w(0) =< e, − x >. w(n + 1) is an immed. subcomp. instruc. of w(n). → Lemma 2. (Moschovakis) The predicate, {e}(− x ) ↑, is ΣZF 1 . Lemma 3. (Moschovakis) There exists a partial E-recursive function g → → such that for all e, − x , d, − y: → − → − → → [{e}( x ) ↓ ∨{d}( y ) ↓] ←→ g(e, − x , d, − y)↓. → → → → → → g(e, − x , d, − y ) ↓−→ g(e, − x , d, − y ) = min(| {e}(− x ) |, | {d}(− x ) |). − → Proof. (A sketch.) g(e, → x , d, − y ) is defined by transfinite E-recursion on → → min(| {e}(− x ) |, | {d}(− y ) |). The essence of the recursion is expressed by: min(| u |, | v |) = max{min(| a |, | b |) | a
page 194

April 23, 2014

17:32

WSPC 9 x 6: BC9154

New˙Erecursion

E-Recursion 2012

195

The definition of f makes use of Lemma 3 with respect to the partial E-recursiveness of the predicate min(| {c}(e, k + 1) |, | {e}}(k) |). By the fixed point theorem ∃d f (d, e, k) = {d}(e, k). Let h be {d}. h(e, k} = h(e, k + 1) + 1 if h(e, k + 1) ↓ ∧ | h(e, k + 1) |≤| {e}}(k) |; h(e, k} = 0 if | {e}}(k) |<| h(e, k + 1) | . ∀k [h(e, k + 1) ↓−→ h(e, k) ↓] and ∀k [{e}(k) ↓−→ h(e, k) ↓]. Hence (1) ∃k{e}(k) ↓−→ h(e, 0) ↓ and (2) h(e, 0) ↓−→ ∃k | {e}}(k) |<| h(e, k + 1) |. Fix k and suppose {e}(k) ↓; then (1), (2) imply (3) h(e, k) = 0 and | {e}}(k) |<| h(e, k + 1) |. Let k0 be the least k that satisfies (3); Then h(e, k0 ) = 0 and ∀i < k0 h(e, i) = h(e, i + 1) + 1. So h(e, 0) = k0 . Note that k0 is the least k such that | {e}}(k) |= min{| {e}(n) || n ∈ ω}. − → Define g(e, → x ) = h(e, − x , 0) (= h(e, 0)). 4.1. Existential number quantifiers Let e be a variable restricted to ω. Lemma 4. If P (e, x) is a E-re predicate, then so is ∃eP (e, x). Proof. By Gandy selection there exists a partial E-recursive function g(x) such that ∃eP (e, x) ←→ g(x) ↓; g(x) ↓−→ g(x) ∈ ω ∨ P (g(x), x). 4.2. E-recursive Skolem functions Say w is E-recursive in z (w ≤E z) iff ∃e[{e}(z) ↓ ∧w = {e}(z)]. The predicate w ≤E z is E-re.

page 195

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

G. E. Sacks

196

Lemma 5. Suppose the predicate P (x, y) is E-re and (∀x ∈ z)(∃y ≤E x)P (x, y). Then there exists a partial E-recursive function f such that (∀x ∈ z)[f (x) ↓ ∧ P (x, f (x)]. Proof. By Gandy selection there exists a partial E-recursive function g(x) such that (∀x ∈ z)[g(x) ↓ ∧ g(x) ∈ ω ∧ {g(x)}(x) ↓ ∧P (x, {g(x)}(x))]. 5. E-Recursion in L 5.1. E-closed sets → Definition 4. Assume b is a transitive set; b is E-closed iff {e}(− x) ∈ b → − → − whenever x ∈ b and {e}( x ) ↓. → − Definition 5. κx = sup{γ | ∃e∃ − a ∈ tc(x) γ = {e}(x, → a )}. − (tc(x) is the transitive closure of x. → a is a1 , . . . , an .) Definition 6. E(x) is the least transitive E-closed set b such that x ∈ b. Proposition 9. tc is an E-recursive function. Problem 1. Proof. By transfinite E-recursion on rank. Proposition 10. E(x) = L(κx , tc({x})). Proof. Suppose δ < κx . By induction on δ, L(δ, tc({x})) ≤E x, L(δ, tc({x})) ∈ E(x), and L(δ, tc({x})) ⊆ E(x). Hence L(κx , tc({x})) ⊆ E(x). → − − → Suppose − a ∈ E(x) and {e}(x, → a ) ↓. Then | {e}(x, → a ) |≤E x, − a , so → − − is first order definable over | {e}(x, a ) |< κx . By induction T<e,x,→ a> → − x L(| {e}(x, a ) |, tc({x})). Hence E(x) ⊆ L(κ , tc({x})). Let Ad1 (x) be the least Σ1 admissible set with x as a member; Ad1 (x) will be of the form L(α, tc({x}). Remark 1. E(x) ⊆ Ad1 (x).

page 196

April 22, 2014

18:46

WSPC 9 x 6: BC9154

New˙Erecursion

E-Recursion 2012

197

Remark 2. E(ω) = L(ω1CK ), hence Σ1 admissible. (ω1CK is the least infinite ordinal not the order type of a recursive wellordering of ω.) Remark 3. E(ω1 ) is not Σ1 admissible, but E(ωω ) is Σ1 admissible (Moschovakis). 5.2. Reflection Definition 7. κx0 = sup{γ | γ ≤E x}. Definition 8. δ is x-reflecting iff L(δ, tc({x})) |= F implies L(κx 0 , tc({x})) |= F ZF for every Σ1 sentence F whose only parameter is x. Proposition 11. If a ∈ tc(x) and δ is < x, a >-reflecting, then δ ≤ κx . Proof. Suppose δ > κx . Then E(< x, a >) = E(x) ∈ L(δ, tc({x})). x,a So E(< x, a >) ∈ L(κx,a 0 , tc({x}), an impossibility since L(κ0 , tc({x}) ⊆ E(< x, a >). Definition 9. κx r is the greatest x-reflecting ordinal. Corollary 2. If a ∈ tc(x), then κx,a ≤ κx . r 5.3. Kechris’s basis theorem Kechris’s result is needed to establish a Harringtonesque connection between divergence witnesses and reflection in Subsection 5.5.2. Proposition 12. The predicate, δ is not x-reflecting, with δ as its only free variable and x as its sole parameter, is E-rec in x. sentence with sole parameter x such that Proof. Let F be a ΣZF 1 L(κxr + 1, tc({x})) |= F and L(κxr , tc({x})) |= ¬F . (Note that the choice of F depends on x. Slaman has shown there is no way to choose F uniformly in x.) Then δ is not x-reflecting ←→ L(δ, tc({x})) |= F . Theorem 3. Assume y ≤E x, A is E-re in x, and (y − A) = ∅. x Then ∃b b ∈ (y − A) ∧ κx,b r ≤ κr .

page 197

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

G. E. Sacks

198

Proof. Let F be ΣZF with sole parameter x. Then F reflects down from 1 x,b to κ . Thus κx,b r 0 x x,b x κx,b 0 ≤ κr −→ κr ≤ κr . x So it suffices to find a b ∈ (y − A) such that κx,b 0 ≤ κr . Suppose there is no such b. Then

y ⊆ A ∪ {b | κxr < κx,b 0 }. By Proposition 12 the predicate, κxr < κx,b 0 , is E-re in x because it is equivalent to ∃δ δ ≤E x, b ∧ δ is not x-reflecting. It follows that ∀b ∈ y ∃δb ≤E x, b such that (i) or (ii) holds. (i) δb is the length of a computation that puts b in A. (ii) δb is not x-reflecting. By Gandy selection δb is a partial E-rec function of b (with x as a parameter) convergent for all b ∈ y, hence bounded on y by some δx ≤E x. If some b ∈ y satisfies (ii) then it satisfies δb > κxr > δ x . Hence (ii) is false for all b ∈ y and so (y − A) = ∅. 5.4. From Gandy to Kechris Definition 10. For z ⊆ ω: δ is recursive in z iff δ is finite or δ is the order type of a recursive wellordering of ω. Kechris: Suppose y ≤E x, A is E-re in x, and (y − A) = ∅. x Then ∃b b ∈ (y − A) ∧ κx,b r ≤ κr .

Gandy: Suppose b, x ⊆ ω, ∅ = y ⊆ 2ω , and y is Σ11 with parameter x. Then ∃b b ∈ y ∧ ω1x,b = ω1x . Assume z ⊆ ω: ω1z is the least ordinal not recursive in z; ω1z = κz0 = κzr ; for all y ⊆ 2ω : y is Σ11 with parameter z iff (2ω − y) is E-re in z. Thus Kechris’s basis theorem can be viewed as a generalization of Gandy’s. 5.5. Divergence witness definability sketch Suppose {e}(x) ↑. A divergence witness for {e}(x) is an infinitely long branch w of T<e.x> .

page 198

April 22, 2014

16:44

WSPC 9 x 6: BC9154

E-Recursion 2012

New˙Erecursion

199

w = {< et , xt >| t ∈ ω}, e0 = e, and x0 = x. < et+1 , xt+1 > is an immediate subcomputation instruction (isi) of < et , xt >. 5.5.1. Scheme T It is safe to pretend in inductive arguments and recursive definitions involving computations that there is only one computation scheme, namely scheme T , a derived scheme that incorporates the principal features of the Normann schemes. The code number of scheme T is < 2m · 3n >. The isi’s of < 2m · 3n , x > are: < m, x >; < n, y > iff {m}(x) ↓ and y ∈ {m}(x). 5.5.2. Definition of w by recursion on t Assume et = < 2mt · 3nt > and {et }(xt ) ↑. If {mt }(xt ) ↑, then < et+1 , xt+1 > = < mt , xt >. If {mt }(xt ) ↓ then et+1 = nt and xt+1 ∈ {mt }(xt ). (x0 = x.) The choice of xt+1 is guided by Kechris’s basis theorem to insure that κxr 0 ,...,xt+1 ≤ κxr

(x0 = x)

and that consequently w will be first order definable over L(κxr , tc({x})). It turns out that xt+1 is the “least” member of {mt }(xt ) and w is the “left-most” infinite branch of T<e,x> . Assume there is a wellordering of x E-recursive in x. 5.6. Divergence witness details Assume xt ∈ L(κxr , tc({x})), et = < 2mt · 3nt >, {et }(xt ) ↑, and κxr 0 ,...,xt ≤ κxr . Then κx0 t ≤ κxr . If {mt }(xt ) ↓, then T<mt,xt > ∈ L(κx0 t , tc({x})). Hence examination of L(κxr , tc({x})) reveals whether or not {mt }(xt ) converges; if it diverges, then < et+1 , xt+1 >=< mt , xt >. Assume {mt }(xt ) ↓ Define et+1 = nt . The assumed wellordering x Erecursive in x induces a wellordering v of {mt }(xt ) E-recursive in x. Define xt+1 = v-least u[u ∈ {mt }(xt )∧ | {nt }(u) ≥ κxr |]. It must be seen that {nt }(xt+1 ) ↑ and κrx0 ,...,xt +1 ≤ κxr 0 ,...,xt . By Kechris’s basis theorem there is a z such that z ∈ {m}(xt ) ∧ {nt }(z) ↑ ∧ (0) κxr 0 ,...,xt ,z ≤ κxr 0 ,...,xt .

page 199

April 23, 2014

17:32

WSPC 9 x 6: BC9154

New˙Erecursion

G. E. Sacks

200

Let z0 be the v-least such z. Only z0 = xt+1 remains to be shown: {nt }(z0 ) ↑, so xt+1 ≤v z0 . ∀z κxr 0 ,...,xt . As in the proof of Kechris’s basis theorem, (1) is equivalent to (2) κ0x0 ,...,xt ,z > κxr 0 ,...,xt , and (2) is equivalent to ∃e {e}(x0 , . . . , xt , z) ↓ ∧ {e}(x0 , . . . , xt , z) > κxr 0 ,...,xt . By Gandy selection there is a partial E-recursive function f (x0 , . . . , xt , z) defined for all z κxr 0 ,...,xt . Let γ = sup{f (x0 , . . . , xt , z) | z
5.8. The divergence-admissibility split Assume L(κ) is E-closed. Definition 11. L(κ) admits divergence witnesses iff ∀e∀x ∈ L(κ); if {c}(x) ↑, then ∃w ∈ L(κ) that witnesses the divergence of {c}(x).

page 200

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

E-Recursion 2012

201

Theorem 4. (a) ←→ (b). (a) L(κ) does not admit divergence witnesses. L(κ) (b) L(κ) is Σ1 admissible and for all A ⊆ L(κ), A is Σ1 iff A is E-re on L(κ). If (a) is false then L(κ), as will be seen, is a suitable platform for Erecursion theory. The presence of divergence witnesses is just what is needed to develop unfamiliar proofs of familiar statements about degrees of E-re sets. If (a) is true, then the fundamental concepts of Σ1 recursion (i.e. alpha recursion) and E-recursion coincide. The proof of Theorem 4 is in the same spirit as that of Lemma 6. 5.9. Relativization and reducibility Relativizing E-recursion to B means adding the scheme → x)=B∩x (e =< 8, n, i >) {e}B (− i

to the seven Normann schemes. Assume for the remainder of this Subsection that L(κ) is E-closed. Definition 12. Assume A, B ⊆ L(κ). A is E-reducible to B (A ≤κ B) iff ∃e∃p ∈ L(κ) ∀x ∈ L(κ) : {e}B (x, p) ↓, T<e,x,p;B> ∈ L(κ), and [x ∈ A ←→ {e}B (, p) = 1]. Definition 13. A is E-recursively enumerable on L(κ) iff ∃e∃p ∈ L(κ) A = {x | x ∈ L(κ) ∧ {e}(x, p) ↓}. E-reducibility is inspired by Turing reducibility. It is transitive for sets E-re on an E-closed L(κ), but fails to be transitive in general. 5.10. Transitivity of E-reducibility Assume L(κ) is E-closed throughout this subsection. Definition 14. D ⊆ L(κ) is subgeneric iff ∀e ∀x ∈ L(κ) {e}D (x) ↓−→ {e}D (x) ∈ L(κ). If C is subgeneric and B ≤κ C, then B is subgeneric. Definition 15. D ⊆ L(κ) is regular iff ∀z ∈ L(κ) (D ∩ z) ∈ L(κ). Note that subgenericity implies regularity.

page 201

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

G. E. Sacks

202

Lemma 7. Suppose A, B, C ⊆ L(κ) are E-re on L(κ). If A ≤κ B and B ≤κ C, then A ≤κ C. Proof. If C is subgeneric, then A ≤κ C by composition. Assume C is not subgeneric. Suppose (1) L(κ) = E(x) for some set x of ordinals. If E(x) is Σ1 admissible, then by Theorem 6 ≤κ defined for E-recursion agrees with its Σ1 admissible recursion counterpart and hence is transitive. If E(x) is not Σ1 admissible, or if (1) is false, then E(x) admits divergence witnesses. Thus it is safe to assume L(κ) admits divergence witnesses. Assume C is regular. Then every computation relative to C of height less than κ belongs to L(κ). Since C is not subgeneric there is a computation relative to C of height κ. Then ∃e∃y ∈ L(κ) ∩ 2κ : (∀x ∈ y) T<e,x;C> ∈ L(κ), and κ = sup{| {e}(x) || x ∈ y}; suppose ∃p ∈ L(κ) ∀z ∈ L(κ) [z ∈ A ←→ {d}(z, p) ↓]; then z ∈ / A iff ∃x∃w (2) x ∈ y ∧ w ∈ L(| {e}C (x) |) ∧ w witnesses {d}(z, p) ↑]. It follows from (2) and the regularity of C that A ≤κ C. The least x satisfying (2) is computable from C by effective transfinite recursion. Assume C is not regular. Suppose ∃q ∈ L(κ) ∀x ∈ L(κ) x ∈ C ←→ {f }(x, q) ↓. Choose y ∈ L(κ) so that (y ∩ C) ∈ / L(κ). Then κ = sup{| {f }(x, q) || x ∈ (y ∩ C)}. Now proceed as before (when C was regular) to show A ≤κ C. 5.11. Regularity of E-re degrees Assume L(κ) is E-closed. Definition 16. Suppose A, B ⊆ L(κ) are E-re on L(κ). A and B are of the same degree (A ≡L(κ) B) iff each is E-reducible to the other. By Lemma 7, ≡L(κ) is an equivalence relation. Definition 17. Suppose B is E-re on L(κ). B is complete iff every set E-re on L(κ) is E-reducible to B. Theorem 5. Suppose C ⊆ L(κ) is E-re on L(κ). Then there exists a regular B ⊆ L(κ), E-re on L(κ), such that B ≡L(κ) C.

page 202

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

E-Recursion 2012

203

Proof. (sketch) If L(κ) is not of the form E(x) for some x ∈ L(κ), then C is regular. Assume L(κ) = E(x). If E(x) is Σ1 admissible, then the “regular sets theorem” of α-recursion theory yields the desired B. Assume that E(x) is Σ1 admissible and C is not regular. As in the proof of Lemma 7, C is complete. Thus there is a computation relative to C of height κ; that computation is used to define B. 5.12. Projecta Assume L(κ) is E-closed. Definition 18. Suppose domain(f ) ⊆ κ. f is partial E-recursive on L(κ) iff ∃e p ∈ L(κ) ∀b < κ f (b) = c ←→ {e}(b, p) = c. Definition 19. Suppose γ ≤ κ, domain f ⊆ γ, and range f = L(κ). f is a partial E-recursive-on-L(κ) map of γ onto L(κ) iff f is partial E-recursive on L(κ). Definition 20. ρ = least γ ≤ κ ∃f f is a partial E-rec-on-L(κ) map of γ onto L(κ). Definition 21. η = least γ ≤ κ ∃A A ∈ (2γ − κ) and A is E-re on L(κ). Lemma 8. η = ρ. Proof. Let f be a be a partial E-recursive-on-L(κ) map of ρ onto L(κ). If ρ < η, then κ ∈ L(κ). To show η ≥ κ, fix γ < ρ and let A ⊆ γ be E-re in τ ∈ κ via e. Let g be a universal partial E-recursive function. g[x] = {{c}(z1 , . . . , zn ) | {c}(z1 , . . . , zn ) ↓; c, n ∈ ω; zi ∈ x}. Let H = g[γ ∪ {τ }]. By induction on complexity, H 0 L(κ) : Every ΔZF 0 sentence with parameters in H is true in L(κ) if true in H. H is isomorphic to a transitive set H0 via a collapsing map t. (∀z ∈ H)[O(z) ∈ H]. (O(z) = least δ[z ∈ L(δ + 1) − L(δ)].) Hence H0 = L(β) for some β < κ, and L(β) = g[γ ∪ {t(τ )}]. A slight alteration of g maps a bounded initial segment of ρ onto L(β). ρ is a cardinal in the sense of L(κ), so β < ρ. A = {x | x < γ ∧ {e}(x, t(τ }}, so A ∈ L(β + 1) ⊆ L(κ). Suppose B ⊆ L(κ). Let ρB be the least γ ≤ κ such that for some p ∈ L(κ; B), there exists a partial map {e}(x, p) of γ onto L(κ; B) via computations in L(κ; B); “partial” means {e}(x, p) might not converge for all x ∈ γ.

page 203

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

G. E. Sacks

204

Let ηB be the least γ ≤ κ such that for some R ∈ 2γ − L(κ; B), R is E-re on L(κ; B) relative to B. This means ∃e∃q ∈ L(κ; B) such that R ∩ γ is {x | {e}B (x, q) ↓ via computations in L(κ; B)}. Lemma 9. (Slaman) Assume L(κ) is E-closed. If B ⊆ κ is regular and E-re-on-L(κ), then η B = ρB . Proposition 13. Assume L(κ) is E-closed and admits divergence witnesses, If A ⊆ κ is E-re-on-L(κ) and not complete, then A is subgeneric. The next theorem is needed for priority constructions in which requirements are indexed by ordinals less than ρ. Theorem 6. Assume L(κ) is E-closed and admits Moschovakis witnesses. If p ∈ κ and γ < ρ, then sup{κp,δ r | δ < γ} < κ. 5.13. Post’s problem Assume L(κ) is E-closed and A, B, C ⊆ L(κ). A is E-reducible to B iff ∃e∃p ∈ L(κ) ∀x ∈ L(κ): {e}B (x, p} ↓, T<e,x,p;B> ∈ L(κ), and x ∈ A ←→ {e}B (x, p} = 1. C is E-re on L(κ) iff ∃e∃p ∈ L(κ) ∀x ∈ L(κ): x ∈ C ←→ {e}(x, p} ↓. Theorem 7. There exist two subsets of L(κ), both E-re on L(κ), such that neither is E-reducible to the other. Proof. (A sketch) If L(κ) lacks divergence witnesses, then L(κ) is Σ1 admissible and the solution to Post’s problem supplied by α-recursion theory is also solution in the sense of E-recursion. Suppose L(κ) admits divergence witnesses. An inequality requirement is handled by waiting for a computation c to converge. If it does, then an inequality is created and preserved forever. The creation may add some x to B, one of the two sets, but x cannot wait forever. It must be added prior of the construction. (p is a parameter needed to enumerate to stage κx,p 0 B.) If c does not converge, then witness to the divergence of c eventually appears and is preserved forever. Requirements are indexed by ordinals less than ρ. If v < ρ and p ∈ κ, | y < v} is less than κ. It follows that a set of requirements then sup{κp,y r indexed by ordinals less than v can be satisfied by some stage less than κ. Define recf (λ), the E-re-on-L(κ) cofinality of λ, to be (least γ) ∃B ⊆ κ [sup B = λ = order type of B and B is E-re on L(κ)]. Requirements are

page 204

April 22, 2014

16:44

WSPC 9 x 6: BC9154

E-Recursion 2012

New˙Erecursion

205

met by a Shore blocking argument based on the fact that recf (ρ) = recf (κ). No requirements are injured. 5.14. Splitting and density A typical requirement in the following result is injured κ-finitely often. As yet there is no priority argument in E-recursion theory in which a requirement is injured unboundedly often. Theorem 8. (Slaman) Suppose C ⊆ L(κ) is regular and E-re, but not E-recursive, on L(κ). Then there exist A and B, each E-re on L(κ), such that A ∪ B = C and A ∩ B = ∅. (Splitting) Suppose A, B ⊆ L(κ) are regular and E-re on L(κ), and A
page 205

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

G. E. Sacks

206

(2) ∃δ[| {e}(a, G) |= δ]; and (3) x ∈ {e}(a, G). Formula (1) is ranked. (2) is unranked; “∃δ” means ∃δ < κ. The meaning of (3) is conditional. If {e}(a, G) ↓ via a computation of height less than κ, then {e}(a, G) names {e}(a, G). 6.3. Forcing relations Let P ∈ L(κ) be a notion of set forcing designed to force sentences of L(κ, G) and to create generic G’s that are subsets of gc(κ). P is < P, ≥ >. The elements of P , denoted by p, q, r, . . . , are called forcing conditions; each one says something about G. ≥ is a binary relation on P . If p ≥ q (p is extended by q), then q says as much as, or more than, p does about G. G ∈ p means: what p says about G is true. P determines a forcing relation  defined by recursion on σ < κ. Three formal entities are defined simultaneously: (i) p | {e}(a, G) |= σ, (ii) T (p, e, a, G), and (iii) q  s ∈ {e}(a, G). If G is generic, (i) holds and G ∈ p, then (ii) is a set of terms that name the elements of {e}(a, G). In (iii) q extends the p of (i) and s ∈ (ii). Assume that P specifies all ground zero forcing facts of the form p  δ ∈ G and q  δ∈ / G and that these facts satisfy standard consistency and completeness assumptions about forcing relations. σ = 0. Take {e}(a, G) to be G. Then ∀p ∀qp≥q p | {e}(a, G) |=0, T (p, e, a, G) = {δ | δ < gc(κ)}, and q  δ ∈ {e}(a, G) iff “q  δ ∈ G” is a ground zero forcing fact. σ > 0. Let e be 2m · 3n (Scheme T of 5.5.1). Then ∀p ∀qp≥q p | {e}(a, G) |= σ iff ∃γ < σ p | {m}(a, G) |= γ, p  ∀x∃τ < σ[x ∈ {m}(a, G) −→ | {n}(x) |= τ ], p  ∀τ < σ[| {m}(a, G) |= τ ∨ ∃x[x ∈ {m}(a, G)∧ | {n}(x) |≥ τ ]. Keep in mind: p  ∃xF (x) iff p  F (t) (for a suitable term t). p  (F ∨ H) iff p  F or p  H. p  ¬F iff ∀q[p ≥ q −→ q  F ]. p  ∀xF (x) iff ∀t ∀qp≥q ∃rq≥r [r  F (t))].

page 206

April 29, 2014

10:11

WSPC 9 x 6: BC9154

E-Recursion 2012

New˙Erecursion

207

6.4. Definability of forcing From now on replace a, G by t, an arbitrary term of L(κ, G). Proposition 14. The relations p | {e}(t) |= σ, s ∈ T (p, e, t, σ), and q |= s ∈ {e}(t) are E-recursive in P, p, e, t, σ. Proof. By transfinite E-recursion. 7. The Tree of Possibilities The binary relation, >V , is the forcing counterpart of >U . A node on >V is of the form < p, e, t >. Define < p, e, t >>V < q, n, s > by ˙ >U < n, s >]. p ≥ q and q ∗ [< e, t> (Recall that q ∗ F means ∀r[q ≥ r −→ r |= F ].) Suppose e is an instance of scheme T as in Subsection 5.5.1. Let e be 2m · 3n . Define p  [< e, t >>i < n, s >] by p | {m}(t) |= γ, s ∈ T (p, m, t, γ), and p  s ∈ {m}(t). Define p  a >U b by ∃a0 , . . . , ak such that a0 = a, ak = b, and ∀j < k p  aj > aj+1 . 7.1. Effective bounding Define p ∗ F by p  ¬¬F. Suppose P ∈L(κ). P satisfies effective bounding iff ∀p, e, t if p ∗ ∃σ | {e}(t) |= σ, then p ∗ | {e}(t) |= γ for some γ ≤E p, t, P . Lemma 10. If >V is wellfounded below < p, e, t > whenever p ∗ ∃σ | {e}(t) |= σ, then P satisfies effective bounding. Proof. γ is computed by transfinite recursion on >V below < p, e, t >. γ is (approximately) the height of the wellfounded subtree below < p, e, t >. Suppose e is 2m · 3n . By recursion p ∗ | {m}(t) |≤ δ for some δ ≤E p, t, P. Define < p , σ >∈ K by p ≥ p ∧ σ ≤ δ ∧ p | {m}(t) = σ. Note K ≤E p, t, P. Fix < p , σ >∈ K and s ∈ T (p , m, t, σ). Then p  ∃β[s ∈ {m}(t) −→| {n}(s) |= β].

page 207

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

G. E. Sacks

208

Define q ∈ J(p , σ, s) by p ≥ q ∧ q  s ∈ {m}(t). By recursion ∀q ∈ J(p , σ, s) ∃ρ ≤E q, s, P such that q |=∗ | {n}(s) |≤ ρ. ρ is a partial Erecursive function of q, s, P. On J(p , σ, s), ρ is bounded by ρρ ,m,t,σ ≤E ρ , σ, t, P. On K, ρρ ,m,t,σ is bounded by ρK ≤E p, t, P. The desired bound γ is the strict sup of δ and ρK . 7.2. Genericity Assume L(κ) is E-closed and P ∈ L(κ). Definition 22. G is P-generic iff for every sentence F of L(κ, G), ∃p ∈ G such that p  F or p  ¬F . If G is P-generic, then L(κ, G) |= F iff (∃p ∈ G) p  F (by induction on the rank of F ). Lemma 11. If P satisfies effective bounding and G is P-generic, then L(κ, G) is E-closed. Proof. Suppose not. Then ∃e∃a ∈ L(κ) such that | {e)(a, G) |= κ. Assume e is 2m · 3n . ∃p ∈ G such that p | {e)(a, G) |= κ. By effective bounding, p ∗ | {m}(a, G) |≤ γ for some γ ≤E p, a, P. Also p |=∗ [∀x ∈ {m}(a, G)∃β | {n}(x) |= β]; x ranges over ∨{T (p, m, a, G, δ) | δ ≤ γ}, a set E-recursive in p, a, P. Hence p weakly forces a bound on | {n}(x) | E-recursive in p, a, P. 7.3. Enumerable forcing relations Theorem 9. (Slaman) Assume L(κ) is E-closed but not Σ1 admissible. Let P ∈ L(κ) be a set forcing relation, Then (i) if and only if (ii). (i) P satisfies effective bounding. (ii) The relation, p ∗ ∃σ | {e}(t) |= σ, is E-recursively enumerable on L(κ). Proof. Suppose (ii) holds. Suppress P. Then an obscure fixed point argument shows ∀p ∈ P p does not weakly force | {e}(t) |≥ κt,p r . ∗ By reflection, if q ∗ | {e}(t) |< κt,q r , then q  | {e}(t) |≤ γ for some γ ≤E t, q. Now (i) follows from Kechris’s Basis Theorem.

page 208

April 22, 2014

16:44

WSPC 9 x 6: BC9154

E-Recursion 2012

New˙Erecursion

209

7.3.1. An Annoying Assumption (AAF) needed for forcing The study of forcing makes repeated use of a key reflection fact: if some wellordering of x is E-recursive in x, then κxr ≥ κx,u for all u. The latr ter is a consequence of the definability of divergence witnesses proved in Subsections 5.5.2 and 5.6. Assumption AAF. P, all p ∈ P, and all P-generic G are effectively wellorderable. The annoyance is mild because G is usually a set of ordinals. From now on AAF will always be in force. 8. Countably Closed Forcing Definition 23. P is countably closed iff for every sequence {pn | n ∈ ω}, ∀n[pn ≥ pn+1 ] −→ ∃q∀n[pn ≥ q]. Example 2. Suppose L(κ) thinks: there is a greatest cardinal denoted by gc(κ) and its cofinality is greater than ω. In short L(κ) |= gc(κ) > ω. Define P by p ∈ P ←→ ∃δ < gc(κ) p : δ −→ 2. Define p ≥ q by domain p ⊆ domain q and ∀x ∈ domain p p(x) = q(x). The above example satisfies Assumption AAF. Theorem 10. Suppose L(κ) is E-closed but not Σ1 admissible, and P ∈ L(κ) is a countably closed forcing relation. Then every P-generic extension of L(κ) is E-closed. Proof. Assume (1) p ∗ ∃σ | {e}(t) |= σ. It suffices to show (2) >V is wellfounded below < p, e, t >, by Lemmas 10 and 11. (3) Suppose not. An infinite descending path below < p, e, t > will be converted to a q ≤ p that weakly forces a witness to the divergence of {e}(t). There is a g such that if (1) and (2) hold, then {g}(p, e, t) ↓ and p ∗ | {e}(t) |≤ {g}(p, e, t). By (3) {g}(p, e, t) ↑; let z ∈ L(κ) witness {g}(p, e, t) ↑: z0 =< g, < p, e, t >> and ∀n zn > zn+1 . Now an infinite descending sequence w below < p, e, t > in >V is extracted from z. w0 =< p, e, t > wr >U wr+1 wr =< pr , er , tr > {g}(pr , er , tr ) ↑ and ∃n zn =< g, < pr , er , tr >> pr ≥ pr+1 . ∃q ∀r pr ≥ q; q weakly forces w to witness {e}(t) ↑.

page 209

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

G. E. Sacks

210

(The assumption “L(κ) is not Σ1 admissible” is not needed in the above proof.) 8.0.1. Enumerability Kleene proved the set of hyperarithmetic reals is Π11 . His result, restated, says 2ω ∩ E(ω) is E-re in ω. The next two theorems clarify the role of Σ1 admissibility in Kleene’s result. Theorem 11. If z ⊆ Ord and E(z) is Σ1 admissible, then E(z) is E-re in some element of L(κ). Proof. ∃κ E(z) = L(κ, z). The Σ1 admissibility of E(z) mplies κ ≤ κz,y r for some y ∈ E(z). ∃v ∈ L(κ) such that v ⊆ L(κ) and z, y ≤E v. Then κ ≤ κv,x r for all x. Let Oz (x) be the least δ < κ such that x ∈ L(δ + 1, z) − L(δ, z). By reflection, Oz (x) ≤E v, x. So ∃e Oz (x) = {e}(x, v). Enumerate x if {e}(x, v) ↓ and E(z) ∈ L({e}(x, v) + 1, z). Theorem 12. Assume L(κ) is E-closed but not Σ1 admissible and L(κ) |= [gc(κ) is regular]. Then 2gc(κ) ∩ L(κ) is not E-re in any b ∈ L(κ). Proof. Suppose 2gc(κ) ∩ L(κ) = {x | {e}(b, x) ↑} for some b ∈ L(κ). Define P by p ∈ P iff ∃δ < gc(κ) p : δ > 2. First assume gc(κ) > ω. Then P is countably closed, and Theorem 10 implies that L(κ, G) is E-closed. Then L(κ, G) admits divergence witnesses since it is not Σ1 admissible. Fix e and b ∈ L(κ). Then (1) or (2) holds. (1) (2) (a) (b)

∅ ∗ ∃σ | {e}(b, G) |= σ. ∃q, w, δ < κ q  w ∈ L(δ, G) ∧ w witnesses {e}(b, G) ↑. (1) implies ∃G ⊆ gc(κ) G ∈ / L(κ) ∧ {e}(b, G) ↓. (2) implies ∃G ⊆ gc(κ) G ∈ L(κ) ∧ {e}(b, G) ↑.

In both cases if κ is uncountable, then G is generic over L(λ) for some λ < κ. The proof of (a) uses an S ∈ 2gs(κ) − L(κ) such that ∀δ < κ (S ∩ δ) ∈ L(κ). S is coded into G to insure that G ∈ / L(κ). 8.1. A general forcing fact Proposition 15. Assume L(κ) is E-closed but not Σ1 admissible and P ∈ L(κ). If p ∗ | {e}(a, G) |≤ κp,a,P , then p ∗ | {e}(a, G) |≤ γ for some r γ ≤E p, a, P.

page 210

April 22, 2014

16:44

WSPC 9 x 6: BC9154

E-Recursion 2012

New˙Erecursion

211

Proof. Assume e = 2m · 3n (Scheme T ). Then ∀q(p ≥ q) ∃r(q ≥ r) ∃δ r | {m}(a, G) |= δ. Also δ < κp,a,P ≤ κp,a,q,P . ∃r(q ≥ r) ∃δ r r p,a,P p,a,q,P ≤ κr . By reflection r and δ r | {m}(a, G) |= δ and δ < κr are E-recursive in p, a, q, P. By Gandy selection r and δ are partial Erecursive functions of p, a, q, P. For all q ≤ p, δ is bounded above by some δ0 ≤E p, a, P. Thus p ∗ | {m}(a, G) |≤ δ 0 . The above mode of argument yields the desired bound γ on | {e}(a, G) |. 9. Countable Chain Condition Forcing 9.1. Less-than-ω 1 selection Suppose L(κ) = E(ω1 ). By Gandy selection, there is an f partial Erecursive in ω1 such that ∀e ∀β < ω1 ∀a ∈ L(κ): if (∃γ < β) {e}{a, γ) ↓, then f (e, β, a) ↓ ∧f (e, β, a) ∈ ω1 ∧ {e}(a, f (e, β, a)) ↓. 9.2. Effectiveness of c.c.c. forcing P ∈L(κ) is said to be c.c.c. iff for every Q ∈ L(κ) ∩ 2P , if the conditions in Q are pairwise incompatible, then Q is countable in L(κ). (Two conditions are incompatible iff no condition extends them both.) Definition 24. min(p, e, t) = min ∃r p ≥ r r | {e}(t) |= δ. δ

Lemma 12. Assume (for simplicity) L(κ) = E(ω1 ). Then min(p, e, t) ≤E p, t, ω1 . Proof. Assume: e = 2m · 3n and min(p, e, t) is defined by the so-called main recursion on the length of computations. Thus min(p, m, t) = δ0 ≤E p, t, ω1 . Let T (p, m, t, γ0 ) be {ti | i < θ}. Note that θ ≤ ω1 and θ ≤E p, t, ω1 . Let Q denote a set of forcing conditions. The following will be shown: ∀i < ∞ ∃ Qi ⊆ P ∀q ∈ Qi p ≥ q and (1) q  ti ∈ {m}(ti ) or (2) q  ti ∈ {m}(ti ) and if (1) holds, then q  ∃δ | {n}(ti ) |= δ. A recursion of length ω1 will add conditions to the Qi ’s. Initially each Qi = ∅. But finally ∩i<θ Qi = ∅. This last means ∃r ∀r1 (r ≥ r1 ) ∀i < θ ∃q ∈ Qi such that q is compatible with r. Suppose at the end of stage β < ω1 , ∩i<θ Qi = ∅. A subrecursion of length ω1 adds conditions to Qi (i < β). The conditions are computed via less-than-ω1 recursion and the main recursion on min. The subrecursion succeeds because P is c.c.c. The

page 211

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

G. E. Sacks

212

main recursion succeeds for the same reason. Thus ∩i<θ = ∅. In other words ∃r ∀i < θ Qi ≥ r and r | {e}(t) = δ. (Qi ≥ r means: ∀r1 ≤ r ∃q ∈ Qi such that r1 and q are compatible.) 10. Selection Suppose A is E-re in b and A∩x = ∅. Is there a way to compute an element of A ∩ x from b, x? (The answer as usual is yes and no, but more no than yes.) Or a nonempty subset of A ∩ x from b, x? Or an ordinal θ from b, x such that computations of length ≤ θ enumerate elements of A ∩ x? Selection(x) says: ∃ partial E-recursive function f such that ∀e∀b ∃z ∈ x {e}(z, b) ↓←→ ∃z ∈ x | {e}(z, b) |≤ f (e, b) < ∞. Sometimes an additional parameter p is present in f . 10.1. Grilliot selection Let f, g : x → ω × (2x × {2x }). Assume x is transitive, closed under pairing, and ω ⊆ x. Define g < f by ∀z ∈ x g(z) is an immediate subcomputation instruction of f (z). Define min(f ) = min{| f (z) || z ∈ x}. Assume min(f ) < ∞. Grilliot’s recursion equation is min(f ) = max{min(g) | g < f }. His intention was to compute min(f ) from f, 2x . An immediate obstacle is the failure of {g | g < f } to be E-recursive in f, 2x . Harrington and MacQueen worked around the obstacle with effective approximations of {g | g < f }. Define g <β f by ∀z ∈ x g(z) is an immediate subcomputation instruction of f (z) via a computation of length ≤ β. Note that {g | g <β f } ≤E β, f, 2x . 10.2. Harrington-MacQueen selection An instance of the axiom of choice, ACx plays a curious role in the proof of Theorem 13. ACx : ∀f If ∀z ∈ x f (z) = ∅, then ∃g ∀z ∈ x g(z) ∈ f (z). (Assume x = domain(f ).) Theorem 13. Assume f : x → ω × (2x × {2x }) and ∃z ∈ x | f (z) |< ∞. Then min{| f (z) | | z ∈ x} ≤E f, 2x . Proof. Let θ = sup{β | ∃f f maps x onto β}. Note θ ≤E 2x . t(γ) is defined by recursion on γ < θ. min(f ) will be sup{t(γ) | γ < θ}. Let

page 212

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

E-Recursion 2012

213

t− (γ) = sup(t(δ) | δ < γ}. If t− (γ) ≥ min(f ), then t(γ) = min(f ). If t− (γ) < min(f ), then −

t(γ) = sup+ {min(g) | g
(γ)

f ∧ t− (γ) ≤ min(g)}.

−

ACx implies ∃g g . If t− (γ) ≤ |< m, u > |, then g(z) =< m, u > . Otherwise ∃y ∈ {m}(u) such that t− (γ) ≤ |< n, y > |. If there is a γ such that t(γ) = min(f ), then sup{t(γ) | γ < θ} is min(f ), If there is no such γ, then there is a map of x onto θ. 10.3. Selection and admissibility x be {z | {e}(z, x, b) ↑} ∩ tc(x). (tc is For each < e, b >∈ E(x), let Ke,b transitive closure.) f is a Grilliot Selection Function for E(x) iff ∀ < x x = ∅ → f (e, b) ∈ E(x) ∩ 2Ke,b . e, b > ∈ E(x), Ke,b

Lemma 13. If f , a Grilliot selection function for E(x), is partial Erecursive in some b ∈ E(x), then E(x) is Σ1 admissible. E(X)

map with domain d. Admissibility Suppose g is a many-valued Σ1 requires an r ∈ E(x) such that ∀v ∈ d, g(v) has a value in r ∃D ∈ ΔZF 0 such that g(v) = w iff E(x)  ∃yD(v, w, y). Let t be a partial E-recursive (in x) map of ω × tc(x) onto E(x). Then for each v ∈ d, {z | t(z) ↓ ∧ D(v, (t(z))0 , (t(z))1 )} is E-re in v, x, a. The required r is {(t(z))0 | ∃v z ∈ f (v) ∧ v ∈ d}. 10.3.1. Moschovakis selection Theorem 14. (Moschovakis) Suppose E(R(α))  cof inality(α) = ω. Let k ∈ R(α) be an ω-sequence through α. Assume f : R(α) → ω × R(α) × {R(α}. If ∃x ∈ R(α) | f (x) | < ∞, then min{| f (x) || x ∈ R(α)} ≤E f, R(α), k. Corollary 3. If E(R(α))  cof inality(α) = ω, then E(R(α)) is Σ1 admissible.

page 213

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

G. E. Sacks

214

Theorem 15. (Sacks & Slaman) Let x be a set of ordinals. Suppose in E(x): ∀i < ω (ki is a cardinal and ki < ki+1 ) and sup(x) = sup{ki | i < ω}. If A ⊆ x is nonempty and E-re in x, then min A ≤E x, {ki | i < ω}. Corollary 4. If x ⊆ Ord and E(x)  cofinality(gc(κ) = ω), then E(x) is Σ1 admissible. 11. van de Wiel’s Theorem Let f map V into V . Definition 25. f is uniformly Σ1 definable iff ∃ lightf ace F (x, y) ∈ such that ∀ Σ1 admissible A, f [A] ⊆ A and f  A = {< a, b > | < ΣZF 1 A, ∈>  F (a, b). (Lightface means nonnegative integer parameters only.) Theorem 16. Let f : V → V . Then (i)←→(ii) (i) f is E-recursive. (ii) f is uniformly Σ1 definable. Proof. (Slaman) Assume (ii). Fix x and define Kx = L(κxr , tc({x})). The plan is to show: Kx  ∃vF (x, v). Then L(κx0 , tc({x}))  ∃vF (x, v) by reflection, and f (x) ≤E by Gandy selection. Choose p ∈ Kx so that κpr = min{κqr | q ∈ Kx }. Let Kx,p be L(κx,p r , tc({x, p})). Assume tc(x) is countable. Now a hull Z ⊆ Kx,p is constructed whose transitive collapse, Z, is Σ1 admissible; Let y0 denote the collapse of y ∈ Z. Then Z  ∃vF (x0 , v), Z  ∃vF (x, v), f (x) ∈ Kx,p , f (x) ≤E x, p and f (x) ∈ Kx . Let Z be {zi | i < ω}. z0 is < x, p >. Choose zi (i˙ > 0) so that z0 ,...,zi κr = κzr0 . Also make choices so that Z is E-closed. If ∃w ∈ Kx,p such that {e}(zi, w) ↑, then ∃w ∈ Z such that {e}(zi, w) ↑. The choices exist thanks to Kechris’s basis theorem. Now show Z is Σ1 admissible. Suppose Z  (∀u ∈ a0 )∃vG(u, v, c0 ). G is ΔZF 0 . First show Kx,p  (∀u ∈ a)∃vG(u, v, c). If not, then U = {u | u ∈ a ∧ Kx,p  ∀v¬G(u, v, c0 )} = ∅. If ∃v Kx,p  G(u, v, c), then by reflection ∃v v ≤E x, p, c, u ∧ Kx,p  G(u, v, c). Thus U is co-E-re in x, p, c, a, so ∃u ∈ U ∩ Z, hence Z  ∃u ∈ a ∀v¬G(u, v, c) and Z  ∃u ∈ a0 ∀v¬G(u, v, c0 ).

page 214

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

E-Recursion 2012

215

If u ∈ a and ∃v Kx,p  G(u, v, c), then by reflection ∃v v ≤E x, p, c, a, u ∧ Kx,p  G(u, v, c); the range of v restricted to a is some b ∈ Z. Thus Z  ∀u ∈ a ∃v ∈ bG(u, v, c). Then Z  ∀u ∈ a0 ∃v ∈ b0 G(u, v, c0 ). The assumption of countability of tc(x) has to be lifted. At this point ∃e∀x if T V (x)is countable, then f (x) = {e}(x). Suppose ∃y f (y) = {e}(y). The formula ‘f (y) = {e}(y)’ is ΣZF and solvable, hence has a solution y 1 such that tc(y) is countable. 12. Logic on E-Closed Sets Assume L(κ) is E-closed. Let L ∈ L(κ) be a set of atomic symbols. L∞,ω denotes a minimal extension of first order logic. The class of formulas is built from L via the finitary formation rules of first order logic and two infinitary rules, arbitrary conjunctions and arbitrary disjunctions. In addition there is a restrictive proviso: a formula must not contain infinitely many free variables. On the other hand a formula may contain arbitrarily many distinct individual constants. Note that each sequence of quantifiers is finite; that is the intended meaning of the ‘ω’ in the subscript ‘∞, ω’. The axioms and rules of deduction come from first order logic with only one significant addition; if for each i ∈ I, there is a deduction of Fi , then there is a deduction of the conjunction ∧ {Fi | i ∈ I}. Lκ,ω is the restriction of L∞,ω to L(κ). Let F ∈L(κ). Assume Δ is an E-recursive-on-L(κ) set of sentences of Lκ,ω . 12.1. Deductions Definition 26. Δ  F iff F is deducible from Δ via a deduction in V in the sense of L∞,ω . Definition 27. Δ κ F iff Δ  F via a deduction in L(κ). Definition 28. Δ E κ F iff Δ  F via a deduction E-recursive in F . Definition 29. Δ admits effectivization of deductions iff for all F ∈L(κ), if Δ κ F , then Δ E κ F. 12.2. Completeness Definition 30. Δ is κ -consistent iff Δ κ (F ∧ ¬F ).

page 215

April 22, 2014

16:44

WSPC 9 x 6: BC9154

New˙Erecursion

G. E. Sacks

216

Assume below that Δ admits effectivization of deductions. Lemma 14. If Δ, ∨{Fi | i ∈ I} is κ-consistent, then Δ, Fi is κ-consistent for some i ∈ I. Theorem 17. If L(κ) is countable and Δ is κ-consistent, then Δ has a model. Corollary 5. If Δ is κ-consistent, then Δ is consistent in the sense of L∞,ω . Question Is there an interesting application of Theorem 17. References 1. Barwise J (1979) Admissible structures, Springer, Berlin, Heidelberg, New York 2. Fenstad JE (1980) General recursion theory, Springer, Berlin, Heidelberg, New York 3. Gandy RO (1967) General recursive functionals of finite type and hierarchies of functionals, Ann. Fac. Sci. Univ. Clermont-Ferrand 35; 215-223 4. Green J (1974) Σ1 compactness for next admissible sets, J. Symb. Log. 39; 105-116 5. Griffor ER (1980) E-recursively enumerable degrees, Ph.D Thesis, Massachusetts Institute of Technology 6. Harrington L (1973) Contributions to recursion theory in higher types, Ph.D Thesis, Massachusetts Institute of Technology 7. Harrington L, MacQueen D (1976) Selection in abstract recursion theory, J. Symb. Log. 41: 153-158 8. Houle T (1982) Abstract extended 2-sections, Ph.D Thesis, Oxford University (Note: date is approximate) 9. Kleene, SC (1959) Recursive functionals and quantifiers of finite types I, Trans Amer. Math. Soc. 91; 1-52 10. Kleene, SC (1963) Recursive functionals and quantifiers of finite types II, Trans Amer. Math. Soc. 108; 106-142 11. Moldstad J (1972) Computations in Higher Types, Lecture Notes in Math. 574, Springer. Berlin, Heidelberg, New York 12. Moschovakis YN (1967) Hyperanalytic predicates, Trans. Amer. Math. Soc. 129; 249-282 13. Moschovakis YN (1980) Descriptive set theory, North-Holland, Amsterdam 14. Normann D (1975) Degrees of functionals, Preprint Series in Math. 22, Oslo 15. Normann D (1978a) Set recursion, In: Generalized recursion theory II, NorthHolland, Amsterdam, 303-320 16. Normann D (1978b) Recursion in 3 E and a splitting theorem, In: Essays on mathematical and philosophical logic, D. Reidel, Dordrecht, 275-285

page 216

April 22, 2014

16:44

WSPC 9 x 6: BC9154

E-Recursion 2012

New˙Erecursion

217

17. Sacks GE (1974) The 1-section of a type n-object, In: Generalized recursion theory, North-Holland, Amsterdam, 81-96 18. Sacks GE (1977) The k-section of a type n-object, Amer. Jour. Math. 99; 901-917 19. Sacks GE (1985) Post’s problem in E-recursion, In: Proc. of symposia in pure mathematics, Amer. Math. Soc. 42; 177-193 20. Sacks GE (1986) On the limits of E-recursive enumerability, Ann. Pure and Applied Log. 31; 87-120 21. Sacks GE (1990) Higher recursion theory, Springer, Berlin, Heidelberg, New York 22. Slaman TA (1985a) Reflection and forcing in E-recursion, Ann. Pure and Applied Log. 29; 79-106 23. Slaman TA (1985b) The E-recursively enumerable degrees are dense, In: Proc. of symposia in pure mathematics, Amer. Math. Soc. 42; 195-213 24. van de Wiele J (1982) Recursive dilators and generalized recursion, In: Proc. Herbrand symposium, North-Holland, Amsterdam, 325-332

page 217