Fine Structure and Class Forcing [Reprint 2011 ed.] 9783110809114, 9783110167771

Table of contents :
Preface
1 The Σ* Approach to the Fine Structure of L
1.1 The J-Hierarchy
1.2 Fine Structure Theory
1.3 Morasses
2 Forcing
2.1 Set Forcing
2.2 Class Forcing
2.3 Examples
3 Construction of Generic Classes
3.1 Rigidity
3.2 Relevant Forcing
4 The Coding Theorem
4.1 Three Questions of Solovay
4.2 The Coding Theorem without 0#
4.3 The Coding Theorem in the General Case
4.4 Large Cardinal Preservation and Relevance
5 The Genericity Problem
5.1 Jensen’s Example and a New Conjecture
5.2 Perturbing the Indiscernibles
5.3 Generic Saturation
6 The π12-Singleton Problem
6.1 An Absolute Singleton
6.2 David’s Trick
6.3 Other Applications
7 The Admissibility Spectrum Problem
7.1 Killing Admissibles
7.2 Strong Coding
7.3 Other Spectra
8 Further Applications of Class Forcing
8.1 Δ1-Coding
8.2 Iterated Class Forcing
8.3 Minimal Coding
8.4 Further Applications to Descriptive Set Theory
Some Open Problems
References
Index

Citation preview

de Gruyter Series in Logic and Its Applications 3 Editors: W. A. Hodges (London) · R. Jensen (Berlin) S. Lempp (Madison) · M. Magidor (Jerusalem)

Sy D. Friedman

Fine Structure and Class Forcing

Walter de Gruyter Berlin · New York 2000

Author

Sy D. Friedman Institut für formale Logik University of Vienna Währingerstr. 25 1090 Vienna Austria

Sy D. Friedman Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139-4307 USA

Series Editors Wilfrid A. Hodges School of Mathematical Sciences Queen Mary and Westfield College University of London Mile End Road London El 4NS, United Kingdom

Ronald Jensen Institut für Mathematik Humboldt-Universität Unter den Linden 6 10099 Berlin, Germany

Steffen Lempp Department of Mathematics University of Wisconsin 480 Lincoln Drive Madison, WI 53706-1388, USA

Menachem Magidor Institute of Mathematics The Hebrew University Givat Ram 91904 Jerusalem, Israel

Mathematics Subject Classification 2000: 03-02; 03E45, 03E15, 03E35, 03E55 Keywords: Fine structure, Forcing, Admissibility, Large cardinals, Descriptive set theory ©

Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability

Library of Congress — Cataloging-in-Publication

Data

Friedman, Sy D„ 1953Fine structure and class forcing / Sy D. Friedman, p. cm. — (De Gruyter series in logic and its applications; 3) Includes bibliographical references and index. ISBN 3-11-016777-8 (alk. paper) 1. Forcing (Model theory). I. Title. II. Series. QA9.7.F75 2000 511.3'22—dc21 00-023002 CIP

Die Deutsche Bibliothek — Cataloging-in-Publication

Data

Friedman, Sy D.: Fine structure and class forcing / Sy D. Friedman. - Berlin ; New York : de Gruyter, 2000 (De Gruyter series in logic and its applications ; 3) ISBN 3-11-016777-8

ISSN 1438-1893 © Copyright 2000 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typesetting using the Author's T g X files: I. Zimmermann, Freiburg - Printing and binding: WB-Druck GmbH & Co., Rieden/Allgäu — Cover design: Rainer Engel, Berlin.

For Lida and Scooty

Preface

This book provides a detailed analysis of the first two canonical approximations to the set-theoretic universe, given by the inner models L and L[0 # ]. It is intended for the student who has completed one year of study in axiomatic set theory. Thus we assume familiarity with the basic principles of the axiom system ZFC and with the basic properties of Gödel's constructible universe L, but we assume background neither in forcing nor in fine structure theory. L has always played a central role in set theory. It is in the L-context that the detailed study of definability, derived from the field of higher recursion theory (Sacks [90]), can be thoroughly applied. What results is the striking fine structure theory of L (Jensen [72]), which demonstrates that at least in one interpretation of set theory, a nearly complete analysis of the structure of the universe V of all sets is possible. The theory offorcing has taught us however that L need not be a very good approximation to V. In the traditional forcing method we begin with a model Μ of set theory, and choose both a partial-ordering Ρ € Μ and a set G c Ρ which is "P-generic over M". Then the model M[G] obtained by adjoining G to Μ is a new and larger model of the ZFC axioms. In this way, forcing suggests that V should be larger than L, as for some partial-orderings Ρ in L, Ρ-generic sets G are likely to exist. But for which Ρ in L should a Ρ-generic set exist? A possible criterion is for Ρ to be definable in L without parameters. However, we face a more serious challenge when we turn to class forcing, where the partial-ordering Ρ is no longer required to be an element of M, but a class of M. Then we have: Fact 1. There exist class forcings PQ, P\ which are L-definable without parameters such that whenever Go, G\ are PQ, PI-generic over L, respectively: (a) ZFC holds in (L[G 0 ], G 0 ) and in (L[Gi],Gi). (b) ZFC (indeed Replacement) fails in (L[G 0 , Gj], G 0 , Gi). Thus we cannot have generics for all ZFC preserving class forcings which are L-definable without parameters. Finding a suitable generic existence criterion for Ldefinable forcings has become more difficult. Fortunately, a natural such criterion is provided by Silver's theory of 0 # (Silver [71]): We say that L is rigid if there is no elementary embedding from (L, e) to itself, other than the identity. Fact 2. L is rigid in class-generic extensions of L. If L is not rigid then there is a smallest inner model in which L is not rigid, and this inner model is L[0 # ] where 0 # is a real. An L-definable forcing Ρ is relevant if there is a class which is P-generic over L and which is definable in the inner model L[0 # ]. If P 0 and Pi are relevant forcings then

viii

Preface

clearly generics for Po and for P\ can coexist, as they both exist definably over L[0 # ], Relevance thereby provides us with the desired generic existence criterion for forcings over L. An important consequence, and indeed the major theme of this book, is that forcing now becomes not only a tool for obtaining consistency results, but also a technique for proving absolute results, in the theory ZFC + 0 # exists. Thus we use the theory of relevant forcing to construct objects which actually exist (in the inner model L[0 # ]), rather than which may exist in a generic extension of the universe. In this book we provide a thorough analysis of L[0 # ] using the class forcing technique. In addition, we take a close look at the fine structure of L, which not only exposes its deepest combinatorial properties, but also supplies the tools necessary to construct some of the forcing partial-orderings used in our analysis of L[0 # ]. Our emphasis in Chapter 1 (fine structure) is on the combinatorial properties O, • , Scale and Morass. The reader whose primary interest is with forcing rather than fine structure may choose to use this chapter solely as a reference for later chapters. Chapters 2, 3 and 4 provide the basic theory of class forcing, including a discussion of generic existence and Jensen coding. They are prerequisite to the rest of the book. What follows in the remaining four chapters, which are relatively independent of each other, are applications of the class forcing technique to resolve a number of problems concerning genericity, admissibility, descriptive set theory and set-theoretic definability. Included are solutions to the Genericity, n^-Singleton and Admissibility Spectrum problems, which are introduced in Section 4.1. We end with a list of open problems. Acknowledgments. I should like to thank the Rockefeller Foundation for its support during my stay as a Resident Scholar at their center in Bellagio, during which time I wrote the first two chapters of this book. The National Science Foundation has provided me with considerable support through its research grants. I am grateful to the MIT Mathematics Department for providing me with such a congenial place to work, and especially to Bonnie Friedman and Jan Wetzel for their excellent preparation of the TgX files. In addition I thank the students of my MIT class forcing course, Elizabeth Brown, Peter Koellner, Boris Piwinger and Ronan Rudolph, for their helpful comments. Lastly, I wish to acknowledge two intellectual debts; one to Gerald Sacks, who revealed to me the beauty and subtlety of the concept of definability, and the other to Ronald Jensen, the creator of both the fine structure theory and the coding method. It is my strong belief that set theory will be most powerfully understood through the use of these techniques, as they are applied to larger and larger approximations to the set-theoretic universe. Vienna and Cambridge, MA, May 2000

Sy D. Friedman

Contents

Preface

vii

1

The 1.1 1.2 1.3

Σ* Approach to the Fine Structure of L The J-Hierarchy Fine Structure Theory Morasses

2

Forcing 2.1 Set Forcing 2.2 Class Forcing 2.3 Examples

25 25 32 39

3

Construction of Generic Classes 3.1 Rigidity 3.2 Relevant Forcing

49 50 59

4

The 4.1 4.2 4.3 4.4

65 65 67 76 86

5

The Genericity Problem 5.1 Jensen's Example and a New Conjecture 5.2 Perturbing the Indiscernibles 5.3 Generic Saturation

92 92 99 112

6

The 6.1 6.2 6.3

117 117 129 133

7

The Admissibility Spectrum Problem 7.1 Killing Admissibles 7.2 Strong Coding 7.3 Other Spectra

Coding Theorem Three Questions of Solovay The Coding Theorem without 0 # The Coding Theorem in the General Case Large Cardinal Preservation and Relevance

Π 2-Singleton Problem An Absolute Singleton David's Trick Other Applications

1 1 5 14

140 140 144 163

χ 8

Contents Further Applications of Class Forcing 8.1 Δι-Coding 8.2 Iterated Class Forcing 8.3 Minimal Coding

169 169 175 183

8.4

202

Further Applications to Descriptive Set Theory

Some Open Problems

209

References

211

Index

215

Chapter 1

The Σ* Approach to the Fine Structure of L

In this chapter we reformulate the fine structure theory from Jensen [72]. We then use this reformulation to prove the • and Fine Scale Principles, and to construct Morasses. The material in this chapter will only be used in Sections 4.3, 7.2 and 8.3 of this book; for this reason, the reader may wish to begin with Chapter 2, and refer to the present chapter only when needed.

1.1 The /-Hierarchy The most elegant hierarchy for Gödel's L is obtained through iterated first-order definability. For any set χ let Def(*) denote {>> | y c χ, y is definable over (x, e) by a first-order formula with parameters}. Then L is obtained as the union of all LA, where LQ = 0, L\ = U{La I A < λ}, for limit λ and: LA+1 = Def(L a ). Unfortunately LA+\ is not closed under pairing and for this reason, Jensen [72] defined a modified hierarchy to get around this problem. We now present a description of the 7-Hierarchy which, as above, is based on the idea of iterated definability. Recall the Levy hierarchy of formulas: A formula is Σο (= Δο = Πο) if it is built from atomic formulas through the use of logical connectives and bounded quantifiers Vx € y, Ξλ: € y. A formula is Σ„+ι if it is of the form 3χψ where φ is Π„. Dually, a formula is Π η + ι if it is of the form Vx

n +\ by (a).

n

and

(c) Clearly ORD(7 a n ) < ωα + η since χ e Ja,n+1 ==> x Q Ja,n• By induction on n, define en+\ such that X(n + 1, en+\) = ωα + n: For η = 0 we can take e\ so that ωα = [f \ Wfiei,/)}. If en+i is defined take so that { / | W?+2(en+2, /)} = {F(n,g) I W"+l(en+i, g)} U {en+\}, where F is from the proof of (a). Then X(n + 2, en+2) = X(n + 1, en+\) U {X(n + 1, en+\)} = ωα + η + 1. (d) Ja+1 is closed under pairing because all 2-element subsets of Ja,n belong to Ja,n+1· For Σο-Comprehension it suffices to show that if X C Ja n is definable over (Ja,n, e) then X belongs to Ja>m for some m. But {e \ X(n, e) € X) is a definable subset of Ja. Choose m so that this set is Σ^-definable over Ja and using F from the proof of (a), produce a Σι(7 α ) G such that for each e, X(n, e) = X(m, G(e)). Then {G(e) I X(n, e) e X} is Σ^,-definable over Ja and X = {X(m, G(e)) \ X(n, e) e X) belongs to Ja,m+1. (e) We get Def(7 a ) c Ja+l by (d). Conversely, if X(n, e) c Ja then {/ | / e X(n, e)} = X(n, e) is a definable subset of Ja, using the definition of X(n,e). • Of course now we may define (e,x), using (d) of Lemma 1.2, thereby completing the definition of the /-hierarchy. In the future we will sometimes refer to the refined hierarchy {Ja \ a e ORD) defined by: Jwa+n = J