Volume 236, Number 1111 Hod Mice and the Mouse Set Conjecture 9781470416928

Table of contents :
Cover
Title page
Introduction
0.1. Why analyze Hod
0.2. A crash course on hod mice
0.3. The mouse set conjecture
0.4. The proof of MSC
0.5. The comparison theory of hod mice
0.6. Hod is a hod premouse
Chapter 1. Hod mice
1.1. Hybrid \J-structures
1.2. Some fine structure
1.3. Iteration trees and iteration strategies
1.4. Layered strategy premice
1.5. Iterations of Σ-mice
1.6. Hull condensation
1.7. Hod mice
Chapter 2. Comparison theory of hod mice
2.1. Hod pair constructions
2.2. Iterability of hod pair constructions
2.3. Universality of the fully backgrounded constructions
2.4. Coarse Γ-Woodin mice
2.5. Comparison under 𝐴𝐷⁺
2.6. Positional and commuting iteration strategies
2.7. The diamond comparison argument
Chapter 3. Hod mice revisited
3.1. The internal theory of hod premice
3.2. OD-full pointclasses
3.3. The derived models of hod mice
3.4. An anomaly
3.5. Getting branch condensation
3.6. Generic comparisons
3.7. Reorganizing hod mice
3.8. 𝑆-constructions
Chapter 4. Analysis of HOD
4.1. Suitability
4.2. 𝐵-iterability
4.3. The direct limit of iterates of hod mice
4.4. The computation of Hod
Chapter 5. Hod pair constructions
5.1. Stacking mice
5.2. Clause 4
5.3. Fullness preservation
5.4. The comparison argument revisited
5.5. Branch condensation
5.6. Γ(\P,Σ) when ł^{\P} is successor
5.7. 𝐵-iterability
5.8. Strongly ⃗𝐵-guided strategies
5.9. Summary
Chapter 6. A proof of the mouse set conjecture
6.1. The generation of the mouse full pointclasses
6.2. An analysis of stacks
6.3. Capturing of hod pairs
6.4. The mouse set conjecture
6.5. A last word
Appendix A. Descriptive set theory primer
A.1. Pointclasses
A.2. 𝐸𝑛𝑣(Γ)
A.3. 𝐴𝐷⁺
A.4. The derived model theorem
Index
Bibliography
Back Cover

Citation preview

EMOIRS M of the American Mathematical Society Volume 236 • Number 1111 • Forthcoming

Hod Mice and the Mouse Set Conjecture Grigor Sargsyan

ISSN 0065-9266 (print)

ISSN 1947-6221 (online)

American Mathematical Society Licensed to Rutgers Univ-New Brunswick. Prepared on Mon Dec 29 14:55:36 EST 2014for download from IP 192.12.88.137. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

EMOIRS M of the American Mathematical Society Volume 236 • Number 1111 • Forthcoming

Hod Mice and the Mouse Set Conjecture Grigor Sargsyan

ISSN 0065-9266 (print)

Licensed to Rutgers Univ-New Brunswick.

ISSN 1947-6221 (online)

American Mathematical Society Providence, Rhode Island Prepared on Mon Dec 29 14:55:36 EST 2014for download from IP 192.12.88.137.

License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

Library of Congress Cataloging-in-Publication Data Cataloging-in-Publication Data has been applied for by the AMS. See http://www.loc.gov/publish/cip/. DOI: http://dx.doi.org/10.1090/memo/1111

Memoirs of the American Mathematical Society This journal is devoted entirely to research in pure and applied mathematics. Subscription information. Beginning with the January 2010 issue, Memoirs is accessible from www.ams.org/journals. The 2015 subscription begins with volume 233 and consists of six mailings, each containing one or more numbers. Subscription prices for 2015 are as follows: for paper delivery, US$860 list, US$688.00 institutional member; for electronic delivery, US$757 list, US$605.60 institutional member. Upon request, subscribers to paper delivery of this journal are also entitled to receive electronic delivery. If ordering the paper version, add US$10 for delivery within the United States; US$69 for outside the United States. Subscription renewals are subject to late fees. See www.ams.org/help-faq for more journal subscription information. Each number may be ordered separately; please specify number when ordering an individual number. Back number information. For back issues see www.ams.org/bookstore. Subscriptions and orders should be addressed to the American Mathematical Society, P. O. Box 845904, Boston, MA 02284-5904 USA. All orders must be accompanied by payment. Other correspondence should be addressed to 201 Charles Street, Providence, RI 02904-2294 USA. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes.

Memoirs of the American Mathematical Society (ISSN 0065-9266 (print); 1947-6221 (online)) is published bimonthly (each volume consisting usually of more than one number) by the American Mathematical Society at 201 Charles Street, Providence, RI 02904-2294 USA. Periodicals postage paid at Providence, RI. Postmaster: Send address changes to Memoirs, American Mathematical Society, 201 Charles Street, Providence, RI 02904-2294 USA. c 2014 by the American Mathematical Society. All rights reserved.  R , Zentralblatt MATH, Science Citation This publication is indexed in Mathematical Reviews  R , Science Citation IndexT M -Expanded, ISI Alerting ServicesSM , SciSearch  R , Research Index  R , CompuMath Citation Index  R , Current Contents  R /Physical, Chemical & Earth Alert  Sciences. This publication is archived in Portico and CLOCKSS. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

20 19 18 17 16 15

Licensed to Rutgers Univ-New Brunswick. Prepared on Mon Dec 29 14:55:36 EST 2014for download from IP 192.12.88.137. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

To Lilit

Licensed to Rutgers Univ-New Brunswick. Prepared on Mon Dec 29 14:55:36 EST 2014for download from IP 192.12.88.137. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

Licensed to Rutgers Univ-New Brunswick. Prepared on Mon Dec 29 14:55:36 EST 2014for download from IP 192.12.88.137. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

Contents Introduction 0.1. Why analyze Hod 0.2. A crash course on hod mice 0.3. The mouse set conjecture 0.4. The proof of MSC 0.5. The comparison theory of hod mice 0.6. Hod is a hod premouse

1 2 6 8 8 10 11

Chapter 1. Hod mice 1.1. Hybrid J -structures 1.2. Some fine structure 1.3. Iteration trees and iteration strategies 1.4. Layered strategy premice 1.5. Iterations of Σ-mice 1.6. Hull condensation 1.7. Hod mice

15 15 17 19 22 23 25 29

Chapter 2. Comparison theory of hod mice 2.1. Hod pair constructions 2.2. Iterability of hod pair constructions 2.3. Universality of the fully backgrounded constructions 2.4. Coarse Γ-Woodin mice 2.5. Comparison under AD+ 2.6. Positional and commuting iteration strategies 2.7. The diamond comparison argument

35 35 38 40 44 47 52 60

Chapter 3. Hod mice revisited 3.1. The internal theory of hod premice 3.2. OD-full pointclasses 3.3. The derived models of hod mice 3.4. An anomaly 3.5. Getting branch condensation 3.6. Generic comparisons 3.7. Reorganizing hod mice 3.8. S-constructions

71 71 79 82 86 91 95 99 100

Chapter 4. Analysis of HOD 4.1. Suitability 4.2. B-iterability 4.3. The direct limit of iterates of hod mice

105 105 110 111

v Licensed to Rutgers Univ-New Brunswick. Prepared on Mon Dec 29 14:55:36 EST 2014for download from IP 192.12.88.137. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

vi

CONTENTS

4.4. The computation of Hod

113

Chapter 5. Hod pair constructions 5.1. Stacking mice 5.2. Clause 4 5.3. Fullness preservation 5.4. The comparison argument revisited 5.5. Branch condensation 5.6. Γ(P, Σ) when λP is successor 5.7. B-iterability  5.8. Strongly B-guided strategies 5.9. Summary

119 119 123 125 126 127 129 134 136 137

Chapter 6. A proof of the mouse set conjecture 6.1. The generation of the mouse full pointclasses 6.2. An analysis of stacks 6.3. Capturing of hod pairs 6.4. The mouse set conjecture 6.5. A last word

141 141 144 146 155 159

Appendix A. Descriptive set theory primer A.1. Pointclasses A.2. Env(Γ) A.3. AD+ A.4. The derived model theorem

161 161 163 163 165

Index

167

Bibliography

169

Licensed to Rutgers Univ-New Brunswick. Prepared on Mon Dec 29 14:55:36 EST 2014for download from IP 192.12.88.137. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

Abstract We develop the theory of hod mice below ADR + “Θ is regular”. We use this theory to show that HOD of the minimal model of ADR + “Θ is regular” satisfies GCH. Moreover, we show that the Mouse Set Conjecture is true in the minimal model of ADR + “Θ is regular”.

Received by the editor Received June 7, 2011 and, in revised form, March 27, 2014. Article electronically published on November 6, 2014. DOI: http://dx.doi.org/10.1090/memo/1111 2010 Mathematics Subject Classification. Primary 03E15, 03E45, 03E60. Key words and phrases. Mouse, inner model theory, descriptive set theory, hod mouse. c 2014 American Mathematical Society

vii Licensed to Rutgers Univ-New Brunswick. Prepared on Mon Dec 29 14:55:36 EST 2014for download from IP 192.12.88.137. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

Licensed to Rutgers Univ-New Brunswick. Prepared on Mon Dec 29 14:55:36 EST 2014for download from IP 192.12.88.137. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

Introduction The interplay between Levy’s hierarchy and the canonical models of fragments of ZF C has been known for many years. For instance, a real x is Δ12 definable from a real y and a countable ordinal iff x ∈ L[y] (Solovay, [8]). Assuming, Δ12 determinacy, a real x is Δ13 definable from a real y and a countable ordinal iff 1 x ∈ M# 1 (y) (Steel-Woodin, see [39]) . That this phenomenon is always true is the conclusion of the Mouse Set Conjecture, the primary topic of this paper. The Mouse Set Conjecture, MSC. Assume AD+ + V = L(P(R))2 . Then for all reals x and y, x is ordinal definable from y if and only if there is a mouse M over y such that x ∈ M. Below Mouse Capturing (M C) stands for the concluding statement of the Mouse Set Conjecture. Notice that ordinal definability is the most robust form of definability and hence, M SC can be viewed as the ultimate generalization of the well-known phenomenon mentioned in the opening paragraph. AD+ in the statement of MSC is an axiomatic system extending AD. It was originally formulated by Woodin. See Definition A.8 for its statement. One of the consequences of AD+ is that if in addition one assumes that V = L(P(R)) then the universe has many of the properties that L(R) has under AD. For instance, under AD+ + V = L(P(R)), it is the case that LΘ (P(R)) ≺Σ1 V (Θ here and below is the least ordinal which is not a surjective image of the reals). More is also true: the fragment of V coded by Suslin, co-Suslin sets is Σ1 elementary in V . See Theorem A.10 for the fundamental consequences of AD+ . We note that assuming V = L(R), AD is equivalent to AD+ . Whether the equivalence is always true is an important open problem. Finally, notice that one cannot hope to prove M C from ZF C as it can be easily arranged by forcing that there are more than ω1 many ordinal definable reals while ZF C + M C implies that there are ω1 many ordinal definable reals (this follows from the comparison theorem for mice). We will prove that M SC holds in the minimal model of ADR + “Θ is regular”. Theorem 6.26 implies that ADR + “Θ is regular” is weaker than the existence of an ω1 +1-iterable mouse with a superstrong cardinal. The proof is based on the theory of hod mice below the theory ADR + “Θ is measurable”. This theory is developed in Chapter 3. Our main theorem can be summarized in the following. Main Theorem. Each of the following statements implies that there is a proper class model containing the reals and satisfying ADR + “Θ is regular”. 1 M# (y) is the minimal sound active y-mouse that has a Woodin cardinal. 1 2 Often times MSC is stated under the additional hypothesis that “there is no

ω1 + 1-iterable mouse with a superstrong cardinal”. This is because currently the general notion of mouse is not well-developed. See Section 0.3 for some more comments on the hypothesis of MSC. 1 Licensed to Rutgers Univ-New Brunswick. Prepared on Mon Dec 29 14:55:36 EST 2014for download from IP 192.12.88.137. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

2

INTRODUCTION

(1) AD+ + ¬M C. (2) There are divergent models of AD+ , i.e., there are Ai ⊆ R (i = 0, 1) such that L(Ai , R)  AD+ and L(A0 , A1 , R)  ¬AD+ . (3) There is an ω1 + 1-iterable mouse with a Woodin limit of Woodins. The following is a restatement of clause 1 of the Main Theorem in a somewhat different language. Corollary 0.1. M C holds in the minimal model of ADR + “Θ is regular”. Woodin showed that V = L(P(R)) + ADR + “Θ is regular” implies AD+ . We say M is the minimal model of ADR + “Θ is regular” if M is a transitive model of ADR + “Θ is regular” containing the ordinals and reals, and any other model of ADR + “Θ is regular” containing the ordinals and reals also contains M . It follows from clause 2 of the Main Theorem that if there is a model of ADR + “Θ is regular” then there is a minimal one. Prior to Corollary 0.1, Woodin, in unpublished work, showed that MSC holds if there is no inner model of AD+ + θω1 < θ. Here, θω1 is the ω1 th member of the Solovay sequence (see Definition 0.9). Neeman and Steel found a technical strengthening of Woodin’s result and Steel obtained equiconsistencies using this work (see [42]). Both of these hypothesis are more stringent than the hypothesis of Theorem 0.1. Our proof of the Main Theorem heavily relies on the analysis of HOD. Part of this analysis is to show that HOD is a hod premouse. The problem of analyzing HOD is the oldest and the most important project of descriptive inner model theory. In the next section we will explain its role and importance in the modern descriptive inner model theory. For an account of descriptive inner model theory aimed at non-experts see [27]. Acknowledgments. I would like to thank John Steel and Hugh Woodin whose prior work was the main source of inspiration for the work presented in this paper. Special thanks to John Steel for many wonderful conversations on inner model theory and descriptive set theory. I also would like to thank Dominik Adolf, Paul Larson, Farmer Schlutzenberg, John Steel, Nam Trang, Trevor Wilson and Yizheng Zhu for reading earlier versions of this paper and for providing many corrections and suggestions. I am hugely indebted to the referee for a very long list of crucial improvements and suggestions. The task assigned to the referee has been quite unreasonable yet the suggested improvements have been invaluable. I am very grateful to the National Science Foundation for supporting parts of this work through Grant No DMS-0902628 and through DMS-1201348. Finally, I would like to thank the Mathematisches Forschungsinstitut Oberwolfach for hosting me as a Leibnitz Fellow for several weeks in the Spring of 2012. Several parts of this book were written during this period. 0.1. Why analyze Hod The study of HOD under AD and later under AD+ was initiated by the Cabal group and early results on HOD can be found in Cabal Volumes ([15], [14], [9] and [10]). Nowadays the study of HOD under AD+ lies in the crossroads of two different subjects, pure descriptive set theory and inner model theory. It is by

Licensed to Rutgers Univ-New Brunswick. Prepared on Mon Dec 29 14:55:36 EST 2014for download from IP 192.12.88.137. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

0.1. WHY ANALYZE HOD

3

far the most important project of descriptive inner model theory. Over the years, number of deep results have been proved on the structure of HOD under AD and we take the following list of theorems as our starting point. Theorem 0.2. Assume AD. Then the following hold. (1) (Folklore) Suppose V = L(R). Then HOD  CH. (2) (Solovay, [8]) ω1 is measurable in HOD. (3) (Becker, [2]) ω1 is the least measurable cardinal of HOD. The theorem suggests that under AD, HOD has a rich structure. Woodin’s Derived Model Theorem, Theorem A.11, opened many doors for further explorations, and the following two theorems were proved soon after. Given a set X, we let trc(X) be the ⊆-least transitive set containing X. A set X is called self-wellordered if there is a well-ordering of trc(X) in Jω (trc(X)).3 Given a self-wellordered set X   is a coherent we say M is a mouse over X if M has the form Jα [E][X] where E extender sequence (see [43]). The distinction between “a mouse” and “a mouse over X” is the same as the distinction between J and J [X]. Notice that every real is self-wellordered. Let Mω (y) be the least4 class size y-mouse with ω Woodin cardinals. Given a real y, we say “Mω (y) exists” if Mω (y) exists as a class and it is κ-iterable for all κ. Theorem 0.3 (Woodin, [43]). Suppose Mω exists. Then AD holds in L(R). Theorem 0.4 (Steel-Woodin, [43]). Suppose Mω exists. Then in L(R), x is ordinal definable iff x is in some mouse. Moreover, the following statements are equivalent where x, y ∈ R. (1) L(R)  “x is ordinal definable from y”. (2) x ∈ Mω (y). Corollary 0.5. Assume Mω exists. Let H = HODL(R) . Then R H = R Mω . Notice that Theorem 0.4 is an instance of capturing definability via mice. Since CH holds in every mouse, Corollary 0.5 gives another proof of 1 of Theorem 0.2. In fact it gives a deeper structural characterization of the reals of HOD prompting the question: is there such a characterization for the entire HOD? Below we collect some questions motivated by the theorems stated so far. Question 0.6. Assume AD+ + V = L(P(R)). (1) (2) (3) (4) (5)

Does HOD  CH? Does HOD  GCH? Is it true that the reals of HOD are the reals of some mouse? What kind of large cardinals does HOD have? What is the structure of HOD?

3 We warn the reader that often, especially when discussing mice, we will use Jensen’s hierarchy J rather then the customary constructible hierarchy L. However, the reader can easily substitute L for J without losing much. 4 Here “least” means that it is the hull of a club of indiscernibles.

Licensed to Rutgers Univ-New Brunswick. Prepared on Mon Dec 29 14:55:36 EST 2014for download from IP 192.12.88.137. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

4

INTRODUCTION

The second question is an important open problem, and the third question is just a different way of asking if MSC is true. Questions 4 and 5 are somewhat vague and are part of ongoing research. Theorem 0.10 is a partial answer to the fourth question. We refer the reader to [27] for more on these two questions. The following surprising theorem of Woodin and the proof of Theorem 0.2 imply that the answer to the first question is yes. Theorem 0.7 (Woodin). Assume AD+ + V = L(P(R)). Then the set A = {(x, y) ∈ R2 : x ∈ OD(y)} is universal Σ21 . Moreover, for some κ there is a tree T ∈ HOD on ω × κ such that A = p[T ]. Corollary 0.8. Assume AD+ + V = L(P(R)). Then HOD  CH. Proof. Let A = {x ∈ R : x ∈ OD}. For x ∈ A, let (φx , α  x ) be the lexico x via φx . Let then ≤∗ be the graphically (≤lex ) least such that x is definable from α OD wellordering of A given by y ≤∗ x iff (φy , αy ) ≤lex (φx , αx ). It is a consequence of Theorem 0.7 that there is T ∈ HOD such that p[T ] =≤∗ . But then by a result of Mansfield and Solovay (see Theorem 14.7 of [8]), in HOD, for every x ∈ A, the set {y ∈ A : y ≤∗ x} has the perfect set property. This then easily gives that HOD  CH.  Questions 2 and 3 above led the way towards analysis of HOD under AD+ +V = L(P(R)). Section 3 of [27] gives a historical account of what was proved next. In particular, we recommend that the interested reader consult Theorem 3.11 of [27]. In what follows we will give some modern motivations for analyzing HOD of models of AD+ + V = L(P(R)). The inner model problem: The project of analyzing HOD is embedded so deeply in inner model theory that it is hard to isolate the most significant impact that it will have when it is fully carried out. The following is one possible chain of implications that lead to analysis of HOD. Suppose we would like to attack the inner model problem for superstrong cardinals5 , i.e., we would like to construct an ω1 + 1-iterable mouse that satisfies “there is a superstrong cardinal”. It follows from the work of Mitchell and Steel (see [20]) that to solve the inner model problem for a superstrong cardinal it is enough to solve some form of the iterability problem, one form of which says that countable submodels of rank initial segments of V are ω1 + 1-iterable. Next, to solve the iterability problem we need to describe a method of choosing branches for iteration trees of length ω1 . It is a well-known theorem that just in ZF C there are ω1 -trees with no ω1 -branch. One natural idea for doing this is to construct absolutely definable ω1 -iteration strategies for countable submodels of rank initial segments of V , since then we can use genericity to construct branches for uncountable trees. A natural attempt is to guarantee that the resulting ω1 strategy is universally Baire. To keep track of the complexity of arising iteration strategies and to guarantee that the resulting strategies are universally Baire, we need to study the descriptive set theoretic properties of these ω1 -strategies, and this is how descriptive set theory gets into the picture. 5 We only mention superstrong cardinals as the general theory of mice for supercompact cardinals at the moment isn’t well understood.

Licensed to Rutgers Univ-New Brunswick. Prepared on Mon Dec 29 14:55:36 EST 2014for download from IP 192.12.88.137. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

0.1. WHY ANALYZE HOD

5

Thus, we have identified the problem we want to solve. Fix an ordinal γ and let π : M → Vγ be a countable substructure of Vγ . We would like to construct an ω1 iteration strategy for M in such a way that the set of reals coding it is universally Baire. One particularly strong intuition is that, assuming proper class of Woodin cardinals, we should be able to construct an iteration strategy for M by analyzing the kind of mice that exist in the determinacy model given by the universally Baire sets, i.e., in the model L(ΓuB , R) where ΓuB is the collection of the universally Baire sets of reals. This intuition is based on the idea that mice in L(ΓuB , R) can serve as Q-structures for countable iteration trees that are based on M (see Definition 1.22 for the definition of a Q-structure). And this later idea is based on the fact that the ω1 fragments of the iteration strategies of all iterable mice are in L(ΓuB , R). Here is then a plausible line of thought. Assume that M is not ω1 + 1-iterable. Attempt to iterate it via a strategy that is guided by canonical information, such as Q-structures, that is coded into L(ΓuB , R). The failure to define such a strategy should translate into the fact that L(ΓuB , R) has complicated descriptive set theoretic structure and hence, also complicated ω1 -iteration strategies. In particular, it has an ω1 -iteration strategy for a mouse with a superstrong. To complete this vague idea we need to have methods that translate descriptive set theoretic strength into inner model theoretic strength, and this is where the Mouse Set Conjecture makes its entrance. It essentially says that complicated models of determinacy have complicated mice. All known proofs of MSC, including the one given in this paper, are by induction. Assume AD+ + V = L(P(R)). We let Γ be the largest Wadge initial segment of P(R) which satisfies MC, i.e., given any x, y ∈ R, x is OD(y) as witnessed by a structure coded by a set of reals in Γ if and only if there is a sound y-mouse M such that x ∈ M and M has an ω1 -iteration strategy coded by a set in Γ. Clearly we want to show that Γ = P(R). Towards a contradiction assume not. Let then A be a set of reals of Wadge rank sup{w(B) : B ∈ Γ} where w(B) is the Wadge ordinal of B. The idea now is to “capture” A inside some mouse. Here “capture” is used in the same sense as when one captures a Σ12 set by a tree in L. It is hard to work with A as it is just devoid of any meaningful structure. Instead, we would like to produce a pair (P, Σ) such that P is some fine structural model, in our case a hod mouse, Σ is an ω1 -iteration strategy for P and if C ⊆ R is a set coding Σ in some natural way then w(C) ≥ w(A). This step is what is called the generations of pointclasses (see the discussion before Conjecture 0.14). Next, as now C is a very meaningful set, we attempt to identify it in various mice and eventually show that it is captured by a mouse. This step is called capturing of hod pairs (see the discussion before Conjecture 0.15). To produce the pair (P, Σ) we use the analysis of HOD of Γ (i.e., the initial segment of HOD coded by sets of reals in Γ). We show that there is a pair (P, Σ) as above with the property that the direct limit of all iterates of P converges to HOD of Γ. It then easily follows that Σ ∈ Γ as Σ gives a surjection f : R → ΘΓ where ΘΓ is the least ordinal that is not a surjective image of the reals. The map f is constructed by considering the various iteration embeddings arising from the direct limit construction (for more details see the discussion after Theorem 0.23). This step is done by proving Conjecture 0.13. Section 0.4 has some of the technical

Licensed to Rutgers Univ-New Brunswick. Prepared on Mon Dec 29 14:55:36 EST 2014for download from IP 192.12.88.137. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

6

INTRODUCTION

details that arise while carrying out the arguments presented in this outline. Core model induction: This is a method, originally due to Woodin, for constructing determinacy models from various hypothesis such as P F A, ¬κ for singular κ, the unique branch hypothesis and etc. It is the most successful tool used to calibrate the set theoretic strength of such statements. We refer the reader to [24], [29], [31] and [41] for some applications of the core model induction. While doing core model induction one encounters two steps that can be characterized as internal and external. In the internal step of the induction, there is a concrete model M and our goal is to show that M  AD+ . M is usually a fine structural model over R. In [41], it is just L(R) while in [24], it is the stack of all countably iterable6 mice over R. While doing this step of the induction, at various stages, we have to translate coarse sets of reals into iteration strategies. Just like in the case of the inner model problem, we use MSC and some weaker form of generation of pointclasses to do this. The details can be found in [31]. The external step of the core model induction amounts to building a new set that is beyond the sets constructed thus far. Let M be the model of the determinacy constructed thus far. The goal is to produce a set of reals A that is not in M and such that L(A, R)  AD+ . To do this, we attempt to construct A as a pair (P, Σ) that has the same properties as (P, Σ) used above to generate Γ (in our current case Γ = P(R)). This is exactly where the analysis of HOD is used in core model induction applications. A reader interested in the technical details may consult [24], [29] or [31]. Next, we give a crash course on hod mice to give some flavor of what to expect in the next few chapters. We then use our exposition to outline the proof of MSC (see Section 0.4). 0.2. A crash course on hod mice Hod mice, which are specifically designed to compute HOD of models of AD+ , feature prominently in the proof of the Main Theorem. One of the motivations behind their definition is Theorem 0.10. A hod mouse, besides having an extender sequence, is also closed under the iteration strategies of its own initial segments.   is a fine extender Thus it is a model of the form P = JαE,Σ where α is an ordinal, E sequence and Σ is a strategy of the initial segments of P. These initial segments, or rather the ordinals defining these initial segments, are called layers. Layers are used to keep track of the stages where new strategies appear in the model. More precisely, given a hod premouse P, η is called a layer of P if the strategy of P|η  appears in the model after stage η. Here P|η = JηE,Σ . There is one important exception. All hod mice have a last layer for which no strategy is activated. All hod mice satisfy ZF C − Replacement and they have exactly ω-more cardinals above the last layer. See Figure 0.2.1 for a generic picture of a hod premouse. Unlike ordinary mice, the hierarchy of hod mice grows according to the Solovay hierarchy, which is a determinacy hierarchy. First we let Θ = sup{α : there is a surjection f : R → α}. It is not hard to show that the length of Wadge order is Θ. We can now define the Solovay sequence as follows: 6 This

means that countable substructures are ω1 + 1-iterable.

Licensed to Rutgers Univ-New Brunswick. Prepared on Mon Dec 29 14:55:36 EST 2014for download from IP 192.12.88.137. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

0.2. A CRASH COURSE ON HOD MICE

7

δλ ** \ **  Σ