Hod Mice and the Mouse Set Conjecture

185 Pages·2014·1.912 MB·English

by  Grigor Sargsyan

MEMOIRS
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.
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MEMOIRS
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
Providence, Rhode Island
Licensed to Rutgers Univ-New Brunswick. Prepared on Mon Dec 29 14:55:36 EST 2014for download from IP 192.12.88.137.
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DOI:http://dx.doi.org/10.1090/memo/1111
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To Lilit
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Contents
Introduction 1
0.1. Why analyze Hod 2
0.2. A crash course on hod mice 6
0.3. The mouse set conjecture 8
0.4. The proof of MSC 8
0.5. The comparison theory of hod mice 10
0.6. Hod is a hod premouse 11
Chapter 1. Hod mice 15
1.1. Hybrid J-structures 15
1.2. Some fine structure 17
1.3. Iteration trees and iteration strategies 19
1.4. Layered strategy premice 22
1.5. Iterations of Σ-mice 23
1.6. Hull condensation 25
1.7. Hod mice 29
Chapter 2. Comparison theory of hod mice 35
2.1. Hod pair constructions 35
2.2. Iterability of hod pair constructions 38
2.3. Universality of the fully backgrounded constructions 40
2.4. Coarse Γ-Woodin mice 44
2.5. Comparison under AD+ 47
2.6. Positional and commuting iteration strategies 52
2.7. The diamond comparison argument 60
Chapter 3. Hod mice revisited 71
3.1. The internal theory of hod premice 71
3.2. OD-full pointclasses 79
3.3. The derived models of hod mice 82
3.4. An anomaly 86
3.5. Getting branch condensation 91
3.6. Generic comparisons 95
3.7. Reorganizing hod mice 99
3.8. S-constructions 100
Chapter 4. Analysis of HOD 105
4.1. Suitability 105
4.2. B-iterability 110
4.3. The direct limit of iterates of hod mice 111
v
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vi CONTENTS
4.4. The computation of Hod 113
Chapter 5. Hod pair constructions 119
5.1. Stacking mice 119
5.2. Clause 4 123
5.3. Fullness preservation 125
5.4. The comparison argument revisited 126
5.5. Branch condensation 127
5.6. Γ(P,Σ) when λP is successor 129
5.7. B-iterability 134
5.8. Strongly B(cid:3)-guided strategies 136
5.9. Summary 137
Chapter 6. A proof of the mouse set conjecture 141
6.1. The generation of the mouse full pointclasses 141
6.2. An analysis of stacks 144
6.3. Capturing of hod pairs 146
6.4. The mouse set conjecture 155
6.5. A last word 159
Appendix A. Descriptive set theory primer 161
A.1. Pointclasses 161
A.2. Env(Γ) 163
A.3. AD+ 163
A.4. The derived model theorem 165
Index 167
Bibliography 169
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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”.
ReceivedbytheeditorReceivedJune7,2011and,inrevisedform,March27,2014.
ArticleelectronicallypublishedonNovember6,2014.
DOI:http://dx.doi.org/10.1090/memo/1111
2010 MathematicsSubjectClassification. Primary03E15,03E45,03E60.
Key wordsand phrases. Mouse,innermodeltheory,descriptivesettheory,hodmouse.
(cid:2)c2014 American Mathematical Society
vii
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Introduction
The interplay between Levy’shierarchy and the canonical models of fragments
of ZFC has been known for many years. For instance, a real x is Δ1 definable
2
from a real y and a countable ordinal iff x ∈ L[y] (Solovay, [8]). Assuming, Δ1-
(cid:2)2
determinacy, a real x is Δ1 definable from a real y and a countable ordinal iff
3
x∈M#(y) (Steel-Woodin, see [39])1. That this phenomenon is always true is the
1
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 (MC) stands for the concluding statement of the
Mouse Set Conjecture. Notice that ordinal definability is the most robust form
ofdefinabilityandhence,MSC canbeviewedastheultimategeneralizationofthe
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 L (P(R)) ≺ V (Θ here and
Θ Σ1
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 Σ elementary in V. See
1
Theorem A.10 for the fundamental consequences of AD+. We note that assuming
V =L(R),AD is equivalent toAD+. Whether the equivalenceis always true is an
important open problem.
Finally,noticethatonecannothopetoproveMC fromZFC asitcanbeeasily
arrangedby forcingthatthereare morethanω many ordinal definable realswhile
1
ZFC +MC implies that there are ω many ordinal definable reals (this follows
1
from the comparison theorem for mice).
We will prove that MSC holds in the minimal model of ADR+“Θ is regular”.
Theorem6.26implies thatADR+“Θis regular”isweaker thanthe existenceof an
ω +1-iterablemousewithasuperstrongcardinal. Theproofisbasedonthetheory
1
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.
MainTheorem. Eachofthefollowingstatementsimpliesthatthereisaproper
class model containing the reals and satisfying ADR+“Θ is regular”.
1M#(y)istheminimalsoundactivey-mousethathasaWoodincardinal.
1
2OftentimesMSCisstatedundertheadditionalhypothesisthat“thereisnoω1+1-iterable
mousewithasuperstrongcardinal”. Thisisbecausecurrentlythegeneralnotionofmouseisnot
well-developed. SeeSection0.3forsomemorecommentsonthehypothesisofMSC.
1
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