Contributions to Descriptive Inner Model Theory by Trevor Miles Wilson A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor John R. Steel, Chair Professor W. Hugh Woodin Professor Sherrilyn Roush Fall 2012 Abstract Contributions to Descriptive Inner Model Theory by Trevor Miles Wilson Doctor of Philosophy in Mathematics University of California, Berkeley Professor John R. Steel, Chair Descriptive inner model theory is the study of connections between descript- ive set theory and inner model theory. Such connections form the basis of the core model induction, which we use to prove relative consistency results relating strong forms of the Axiom of Determinacy with the existence of a strong ideal on }!1 (R) having a certain property related to homogeneity. The main innovation is a unified approach to the \gap in scales" step of the core model induction. 1 Contents Introduction iii Acknowledgements v Chapter 1. Forcing strong ideals from determinacy 1 1.1. The theory \ADR + Θ is regular" 1 1.2. Col(!; R)-generic ultrapowers 2 1.3. A covering property for ideals in generic extensions 5 1.4. The covering property for NS!1;R 8 1.5. A c-dense ideal with the covering property 10 Chapter 2. The core model induction 13 2.1. Model operators 14 2.2. F -mice 18 2.3. The Kc;F construction and the KF existence dichotomy 22 F;] 2.4. M1 from a strong pseudo-homogeneous ideal 28 2.5. The coarse and fine-structural mouse witness conditions 33 2.6. Hyperprojective determinacy from a strong pseudo-homogeneous ideal 34 Chapter 3. The next Suslin cardinal in ZF + DCR 37 3.1. A local notion of ordinal definability 37 3.2. The envelope of a pointclass 40 3.3. Digression: gaps in L(R) 44 3.4. The models L[T; x] 47 3.5. Countably complete measures and towers 50 3.6. Semi-scales with norms in the envelope 53 + 3.7. Digression: ADR from divergent models of AD 56 3.8. Digression: ADR in certain derived models 56 Chapter 4. Sealing the envelope 61 4.1. Scales with norms in the envelope 61 4.2. Local term-capturing 64 4.3. Self-justifying systems and condensation 66 4.4. Sealing the envelope with a strong pseudo-homogeneous ideal 69 4.5. AD from a strong pseudo-homogeneous ideal 72 4.6. AD + θ0 < Θ from a strong pseudo-homogeneous ideal 74 i Chapter 5. Sealing the envelope: an alternative method 79 5.1. Quasi-iterable pre-mice and M1 79 5.2. Full hulls 80 5.3. The full factors property 82 5.4. Application to strong pseudo-homogeneous ideals 85 Bibliography 87 Index 91 ii Introduction We will prove the following relative consistency statements. Main Theorem. (1) Assuming ZF + ADR + \Θ is regular," there is a forcing extension where ZFC holds, (a) The nonstationary ideal NS!1;R is strong and pseudo-homogeneous, and (b) There is a c-dense pseudo-homogeneous ideal on }!1 (R). (2) Assuming ZFC and the existence of a strong pseudo-homogeneous ideal on }!1 (R), there is an inner model of ZF + AD + θ0 < Θ containing all the reals and ordinals. The theories ADR + “Θ is regular" and AD + θ0 < Θ are both natural strengthenings of AD, the Axiom of Determinacy. Strength is a property of ideals introduced in [1] that is intermediate between precipitousness and pre-saturation. Pseudo-homogeneity is a property of ideals introduced in Chapter 1 that is similar to homogeneity except that it pertains to the theory of the generic ultrapower rather than to that of the generic extension. The ideals in the conclusions of (1a) and (1b) both satisfy the hypothesis of (2), and in turn the model of AD + θ0 < Θ in the conclusion of (2) is a significant step toward constructing a model of ADR + “Θ is regular" and thereby acheving equiconsistency. Throughout the text we assume the following base theory unless otherwise noted: Assume ZF + DCR: So a theorem is a theorem of ZF + DCR, and AD means AD + ZF + DCR, for example. Part (1) of the Main Theorem, the forcing direction, is established in Chapter 1. We define a fairly general class of forcing notions P that can be used to force the conclusion of (1). This class of forcing notions contains P = Col(!1; R) and P = Pmax, showing respectively that we can add CH or :CH to the conclusion. These forcing extensions satisfy c-DC and we get full AC by a second forcing with Col(Θ;}(R)) whose role in the argument is negligible. We show that these forcing extensions have ideals I with the ordinal covering property, which says that for every function }!1 (R) ! Ord there are densely many I-positive sets on which the function agrees with a function in the ground model. The ordinal covering property in turn can be used to show that I is strong, and together with the homogeneity of P, to show that I is pseudo-homogeneous. The remaining chapters are all devoted to establishing the inner model direction (2) of the Main Theorem. From a strong pseudo-homogeneous ideal on }!1 (R) we construct an inner model of AD + θ0 < Θ via a core model induction. The basic idea of the core model induction is to analyze the extent of determinacy using mice with Woodin cardinals. The mice with Woodin cardinals themselves are obtained by core model theory, namely the Kc constructions of [40] and relativized versions that we call Kc;F constructions. Our core model induction argument is similar to that used to prove the following related theorem: Theorem (Ketchersid [18]). If ZFC + CH holds, the nonstationary ideal NS!1 is !1- dense below a stationary set, and the corresponding generic elementary embedding j Ord iii is independent of the generic filter, then there is an inner model of AD + θ0 < Θ containing all the reals and ordinals. By a theorem whose proof is outlined in Sargsyan's thesis [30], the conclusion of Ketcher- sid's theorem can be strengthened to the existence of an inner model of ADR +“Θ is regular" containing all the reals and ordinals. This produces an equiconsistency because Woodin has shown in unpublished work that the hypothesis can be forced from ADR + “Θ is regular" by a similar method to the one we use to prove (1) of the Main Theorem. The primary difference from Ketchersid's theorem is that our ideal is not assumed to be dense or even pre-saturated but merely strong. A generic ultrapower Ult(V; H) by a strong ideal may fail to contain all the reals of the generic extension V [H]. Although strong ideals were introduced thirty years ago in [1] it was not known how to derive significant large cardinal strength from them until recently. In [3] the existence of a strong ideal is shown to be equiconsistent with the existence of a Woodin cardinal. Another difference is that we take a new approach to the \gap in scales" case of the core model induction, which first occurs when going beyond hyperprojective determinacy. In Chapter 3, which can be read independently of the rest of the paper, we develop some descriptive-set-theoretic tools for analyzing a gap in scales. Then in Chapters 4 and 5 we present two parallel methods for \sealing" the gap; one using weakly homogeneous trees and the other using directed systems of quasi-iterable pre-mice. These methods are inter- changable in our argument except that the latter depends on a conjecture (Conjecture 5.1.2) and so it is not officially part of the proof of the Main Theorem. Changing the approach to the \gap in scales" case is not necessary to prove the Main Theorem but we have chosen to do so because it results in a substantial simplification of the argument. In particular, it erases the distinctions between weak and strong gaps, and between gaps that end inside a premouse over R and those that do not. The Main Theorem is a step toward our original goal, which was to find a theory sat- isfied by the Pmax extension of a model of ADR + “Θ is regular" that is equiconsistent with the theory ADR + “Θ is regular." Originally it was anticipated that this theory would in- volve forcing axioms and properties of the nonstationary ideal on !1 because these are the statements that Pmax was designed to force. For example, under ADR + “Θ is regular" the forcing axiom MM(c), which is Martin's Maximum for posets no larger than the continuum, is forced by Pmax. In turn MM(c) implies that AD holds in L(R) (Steel{Zoble [38]) and is expected to be stronger, possibly equiconsistent with ADR + “Θ is regular" itself. However, determining the consistency strength of MM(c) appears to be a hard problem because the known methods for getting a model of AD + θ0 < Θ, including those of Chapters 4 and 5, do not seem to apply. iv Acknowledgements I would like to thank my advisor, John Steel, for always listening patiently and for gently correcting my frequent mistakes. I also thank Hugh Woodin for explaining his proof of Theorem 3.6.5 and allowing me to include it here, and Paul Larson for several corrections on earlier drafts. Finally, I thank Alekos Kechris for introducing me to research in set theory. v CHAPTER 1 Forcing strong ideals from determinacy In this chapter we prove the forcing direction (1) of the Main Theorem.