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Mice with Finitely Many Woodin Cardinals from Optimal Determinacy Hypotheses

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Mice with Finitely Many Woodin Cardinals from Optimal Determinacy Hypotheses MICE WITH FINITELY MANY WOODIN CARDINALS FROM OPTIMAL DETERMINACY HYPOTHESES SANDRA MULLER,¨ RALF SCHINDLER, AND W. HUGH WOODIN Abstract We prove the following result which is due to the third author. 1 1 Let n ≥ 1. If Πn determinacy and Πn+1 determinacy both hold true and 1 # there is no Σn+2-definable !1-sequence of pairwise distinct reals, then Mn 1 exists and is !1-iterable. The proof yields that Πn+1 determinacy implies # that Mn (x) exists and is !1-iterable for all reals x. A consequence is the Determinacy Transfer Theorem for arbitrary n ≥ 1, 1 (n) 2 1 namely the statement that Πn+1 determinacy implies a (< ! − Π1) de- terminacy. Contents Abstract 1 Preface 2 Overview 4 1. Introduction 6 1.1. Games and Determinacy 6 1.2. Inner Model Theory 7 1.3. Mice with Finitely Many Woodin Cardinals 9 1.4. Mice and Determinacy 14 2. A Model with Woodin Cardinals from Determinacy Hypotheses 15 2.1. Introduction 15 2.2. Preliminaries 16 2.3. OD-Determinacy for an Initial Segment of Mn−1 32 2.4. Applications 40 2.5. A Proper Class Inner Model with n Woodin Cardinals 44 3. Proving Iterability 49 First author formerly known as Sandra Uhlenbrock. The first author gratefully ac- knowledges her non-material and financial support from the \Deutsche Telekom Stiftung". The material in this paper is part of the first author's Ph.D. thesis written under the su- pervision of the second author. 1 2 SANDRA MULLER,¨ RALF SCHINDLER, AND W. HUGH WOODIN 3.1. Existence of n-suitable Premice 50 3.2. Correctness for n-suitable Premice 63 3.3. Outline of the Proof 68 3.4. Making use of the Right Hypothesis 69 # 1 3.5. M2n−1(x) from Boldface Π2n Determinacy 71 # 1 3.6. M2n(x) from Boldface Π2n+1 Determinacy 85 4. Conclusion 117 4.1. Applications 118 4.2. Open problems 118 References 118 Preface In 1953 David Gale and Frank M. Stewart developed a basic theory of infinite games in [GS53]. For every set of reals, that means set of sequences of natural numbers, A they considered a two-player game G(A) of length !, where player I and player II alternately play natural numbers. They defined that player I wins the game G(A) if and only if the sequence of natural numbers produced during a run of the game G(A) is contained in A and otherwise player II wins. Moreover the game G(A) or the set A itself is called determined if and only if one of the two players has a winning strategy, that means a method by which they can win, no matter what their opponent does, in the game described above. Already in [GS53] the authors were able to prove that every open and every closed set of reals is determined using ZFC. But furthermore they proved that determinacy for all sets of reals contradicts the Axiom of Choice. This leads to the natural question of what the picture looks like for definable sets of reals which are more complicated than open and closed sets. After some partial results by Philip Wolfe in [Wo55] and Morton Davis in [Da64], Donald A. Martin was finally able to prove in [Ma75] from ZFC that every Borel set of reals is determined. In the meantime the development of so called Large Cardinal Axioms was proceeding in Set Theory. In 1930 Stanis law Ulam first defined measurable cardinals and at the beginning of the 1960's H. Jerome Keisler [Kei62] and Dana S. Scott [Sc61] found a way of making them more useful in Set Theory by reformulating the statements using elementary embeddings. About the same time, other set theorists were able to prove that Determi- nacy Axioms imply regularity properties for sets of reals. More precisely Banach and Mazur showed that under the Axiom of Determinacy (that means every set of reals is determined), every set of reals has the Baire property. Mycielski and Swierczkowski proved in [MySw64] that under the MICE FROM OPTIMAL DETERMINACY HYPOTHESES 3 same hypothesis every set of reals is Lebesgue measurable. Furthermore Davis showed in [Da64] that under this hypothesis every set of reals has the perfect set property. Moreover all three results also hold if the De- terminacy Axioms and regularity properties are only considered for sets of reals in certain pointclasses. This shows that Determinacy Axioms have a large influence on the structure of sets of reals and therefore have a lot of applications in Set Theory. In 1965 Robert M. Solovay was able to prove these regularity properties 1 for a specific pointclass, namely Σ2, assuming the existence of a measurable cardinal instead of a Determinacy Axiom (see for example [So69] for the per- fect set property). Then finally Donald A. Martin was able to prove a direct connection between Large Cardinals and Determinacy Axioms: he showed in 1970 that the existence of a measurable cardinal implies determinacy for every analytic set of reals (see [Ma70]). Eight years later Leo A. Harrington established that this result is in some sense optimal. In [Ha78] he proved that determinacy for all analytic sets implies that 0#, a countable active mouse which can be obtained from a mea- surable cardinal, exists. Here a mouse is a fine structural, iterable model. Together with Martin's result mentioned above this yields that the two state- ments are in fact equivalent. This of course motivates the question of whether a similar result can be ob- tained for larger sets of reals, so especially for determinacy in the projective hierarchy. The right large cardinal notion to consider for these sets of reals was introduced by the third author in 1984 and is nowadays called a Woodin cardinal. Building on this, Donald A. Martin and John R. Steel were able to prove in [MaSt89] almost twenty years after Martin's result about ana- lytic determinacy that, assuming the existence of n Woodin cardinals and 1 a measurable cardinal above them all, every Σn+1-definable set of reals is determined. In the meantime the theory of mice was further developed. At the level of strong cardinals it goes back to Ronald B. Jensen, Robert M. Solovay, Tony Dodd and Ronald B. Jensen, and William J. Mitchell. Then it was further extended to the level of Woodin cardinals by Donald A. Martin and John R. Steel in [MaSt94] and William J. Mitchell and John R. Steel in [MS94], where some errors were later repaired by John R. Steel, Martin Zeman, and the second author in [SchStZe02]. Moreover Ronald B. Jensen developed another approach to the theory of mice at the level of Woodin cardinals in [Je97]. In 1995 Itay Neeman was able to improve the result from [MaSt89] in [Ne95]. # He showed that the existence and !1-iterability of Mn , the minimal count- able active mouse at the level of n Woodin cardinals, is enough to obtain n 2 1 that every a (< ! − Π1)-definable set of reals is determined. Here \a" is a quantifier which is defined by a game. Neeman's result implies that if for # all reals x the premouse Mn (x) exists and is !1-iterable, then in particular 4 SANDRA MULLER,¨ RALF SCHINDLER, AND W. HUGH WOODIN 1 every Σn+1-definable set of reals is determined. For odd n this latter result was previously known by the third author. The converse of this latter result was announced by the third author in the 1980's but a proof has never been published. The goal of this paper is to finally provide a complete proof of his result. Overview The purpose of this paper is to give a proof of the following theorem, which connects inner models with Woodin cardinals and descriptive set theory at projective levels in a direct level-wise way. This theorem is due to the third author and announced for example in the addendum (x5) of [KW08] and in Theorem 5:3 of [Sch10], but so far a proof of this result has never been published. 1 Theorem 2.1. Let n ≥ 1 and assume Πn+1 determinacy holds. Then # ! Mn (x) exists and is !1-iterable for all x 2 !. The converse of Theorem 2.1 also holds true and is for odd n due to the third author (unpublished) and for even n due to Itay Neeman (see [Ne95]). From this we can obtain the following corollary, which is Theorem 1:10 in [KW08] for odd n. We will present a proof of the Determinacy Transfer Theorem for the even levels n as a corollary of Theorem 2.1 in Section 4.1, using Theorem 2:5 in [Ne95] due to Itay Neeman. 1 Corollary 4.1 (Determinacy Transfer Theorem). Let n ≥ 1. Then Πn+1 (n) 2 1 determinacy is equivalent to a (< ! − Π1) determinacy. In fact we are going to prove the following theorem which will imply Theorem 2.1. 1 Theorem 3.14. Let n ≥ 1 and assume there is no Σn+2-definable !1- 1 sequence of pairwise distinct reals. Moreover assume that Πn determinacy 1 # and Πn+1 determinacy hold. Then Mn exists and is !1-iterable. This is a part of the following theorem. 1 Theorem 3.13. Let n ≥ 1 and assume there is no Σn+2-definable !1- sequence of pairwise distinct reals. Then the following are equivalent. 1 1 (1) Πn determinacy and Πn+1 determinacy, ! # # (2) for all x 2 !, Mn−1(x) exists and is !1-iterable, and Mn exists and is !1-iterable, # (3) Mn exists and is !1-iterable.
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