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Large Cardinals from Determinacy Peter Koellner and W Large Cardinals from Determinacy Peter Koellner and W. Hugh Woodin Contents 1Introduction ....................... 2 1.1 Determinacy and Large Cardinals . 2 1.2 Notation ......................... 16 2 BasicResults....................... 18 2.1 Preliminaries. .. .. .. .. .. .. 18 2.2 BoundednessandBasicCoding . 22 2.3 Measurability ...................... 26 2.4 TheLeastStable .................... 32 2.5 Measurability of the Least Stable . 40 3Coding........................... 44 3.1 CodingLemma ..................... 44 3.2 UniformCodingLemma . 49 3.3 Applications ....................... 55 R 4 A Woodin Cardinal in HODL( ) ............ 58 4.1 Reflection ........................ 59 4.2 StrongNormality . 72 4.3 AWoodinCardinal . 91 5 Woodin Cardinals in General Settings . 95 5.1 FirstAbstraction . 97 5.2 Strategic Determinacy . 100 1 1. Introduction 2 5.3 Generation Theorem . 109 5.4 SpecialCases ......................137 6 Definable Determinacy . 145 6.1 Lightface Definable Determinacy . 146 6.2 Boldface Definable Determinacy . 169 7 Second-Order Arithmetic . 177 7.1 First Localization . 179 7.2 Second Localization . 186 8 FurtherResults ..................... 187 8.1 Large Cardinals and Determinacy . 188 8.2 HOD-Analysis . .192 Bibliography......................... 199 1. Introduction In this chapter we give an account of Woodin’s technique for deriving large cardinal strength from determinacy hypotheses. These results appear here for the first time and for this reason we have gone into somewhat more detail than is customary in a handbook. All unattributed results that follow are either folklore or due to Woodin. 1.1. Determinacy and Large Cardinals In the era of set theory following the discovery of independence a major concern has been the discovery of new axioms that settle the statements left undecided by the standard axioms (ZFC). One interesting feature that has emerged is that there are often deep connections between axioms that spring from entirely different sources. In this chapter we will be concerned with one instance of this phenomenon, namely, the connection between axioms of definable determinacy and large cardinal axioms. In this introduction we will give a brief overview of axioms of definable determinacy and large cardinal axioms (in sections A and B), discuss their 1. Introduction 3 interconnections (in sections C and D), and give an overview of the chapter (in section E). At some points we will draw on notation and basic notions that are explained in fuller detail in sections 1.2 and 2.1. A. Determinacy For a set of reals A ⊆ ωω consider the game where two players take turns playing natural numbers: I x(0) x(2) x(4) . II x(1) x(3) . At the end of a round of this game the two players will have produced a real x, obtained through “interleaving” their plays. We say that Player I wins the round if x ∈ A; otherwise Player II wins the round. The set A is said to be determined if one of the players has a “winning strategy” in the associated game, that is, a strategy which ensures that the player wins a round regardless of how the other player plays. The Axiom of Determinacy (AD) is the statement that every set of reals is determined. It is straightforward to see that very simple sets are determined. For example, if A is the set of all reals then clearly I has a winning strategy; if A is empty then clearly II has a winning strategy; and if A is countable then II has a winning strategy (by “diagonalizing”). A more substantive result is that if A is closed then one player must have a winning strategy. This might lead one to expect that all sets of reals are determined. However, it is straightforward to use the Axiom of Choice (AC) to construct a non- determined set (by listing all winning strategies and “diagonalizing” across them). For this reason AD was never really considered as a serious candidate for a new axiom. However, there is an interesting class of related axioms that are consistent with AC, namely, the axioms of definable determinacy. These axioms extend the above pattern by asserting that all sets of reals at a given 1 level of complexity are determined, notable examples being, ∆∼ 1-determinacy (all Borel sets of reals are determined), PD (all projective sets of reals are R determined) and ADL( ) (all sets of reals in L(R) are determined). One issue is whether these are really new axioms or whether they follow from ZFC. In the early development of the subject the result on the deter- minacy of closed sets was extended to higher levels of definability. These 1 developments culminated in Martin’s proof of ∆∼ 1-determinacy in ZFC. It turns out that this result is close to optimal—as one climbs the hierarchy 1. Introduction 4 1 of definability, shortly after ∆1 one arrives at axioms that fall outside the ∼ R provenance of ZFC. For example, this is true of PD and ADL( ). Thus, R we have here a hierarchy of axioms (including PD and ADL( )) which are genuine candidates for new axioms. There are actually two hierarchies of axioms of definable determinacy, one involving lightface notions of definability (by which we mean notions 1 (such as ∆2) that do not involve real numbers as parameters) and the other involving boldface notions of definability (by which we mean notions (such 1 as ∆∼ 2) that do involve real numbers as parameters). (See Jackson’s chapter in this Handbook for details concerning the various grades of definability and the relevant notation.) Each hierarchy is, of course, ordered in terms of increasing complexity. Moreover, each hierarchy has a natural limit: the natural limit of the lightface hierarchy is OD-determinacy (all OD sets of reals are determined) and the natural limit of the boldface hierarchy is OD(R)- determinacy (all OD(R) sets of reals are determined). The reason these are natural limits is that the notions of lightface and boldface ordinal definability are candidates for the richest lightface and boldface notions of definability. To see this (for the lightface case) notice first that any notion of definability which does not render all of the ordinals definable can be transcended (as can be seen by considering the least ordinal which is not definable according to the notion) and second that the notion of ordinal definability cannot be so transcended (since by reflection OD is ordinal definable). It is for this reason that G¨odel proposed the notion of ordinal definability as a candidate for an “absolute” notion of definability. Our limiting cases may thus be regarded as two forms of absolute definable determinacy. So we have two hierarchies of increasingly strong candidates for new ax- ioms and each has a natural limit. There are two fundamental questions concerning such new axioms. First, are they consistent? Second, are they true? In the most straightforward sense these questions are asked in an ab- solute sense and not relative to a particular theory such as ZFC. But since we are dealing with new axioms, the traditional means of answering such questions—namely, by establishing their consistency or provability relative to the standard axioms (ZFC)—is not available. Nevertheless, one can hope to establish results—such as relative consistency and logical connections with respect to other plausible axioms—that collectively shed light on the origi- nal, absolute question. Indeed, there are a number of results that one can R bring to bear in favour of PD and ADL( ). For example, these axioms lift the structure theory that can be established in ZFC to their respective domains, 1. Introduction 5 namely, second-order arithmetic and L(R). Moreover, they do so in a fash- ion which settles a remarkable number of statements that are independent of ZFC. In fact, there is no “natural” statement concerning their respective domains that is known to be independent of these axioms. (For more on the structure theory provided by determinacy and the traditional considerations in their favour see [10] and for more recent work see Jackson’s chapter in this Handbook.) The results of this chapter figure in the case for PD and R ADL( ). However, our concern will be with the question of relative consis- tency; more precisely, we wish to calibrate the consistency strength of axioms of definable determinacy—in particular, the ultimate axioms of lightface and boldface determinacy—in terms of the large cardinal hierarchy. There are some reductions that we can state at the outset. In terms of consistency strength the two hierarchies collapse at a certain stage: Kechris and Solovay showed that ZF + DC implies that in the context of L[x] for x ∈ ω 1 ω , OD-determinacy and ∆2-determinacy are equivalent (see Theorem 6.6). R And it is a folklore result that ZFC+OD(R)-determinacy and ZFC+ADL( ) are equiconsistent. Thus, in terms of consistency strength, the lightface 1 hierarchy collapses at ∆2-determinacy and the boldface hierarchy collapses R at ADL( ). So if one wishes to gauge the consistency strength of lightface 1 and boldface determinacy it suffices to concentrate on ∆2-determinacy and R ADL( ). 1 Now, it is straightforward to see that if ∆2-determinacy holds then it R holds in L[x] for some real x and likewise if ADL( ) (or AD) holds then it holds in L(R). Thus, the natural place to study the consistency strength of lightface definable determinacy is L[x] for some real x and the natural place to study the consistency strength of boldface definable determinacy (or full determinacy) is L(R). For this reason these two models will be central in what follows. To summarize: We shall be investigating the consistency strength of light- 1 L(R) face and boldface determinacy. This reduces to ∆2-determinacy and AD . The settings L[x] and L(R) will play a central role.
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