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Extensions of the Axiom of Determinacy Paul B. Larson September 22, 2021 2 Contents 0.1 Introduction . 5 0.2 Notation . 7 0.3 Prerequisites . 9 0.4 Forms of Choice . 9 0.5 Partial orders . 9 I Preliminaries 11 1 The Axiom of Determinacy 13 1.1 Turing determinacy . 15 2 The Wadge hierarchy 19 2.1 Lipschitz determinacy . 19 2.2 Lipschitz degrees and Wadge degrees . 21 2.3 Pointclasses . 26 2.4 Universal sets . 28 2.5 The cardinal Θ . 33 3 Coding Lemmas 37 4 Properties of pointclasses 43 4.1 Separation and reduction . 43 4.2 The prewellordering property . 48 4.3 Prewellorderings and wellfounded relations . 52 4.4 Closure under wellordered unions . 54 5 Strong Partition Cardinals 57 5.1 The Martin Conditions . 59 6 Suslin sets and Uniformization 63 6.1 Uniformization ............................. 68 6.2 The Kunen-Martin property . 69 6.3 A pointclass fact . 69 6.4 The Solovay sequence . 70 3 4 CONTENTS II AD+ 73 7 <Θ-determinacy 75 8 Cone measures 81 8.1 Measurable cardinals from cone measures . 85 8.2 The degree order on sets of ordinals . 86 8.3 Pointed trees . 91 8.4 Coding ultrapowers . 92 8.5 Forcing with positive sets . 93 9 1-Borel sets 97 9.1 Local 1-Borel codes . 103 10 Vopˇenka algebras 107 10.1 Codes for projections, and Uniformization . 111 10.2 Vopˇenka algebras and 1-Borel sets . 116 10.3 1-Borel subsets of P(δ) . 120 11 Applications 125 11.1 Strong 1-Borel codes . 125 11.2 Producing strong 1-Borel codes . 127 11.3 1-Borel representations from Uniformization . 130 11.4 Closure of the Suslin cardinals . 132 12 Scales from Uniformization 137 12.1 Ordinal determinacy in the codes . 137 12.2 Boundedness for 1-Borel relations . 141 12.3 Becker's argument . 142 13 ADR 149 13.1 Weakly homogeneous trees . 150 13.2 Normal measures . 153 13.3 AD implies ADR if all sets are Suslin . 157 14 Questions 161 0.1. INTRODUCTION 5 This draft includes notes to myself, in footnotes. These should probably be ignored. Comments and corrections are more than welcome, although some parts are obviously very rough and these may not be worth commenting on, since they'll probably we rewritten. Chapters 1, 2 and 3 should be essentially in their final form, so please let me know when you see issues there. Ideally, I'll remember to post frequent updates. 0.1 Introduction The Axiom of Determinacy (AD) is the statement that all length-! integer games of perfect information are determined. The beginning of Chapter 1 contains a more precise definition, but we expect the reader to be familiar with the classical theory of determinacy, as found in [7, 9], for instance. The axiom AD+ is a generalization, due to W. Hugh Woodin, of the Axiom of Determinacy. We define it as the conjunction of three statements. 0.1.1 Definition. The axiom AD+ is the conjunction of the following three statements. 1. DCR 2. Every subset of !! is 1-Borel. 3. (<Θ-Determinacy) For all λ < Θ, every A ⊆ !! and every continuous function π : !λ ! !!, π−1(A) is determined (as a subset of !λ). The 1-Borel sets are defined in Definition 9.0.1 for subsets of 2!, and in Remark 9.0.8 for subsets of !!. The notion of continuity in <Θ-Determinacy refers to the discrete topology on λ, not the interval topology. The restriction of <Θ-Determinacy to the case where π is the identity function on !! is exactly AD. It is an open question whether AD implies any or all of the parts of AD+, and in fact whether AD plus any two parts of AD+ implies the third. It also + open whether ADR implies AD . The issue in this case is whether ADR implies <Θ-Determinacy, as DCR is easily seen to follow from ADR, and the second part + of AD follows from ADR (moreover, from AD + Uniformization) by Theorem 11.3.2. If M ⊆ N are models of AD with the same reals, and every set of reals in M is Suslin in N, then M j= AD+.1 In fact, this is the context which the axiom AD+ was designed to describe; its original name was \within scales". Here is a brief2 outline of what we aim to prove is this book (all due to Woodin): + • In L(R), AD implies AD . 1Reference the relevant theorems. 2and incomplete 6 CONTENTS • If AD+ holds then every inner model containing the reals satisfies AD+. • If AD+ holds then the set of Suslin cardinals is closed below Θ. This follows from Theorem 11.4.1.3 • Assuming DCR and <Θ-Determinacy, ADR is equivalent to the assertion that the Solovay sequence has limit length. This is shown in Section ??.4 + 2 • AD implies Σ1 reflection into the Suslin, co-Suslin (i.e., hom) sets • AD+ implies that the ultrapower of V by the Turing measure is well- founded. ! • AD + DCR implies that for all ordinals χ, every subset of ! in L(P(χ)) is 1-Borel. Theorem 0.1.2 (Woodin). If <Θ-Determinacy and Uniformization hold, then every subset of !! is Suslin. Proof. By Theorem 11.3.2, the hypotheses give that every subset of !! is 1- Borel. The theorem then follows from Corollary 10.1.7 and Theorem 11.2.1. The first sentence of the following theorem is due to Woodin. The implication from (4) to (1) under DC combines results of Woodin and Becker (see Chapter 12) Theorem 0.1.3. Each of the following statements implies the ones below it, and the first two statements are equivalent. If DC holds, then all four statements are equivalent. 1. AD + \every subset of !! is Suslin" 2. AD+ + \every subset of !! is Suslin" 3. ADR 4. AD + Uniformization Proof. To see that (3) implies (4), note that ADR implies Uniformization, via a game in which each player plays once. That (2) implies (1) follows from applying Ordinal Determinacy in the case λ = !. As discussed in Chapter 6, Suslin sets can be uniformized, and Uniformization implies DCR. The implication from (1) to (2) then follows from Theorems 5.0.2 and 7.0.3, and Remark 6.0.9. Theorem 13.0.1 says that (1) implies (3). That (4) implies (1) under DC is Theorem 12.3.1 (the cofinality of Θ being uncountable is the relevant consequence of DC). 5 3Get closure below the supremum just from AD; comment on the reversal. 4 Does this need DCR? 5I should also introduce this theorem properly. 0.2. NOTATION 7 ! Theorem 0.1.4. Assume that AD + DCR holds. Let Γ be the set of A ⊆ ! for + which L(A; R) j= AD . Then + L(Γ; R) j= AD ; ! 6 and, if Γ 6= P(! ), then L(Γ; R) j= DC + ADR. Proof78 The following theorem follows from part (1) of Corollary 10.3.7. ! Theorem 0.1.5. Assuming AD+DCR, a subset of ! is 1-Borel if and only if there is a set of ordinals S such that A 2 L(S; R). The following is Theorem 11.4.8. + Theorem 0.1.6. If AD holds, then ADR fails if and only if there is a set of ordinals T such that L(P(R)) = L(T; R). + Theorem 0.1.7. Assume that V = L(P(R)) and that AD holds. Then for all A ⊆ !!, either V = L(A; R), or A# exists.9 Theorem 0.1.8. If AD and DCR hold, then for every ordinal χ, every subset of !! inL (P(χ)) is 1-Borel. Part I of the book collects various facts (primarily from the Cabal seminar) which will be needed in Part II. A natural way to read the book would be to start at the beginning of Part II and refer back to Part I as needed. This book has a great deal of overlap with many sources, notably [6, 18, 19]. 0.2 Notation We reserve the symbol R for the real line, which is never used directly. However, we use traditional notation such as ADR, DCR, L(R) and so on, as these terms ! are equivalent to more relevant forms such as AD!! , DC!! and L(! ). We informally use the word \real" to mean an element of the Baire space !!. We use the following notational conventions. • We write Ord for the class of ordinals. • We write XY to mean the set of functions from Y to X, and, when γ is <γ S α an ordinal X to mean α2γ X . 6 ! What is the proof? Let χB be the 1-Borel threshhold. Then P(! ) \ L(P(χB)) is the 1-Borel sets. Look at a limit of Suslin cardinals; all sets are < Borel, so also Suslin? 7 ! Let χB be the universally Baire threshold. Then P(! ) \ L(P(χB)) = 1-Borel. 8Also : let λ be a limit of Suslin cardinals. All sets are < λ-Borel, so also Suslin (what did mean by this?? 9 Another statement to consider : The axiom ADR implies that the sharp of each subset of !! exists. 8 CONTENTS • For an ordinal α and a set of ordinals X,[X]α denotes the collection of subsets of X of ordertype α, and [X]<α denotes the collection of subsets of X of ordertype less than α. • Given a set X, we write 9X and 8X for existential and universal quantifi- cation over X, respectively. • A preorder on a set is a binary relation which is reflexive and transitive.
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