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The Envelope of a Pointclass Under a Local Determinacy Hypothesis

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The Envelope of a Pointclass Under a Local Determinacy Hypothesis THE ENVELOPE OF A POINTCLASS UNDER A LOCAL DETERMINACY HYPOTHESIS TREVOR M. WILSON Abstract. Given an inductive-like pointclass Γ and assuming the Ax- iom of Determinacy, Martin identified and analyzede the pointclass con- taining the norm relations of the next semiscale beyond Γ, if one exists. We show that much of Martin's analysis can be carriede out assuming only ZF + DCR + Det(∆Γ). This generalization requires arguments from Kechris{Woodin [10] ande e Martin [13]. The results of [10] and [13] can then be recovered as immediate corollaries of the general analysis. We also obtain a new proof of a theorem of Woodin on divergent models of AD+, as well as a new result regarding the derived model at an inde- structibly weakly compact limit of Woodin cardinals. Introduction Given an inductive-like pointclass Γ, Martin introduced a pointclass (which we call the envelope of Γ, following [22])e whose main feature is that it con- tains the prewellorderingse of the next scale (or semiscale) beyond Γ, if such a (semi)scale exists. This work was distributed as \Notes on the nexte Suslin cardinal," as cited in Jackson [3]. It is unpublished, so we use [3] as our reference instead for several of Martin's arguments. Martin's analysis used the assumption of the Axiom of Determinacy. We reformulate the notion of the envelope in such a way that many of its essen- tial properties can by derived without assuming AD. Instead we will work in the base theory ZF+DCR for the duration of the paper, stating determinacy hypotheses only when necessary. With the exception of Section 9 this paper is mainly expository. It places results of Martin, Kechris, Woodin, and others in a unified framework and in some cases substantially simplifies the original proofs. Section 1 contains background information on descriptive set theory in- cluding the definition of \inductive-like pointclass." In Section 2 we define the envelope of an inductive-like pointclass in a special case that will suffice for results in L(R) and we prove some results about it, most notably its determinacy and its closure under real quantifiers. In Sections 3 and 4 we apply these results to the envelope of the pointclass of inductive sets and Date: May 4, 2014. 2010 Mathematics Subject Classification. Primary 03E15; Secondary 03E60. Key words and phrases. determinacy, pointclass, scale, derived model. The author gratefully acknowledges support from NSF grant DMS-1044150. 1 2 TREVOR M. WILSON then more generally to the envelopes of inductive-like pointclasses in L(R). In particular we show that the Kechris{Woodin determinacy transfer theo- rem and a theorem of Martin on reflection of ordinal-definability for reals can be obtained as immediate consequences. In Section 6 we give a more general definition of the envelope for boldface inductive-like pointclasses, which is phrased in terms of Moschovakis's notion of \companion" for such pointclasses (see Section 5.) We then show that the results of Section 2 can readily be adapted to this more general setting. In Section 7 we give an equivalent condition for the existence of semiscales that is phrased in terms of games. In the later sections these games are used to construct (under additional assumptions) semiscales on a universal Γˇ set with prewellorderings in the envelope of Γ. In Section 8 we give a new proofe of Woodin's theorem on divergent modelse of AD+. Finally, in Section 9 we show that the derived model at an indestructibly weakly compact limit of Woodin cardinals satisfies \every set of reals is Suslin." The results in this paper are mostly taken from the author's PhD thesis [25, Ch. 3]. The author wishes to thank John Steel, who supervised this thesis work, for his guidance; Hugh Woodin, for explaining his argument for the result stated as Theorem 8.2 below; and Martin Zeman, for suggesting several corrections. 1. Pointclasses We begin by recalling some standard notions from descriptive set theory. ! By convention we denote the Baire space ! by R and call its elements \reals." A product space is a space of the form X = X1 × · · · × Xn;Xi = R or Xi = ! for all i ≤ n: A pointset is a subset of a product space. A pointclass is a collection of pointsets, typically an initial segment of some complexity hierarchy for pointsets. We say that a pointclass Γ is !-parameterized (respectively, R-parameter- ized) if for every product space X there is a Γ subset of !×X (respectively, of R × X ) that is universal for Γ subsets of X . In this paper we consider the following types of pointclasses, which are named for their resemblance to the pointclasses IND and IND of induc- tive sets and boldface inductive sets respectively. The inductive] pointsets are those that are definable without parameters by positive elementary in- duction on R (sometimes called \absolutely" or \lightface" inductive.) By \boldface" we mean that real parameters are allowed. Definition 1.1. A pointclass Γ is (lightface) inductive-like if it is !-parameterized, closed under 9R, 8R, and recursive substitution, and has the prewellordering property. THE ENVELOPE UNDER LOCAL DETERMINACY 3 Definition 1.2. A pointclass Γ is boldface inductive-like if it is R-parameterized, closed under 9R, 8R, and continuouse reducibility, and has the prewellorder- ing property. Note that some authors strengthen the prewellordering property to the scale property in these definitions. In this paper we will typically use the following notational conventions. By Γ and Γ we will denote an inductive-like or boldface-inductive-like pointclass respectively.e As usual we denote the dual pointclass of Γ by Γˇ = f:A : A 2 Γg and the ambiguous part of Γ by ∆Γ = Γ \ Γ.ˇ (Note that the existence of universal sets implies non-self-duality by a diagonal argument.) Similarly for a boldface pointclass we define the dual and ambiguous pointclasses Γˇ and ∆Γ. e Givene e a lightface pointclass Γ we can define the corresponding relativiza- tions Γ(x) for x 2 and also the boldface pointclasses Γ = S Γ(x), R x2R Γˇ = S Γ(ˇ x), and ∆ = S ∆ (x). Naturally, if Γe is inductive-like x2R Γ x2R Γ thene the correspondinge boldface pointclass Γ is boldface inductive-like. Typically given a pointclass Γ we will assumee the local determinacy hypo- thesis Det(∆Γ), meaning that every two-player game of perfect information on ! with ae∆Γ payoff set is determined, although some of our results can proved undere weaker hypotheses. The prewellordering ordinal of an inductive-like pointclass Γ (or more generally of a pointclass Γ with the prewellordering property) is the supre- mum of the ordinal lengths of all ∆Γ prewellorderings of R, or equivalently the range of any regular Γ-norm one a complete Γ pointset. We will often denote the prewellordering ordinal of a pointclass by κ. The prototypical example of an inductive-like pointclass is, as noted above, the pointclass IND of inductive sets. Other examples of inductive-like 2 L(R) pointclasses include the pointclass (Σ1) and more generally the point- 2 + class Σ1 under AD . Examples of boldface inductive-like pointclasses include, in addition to the boldface versions of the above lightface pointclasses, the pointclass S(κ) of κ-Suslin sets when AD holds, κ is a Suslin cardinal, and S(κ) is closed under 8R. The following notion of countable approximation for pointsets is central to the paper. Definition 1.3 (Martin, see [3, Defn. 3.9]). Let X be a product space, let κ be an ordinal, and let (Aα : α < κ) be a sequence of subsets of X . For a pointset A ⊂ X , we say A 2 (Aα : α < κ) if for every countable set σ ⊂ X there is an ordinal α < κ such that A \ σ = Aα \ σ. Martin then made the following definition. It is similar to the definition of \envelope" that we will use in this paper, but we will need to make a significant modification. We include the original definition below in order to provide historical context. 4 TREVOR M. WILSON Definition 1.4 (Martin, see [3, Defn. 3.9]). Let ∆ be a pointclass and let κ be an ordinal. For a product space X and a pointsete A ⊂ X , we say A 2 Λ(∆; κ) if A 2 (Aα : α < κ) for some sequence (Aα : α < κ) of ∆ subsets ofe X . e If AC holds then the pointclass Λ(∆; κ) is not likely to be very useful because there will be too many well-orderede sequences of pointsets. For example, if ∆ is a nontrivial boldface pointclass (so it contains every count- able pointset)e and κ ≥ c then any pointset whatsoever is in Λ(∆; κ). To get a useful definition without assuming AD we need to require somee kind of uniformity for the sequence of pointsets (Aα : α < κ). First we deal with a special case. Jκ(R) 2. The envelope of Σ1 e We will need some background on the fine structure of L(R) that can be found, for example, in [23, x1]. We use the Jensen hierarchy (Jα : α 2 Ord) for L(R) but the reader would not miss much by reading \Lκ(R)" in place of \Jκ(R)." However we should note that the ordinal height of Jα(R) is !α rather than α. By convention we allow R as an unstated parameter in our formulas. For example, by Σ1 we mean Σ1(fRg). Therefore quantification over the reals always counts as bounded quantification. R Given ordinals κ and β with κ ≤ β, we say Jκ(R) ≺1 Jβ(R) if Jκ(R) is a Σ1(R [ fRg)-elementary substructure of Jβ(R).
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