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Determinacy-Related Consequences on Limit Superiors

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Determinacy-Related Consequences on Limit Superiors DETERMINACY-RELATED CONSEQUENCES ON LIMIT SUPERIORS Daniel Walker, B.S. Dissertation Prepared for the Degree of DOCTOR OF PHILOSOPHY UNIVERSITY OF NORTH TEXAS May 2013 APPROVED: Stephen Jackson, Major Professor Douglas Brozovic, Committee Member Su Gao, Committee Member and Chair of the Department of Mathematics Mark Wardell, Dean of the Toulouse Graduate School Walker, Daniel. Determinacy-Related Consequences on Limit Superiors. Doctor of Philosophy (Mathematics), May 2013, 26 pp., bibliography, 13 titles. Laczkovich proved from ZF that, given a countable sequence of Borel sets on a perfect Polish space, if the limit superior along every subsequence was uncountable, then there was a particular subsequence whose intersection actually contained a perfect subset. Komjath later expanded the result to hold for analytic sets. In this paper, by adding AD and sometimes V=L(R) to our assumptions, we will extend the result further. This generalization will include the increasing of the length of the sequence to certain uncountable regular cardinals as well as removing any descriptive requirements on the sets. Copyright 2013 by Daniel Walker ii TABLE OF CONTENTS Page CHAPTER 1. INTRODUCTORY REMARKS ............................................................... 1 CHAPTER 2. THE COUNTABLE CASE ...................................................................... 9 CHAPTER 3. THE GENERAL CASE .......................................................................... 16 CHAPTER 4. GENERALIZING THE SELECTION LEMMA ...................................... 19 BIBLIOGRAPHY ........................................................................................................... 25 iii CHAPTER 1 INTRODUCTORY REMARKS Throughout this paper we are assuming the axioms of ZF. We also equivocate between R and !!. Definition 1. Given a limit ordinal λ, an unbounded S ⊆ λ, and a sequence of sets 0 0 hAαiα2S, define the limit superior of hAαiα2S by lim sup Aα = fx : 8α < λ 9α > α (α 2 S α2S ^ x 2 Aα0 )g. That is, the limit superior is simply all those objects that are in cofinally many Aα's. Definition 2. Let X be a set of ordinals and κ a cardinal. Define [X]κ = ff : κ ! X : 8α < β f(α) < f(β)g. The elements of [X]κ can also be identified with their ranges. We are very abusive of this ambiguity. When viewing the elements as subsets of X we denote them with capital letters. When viewing them as functions we use standard 0 κ functions variables, such as f, g, etc. We say two functions f; f 2 [X] are E0-equivalent κ if their ranges are the same for some point in X on. A set C ⊆ [X] is E0-invariant if it is closed under E0 equivalence. We first start with the already established results on limit superiors of sets of reals. Theorem 1 (Laczkovich, 1977). [1] If hAnin2! is a sequence of Borel sets of reals ! ! T such that 8f 2 [!] (lim sup Af(n) is uncountable), then for some f 2 [!] ( Af(n) has a n2! n2! perfect subset). In 1982, Komjath extended Laczkovich's result to hold for analytic sets (as well as to all sets when assuming Martin's Axiom) [2]. Our main goal in this paper is, by strength- ening the axiomatic assumptions, to generalize this phenomenon. Before we state our goal, however, we must first introduce some more preliminary concepts. 1 Definition 3. Consider a game between two players, traditionally referred to as I and II, with the following rules: Before the game starts, a set A ⊆ !!, known as the payoff set, is fixed. I goes first, playing an integer n0. II then plays an integer n1, which is followed by I playing an integer n2, etc. This continues for !-many turns. At the end of time, the two players have together built up a real x = hn0; n1; n2; :::i. We say that I wins the game if x 2 A and II wins otherwise. We say that the game is determined, or more often we refer to the set A itself being determined, if one of the players has a winning strategy. To be clear, a winning strategy for I would be a function σ from the set of all even length, finite ! sequences of integers into ! such that 8hn0; n1; n2; :::i 2 ! if 8k σ(hn0; :::; n2k−1i) = n2k, then hn0; n1; n2; :::i 2 A. The notion of a winning strategy for II is defined similarly. The Axiom of Determinacy, abbreviated AD, is the statement that every set of reals is determined. A winning strategy for one of the players can also be viewed as a continuous function. That is, if σ is a winning strategy for I and y 2 !!, let σ(y) = the real that σ demands I play when II plays y. A similar notion is defined for winning strategies for II. Definition 4. Define the ordinal (actually, the cardinal) Θ as the supremum of the ranks of the prewellorderings on subsets of R. If AC and CH hold, then Θ = !2. Once we assume AD, however, Θ becomes quite a large cardinal. Theorem 2. Assume AD. (i) Every set of reals has the perfect set property. That is, if A ⊆ !! is uncountable, then A has a perfect subset. [3] (ii) The well-ordered union of countable sets of reals is countable. (iii) There is no uncountable, well-ordered sequence of distinct, countable sets of reals. (iv) !1 and !2 are regular, but if n ≥ 3, then cof(!n) = !2. [4] (v) If κ < Θ has uncountable cofinality, then there is some S ⊆ R and a prewellordering 1 ≺ on S with length κ that has the Σ1-boundedness property. That is, if A ⊆ S is 1 Σ1, then 9α < κ 8y 2 A jyj < α, where jyj is y's rank in ≺. [5] 2 (vi) Suppose X ; Y are perect Polish spaces and R ⊆ X × Y is such that 8x 2 X 9y 2 Y R(x; y). Then R has a continuous uniformizing function on some comeager subset of X . (vii) The Axiom of Countable Choice holds for sets of reals. Definition 5. Informally, Godel's constructible universe, denoted by L, is the smallest inner model of set theory. That is, it is the smallest model of set theory that contains ON, the class of ordinal numbers. There is another model that is of great inerest to descriptive set theorists, namely L(R), which is the smallest inner model that contains all of the reals. Definition 6. A predicate R ⊆ !n1 × (!!)n2 × (P(!!))n3 is projective if is of the ~ ! ! ! ~ form R(~n;~x; A) $ 9f1 2 ! 8f2 2 ! ... 9fm 2 ! 8k 2 !'(~n;~x; A; f1; :::; fm; k), where ' is recursive with respect to some oracle machine. To make the definition clear, the predicate n1 ! n2 ! n3 2 (x; A) $ x 2 A is considered recursive. A predicate R ⊆ ! × (! ) × (P(! )) is Σ1 if it is of the form R(~n;~x; A~) $ 9B 2 P(!!) '(~n;~x; A;~ B), where ' is projective. A predicate 2 2 is ∆1 if both it and its complement are Σ1. The model L(R) has some fascinating descriptive properties: Theorem 3. Assume AD and V=L(R). (i) Let R ⊆ P(R) be a projective relation (that is, a relation that only allows for real 2 quantification). If R is a non-empty relation, then there is some ∆1 set A ⊆ R such that R(A). 2 2 (ii) Let X ; Y be perfect Polish spaces and R ⊆ X × Y a ∆1 (Σ1) relation such that 2 2 8x 2 X 9y 2 Y R(x; y). Then R possesses a ∆1 (Σ1) uniformizing function. [6] ! 2 (iii) A ⊆ ! is Σ1 iff A is κ-Suslin for some κ < Θ. [6] (iv) The Axiom of Dependent Choice, DC, is true. [4] 3 Definition 7. One of the topological spaces that we will make use of in chapter 2 is ! ! the Ellentuck space, h[!] ;"i. The basic open sets are of the form [t; Z] = fX 2 [!] : X `(t) = t ^ Xnt ⊆ Zg, where t 2 2<! and Z 2 [!]!. Definition 8. A ⊆ [!]! is completely Ramsey if for every [t; Z] there is some Y 2 [Z]! such that either [t; Y ] ⊆ A or [t; Y ] \ A = ;. The following foundational facts are known about this space: Theorem 4. (i)[ !]! is a Baire space. [7] (ii) A ⊆ [!]! is "-Baire iff it is completely Ramsey. [7] 1 (iii) (AD + V=L(R)) All subsets of [!]! are completely Ramsey. [8] 1 1 Definition 9. For n ≥ 1 let δn = supfj ≺ j :≺ is a ∆n prewellordering on a subset of Rg. Definition 10. A regular cardinal κ has the strong partition property if, given any function h :[κ]κ ! f0; 1g there is an H 2 [κ]κ and an i 2 f0; 1g such that 8f 2 [H]κ h(f) = i. Definition 11. For this purposes of this paper, when referring to a measure µ on a regular, uncountable cardinal κ we will assume that µ only takes values in f0; 1g and that its domain is P(κ). If ' represents some unary formula on κ, then the statement µ(fα < ∗ 0 0 κ : '(α)g) = 1 will be written as 8 α '(α). We write f =µ f if µ(fα : f(α) = f (α)g) = 1. Let [κ]κ/µ be the naturally induced ultrapower, which will be another cardinal. A measure is normal if, given any function f : κ ! κ such that 8∗α f(α) < α (a "pressing down" function), there is some γ < κ such that 8∗α f(α) = γ. A few introductory known results: Theorem 5. Suppose AD is true. For n ≥ 0, 1It is still an open problem as to whether this fact follows from AD alone. 4 1 1 + (i) δ2n+2 = (δ2n+1) [9] 1 (ii) δ2n+1 has the strong partition property.
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