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Effective Cardinals of Boldface Pointclasses EFFECTIVE CARDINALS OF BOLDFACE POINTCLASSES ALESSANDRO ANDRETTA, GREG HJORTH, AND ITAY NEEMAN Abstract. Assuming AD + DC(R), we characterize the self-dual boldface point- classes which are strictly larger (in terms of cardinality) than the pointclasses contained in them: these are exactly the clopen sets, the collections of all sets of ξ ξ Wadge rank ≤ ω1, and those of Wadge rank < ω1 when ξ is limit. 1. Introduction A boldface pointclass (for short: a pointclass) is a non-empty collection Γ of subsets of R such that Γ is closed under continuous pre-images and Γ 6= P(R). 0 0 0 Examples of pointclasses are the levels Σα, Πα, ∆α of the Borel hierarchy and the 1 1 1 levels Σn, Πn, ∆n of the projective hierarchy. In this paper we address the following Question 1. What is the cardinality of a pointclass? Assuming AC, the Axiom of Choice, Question 1 becomes trivial for all pointclasses Γ which admit a complete set. These pointclasses all have size 2ℵ0 under AC. On the other hand there is no obvious, natural way to associate, in a one-to-one way, 0 an open set (or for that matter: a closed set, or a real number) to any Σ2 set. This 0 0 suggests that in the realm of definable sets and functions already Σ1 and Σ2 may have different sizes. Indeed the second author in [Hjo98] and [Hjo02] showed that AD + V = L(R) actually implies 0 0 (a) 1 ≤ α < β < ω1 =⇒ |Σα| < |Σβ|, and 1 1 1 (b) |∆1| < |Σ1| < |Σ2| < . (Since we do not assume Choice, cardinal inequalities are to be understood as follows: for any sets X and Y , |X| ≤ |Y | means that there is an injection of X into Y , |X| < |Y | means that |X| ≤ |Y | & |Y | |X|, and |X| = |Y | means that |X| ≤ |Y | ≤ |X| or, equivalently (by the Shroeder-Bernstein Theorem), that there is a bijection between X and Y .) The third author strengthened the result in (a) by 0 0 showing that |∆α+1| < |Σα+1|, all 1 ≤ α < ω1. Therefore, in the AD-world, the answer to Question 1 is far from being trivial. The results mentioned above did not characterize completely the cardinality point- classes, that is those Γ such that |Γ0| < |Γ|, for any Γ0 ⊂ Γ. For example they said 0 0 nothing about the existence of cardinality pointclasses strictly between Σ2 ∪Π2 and 0 ∆3. The main result of this paper is a complete characterization, under AD, of all cardinality pointclasses in terms of their Wadge rank. (The notion of Wadge rank 1991 Mathematics Subject Classification. 03E15. Key words and phrases. Determinacy, Wadge hierarchy. 1 2 ALESSANDRO ANDRETTA, GREG HJORTH, AND ITAY NEEMAN and other relevant facts from Descriptive Set Theory are recalled in sections 2 and 3.) In order to state our characterization of cardinality pointclasses, we need an ˘ 0 auxiliary definition. Since |Γ| = |Γ ∪ Γ| for any pointclass Γ containing ∆1 (see Corollary 19 below), it is enough to restrict our attention to self-dual pointclasses. By AD the self-dual pointclasses are exactly the levels of the Wadge hierarchy—that is, they are of the form (α) P (R) = the collection of all sets of Wadge rank < α. Say that a self-dual ∆ is a cardinality level just in case |∆0| < |∆| for any self-dual ∆0 ⊂ ∆. Since cardinality pointclasses correspond to cardinality levels, determining all cardinality pointclasses amounts to pinning-down those ordinals α for which P(α)(R) is a cardinality level. In this paper a description of such ordinals is given, providing thus a complete answer to Question 1. Theorem 1. Assume AD + DC(R). Then P(α)(R) is a cardinality level iff α = 3 or ξ ξ ∃ξ < Θ α = ω1 + 1 ∨ (α = ω1 & ξ is limit) . DC(R) is the Axiom of Dependent Choices over the reals, and Θ is the supremum of the ordinals which are the surjective image of R, (1) Θ = sup{α | ∃f : R α} , and it is also the length of the Wadge hierarchy. Theorem 1 subsumes all previous results—see Corollary 27. For example, by 0 ξ [Wad83] the Wadge rank of a complete Σα set is ω1, with ξ limit of uncountable cofinality if α ≥ 3, and ξ = 1 when α = 2. Notice that ξ ξ (ω1+1) 0 0 (ω1) 0 P (R) = Σα ∪ Πα and P (R) = ∆α for this ξ. Using Theorem 1 it follows that for 2 ≤ β < α < ω1, 0 0 0 0 0 |Σβ| < |∆α| < |Σα| = |Σα ∪ Πα| . 0 0 (The case when β = 1 and α = 2 is a bit different as |Σ1| = |∆2| = |R|.) Assuming AD, each pointclass Γ is the surjective image of R, so it is in bijection with R/E for some equivalence relation E on R. Conversely, suppose E is an equivalence relation on R, and let α be the Wadge rank of E (after identifying R × R with R). Then every equivalence class [x]E is the continuous pre-image of E ∩ ({x} × R), which is in P(α+1)(R) (if α ≥ ω—see Lemma 6 below), since it is the intersection of a set of Wadge rank α and a closed set, and hence R/E ⊆ P(α+1)(R). Therefore the cardinalities of the pointclasses P(α)(R) as in Theorem 1 are cofinal in the cardinalities of quotients of R. Theorem 1 is stated and proved under AD, but since determinacy is used in a “local” way, weaker versions of the theorem—compatible with AC—can be extracted from its proof. For example, suppose we restrict ourselves to Borel pointclasses, 1 that is pointclasses contained in ∆1. Thus a self-dual Borel pointclass is of the (α) 1 form P (R) with α ≤ η, where η is the Wadge rank of a complete Σ1 set or, EFFECTIVE CARDINALS OF BOLDFACE POINTCLASSES 3 1 equivalently, of a complete Π1 set. Each Borel pointclass is the surjective image of R, i.e., there is a πΓ : R Γ and moreover such πΓ can be taken so that the equivalence relation x ∼Γ y ⇐⇒ πΓ(x) = πΓ(y) 1 is Π1. A function F : Γ → Λ between Borel pointclasses is said to be Borel-in-the- codes if there is a Borel Fˆ : R → R such that ˆ ˆ x ∼Γ y =⇒ F (x) ∼Λ F (y) ˆ and F (A) = πΛ(F (x)), for some/any x ∈ R such that πΓ(x) = A. Let us say that (α) 1 a self-dual Borel pointclass P (R) is a ∆1-cardinality level if it does not admit a Borel-in-the-codes injection into some P(β)(R), with β < α. Then the proof of Theorem 1 yields: 0 (α) 1 (α) 1 Theorem 1 (ZFC). If P (R) ⊆ ∆1, then P (R) is a ∆1-cardinality level iff α = 3 or ξ ξ ∃ξ α = ω1 + 1 ∨ (α = ω1 & ξ is limit) . Similarly, if we assume Projective Determinacy and restrict the injections between projective pointclasses to be projective-in-the-codes, we have a characterization of all projective pointclasses which are projective-cardinality level. The paper is organized as follows: after presenting the notations and conventions in §2, we summarize the relevant results on the Wadge hierarchy in §3. The basic facts about cardinalities of pointclasses are proved in §4, while the proof of the key result (Theorem 29) for showing Theorem 1 is in §5. In the Appendix a proof of a 0 special case of Theorem 1 is sketched, stating that there are more 2-Σ2 sets (i.e., 0 0 sets of the form F \ G with F, G ∈ Σ2) than Σ2 sets. This was our first result and its proof—properly generalized—yielded the main result of this paper. We hope that this proof (which can be read independently from the rest of the paper) will be helpful to get a better understanding of the arguments in Sections 4 and 5. 2. Preliminaries As customary in set theory, R denotes the Baire space ωω, with the topology generated by the metric ( 0 if x = y, d(x, y) = 2−n if n is least such that x(n) 6= y(n). <ω Thus the basic open sets are of the form Ns = {x ∈ R | x ⊃ s}, with s ∈ ω. For A ⊆ R, the interior of A is denoted by Int(A), and the complement R \ A of A is denoted by ¬A. A tree on ω is a T ⊆ <ωω which is closed under initial segments; it is pruned iff ∀t ∈ T ∃s ∈ T (t ⊂ s). The body of T is the set [T ] = {x ∈ R | ∀n(x n ∈ T )} of all infinite branches of T ;[T ] is a closed subset of R, and all closed subsets of R are 4 ALESSANDRO ANDRETTA, GREG HJORTH, AND ITAY NEEMAN of this form with T pruned. The boundary of T is the set of all sequences that just left T , that is <ω ∂T = {s ∈ ω | s∈ / T & s lh(s) − 1 ∈ T } . If s, t ∈ <ωω, then sat is the concatenation of s and t; when t = hni we will often write san instead of sahni. The definition of sat is extended in the obvious way to the case when t is an infinite sequence, i.e., when t ∈ R. For x ∈ <ωω ∪ R let x + 1 = hx(n) + 1 | n < lh(x)i and x −˙ 1 = hx(n) −˙ 1 | n < lh(x)i, where, for k ∈ ω, ( k − 1 if k ≥ 1, k −˙ 1 = 0 otherwise.
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