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 Wadge rank ≤ ω ξ 1 , and those of Wadge rank <ω ξ 1 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 = P(R). Examples of pointclasses are the levels Σ 0 α , Π 0 α , Δ 0 α of the Borel hierarchy and the levels Σ 1 n , Π 1 n , Δ 1 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 2 ℵ 0 under AC. On the other hand there is no obvious, natural way to associate, in a one-to-one way, an open set (or for that matter: a closed set, or a real number) to any Σ 0 2 set. This suggests that in the realm of definable sets and functions already Σ 0 1 and Σ 0 2 may have different sizes. Indeed the second author in [Hjo98] and [Hjo02] showed that AD + V = L(R) actually implies (a) 1 ≤ α<β<ω 1 = ⇒|Σ 0 α | < |Σ 0 β |, and (b) |Δ 1 1 | < |Σ 1 1 | < |Σ 1 2 | <... (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 showing that |Δ 0 α+1 | < |Σ 0 α+1 |, all 1 ≤ α<ω 1 . 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 |Γ ′ | < |Γ|, for any Γ ′ ⊂ Γ. For example they said nothing about the existence of cardinality pointclasses strictly between Σ 0 2 ∪ Π 0 2 and Δ 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