If the An's belong to some class r we talk about a "r separating family." Of particular interest of course is the case when the An's are Borel. If E has a Borel separating family we say that E is smooth. Notice that this is equivalent to saying that there is a Borel map f: X-+ Y, Y some Polish space, such that xEy ¢} f(x) = f(y); so this means that there exist Borel calculable "invariants" f(x), belonging to some Polish space, thus of a fairly "concrete" nature, associated with each x E X which classify x up to E-equivalence. A typical example of such 2 classification is the case X = the n x n complex matrices (= en), E = the equivalence relation of similarity between n x n matrices, and f(A) = the Jordan canonical form of A.

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Martin [12] has shown that the determinacy of analytic games is a consequence of the existence of sharps. Our main result is the converse of this:Theorem. If analytic games are determined, then x2 exists for all reals x.This theorem answers question 80 of Friedman [5]. We actually obtain a somewhat sharper result; see Theorem 4.1. Martin had previously deduced the existence of sharps from 3 − Π11-determinacy (where α − Π11 is the αth level of the difference hierarchy based on − Π11 see [1]). Martin has also shown that the existence of sharps implies < ω2 − Π11-determinacy.Our method also produces the following:Theorem. If all analytic, non-Borel sets of reals are Borel isomorphic, then x* exists for all reals x.The converse to this theorem had been previously proven by Steel [7], [18].We owe a debt of gratitude to Ramez Sami and John Steel, some of whose ideas form basic components in the proofs of our results.For the various notation, definitions and theorems which we will assume throughout this paper, the reader should consult [3, §§5, 17], [8], [13] and [14, Chapter 16].Throughout this paper we will concern ourselves only with methods for obtaining 0# (rather than x# for all reals x). By relativizing our arguments to each real x, one can produce x2.

The extreme interest of set theorists in the notion of “closed unbounded set” is epitomized in the following well-known theorem:Theorem A. For any regular cardinal κ > ω, the intersection of any two closed unbounded subsets of κ is closed and unbounded.The proof of this theorem is easy and in fact yields a stronger result, namely that for any uncountable regular cardinal κ the intersection of fewer than κ many closed unbounded sets is closed and unbounded. Thus, if, for κ a regular uncountable cardinal, we let denote {A ⊆ κ ∣ A contains a closed unbounded subset}, then, for any such κ, is a κ-additive nonprincipal filter on κ.Now what about the possibility of being an ultrafilterκ It is routine to see that this is impossible for κ > ℵ1. However, for κ = ℵ1 the situation is different. If were an ultrafilter, ℵ1 would be a measurable cardinal. As is well-known this is impossible if we assume the axiom of choice; however if ZF + “there exists a measurable cardinal” is consistent, then so is ZF + “ℵ1 is a measurable cardinal” [2]. Furthermore, under the assumption of certain set theoretic axioms (such as the axiom of determinateness or various infinite exponent partition relations) can be proven to be an ultrafilter. (See [3] and [5].)

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