Two Consequences of Determinacy Consistent with Choice

We begin with a ground model satisfying ZF + AD + ACR, and from it construct a generic extension satisfying ZFC + ô2 = w2 + "me nonstationary ideal on w | is w2-saturated". 0. Introduction. The "axiom" of determinacy (henceforth AD) implies many interesting propositions about small cardinal numbers, for example 2, and the second that the nonstationary ideal on w, is w2-saturated. We shall show that the theory consisting of ZFC + ADL(R) together with these two propositions is consistent. (ADL(R) is the assertion that all games in £(R), a large class of definable games, are determined.) In consonance with our working hypothesis, we would like to prove our result assuming only the consistency of ZFC + ADL(R), or equivalently, that of ZF + AD + DC (cf. §1). At present we cannot do this. We instead assume the consistency of ZF + AD + ACR, where ACR is the axiom of choice for families indexed by the reals. Consistency-wise, ZF + AD + ACR is one of the strongest theories known to man; we show in §1 that it proves the consistency of ZF + AD + DC. The only upper bound we know on the consistency strength of ZF + AD + ACR is that of ZF + ADR + "Ö is regular". We offer some partial justification for our use of such a strong hypothesis at the end of the paper. Our consistency proof is by forcing. We start with a ground model M satisfying ZF + AC + ACR. By a two-step iteration we construct a generic extension M[C7] „onto M so that z\h (h: w2 -* R ). The forcing will add no new reals and preserve w2, i.e., RA/ _ RA/[C] an(J UM _ UM[C] We then pass t0 the inner model N _ £(RM ^ an(J show it has the desired properties. The paper is organized as follows: In §1 we give some preliminaries and background information. In particular, we set forth those consequences of AD whose Received by the editors February 18, 1980. 1980 Mathematics Subject Classification. Primary 03E35; Secondary 03E60. ©1982 American Mathematical Society 0002-9947/81 /0000-1000/$05.75