AD and the supercompactness of ℵ1

Abstract

Since the discovery of forcing in the early sixties, it has been clear that many natural and interesting mathematical questions are not decidable from the classical axioms of set theory, ZFC. Therefore some mathematicians have been studying the consequences of stronger set theoretic assumptions. Two new types of axioms that have been the subject of much research are large cardinal axioms and axioms asserting the determinacy of definable games. The two appear at first glance to be unrelated; one of the most surprising discoveries of recent research is that this is not the case.In this paper we will be assuming the axiom of determinacy (AD) plus the axiom of dependent choice (DC). AD is false, since it contradicts the axiom of choice. However every set in L[R] is ordinal definable from a real. Our axiom that definable games are determined implies that every game in L[R] is determined (in V), and since a strategy is a real, it is determined in L[R]. That is, L[R] ⊨ AD. The axiom of choice implies L[R] ⊨ DC. So by embedding ourselves in L[R], we can assume AD + DC and begin proving theorems. These theorems true in L[R] imply corresponding theorems in V, by e.g. changing “every set” to “every set in L[R]”. For more information on AD as an axiom, and on some of the points touched on here, the reader should consult [14], particularly §§7D and 8I. In this paper L[R] will no longer even be mentioned. We just assume AD for the rest of the paper.