In set theory without the axiom of regularity, we consider a game in which two players choose in turn an element of a given set, an element of this element, etc.; a player wins if its adversary cannot make any next move. Sets that are winning, i.e. have a winning strategy for a player, form a natural hierarchy with levels indexed by ordinals. We show that the class of hereditarily winning sets is an inner model containing all well-founded sets, and that all four possible relationships between the universe, the class of hereditarily winning sets, and the class of well-founded sets are consistent. We describe classes of ordinals for which it is consistent that winning sets without minimal elements are exactly in the levels indexed by ordinals of this class. For consistency results, we propose a new method for getting non-wellfounded models. Finally, we establish a probability result by showing that on hereditarily finite well-founded sets the first player wins almost always.We work in ZF, the Zermelo -Fraenkel set theory, minus AR, the Axiom of Regularity; we use AC, the Axiom of Choice, nowhere, for one exception. Let ZF − and ZFC − be ZF and ZFC minus AR. The notations are standard in set theory (see [Jech]): V is the universe, i.e. the class of sets, On is the class of ordinals, V α is the αth level of the cumulative hierarchy, tc is the transitive closure, cf is the cofinality, etc. Let also V ∞ be the class of well-founded sets and r the rank function.Suppose two players, I and II, starting from a given set x, try to construct an ∈-decreasing sequence of sets x n such that the adversary cannot * The author was partially supported by RFFI 06-01-00608-a.
