A game on the universe of sets

Abstract

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-well-founded models. Finally, we establish a probability result by showing that on hereditarily finite well-founded sets the first player wins almost always.