The Problem of Determinacy of Infinite Games from an Intuitionistic Point of View

Abstract

Let A be a subset of Baire space N, that is, the set of all infinite sequences α(0), α(1), ... of natural numbers. We define the game for A, G(A). There are two players I, II, who together build an element of N, as follows: player I chooses α(0), player II chooses α(1), player I chooses α(2), and so on. Player I wins if α belongs to A, and player II wins if α does not belong to A. The set A is called determinate if either player I or player II has a winning strategy, that is, a method to ensure that he will always win the game. The problem which subsets of N are determinate came up in the 1930's in discussion between Polish mathematicians and has a fascinating history. In 1964, Jan Mycielski proposed to take the statement that all subsets of N are determinate as an axiom of set theory and an alternative to E. Zermelo's Axiom of Choice.Taking Brouwer's intuitionistic standpoint, we re-examine the problem of the determinacy of subsets of N and also of subsets of certain subspaces of N, for instance Cantor space, the set of all α in N that assume no other values than 0,1. If one understands the disjunction occurring in the classical notion of determinacy constructively, even finite games are not always determinate. We suggest a different intuitionistic notion of determinacy and prove that every subset of Cantor space is determinate in the proposed sense. In Cantor space both player I and player II have two alternative possibilities for each move. More generally, every subset of a space where player II has, for each one of his moves, no more than a finite number of alternative possibilities, is determinate according to our definition. If we allow player II to have countably many alternatives for only one of his moves, there are non-determinate subsets of the space.