A few results
نویسندگان
چکیده
Consider a set A of binary sequences, or equivalently, a subset of Cantor set. Players I and II play binary bits alternatively for infinitely many turns, forming an infinite binary sequence; player I wins if the sequence is in A, and, otherwise, player II wins. We say that A is determined if one of the players has a winning strategy for this game. Statements about determinacy of games have attracted logicians for many decades because of the high complexity of the winning strategies, and also because they have been a useful combinatorial tool in a wide range of areas. Martin’s celebrated theorem says whenever A is a Borel subset of Cantor set, A is determined. But, as noticed by Friedman, Martin’s proof requires unusually large cardinals almost never used in mathematics. Second-order arithmetic, a widely studied logical system, captures all mathematics that can be done in the realm of countable objects. Countable objects include real numbers, continuous functions, real analysis, countable algebra, etc.. Almost all of mathematics that can be modeled with, or coded by, countable objects can be carried out in second-order arithmetic. For the statement above we also need the following definitions: A subset of Cantor space is Gδσ if it is the countable union of countable intersections of open sets (i.e. it is in the third level of the Borel hierarchy); a set is a Boolean combination of n sets A1, ..., An if it can be defined form them using (finite) intersections, unions and complements. The study of the determinacy of Gδσ sets seemed intractable for a long time, and so did the study of results at the boundary of second-order arithmetic. Our result broke those two barriers.
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تاریخ انتشار 2010