Large Cardinals and the Determinacy of Long Games

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This project concerns the determinacy of games of variable countable length, with definable payoff. Recent progress in the area of large cardinals indicates that it should be possible to prove the determinacy of such games using large cardinals in the region of Woodin cardinals. The principal investigator has already proved results in this direction which introduce a basic technique for converting a general long game to an iteration game on a given model M, provided that M has enough large cardinals. (The precise large cardinal needed of course depends on the length of the game in question.) If M is an iterable model then the iteration game is determined, implying the determinacy of the original long game. The particular result proved is that continuously coded games with Pi 1-1 payoff are determined, assuming the existence of an iterable inner model with a cardinal which is `strong past a Woodin cardinal.' Using the techniques of this proof, as a main building block we propose to study more closely the correlations between large cardinals and the determinacy of long games. Games of infinite length play a major rule in current study of Descriptive Set Theory. Typically such games are played as follows: Fix a set A contained in the interval of reals between 0 and 1. We view those reals as written in binary base, so that a real has the form 0. a0, a1, a2, a3, a4, a5 ...... where a0, a1, etc. are "digits" namely either a 0 or a 1. Consider now a game played between two players (player I and player II) who take turns playing digits. I begins playing the digit a0, II follows playing the digit a1, then I plays the digit a2, II plays a3, etc. The two players continue playing ad-infinitum eventually producing the real number x = 0. a0, a1, a2, a3, ..... We call x a "run" of the game G(A), and say that the run x is won by player I in the case that x is in the set A. We say that the game G(A) is "determined" if one of the players has a winning strategy, or a set of instructions telling this play er precisely what to do in each round, and as long as these instructions are followed, the player is guaranteed to win. For a general set A, the game G(A) need not be determined. However, it turns out that G(A) is determined if the set A is definable in certain ways. For example, Martin showed in 1970 that if A is the projection of a closed set then G(A) is determined. Recent progress in the study of large cardinals allows us to prove determinacy of longer games, and the purpose of this project is to study further the relationships between large cardinal axioms and the determinacy of several long games. We expect this to help our understanding of large cardinals, since long games occur naturally in the study of large cardinals.