Large Cardinals and the Determinacy of Long Games

本课题研究具有可定义收益的可变可数长度博弈的确定性问题。最近在大型红衣主教领域的进展表明，应该有可能在Woodin红衣主教地区使用大型红衣主教来证明这种游戏的确定性。首席研究员已经证明了这个方向的结果，它引入了一种基本技术，可以在给定模型M上将一般的长博弈转换为迭代博弈，前提是M有足够大的基数。（确切的大基数当然取决于游戏的长度。）如果M是一个可迭代模型，则迭代博弈是确定的，这意味着原始长博弈的确定性。证明的特殊结果是，假设存在一个具有“强于Woodin基数”的可迭代内部模型，则具有Pi -1收益的连续编码博弈是确定的。使用这个证明的技术，作为一个主要的构建块，我们建议更密切地研究大基数和长博弈的确定性之间的相关性。无限长博弈是当前描述性集合论研究的一个重要内容。通常这样的博弈是这样进行的：固定一个集合a，它包含在0到1之间的实数区间内。我们把这些实数写成二进制，所以一个实数的形式是0。A0 a1 a2 a3 a4 a5 ......其中a0， a1等为“数字”，即0或1。现在考虑两个玩家（玩家I和玩家II）轮流玩数字的游戏。I开始玩数字a0， II接着玩数字a1，然后I玩数字a2， II玩数字a3，等等。两个参与人继续无限博弈，最终得到实数x = 0。A0 a1 a2 a3 .....我们称x为游戏G(a)的一次“运行”，并说在x位于集合a的情况下，玩家I赢得了x的运行。我们说游戏G(a)是“确定的”，如果其中一个玩家有一个获胜策略，或者一组指示告诉这个玩家在每一轮中精确地做什么，只要遵守这些指示，玩家就一定会赢。对于一般集合a，博弈G(a)不需要确定。然而，如果集合A以某种方式可定义，则G(A)是确定的。例如，Martin在1970年证明，如果A是一个闭集的投影，那么G(A)是确定的。大基数公理研究的最新进展使我们能够证明长博弈的确定性，本项目的目的是进一步研究大基数公理与几个长博弈的确定性之间的关系。我们希望这能帮助我们理解大型枢机，因为在大型枢机的研究中，长博弈是自然发生的。

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.