Large Cardinals

PI正在研究大基数理论及其在确定性中的应用，重点关注以下主题：（1）长博弈和迭代性;（2）确定性模型上的超滤强迫;（3）序数的一元理论。1 -PI以前的工作确定了一类特定的不可数长度的游戏，因此相关的游戏量词足够强大，可以定义最小的可迭代内部模型，外部度量集中在Woodin基数上。这是最小的水平，在大型基数层次，不能被捕获的游戏的可数长度。本项目的目的是扩展和研究这种连接之间的大型基数层次和游戏的不可数长度。2 -使用大基数的内部模型，可以在包含所有实数的集合论的最小模型中确定可数序列的特定集合上的超滤子。这些超滤子引起有趣的强迫扩展的模型。PI正在研究从大基数中构造超滤子，目的是将它们推广到不可数序列的情况，并研究由此产生的强迫扩展。3 -PI正在研究一元二阶语言在序数结构中的表达能力，无论是在选择公理（对于第二不可数基数以上的序数）下还是在确定性公理下。大基数公理陈述作用于整个集合论域的函数的存在，并保持集合成员的结构。这是现代集合论最惊人的发现之一，这些函数，在表面上应该只影响极大的集合（大到不能使用集合成员结构从较小的集合定义），具体影响真实的数字的属性。连接大基数和真实的数的中介是确定性公理，它说明了在完全信息的无限博弈中获胜策略的存在。目前的项目是大基数和确定性之间的联系的研究的一部分。它解决了游戏的不可数无限长，二值措施的序列不可数长度下公理的确定性，以及表达能力的声明，涉及集，但不是功能，在有序结构。

The PI is investigating the theory of large cardinals, and their applications to determinacy, with emphasis on the followingtopics: (1) long games and iterability; (2) forcing with ultrafilters over models of determinacy; and (3) monadic theories of ordinals. 1 -- Previous work by the PI identified a specific class of games of uncountable length, so that the associated game quantifier is precisely strong enough to define the minimal iterable inner model with an external measure concentrating on Woodin cardinals. This is the least level in the large cardinal hierarchy which cannot be captured by games of countable length.The present project aims to extend and study this connection between levels of the large cardinal hierarchy and games of uncountable length. 2 -- Using inner models for large cardinals it is possible to identify ultrafilters on specific sets of countable sequences in the smallest model of set theory containing all the reals. These ultrafilters give rise to interesting forcing extensions of the model. The PI is investigating the constructions of ultrafilters from large cardinals with the aim of generalizing them to the case of uncountable sequences, and studying the resulting forcing extensions. 3 --- The PI is studying the expressive power of the monadic second order language in the structure of the ordinals, both under the axiom of choice (for ordinals above the second uncountable cardinal) and under the axiom of determinacy.Large cardinal axioms state the existence of functions which act on the entire universe of sets, and preserve the structure of set membership. It is one of the most amazing discoveries of modern set theory that these functions, which at face value should only affect extremely large sets (large enough to not be definable from smaller sets using the structure of set membership), concretely affect the properties of real numbers. The intermediary connecting large cardinals to real numbers is the axiom of determinacy, stating the existence of winning strategies in infinite games of perfect information. The present project is part of the study of the ties between large cardinals and determinacy. It addresses games of uncountable infinite length, two-valued measures on sequences of uncountable length under the axiom of determinacy, and the expressive power of statements involving sets, but not functions, over wellordered structures.