Large Cardinals

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

The PI is investigating the theory of large cardinals, and their applications to determinacy, with emphasis on the following topics: (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.