Singular Combinatorics

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It turns out that standard axioms of set theory do not settle many classical questions. For example, Gödel and Cohen showed that the Continuum Hypothesis (that there is no set whose cardinality is strictly between that of the integers and that of the real numbers) is independent of this axiom system. Since then, a long standing project in set theory has been to find the "right" strengthening of the axioms. There are several candidates, and this project contributes to understanding of the nature of these extensions. This project explores various aspects of combinatorial set theory. The main goal is to investigate the interplay between large cardinals, forcing, and principles such as square, the tree property, and Shelah's theory of possible cofinalities and their applications to singular combinatorics. The work is part of a project to determine the canonical structures that exist at singular cardinals and their successors in extensions of ZFC by large cardinals or strong forcing axioms. The long term goal is understanding what is possible relative to large cardinals, what can be obtained as remnants of large cardinals, and developing the theory of certain forcing posets. Forcing is used to test both the power and limitations of these strengthenings of ZFC, and combinatorial principles like the tree property provide the key test questions.