CAREER: Forcing and Large Cardinals

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

The standard axioms of set theory, Zermelo-Fraenkel set theory with the axiom of choice (ZFC), do not decide many natural questions. For example, 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 the standard axioms, as shown in 1963 by Paul Cohen through the breakthrough method of forcing. Since then, a longstanding project in set theory has been to use forcing for relative consistency results and to study strengthening of the ZFC axioms. This constitutes the broad motivation of the project: What is possible in strengthening of ZFC, versus what constraints are imposed by ZFC itself? The educational component of the project features organizing two workshops and support for undergraduate and graduate student research. The main candidates for ZFC strengthenings are large cardinal axioms and strong forcing axioms. Forcing over a model with large cardinals is also the most powerful tool for showing consistency results. Combinatorial principles, especially at singular cardinals are used to understand both the nature of these extensions, and how much we can do with forcing and large cardinals. Jensen's square, Shelah's approachability property are "anti-compactness" type principles that hold in models that sufficiently resemble L. In contrast, the tree property is a reflection property that resembles large cardinal properties, but can hold at successor cardinals. The project will focus on the interplay between these principles and forcing extensions constructed from large cardinals. The PI will also investigate how they interact with the singular cardinal hypothesis and Shelah's PCF theory. The latter is mostly decided in ZFC, and provides certain "canonical invariants," against which one can test new axioms.