I was invited to give a talk in the Logic Colloquium of the University of Vienna on March 10, 2022.
The Interplay of Determinacy, Large Cardinals, and Inner Models
The standard axioms of set theory, Zermelo-Fraenkel set theory with Choice (ZFC), do not suffice to answer all questions in mathematics. While this follows abstractly from Kurt Gödel’s famous incompleteness theorems, we nowadays know numerous concrete examples for such questions. In addition to a large number of problems in set theory, even many problems outside of set theory have been showed to be unsolvable, meaning neither their truth nor their failure can be proven from ZFC. A major part of set theory is devoted to attacking this problem by studying various extensions of ZFC and their properties with the overall goal to identify the “right” axioms for mathematics that settle these problems.
Determinacy assumptions are canonical extensions of ZFC that postulate the existence of winning strategies in natural infinite two-player games. Such assumptions are known to enhance sets of real numbers with a great deal of canonical structure. Other natural and well-studied extensions of ZFC are given by the hierarchy of large cardinal axioms. Inner model theory provides canonical models for many large cardinal axioms. Determinacy assumptions, large cardinal axioms, and their consequences are widely used and have many fruitful implications in set theory and even in other areas of mathematics. Many applications, in particular, proofs of consistency strength lower bounds, exploit the interplay of determinacy axioms, large cardinals, and inner models. In this talk I will survey recent developments as well as my contribution to this flourishing area.