Large infinities and definable sets

Abstract

Large cardinal axioms extend the standard set of axioms for mathematics by asserting that very large infinite sets exist. A prominent line of current research in mathematical logic is identifying ever stronger principles of this kind; this serves as an avenue toward mitigating the phenomenon of Gödel incompleteness. However, there is a tension between large cardinal axioms and principles asserting global simplicity of the mathematical universe, as well as forms of the Axiom of Choice. New kinds of infinity recently identified shed light into this tension and raise important mathematical and philosophical questions.