Joan Bagaria: Large cardinals as reflection principles: An overview

Abstract: The so-called Gödel's program in the foundation of mathematics consists on the identification and study of new set-theoretic axioms that would settle natural mathematical questions that cannot be decided on the basis of the standard ZFC (Zermelo-Fraenkel with Choice) system of set theory. Gödel proposed that any new axioms should take the form of strong reflection principles about the set-theoretic universe. This kind of principles are known as axioms of large cardinals and form a hierarchy of increasing consistency strength. In this talk we'll give an overview of the world of large cardinals under the general unifying principle of structural reflection, stopping at key milestones in the large-cardinal hierarchy, such as Woodin cardinals, supercompact cardinals, or Vopěnka's Principle. The talk will also include a brief presentation of a recently discovered new kind of large cardinals which imply the failure of strong forms of the Axiom of Choice, which creates a tension between large cardinal axioms and principles asserting global simplicity of the mathematical universe.