The many universes of modern set theory

This is a talk at the Mathematics Colloquium, University of Warwick, June 25, 2021 (virtual).

All mathematical objects can be reduced down to sets and properties of sets underlie their fundamental behavior. A universe of set theory is a mathematical structure consisting of sets together with a membership relation telling us when a set is an element of another set. We think of mathematics as taking place within some such set-theoretic universe. Mathematicians of the early 20th century had over-optimistically hoped that a cleverly chosen collection of axioms would determine the unique set-theoretic universe to serve as the foundation of mathematics. This naive foundational view was crushed by Gödel's incompleteness theorems. In the decades that followed set theorists worked to construct a myriad of extremely different universes of set theory all satisfying the Zermelo-Fraenkel axioms (with the Axiom of Choice) ZF(C). Godel discovered the constructible universe $L$, a minimal universe sitting inside any other universe. Cohen developed the technique of forcing for expanding a given universe of set theory to a larger universe - called a forcing extension- satisfying some desired property such as the failure of the continuum hypothesis. Set theorists began strengthening ZFC by adding large cardinal axioms asserting the existence of large infinite objects beyond those provable in ZF(C). Remarkably the existence of these extreme infinities was shown to affect the properties 'small' mathematical objects such as the reals. Interestingly, large cardinals are also intimately connected with the existence of elementary embeddings of a universe of set theory into a sub-universe of itself. Set theory, having been around for a little longer than a century, is a relatively young field. In this talk, I will try to give an overview of its history proceeding into the modern times. If I have time, I will mention some very recent research connecting large cardinals with forcing, where we introduce fruitful axioms asserting that elementary embeddings following from a given large cardinal exist in a forcing extension, but not necessarily in the universe itself.