A primer on the set-theoretic multiverse
This is a talk at the ALPS Seminar, Virginia Commonwealth University , April 12, 2019.
Abstract: Set Theory is the study of properties of sets, which are the building blocks of all mathematical objects. It is a fairly young discipline that, in the early 19th century, arose out of attempts by mathematicians to provide a formal foundation for their work. Building on pioneering work of Georg Cantor, mathematicians codified elementary properties of sets by the Zermelo-Fraenkel axioms ZFC, but establishing even such basic properties of sets as the size of the continuum from the ZFC axioms proved difficult. This was soon partially explained by Godel's Incompleteness Theorem, which showed that no sufficiently complex axiomatic system can decide the truth of all assertions. Sophisticated techniques, such as forcing, developed by set theorists over the course of the 20th century showed that most interesting properties of sets are not decided by ZFC, so that we can equally well consider both ZFC augmented by such a property or by its negation. At the same time, set theorists introduced a hierarchy of strong axiomatic systems extending ZFC that impose an elegant structure on the mathematical universe. If we believe that to each of these various axiomatic systems corresponds a mathematical universe satisfying them, then there is a multiverse of mathematical universes, each with its own version of mathematics. In this talk, I will give an overview of modern set theoretic research and argue that it leads naturally to the conception of set theory as the study of the multiverse of mathematical universes, and that this view in turn leads to fruitful avenues of mathematical research.