What are Models of Set Theory?

We are already coming up to the third installment of the new what is....? seminar. This time David Schrittesser will explain what models of set theory are.

Most mathematicians phrase their thoughts in terms of sets. Thus it is probably a good idea to understand models of the theory of sets, right? 

Abstract:

The definition of a `model of set theory', is not hard to give - it is a structure that satisfies the axioms of set theory (a.k.a. ZFC), in the same way that a group is a structure satisfying certain axioms.

Unfortunately, this does not give us a lot of information about what such models might look like. Therefore, in this talk, I will mention different ways of how different kinds of such models might be obtained and how we can manipulate them. This is also why in the title of the talk, the words `models of set theory' is cast in plural form.