A master’s thesis by Samuel Dworetzky, Spring 2017 (link)

Abstract: Models of set theory are ubiquitous in modern day set theoretic research. There are many different constructions that produce countable models of ZFC via techniques such as forcing, ultraproducts, and compactness. The models that these techniques produce have many different characteristics; thus it is natural to ask whether or not models of ZFC are classifiable. We will answer this question by showing that models of ZFC are unclassifiable and have maximal complexity. The notions of complexity used in this thesis will be phrased in the language of Borel complexity theory.

In particular, we will show that the class of countable models of ZFC is Borel complete. Most of the models in the construction as it turns out are ill-founded. Thus, we also investigate the sub problem of identifying the complexity for well-founded models. We give partial results for the well-founded case by identifying lower bounds on the complexity for these models in the Borel complexity hierarchy.