The classification of countable models of set theory

TL;DR

This paper investigates the complexity of classifying countable models of set theory, establishing that the general classification problem is maximally complex, with partial results for well-founded models.

Contribution

It proves that the classification of all countable models of ZFC is Borel complete and provides partial results for well-founded models.

Findings

Classification of countable models of ZFC is Borel complete

Partial classification results for well-founded models

Highlights the complexity of set-theoretic model classification

Abstract

We study the complexity of the classification problem for countable models of set theory (ZFC). We prove that the classification of arbitrary countable models of ZFC is Borel complete, meaning that it is as complex as it can conceivably be. We then give partial results concerning the classification of countable well-founded models of ZFC.