# Classification of countable models of PA and ZFC

Boise Set Theory Seminar, November 2017

*Abstract*: In 2009 Roman Kossak and I showed that the classification problems for countable models of arithmetic (PA) is Borel complete, which means it is complex as possible. The proof is elementary modulo Gaifman’s construction of so-called canonical I-models. Recently Sam Dworetzky, John Clemens, and I adapted the method to show that the classification problem for countable models of set theory (ZFC) is Borel complete too. In this talk I’ll give the background needed to state such results, and then give an outline of the two very similar proofs.