# Computable models of finite set theory

## Michał Tomasz Godziszewski

In 2001, Mancini and Zambella investigated computable models of fragments of set theory. They emplyed the Bernays-Rieger method of permutations to construct a computable model of finite set theory (i.e. $ZF_{fin}$ – the theory obtained from ZF by replacing the axiom of infinity by its negation). In 2009, Enayat, Schmerl and Visser showed how to build computable nonstandard models of this theory without the use of permutations. Furthermore, they demonstrated that in every computable model of ZF_{fin} every set (as viewed externally) has only finitely many elements (such models are called $\omega$-models). The corollaries of these results are, among others, that there are continuum-many nonisomorphic pointwise definable $\omega$-models of $ZF_{fin}$ and that $PA$ and $ZF_{fin}$ are not bi-interpretable. The purpose of the talk is to present proofs of the results of Macinini, Zambella and Enayat, Schmerl, Visser together with the corollaries.

Michał Tomasz Godziszewski is visiting CUNY from September 2016 until March 2017.