# The model companions of set theory

#### Matteo Viale, University of Turin. Part of the logic seminar series.

It has been proved in the seventies of the past century that no recursive extension of ZFC has a model companion.  In this seminar we will argue that (modulo some technicalities) the first-order theory (with certain parameters) of any model of ZFC + large cardinals has a model companion (which is the first-order theory of the hereditarily countable sets according to the model).

This exploits the generic absoluteness results of Woodin for second-order number theory as well as an analysis (inspired by Robinson’s notion of infinite forcing) of the generic multiverse given by all forcing extensions of V.

We will give all necessary background information.

This work is joint with Giorgio Venturi.