The model companions of set theory

Giorgio Venturi, Matteo Viale

Published 2019-09-29Version 1

We show that the first order theory of $H_{\omega_1}$ is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for all universally Baire sets of reals. We also outline some basic conditions granting the model completeness of the first order theory of $H_{\omega_2}$ and of the axiom system $\mathsf{ZF}+V=L$ in an appropriate language.