The model companions of set theory

This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin's generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a $\Pi_2$-property formalized in an appropriate language for second order number theory is forcible from some $T\supseteq\mathsf{ZFC}+$large cardinals if and only if it is consistent with the universal fragment of $T$ if and only if it is realized in the model companion of $T$. In particular 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 $\Delta_0$-properties and for all universally Baire sets of reals. We will extend these results also to the theory of $H_{\aleph_2}$ in a follow up of this paper.