The model companions of set theory

TL;DR

This paper explores the connection between generic absoluteness results in set theory and the model theoretic concept of model companionship, establishing equivalences and characterizations involving large cardinals and universally Baire sets.

Contribution

It develops a framework linking Woodin's generic absoluteness results to model companionship, showing equivalences between forcibility, consistency, and realization in model companions.

Findings

The first order theory of H_{ω_1} is the model companion of the universe's theory under large cardinal assumptions.

A Pi_2-property in second order number theory is forcible from some T extending ZFC with large cardinals if and only if it is consistent with the universal fragment of T.

Results will be extended to the theory of H_{ℵ_2} in future work.

Abstract

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.