Tameness for set theory $I$

TL;DR

This paper demonstrates that, under large cardinal assumptions, set theory can be viewed as a tame first-order theory when formalized with natural predicates, linking generic absoluteness to model companionship.

Contribution

It develops a framework connecting generic absoluteness results to model companionship, showing equivalences involving forcibility, consistency, and realization in model companions.

Findings

Set theory is tame under large cardinal assumptions.

Equivalence between forcibility, consistency, and realization in model companions.

Framework applies to second and third order arithmetic.

Abstract

The paper is a first of two and aims to show that (assuming large cardinals) set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic definable concepts of second and third order arithmetic, and appealing to the model-theoretic notions of model completeness and model companionship. Specifically we develop a general framework linking generic absoluteness results to model companionship and show that (with the required care in details) a $\Pi_2$-property formalized in an appropriate language for second or third 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 $T$. The paper is accessible to any person who has a fair acquaintance with set theory and first order logic at the level of an under-graduate course in both topics; however bizarre this may appear (given the results we aim to prove) no knowledge of forcing or large cardinals is required to get the proofs of its main results (if one accepts as black-boxes the relevant generic absoluteness results). On the other hand familiarity with the notions of model completeness and model companionship is essential. All the necessary model-theoretic background will be given in full detail. The present work expands and systematize previous results obtained with Venturi.