What model companionship can say about the Continuum problem

TL;DR

This paper explores the model companions of set theory, linking them to the Continuum problem and proposing that certain theories of $H_{eth}$ structures relate to solutions of the Continuum hypothesis, with a focus on maximality principles.

Contribution

It analyzes how model companionship depends on the signature in set theory and identifies specific theories of $H_{eth}$ structures as natural model companions, connecting to the Continuum problem.

Findings

Model companions of set theory relate to theories of $H_{eth}$ structures.

The unique solution $2^{eth_0}=eth_2$ can belong to a model companion.

A maximality principle inspired by Hilbert's completeness is used to justify the approach.

Abstract

We present recent results on the model companions of set theory, placing them in the context of the current debate in the philosophy of mathematics. We start by describing the dependence of the notion of model companionship on the signature, and then we analyze this dependence in the specific case of set theory. We argue that the most natural model companions of set theory describe (as the signature in which we axiomatize set theory varies) theories of $H_{\kappa^+}$, as $\kappa$ ranges among the infinite cardinals. We also single out $2^{\aleph_0}=\aleph_2$ as the unique solution of the Continuum problem which can (and does) belong to some model companion of set theory (enriched with large cardinal axioms). Finally this model-theoretic approach to set-theoretic validities is explained and justified in terms of a form of maximality inspired by Hilbert's axiom of completeness.