The model-companionship spectrum of set theory, generic absoluteness, and the Continuum problem

TL;DR

This paper explores the relationship between forcing, generic absoluteness, and model companionship in set theory, providing insights into longstanding problems like the Continuum hypothesis through advanced logical frameworks.

Contribution

It establishes a connection between generic absoluteness, forcing axioms, and model companionship, offering a new perspective on the Continuum hypothesis and related set-theoretic properties.

Findings

Forcibility overlaps with consistency under large cardinal assumptions.

Results relate generic absoluteness and forcing axioms to model companionship.

Provides arguments supporting the refutation of the Continuum hypothesis.

Abstract

We show that for $\Pi_2$-properties of second or third order arithmetic as formalized in appropriate natural signatures the apparently weaker notion of forcibility overlaps with the standard notion of consistency (assuming large cardinal axioms). Among such $\Pi_2$-properties we mention: the negation of the Continuum hypothesis, Souslin Hypothesis, the negation of Whitehead's conjecture on free groups, the non-existence of outer automorphisms for the Calkin algebra, etc... In particular this gives an a posteriori explanation of the success forcing (and forcing axioms) met in producing models of such properties. Our main results relate generic absoluteness theorems for second order arithmetic, Woodin's axiom $(*)$ and forcing axioms to Robinson's notion of model companionship (as applied to set theory). We also briefly outline in which ways these results provide an argument to refute the Continuum hypothesis.