Canonical Truth

TL;DR

This paper explores variants of canonical set-theoretic truth in models of ZFC, identifying statements that hold universally in such models but are independent of ZFC.

Contribution

It introduces new notions of canonical truth in set theory and demonstrates their implications beyond ZFC's provability, including properties of models and large cardinals.

Findings

Certain statements like the ground model axiom hold in all canonical models.

Canonical models can exist without implying the existence of measurable cardinals.

The size of the continuum is not fixed by the canonical truth notions studied.

Abstract

We introduce and study some variants of a notion of canonical set theoretical truth. By this, we mean truth in a transitive proper class model $M$ of ZFC that is uniquely characterized by some $\in$-formula. We show that there are interesting statements that hold in all such models, but do not follow from ZFC, such as the ground model axiom and the nonexistence of measurable cardinals. We also study a related concept in which we only require M to be fixed up to elementary equivalence. We show that this theory-canonicity also goes beyond provability in ZFC, but it does not rule out measurable cardinals and it does not fix the size of the continuum.