The Copernican Multiverse of Sets

TL;DR

This paper introduces a flexible, semantically motivated framework for the multiverse of set theory, incorporating new axioms and principles that align with philosophical multiverse ideas and demonstrate low consistency strength.

Contribution

It develops a novel untyped, semantically motivated multiverse framework for set theory, linking philosophical principles with formal logical structures and consistency results.

Findings

The framework is compatible with various multiverse conceptions.

Extensions have consistency strength just above ZF.

Applied to arithmetic absoluteness and Hamkins' multiverse theory.

Abstract

We develop an untyped framework for the multiverse of set theory. $\mathsf{ZF}$ is extended with semantically motivated axioms utilizing the new symbols $\mathsf{Uni}(\mathcal{U})$ and $\mathsf{Mod}(\mathcal{U, \sigma})$, expressing that $\mathcal{U}$ is a universe and that $\sigma$ is true in the universe $\mathcal{U}$, respectively. Here $\sigma$ ranges over the augmented language, leading to liar-style phenomena that are analysed. The framework is both compatible with a broad range of multiverse conceptions and suggests its own philosophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation expressing that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as a Copernican principle that the background theory does not hold a privileged position over the theories of its internal universes. Our main mathematical result is a lemma encapsulating a technique for locally interpreting a wide variety of extensions of our basic framework in more familiar theories. We apply this to show, for a range of such semantically motivated extensions, that their consistency strength is at most slightly above that of the base theory $\mathsf{ZF}$, and thus not seriously limiting to the diversity of the set-theoretic multiverse. We end with case studies applying the framework to two multiverse conceptions of set theory: arithmetic absoluteness and Joel D. Hamkins' multiverse theory.