The Copernican Multiverse of Sets
Paul K. Gorbow, Graham E. Leigh

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. is extended with semantically motivated axioms utilizing the new symbols and , expressing that is a universe and that is true in the universe , respectively. Here 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…
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