THE COPERNICAN MULTIVERSE OF SETS

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