What David Teaches Goliath: An Inevitable Extension of ZFC

Abstract

We examine the metamathematical significance of the Levy scheme $Λ$, which asserts, for each natural number $n$, the existence of an $n$-Mahlo cardinal $κ$ such that $V_κ\prec_{Σ_n}V$. Several initially unrelated constructions single out $\mathsf{ZFC}+Λ$: comparisons with the model theory of arithmetic and finite set theory; class theories in which the class of ordinals has the tree property; set theories with a proper class of indiscernibles; and interpretations of ordinary set theory in Cantorian parts of models of the extension $\mathsf{NFUA}$ of Quine--Jensen set theory $\mathsf{NFU}$. Taken together, these results provide a precise sense in which $\mathsf{ZFC}+Λ$ is an inevitable extension of $\mathsf{ZFC}$ if one attempts to generalize certain metamathematical results about the universe of all finite sets to the entire universe of sets.