What David Teaches Goliath: An Inevitable Extension of ZFC

Abstract

We examine the metamathematical significance of the Levy scheme $\Lambda$, which asserts, for each natural number $n$, the existence of an $n$-Mahlo cardinal $\kappa$ such that $V_{\kappa }{\prec }_{{\Sigma }_n}V$. Several initially unrelated constructions single out $\mathsf{ZFC}+\Lambda$: 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, the results surveyed here provide a precise sense in which $\mathsf{ZFC}+\Lambda$ is an inevitable extension of $\mathsf{ZFC}$ if one attempts to generalize certain metamathematical results about the microcosm of hereditarily finite sets to the entire macrocosm of sets.

arXiv comment12 pages. No substantive changes since the previous draft (mostly fixed historical references)