What David Teaches Goliath: An Inevitable Extension of ZFC

We examine the metamathematical significance of the Levy scheme $\mathrm{\Lambda }$, which asserts, for each natural number $n$, the existence of an $n$-Mahlo cardinal $\kappa$ such that $V_{\kappa }{\prec }_{{\mathrm{\Sigma }}_n}V$. Several initially unrelated constructions single out $\mathsf{Z}\mathsf{F}\mathsf{C}+\mathrm{\Lambda }$: 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 $\mathsf{N}\mathsf{F}\mathsf{U}\mathsf{A}$ of Quine–Jensen set theory $\mathsf{N}\mathsf{F}\mathsf{U}$. Taken together, these results provide a precise sense in which $\mathsf{Z}\mathsf{F}\mathsf{C}+\mathrm{\Lambda }$ is an inevitable extension of $\mathsf{Z}\mathsf{F}\mathsf{C}$ if one attempts to generalize certain metamathematical results about the universe of all finite sets to the entire universe of sets.