A note on the asymptotic expressiveness of ZF and ZFC

Abstract

We investigate the asymptotic densities of theorems provable in Zermelo-Fraenkel set theory ZF and its extension ZFC including the axiom of choice. Assuming a canonical De Bruijn representation of formulae, we construct asymptotically large sets of sentences unprovable within ZF, yet provable in ZFC. Furthermore, we link the asymptotic density of ZFC theorems with the provable consistency of ZFC itself. Consequently, if ZFC is consistent, it is not possible to refute the existence of the asymptotic density of ZFC theorems within ZFC. Both these results address a recent question by Zaionc regarding the asymptotic equivalence of ZF and ZFC.