A note on the asymptotic expressiveness of ZF and ZFC

TL;DR

This paper explores the asymptotic densities of theorems in ZF and ZFC set theories, revealing differences and linking theorem density to the provability of ZFC's consistency, thus addressing questions about their asymptotic equivalence.

Contribution

It constructs large sets of sentences unprovable in ZF but provable in ZFC and connects theorem density with ZFC's provable consistency.

Findings

Existence of large unprovable sentences in ZF that are provable in ZFC

Asymptotic density of ZFC theorems linked to ZFC's consistency

If ZFC is consistent, its theorem density cannot be refuted within 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.