  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.

Logic;Combinatorics