Thus, \(V\) cannot be characterized by any sentence, as any sentence that is true in \(V\) must be also true in some initial segment \(V_\alpha\). In particular, ZFC is not finitely axiomatizable, for otherwise ZFC would prove that, for unboundedly many ordinals \(\alpha\), \(V_\alpha\) is a model of ZFC, contradicting Gödel’s second incompleteness theorem (see  Section 5.2).