Truth in V for ∃*∀∀-sentences is decidable

Abstract

Let V be the cumulative set theoretic hierarchy, generated from the empty set by taking powers at successor stages and unions at limit stages and, following [2], let the primitive language of set theory be the first order language which contains binary symbols for equality and membership only. Despite the existence of ∀∀-formulae in the primitive language, with two free variables, which are satisfiable in V but not by finite sets ([5]), and therefore of ∃∃∀∀ sentences of the same language, which are undecidable in ZFC without the Axiom of Infinity, truth in V for ∃^*∀∀-sentences of the primitive language, is decidable ([1]). Completeness of ZF with respect to such sentences follows.