Truth in V for Ǝ ∀∀-Sentences Is Decidable

Journal of Symbolic Logic 71 (4):1200 - 1222 (2006)
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 ƎƎ∀∀ sentences of the same language, which are undecidable in ZFC without the Axiom of Infinity, truth in V for Ǝ*∀∀-sentences of the primitive language, is decidable ([1]). Completeness of ZF with respect to such sentences follows
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