Journal of Symbolic Logic

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

D. Bellé and F. Parlamento

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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.

Article information

J. Symbolic Logic, Volume 71, Issue 4 (2006), 1200-1222.

First available in Project Euclid: 20 November 2006

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: Primary 03B25, Secondary 03E30, 03C62


Bellé, D.; Parlamento, F. Truth in V for ∃ * ∀∀-sentences is decidable. J. Symbolic Logic 71 (2006), no. 4, 1200--1222. doi:10.2178/jsl/1164060452.

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