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Truth In V for ∃*∀∀-Sentences is Decidable

Published online by Cambridge University Press:  12 March 2014

D. Bellé  and
F. Parlamento
D. Bellé
Affiliation:
Department of Mathematics and Computer Science, University of Udine, Via Delle Scienze 206, 33100 Udine, Italy, E-mail: [email protected]
F. Parlamento
Affiliation:
Department of Mathematics and Computer Science, University of Udine, Via Delle Scienze 206, 33100 Udine, Italy, E-mail: [email protected]
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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 ∀ 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 4 , December 2006 , pp. 1200 - 1222
Copyright
Copyright © Association for Symbolic Logic 2006

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References

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