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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: belle@dimi.uniud.it
F. Parlamento
Affiliation:
Department of Mathematics and Computer Science, University of Udine, Via Delle Scienze 206, 33100 Udine, Italy, E-mail: parlamen@dimi.uniud.it
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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

REFERENCES

[1]Bellè, D., The decision problem for extensional set theories, Ph.D. thesis, Universitàdi Siena, 1998.Google Scholar
[2]Friedman, H., Three quantifier sentences, Fundamenta Mathematicae, vol. 177 (2003), pp. 213–240.CrossRefGoogle Scholar
[3]Gogol, D., The ∀n∃-completeness of Zermelo-Fraenkel set theory, Zeitschrift für mathematische Logik und die Grundlagen der Mathematik, vol. 24 (1978), pp. 289–290.CrossRefGoogle Scholar
[4]Omodeo, E., Parlamento, F., and Policriti, A., Decidability of ∃*∀-sentences in membership theories, Mathematical Logic Quarterly, vol. 42 (1996), pp. 41–58.CrossRefGoogle Scholar
[5]Parlamento, F. and Policriti, A., Note on: The logically simplest form of the infinity axiom, Proceedings of the American Mathematical Society, vol. 108 (1990), pp. 285–286.Google Scholar
[6]Parlamento, F. and Policriti, A., Expressing infinity without foundation, this Journal, vol. 56 (1991), no. 4, pp. 1230–1235.Google Scholar
[7]Parlamento, F., Policriti, A., and Rao, K. P. S., Witnessing differences without redundancies, Proceedings of the American Mathematical Society, vol. 125 (1997). pp. 587–594.CrossRefGoogle Scholar