The Decidability of the
Class and the Axiom of Foundation
Dorella Bellè and Franco Parlamento
Source: Notre Dame J. Formal Logic Volume 42, Number 1
(2001), 41-53.
Abstract
We show that the Axiom of Foundation, as well as the Antifoundation Axiom
AFA, plays a crucial role in determining the decidability of the following problem. Given a first-order theory T over the language , and a sentence
F of the form
with
quantifier-free in the
same language, are there models of T in which F is true?
Furthermore we show that the Extensionality Axiom is quite irrelevant in
that respect.
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1054301354
Digital Object Identifier: doi:10.1305/ndjfl/1054301354
Mathematical Reviews number (MathSciNet): MR1993389
Zentralblatt MATH identifier: 1023.03007
References
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Mathematical Reviews (MathSciNet): MR89j:03039
Zentralblatt MATH: 0668.04001
[2] Bellè, D., and F. Parlamento, "Decidability and completeness for open formulas of membership theories", Notre Dame Journal of Formal Logic, vol. 36 (1995), pp. 304--18.
Mathematical Reviews (MathSciNet): MR96k:03114
Zentralblatt MATH: 0837.03007
Digital Object Identifier: doi:10.1305/ndjfl/1040248461
Project Euclid: euclid.ndjfl/1040248461
[3] Bellè, D., and F. Parlamento, "Undecidability in weak membership theories", pp. 327--37 in Logic and Algebra, Lecture Notes in Pure and Applied Mathematics, edited by A. Ursini and P. Aglianò, Dekker, New York, 1996.
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[6] Parlamento, F., A. Policriti, and K. P. S. B. Rao, "Witnessing differences without redundancies", Proceedings of the American Mathematical Society, vol. 125 (1997), pp. 587--94.
Mathematical Reviews (MathSciNet): MR97d:04003
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Digital Object Identifier: doi:10.1090/S0002-9939-97-03630-7
[7] Parlamento, F., and A. Policriti, "Note on: `The logically simplest form of the infinity axiom'", Proceedings of the American Mathematical Society, vol. 108 (1990), pp. 285--86.
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[8] Parlamento, F., and A. Policriti, "Expressing infinity without foundation", The Journal of Symbolic Logic, vol. 56 (1991), pp. 1230--35.
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