Notre Dame Journal of Formal Logic

The Decidability of the $ \forall^*\exists$ Class and the Axiom of Foundation

Dorella Bellè and Franco Parlamento

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 $ =,\in$, and a sentence F of the form $ \forall x_1, \ldots, x_n \exists y F^M$ with $ F^M$ 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.

Article information

Source
Notre Dame J. Formal Logic, Volume 42, Number 1 (2001), 41-53.

Dates
First available in Project Euclid: 30 May 2003

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1054301354

Digital Object Identifier
doi:10.1305/ndjfl/1054301354

Mathematical Reviews number (MathSciNet)
MR1993389

Zentralblatt MATH identifier
1023.03007

Subjects
Primary: 03B25: Decidability of theories and sets of sentences [See also 11U05, 12L05, 20F10] 03C62: Models of arithmetic and set theory [See also 03Hxx]
Secondary: 03E50: Continuum hypothesis and Martin's axiom [See also 03E57]

Keywords
decidability undecidability foundation extensionality

Citation

Bellè, Dorella; Parlamento, Franco. The Decidability of the $ \forall^*\exists$ Class and the Axiom of Foundation. Notre Dame J. Formal Logic 42 (2001), no. 1, 41--53. doi:10.1305/ndjfl/1054301354. https://projecteuclid.org/euclid.ndjfl/1054301354


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References

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