Notre Dame Journal of Formal Logic
- Notre Dame J. Formal Logic
- Volume 42, Number 1 (2001), 41-53.
The Decidability of the $ \forall^*\exists$ Class and the Axiom of Foundation
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.
Notre Dame J. Formal Logic, Volume 42, Number 1 (2001), 41-53.
First available in Project Euclid: 30 May 2003
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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