Notre Dame Journal of Formal Logic

Semantic Completeness of First-Order Theories in Constructive Reverse Mathematics

Christian Espíndola

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Abstract

We introduce a general notion of semantic structure for first-order theories, covering a variety of constructions such as Tarski and Kripke semantics, and prove that, over Zermelo–Fraenkel set theory (ZF), the completeness of such semantics is equivalent to the Boolean prime ideal theorem (BPI). Using a result of McCarty (2008), we conclude that the completeness of Kripke semantics is equivalent, over intuitionistic Zermelo–Fraenkel set theory (IZF), to the Law of Excluded Middle plus BPI. Along the way, we also prove the equivalence, over ZF, between BPI and the completeness theorem for Kripke semantics for both first-order and propositional theories.

Article information

Source
Notre Dame J. Formal Logic, Volume 57, Number 2 (2016), 281-286.

Dates
Received: 10 July 2013
Accepted: 15 February 2014
First available in Project Euclid: 9 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1455030207

Digital Object Identifier
doi:10.1215/00294527-3470433

Mathematical Reviews number (MathSciNet)
MR3482747

Zentralblatt MATH identifier
06585188

Subjects
Primary: 03F50: Metamathematics of constructive systems 03E35: Consistency and independence results
Secondary: 03E25: Axiom of choice and related propositions

Keywords
completeness constructive reverse mathematics Kripke semantics algebraic semantics

Citation

Espíndola, Christian. Semantic Completeness of First-Order Theories in Constructive Reverse Mathematics. Notre Dame J. Formal Logic 57 (2016), no. 2, 281--286. doi:10.1215/00294527-3470433. https://projecteuclid.org/euclid.ndjfl/1455030207


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