Notre Dame Journal of Formal Logic

Semantic Completeness of First-Order Theories in Constructive Reverse Mathematics

Christian Espíndola

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

completeness constructive reverse mathematics Kripke semantics algebraic semantics


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.

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