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2016 Semantic Completeness of First-Order Theories in Constructive Reverse Mathematics
Christian Espíndola
Notre Dame J. Formal Logic 57(2): 281-286 (2016). DOI: 10.1215/00294527-3470433


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.


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Christian Espíndola. "Semantic Completeness of First-Order Theories in Constructive Reverse Mathematics." Notre Dame J. Formal Logic 57 (2) 281 - 286, 2016.


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

zbMATH: 06585188
MathSciNet: MR3482747
Digital Object Identifier: 10.1215/00294527-3470433

Primary: 03E35, 03F50
Secondary: 03E25

Rights: Copyright © 2016 University of Notre Dame


Vol.57 • No. 2 • 2016
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