Journal of Symbolic Logic

Completeness and incompleteness for intuitionistic logic

Charles McCarty

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Abstract

We call a logic regular for a semantics when the satisfaction predicate for at least one of its nontheorems is closed under double negation. Such intuitionistic theories as second-order Heyting arithmetic HAS and the intuitionistic set theory IZF prove completeness for no regular logics, no matter how simple or complicated. Any extensions of those theories proving completeness for regular logics are classical, i.e., they derive the tertium non datur. When an intuitionistic metatheory features anticlassical principles or recognizes that a logic regular for a semantics is nonclassical, it proves explicitly that the logic is incomplete with respect to that semantics. Logics regular relative to Tarski, Beth and Kripke semantics form a large collection that includes propositional and predicate intuitionistic, intermediate and classical logics. These results are corollaries of a single theorem. A variant of its proof yields a generalization of the Gödel—Kreisel Theorem linking weak completeness for intuitionistic predicate logic to Markov's Principle.

Article information

Source
J. Symbolic Logic, Volume 73, Issue 4 (2008), 1315-1327.

Dates
First available in Project Euclid: 27 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1230396921

Digital Object Identifier
doi:10.2178/jsl/1230396921

Mathematical Reviews number (MathSciNet)
MR2467219

Zentralblatt MATH identifier
1165.03050

Citation

McCarty, Charles. Completeness and incompleteness for intuitionistic logic. J. Symbolic Logic 73 (2008), no. 4, 1315--1327. doi:10.2178/jsl/1230396921. https://projecteuclid.org/euclid.jsl/1230396921


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