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Winter 1994 On Theorems of Gödel and Kreisel: Completeness and Markov's Principle
D. C. McCarty
Notre Dame J. Formal Logic 35(1): 99-107 (Winter 1994). DOI: 10.1305/ndjfl/1040609297


In 1957, Gödel proved that completeness for intuitionistic predicate logic HPL implies forms of Markov's Principle, MP. The result first appeared, with Kreisel's refinements and elaborations, in Kreisel. Featuring large in the Gödel-Kreisel proofs are applications of the axiom of dependent choice, DC. Also in play is a form of Herbrand's Theorem, one allowing a reduction of HPL derivations for negated prenex formulae to derivations of negations of conjunctions of suitable instances. First, we here show how to deduce Gödel's results by alternative means, ones arguably more elementary than those of Kreisel. We avoid DC and Herbrand's Theorem by marshalling simple facts about negative translations and Markov's Rule. Second, the theorems of Gödel and Kreisel are commonly interpreted as demonstrating the unprovability of completeness for HPL, if means of proof are confined within strictly intuitionistic metamathematics. In the closing section, we assay some doubts about such interpretations.


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D. C. McCarty. "On Theorems of Gödel and Kreisel: Completeness and Markov's Principle." Notre Dame J. Formal Logic 35 (1) 99 - 107, Winter 1994.


Published: Winter 1994
First available in Project Euclid: 22 December 2002

zbMATH: 0801.03038
MathSciNet: MR1271701
Digital Object Identifier: 10.1305/ndjfl/1040609297

Primary: 03B20
Secondary: 03F55

Rights: Copyright © 1994 University of Notre Dame


Vol.35 • No. 1 • Winter 1994
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