Notre Dame Journal of Formal Logic

On Theorems of Gödel and Kreisel: Completeness and Markov's Principle

D. C. McCarty

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Abstract

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.

Article information

Source
Notre Dame J. Formal Logic, Volume 35, Number 1 (1994), 99-107.

Dates
First available in Project Euclid: 22 December 2002

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1040609297

Digital Object Identifier
doi:10.1305/ndjfl/1040609297

Mathematical Reviews number (MathSciNet)
MR1271701

Zentralblatt MATH identifier
0801.03038

Subjects
Primary: 03B20: Subsystems of classical logic (including intuitionistic logic)
Secondary: 03F55: Intuitionistic mathematics

Citation

McCarty, D. C. On Theorems of Gödel and Kreisel: Completeness and Markov's Principle. Notre Dame J. Formal Logic 35 (1994), no. 1, 99--107. doi:10.1305/ndjfl/1040609297. https://projecteuclid.org/euclid.ndjfl/1040609297


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