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On the computational content of the axiom of choice

Published online by Cambridge University Press:  12 March 2014

Stefano Berardi
Affiliation:
Torino University, Dip. Informatica, C. So Svizzera 185, 10149 Torino, Italy, E-mail: stefano@di.unito.it
Marc Bezem
Affiliation:
Utrecht University, Department of Philosophy, P.O. Box 80126, 3508 TC Utrecht, The Netherlands, E-mail: bezem@phil.ruu.nl
Thierry Coquand
Affiliation:
Chalmers University of Gothenburg, Department of Computer Sciences, S-41296, Gothenburg, Sweden, E-mail: coquand@cs.chalmers.se

Abstract

We present a possible computational content of the negative translation of classical analysis with the Axiom of (countable) Choice. Interestingly, this interpretation uses a refinement of the realizability semantics of the absurdity proposition, which is not interpreted as the empty type here. We also show how to compute witnesses from proofs in classical analysis of ∃-statements and how to extract algorithms from proofs of ∀∃-statements. Our interpretation seems computationally more direct than the one based on Gödel's Dialectica interpretation.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1998

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On the computational content of the axiom of choice
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