D. Hilbert. The foundations of mathematics. In J. van Heijenoort, editor, From Frege to Godel: A Sourcebook in Mathematical Logic, 1879-1931, pages 464--479. Harvard University Press, Cambridge, Massachusetts, 1967. Home/Search Document Not in Database Summary Related Articles Check |
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On the computational content of the Axiom of Choice - Berardi, Bezem, Coquand (1995) (8 citations) (Correct)
....reals. This, however, appears to be non trivial. The combination of the Axiom of Choice and the Excluded Middle turns out to be extremely problematic from a constructive point of view. To make constructive sense of such a combination can actually be seen as one the main aims of Hilbert s programme [8, 9]. We address here the more modest question of the analysis of the computational content of the Axiom of Choice, by giving a novel realizability interpretation of the negative translation of the Axiom of Choice. Our paper is organized as follows. After a presentation of the formal system under ....
....by trying to understand the computational behaviour of our interpretation. They hint at possible connections with learning theories, such as the one described in [19] where the learning agent may benefit from negative information. Yet another natural connection is with the work of Hilbert [8, 9] and Ackermann [1] Our interpretation can be seen as a variation of Hilbert s epsilon method [8, 9] with a (classical) proof of termination. It will be interesting to compare our algorithm with the one described in Ackermann s paper [1] 4 . Acknowledgement We thank Anne Troelstra for helpful ....
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D. Hilbert. The foundations of mathematics. In van Heijenoort ed., From Frege to Godel, p. 465-479. 4 According to [14], if it is well-known that the proof of convergence of Ackermann's algorithm was defective as presented in [1], it is not known yet, not even non-constructively, if the method converges at all.
Ordinals and Interactive Programs - Hancock (2000) (Correct)
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D. Hilbert. The foundations of mathematics. In J. van Heijenoort, editor, From Frege to Godel: A Sourcebook in Mathematical Logic, 1879-1931, pages 464--479. Harvard University Press, Cambridge, Massachusetts, 1967.
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