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On the computational content of the axiom of choice

  • Stefano Berardi (a1), Marc Bezem (a2) and Thierry Coquand (a3)
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.

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[1]Ackermann, W., Begründung des Tertium non datur mittels der Hilbertschen Theorie der Widerspruchsfreiheit, Mathematische Annalen, vol. 93 (1924), pp. 136.
[2]Bezem, M., Strong normalization of barrecursive terms without using infinite terms, Archiv für mathematische Logik und Grundlagenforschung, vol. 25 (1985), pp. 175182.
[3]Bishop, E., Foundations of constructive analysis, McGraw-Hill, New York, 1967.
[4]Constable, R. and Murthy, C., Finding computational content in classical proofs, Logical frameworks (Huet, G. and Plotkin, G., editors), Cambridge University Press, 1991, pp, 341362.
[5]Coquand, Th., A semantics of evidence for classical arithmetic, this Journal, vol. 60 (1995), pp. 325337.
[6]Gödel, K., Collected work, vol. I and II (Feferman, S., Dawson, J. W., Kleene, S. C., Moore, G. H., Solovay, R. M., and van Heijenoort, J., editors), Oxford, 1986.
[7]Goodman, N., Intuitionistic arithmetic as a theory of constructions, Ph.D. thesis, Stanford University, 1968.
[8]Hilbert, D., Die logischen Grundlagen der Mathematik, Mathematische Annalen, vol. 88 (1923), pp. 151165.
[9]Hilbert, D., The foundations of mathematics, From Frege to Gödel (van Heijenoort, J., editor), Harvard University Press, Cambridge, MA, 1971, pp. 465479.
[10]Howard, W. A., Functional interpretation of bar induction by bar recursion, Compositio Mathematica, vol. 20 (1968), pp. 107124.
[11]Howard, W. A. and Kreisel, G., Transfinite induction and bar induction of types zero and one, and the role of continuity in intuitionistic analysis, this Journal, vol. 31 (1966), pp. 325358.
[12]Kleene, S. C., On the interpretation of intuitionistic number theory, this Journal, vol. 10 (1945), pp. 109124.
[13]Kolmogorov, A. N., On the principle of the excluded middle, From Frege to Gödel (van Heijenoort, J., editor), Harvard University Press, Cambridge, MA, 1971, pp. 465479.
[14]Kreisel, G., On weak completeness of intuitionistic predicate logic, this Journal, vol. 27 (1962), pp. 139158.
[15]Kreisel, G., Mathematical logic, Lectures on modern mathematics (Saaty, , editor), vol. III, Wiley, 1965, pp. 95195.
[16]Moore, G. E., Zermelo's axiom of choice: Its origins, development and influence, Springer-Verlag, 1982.
[17]Murthy, C., Extracting constructive content from classical proofs, Ph.D. thesis, Cornell University, 1990.
[18]Novikoff, P. S., On the consistency of certain logical calculi, Matematiceskij sbornik (Recueil Mathématique), vol. 12 (1943), no. 54, pp. 230260.
[19]Osherbon, D. N., Stob, M., and Weinstein, S., Systems that learn: An introduction to learning theory for cognitive and computer scientists, MIT Press, 1986.
[20]Scott, D. and Tarski, A., The sentential calculus with infinitely long expressions, Colloquium Mathematicum, vol. VI (1958), pp. 165170.
[21]Shoenfield, J. R., Mathematical logic, Addison-Wesley, 1967.
[22]Spector, C., Provably recursive functional of analysis: A consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics, Recursive function theory (Dekker, J. C. E., editor), Proceedings of Symposia in Pure Mathematics, no. V, American Mathematical Society, 1961, pp. 127.
[23]Tait, W. W., Normal derivability in classical logic, Lecture notes in mathematics (Barwise, J., editor), no. 72, Springer-Verlag, Berlin, 1968, pp. 204236.
[24]Tait, W. W., Normal form theorem for bar recursive functions of finite type, Proceedings of the second Scandinavian logic symposium (Fenstad, J. E., editor), North-Holland, Amsterdam, 1971, pp. 353367.
[25]Troelstra, A. S., Realizability, ILLC Prepublication Series for Mathematical Logic and Foundations ML-92-09.
[26]Troelstra, A. S., Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture Notes in Mathematics, no. 344, Springer-Verlag, Berlin, 1973.
[27]Troelstra, A. S., A note an non-extensional operations in connection with continuity and recursiveness, Indagationes Mathematicae, vol. 39 (1977), pp. 455462.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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