Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-20T03:50:56.773Z Has data issue: false hasContentIssue false
  • English
  • Français

On the computational content of the axiom of choice

Published online by Cambridge University Press:  12 March 2014

Stefano Berardi ,
Marc Bezem  and
Thierry Coquand
Stefano Berardi
Affiliation:
Torino University, Dip. Informatica, C. So Svizzera 185, 10149 Torino, Italy, E-mail: [email protected]
Marc Bezem
Affiliation:
Utrecht University, Department of Philosophy, P.O. Box 80126, 3508 TC Utrecht, The Netherlands, E-mail: [email protected]
Thierry Coquand
Affiliation:
Chalmers University of Gothenburg, Department of Computer Sciences, S-41296, Gothenburg, Sweden, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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
Information
The Journal of Symbolic Logic , Volume 63 , Issue 2 , June 1998 , pp. 600 - 622
Copyright
Copyright © Association for Symbolic Logic 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Ackermann, W., Begründung des Tertium non datur mittels der Hilbertschen Theorie der Widerspruchsfreiheit, Mathematische Annalen, vol. 93 (1924), pp. 1–36.Google Scholar
[2] Bezem, M., Strong normalization of barrecursive terms without using infinite terms, Archiv für mathematische Logik und Grundlagenforschung, vol. 25 (1985), pp. 175–182.Google Scholar
[3] Bishop, E., Foundations of constructive analysis, McGraw-Hill, New York, 1967.Google Scholar
[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, 341–362.Google Scholar
[5] Coquand, Th., A semantics of evidence for classical arithmetic, this Journal, vol. 60 (1995), pp. 325–337.Google Scholar
[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.Google Scholar
[7] Goodman, N., Intuitionistic arithmetic as a theory of constructions, Ph.D. thesis , Stanford University, 1968.Google Scholar
[8] Hilbert, D., Die logischen Grundlagen der Mathematik, Mathematische Annalen, vol. 88 (1923), pp. 151–165.Google Scholar
[9] Hilbert, D., The foundations of mathematics, From Frege to Gödel (van Heijenoort, J., editor), Harvard University Press, Cambridge, MA, 1971, pp. 465–479.Google Scholar
[10] Howard, W. A., Functional interpretation of bar induction by bar recursion, Compositio Mathematica, vol. 20 (1968), pp. 107–124.Google Scholar
[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. 325–358.Google Scholar
[12] Kleene, S. C., On the interpretation of intuitionistic number theory, this Journal, vol. 10 (1945), pp. 109–124.Google Scholar
[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. 465–479.Google Scholar
[14] Kreisel, G., On weak completeness of intuitionistic predicate logic, this Journal, vol. 27 (1962), pp. 139–158.Google Scholar
[15] Kreisel, G., Mathematical logic, Lectures on modern mathematics (Saaty, , editor), vol. III, Wiley, 1965, pp. 95–195.Google Scholar
[16] Moore, G. E., Zermelo's axiom of choice: Its origins, development and influence, Springer-Verlag, 1982.Google Scholar
[17] Murthy, C., Extracting constructive content from classical proofs, Ph.D. thesis , Cornell University, 1990.Google Scholar
[18] Novikoff, P. S., On the consistency of certain logical calculi, Matematiceskij sbornik (Recueil Mathématique), vol. 12 (1943), no. 54, pp. 230–260.Google Scholar
[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.Google Scholar
[20] Scott, D. and Tarski, A., The sentential calculus with infinitely long expressions, Colloquium Mathematicum, vol. VI (1958), pp. 165–170.Google Scholar
[21] Shoenfield, J. R., Mathematical logic, Addison-Wesley, 1967.Google Scholar
[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. 1–27.Google Scholar
[23] Tait, W. W., Normal derivability in classical logic, Lecture notes in mathematics (Barwise, J., editor), no. 72, Springer-Verlag, Berlin, 1968, pp. 204–236.Google Scholar
[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. 353–367.Google Scholar
[25] Troelstra, A. S., Realizability, ILLC Prepublication Series for Mathematical Logic and Foundations ML-92-09.Google Scholar
[26] Troelstra, A. S., Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture Notes in Mathematics, no. 344, Springer-Verlag, Berlin, 1973.Google Scholar
[27] Troelstra, A. S., A note an non-extensional operations in connection with continuity and recursiveness, Indagationes Mathematicae, vol. 39 (1977), pp. 455–462.Google Scholar