Notre Dame Journal of Formal Logic

On Goodman Realizability

Emanuele Frittaion

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Goodman’s theorem states that HAω+AC+RDC is conservative over HA. The same result applies to the extensional case, that is, E-HAω+AC+RDC is also conservative over HA. This is due to Beeson. In this article, we modify Goodman realizability and provide a new proof of the extensional case.

Article information

Source
Notre Dame J. Formal Logic, Volume 60, Number 3 (2019), 523-550.

Dates
Received: 8 November 2017
Accepted: 8 March 2018
First available in Project Euclid: 12 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1562918420

Digital Object Identifier
doi:10.1215/00294527-2019-0018

Mathematical Reviews number (MathSciNet)
MR3985625

Zentralblatt MATH identifier
07120754

Subjects
Primary: 03F03: Proof theory, general
Secondary: 03F10: Functionals in proof theory 03F30: First-order arithmetic and fragments 03F35: Second- and higher-order arithmetic and fragments [See also 03B30] 03F50: Metamathematics of constructive systems

Keywords
Goodman realizability axiom of choice extensionality

Citation

Frittaion, Emanuele. On Goodman Realizability. Notre Dame J. Formal Logic 60 (2019), no. 3, 523--550. doi:10.1215/00294527-2019-0018. https://projecteuclid.org/euclid.ndjfl/1562918420


Export citation

References

  • [1] Beeson, M. J., “Goodman’s theorem and beyond,” Pacific Journal of Mathematics, vol. 84 (1979), pp. 1–16.
  • [2] Beeson, M. J., Foundations of Constructive Mathematics, vol. 6 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer, Berlin, 1985.
  • [3] Coquand, T., “About Goodman’s theorem,” Annals of Pure and Applied Logic, vol. 164 (2013), pp. 437–42.
  • [4] Goodman, N. D., “The theory of the Gödel functionals,” Journal of Symbolic Logic, vol. 41 (1976), pp. 574–82.
  • [5] Goodman, N. D., “Relativized realizability in intuitionistic arithmetic of all finite types,” Journal of Symbolic Logic, vol. 43 (1978), pp. 23–44.
  • [6] Kohlenbach, U., “A note on Goodman’s theorem,” Studia Logica, vol. 63 (1999), pp. 1–5.
  • [7] Kohlenbach, U., Applied Proof Theory: Proof Interpretations and Their Use in Mathematics, Springer Monographs in Mathematics, Springer, Berlin, 2008.
  • [8] Renardel de Lavalette, G. R., “Extended bar induction in applicative theories,” Annals of Pure and Applied Logic, vol. 50 (1990), pp. 139–89.
  • [9] Troelstra, A. S., Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, vol. 344 of Lecture Notes in Mathematics, Springer, New York, 1973.
  • [10] van den Berg, B., and L. van Slooten, “Arithmetical conservation results,” Indagationes Mathematicae, vol. 29 (2017), pp. 260–75.