Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T00:14:19.779Z Has data issue: false hasContentIssue false
  • English
  • Français

On the interpretation of intuitionistic number theory

Published online by Cambridge University Press:  12 March 2014

S. C. Kleene
S. C. Kleene*
Affiliation:
The University of Wisconsin, Madison, Wisconsin
Get access Rights & Permissions [Opens in a new window]

Extract

The purpose of this article is to introduce the notion of “recursive realizability.”

Let P be some property of natural numbers. Consider the existential statement, “There exists a number n having the property P.” To explain the meaning which this has for a constructivist or intuitionist, it has been described as a partial judgement, or incomplete communication of a more specific statement which says that a certain given number n, or the number n obtainable by a certain given method, has the property P. The meaning of the existential statement thus resides in a reference to certain information, which it implies could be stated in detail, though the trouble is not taken to do so. Perhaps the detail is suppressed in order to convey a general view of some fact.

The information to which reference is made should be thought of as possibly comprising other items besides the value of n or method for obtaining it, namely such items as may be necessary to complete the communication that that n has the property P.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 10 , Issue 4 , December 1945 , pp. 109 - 124
Copyright
Copyright © Association for Symbolic Logic 1946

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

Church, Alonzo. [1] An unsolvable problem of elementary number theory, American journal of mathematics, vol. 58 (1936), pp. 345–363.CrossRefGoogle Scholar
Gentzen, Gerhard. [2] Die Widerspruchsfreiheit der reinen Zahlentheorie, Mathematische Annalen, vol. 112 (1936), pp. 493–565.CrossRefGoogle Scholar
Gentzen, Gerhard. [3] Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie, Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, new series no. 4 (1938), pp. 19–44, Leipzig (Hirzel).Google Scholar
Gödel, Kurt. [4] Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198.CrossRefGoogle Scholar
Gödel, Kurt. [5] Zur intuitionistischen Arithmetik und Zahlentheorie, Ergebnisse eines mathematischen Kolloquiums, Colloquium 52, June 28, 1932.Google Scholar
Gödel, Kurt. [6] On undecidable propositions of formal mathematical systems, mimeographed lecture notes, Institute for Advanced Study, 1934.Google Scholar
Heyting, Arend. [7] Die formalen Regeln der intuilionistischen Logik, Silzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, 1930, pp. 42–56.Google Scholar
Heyting, Arend. [8] Mathematische Grundlagenforschung. Intuitionismus. Beweistheorie. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 3 no. 4 (1934), Berlin (Springer).Google Scholar
Hilbert, David and Bernays, Paul. [9] Grundlagen der Mathematik, vol. 1 (1934), Berlin (Springer).Google Scholar
Hilbert, David and Bernays, Paul. [10] Grundlagen der Mathematik, vol. 2 (1939), Berlin (Springer).Google Scholar
Kleene, S. C. [11] General recursive functions of natural numbers, Mathematische Annalen, vol. 112 (1936), pp. 727–742.CrossRefGoogle Scholar
Kleene, S. C. [12] On notation for ordinal numbers, this Journal, vol. 3 (1938), pp. 150–155.Google Scholar
Kleene, S. C. [13] On the term ‘analytic’ in logical syntax, The journal of unified science (Erkenntnis), vol. 9 (preprinted for the fifth International Congress for the Unity of Science, Cambridge, Mass., 1939, pp. 189–192).Google Scholar
Kleene, S. C. [14] Recursive predicates and quantifiers, Transactions of the American Mathematical Society, vol. 53 (1943), pp. 41–73.CrossRefGoogle Scholar
Kleene, S. C. [15] On the forms of the predicates in the theory of constructive ordinals, American journal of mathematics, vol. 66 (1944), pp. 41–58. Errata: Page 43, line 12, read “(4)” instead of “(4*)”. Page 54, line 4, read “a” instead of “e”.CrossRefGoogle Scholar
Nelson, David. [16] Recursive functions and intuitionistic number theory, forthcoming.Google Scholar
Turing, A. M. [17] On computable numbers, with an application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, ser. 2 vol. 42 (1937), pp. 230–265.CrossRefGoogle Scholar
Turing, A. M. [18] A correction (to the same), Proceedings of the London Mathematical Society, ser. 2 vol. 43 (1937), pp. 544–546.Google Scholar