Skip to main content
    • Aa
    • Aa

On the interpretation of intuitionistic number theory

  • S. C. Kleene (a1)

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.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

Alonzo. Church [1] An unsolvable problem of elementary number theory, American journal of mathematics, vol. 58 (1936), pp. 345363.

Gerhard. Gentzen [2] Die Widerspruchsfreiheit der reinen Zahlentheorie, Mathematische Annalen, vol. 112 (1936), pp. 493565.

S. C. Kleene [11] General recursive functions of natural numbers, Mathematische Annalen, vol. 112 (1936), pp. 727742.

S. C. Kleene [14] Recursive predicates and quantifiers, Transactions of the American Mathematical Society, vol. 53 (1943), pp. 4173.

S. C. Kleene [15] On the forms of the predicates in the theory of constructive ordinals, American journal of mathematics, vol. 66 (1944), pp. 4158. Errata: Page 43, line 12, read “(4)” instead of “(4*)”. Page 54, line 4, read “a” instead of “e”.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 11 *
Loading metrics...

Abstract views

Total abstract views: 175 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 21st August 2017. This data will be updated every 24 hours.