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On the interpretation of intuitionistic number theory

  • S. C. Kleene (a1)
Abstract

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.

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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”.

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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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