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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
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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

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