On the interpretation of intuitionistic number theory

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 .