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.

Consider an axiomatic set theory in which there is a distinction between “sets” and “classes,” only sets being allowable as elements. How can one define a finite sequence of classes? This problem was proposed to me by A. Tarski, and a solution is given in this note. We shall assume the axiom system Σ used by Godei in his study of the continuum hypothesis, and shall use the same notation.1