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Published online by Cambridge University Press: 12 March 2014
In a simple, applied functional calculus of first order (i.e., one admitting no functional variables but at least one functional constant), abstracts or schematic expressions may be introduced to play the role of variables over designatable sets or classes. The entities or quasi-entities designated or quasi-designated by such abstracts may be called, following Quine, virtual classes and relations. The notion of virtual class is always relative to a given formalism and depends upon what functional constants are taken as primitive. The first explicit introduction of a general notation for virtual classes (relative to a given formalism) appears to be D4.1 of the author's A homogeneous system for formal logic. That paper develops a system admitting only individuals as values for variables and is adequate for the theory of general recursive functions of natural numbers. Numbers and functions are in fact identified with certain kinds of virtual classes and relations.
In the present paper it will be shown how certain portions of the theory of real numbers can be constructed upon the basis of the theory of virtual classes and relations of H.L.
The method of building up the real numbers to be employed is essentially an adaptation of standard procedure. Although the main ideas underlying this method are well known, the mirroring of these ideas within the framework of the restricted concepts admitted here presents possibly some novelty. In particular, a basis for the real numbers is provided which in no way admits classes or relations or other "abstract" objects as values for variables. Presupposing the natural numbers, the essential steps are to construct the simple rationals as virtual dyadic relations between natural numbers, to construct the generalized or signed rationals as virtual tetradic relations among natural numbers, and then to formulate a notation for real numbers as virtual classes (of a certain kind) of generalized rationals. Of course, there are several alternative methods. This procedure, however, appears to correspond more to the usual one.
This paper was read before a meeting of the Association for Symbolic Logic on December 31, 1948 at Columbus, Ohio. A part of the material was contained in a doctoral dissertation at Yale University in 1941.
2 See especially Quine, W. V., On universals, this Journal, vol. 12 (1947), pp. 74–84Google Scholar.
3 This Journal, vol. 8 (1943), pp. 1–23. This paper will subsequently be referred to as H.L.
4 See Martin, R. M., A note on nominalism and recursive functions, this Journal, vol. 14 (1949), pp. 27–31Google Scholar. This paper will subsequently be referred to as N.R.F.
An additional correction to H.L. should be noted. The proof of *T3.34 (p. 18) should make no reference to R8(4). Rather the whole proof must be reconstructed as two inductive proofs, one of the implication read from right to left, the other from left to right. I am indebted to my student, Mr. Leon Robbins, Jr., for calling attention to this error. Also note that R8(4) itself is redundant, being readily provable by means of R8(1)–R8(3). Take G of R8(3) merely as ‘x 1 … xn ϶ (*G(x 1, …, xn, a n+1, …, a 2n))’.