On virtual classes and real numbers1

Publisher: Cambridge University Press

E-ISSN: 1943-5886|15|2|131-134

ISSN: 0022-4812

Source: The Journal of Symbolic Logic, Vol.15, Iss.2, 1950-06, pp. : 131-134

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Abstract

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.