The formal structure of a denumerable system

Abstract

Introduction. A strictly formal mathernatical system comprising definite and variable number and function signs, logical constants and universal and existential operators, and employing the sentential calculus, may be termed classical if the application of the operators to the variables is without restriction; as is well known classical analysis is formalisable in such a system [1](1). If the application of the operators is restricted to variables whose range of values is a finite class, the system is said to be finitist. A limited part only of classical analysis can be formulated in a system finitist in this sense [2]. A denumerable system is one in which the universal and existential operators are restricted to variables whose range of values is a class of cardinal No. The concept of a denumerable system was introduced by Weyl [4]. The purpose of this paper is to indicate how wide a part of classical function theory may be formulated in a denumerable system. Formal characterisation. The primary elements of the system are numerals, numeral variables and numeral functions, the universal, existential, and minimal operators, and the logical constants. The calculus of the primary elements will be taken to be the system Z, of Hilbert-Bernays [1], so that at the first level, a denumerable system differs in no respect from the classical. (In particular the tertium non datur is valid in the form . (Ax) (f(x) 0) . V . (Ex) (f (x)-O0)., where x is a numeral variable.) We use different kinds of variables to distinguish numeral variables from function variables (a denumerable infinity of both kinds being included amongst the primary elements) and we rely on Russell's ramified theory of types [5] for avoiding paradoxes. The secondary elements are integral and rational number and function signs, which may be eliminated from the formulas of the system by means of the following definitions: An integer is a pair of numerals [p, q]; we define [p, q]_ [p', q'], [p, q] p'+q, p+q' q or p <q; it follows that [p, q] is positive or negative according as [p, q] [0, O]. An integer function f([p, q]) is a pair of numeral functions [P(p, q), Q(p, q) ] such that