Chapter 4 - EXPRESSIVE POSSIBILITIES OF THE PRESENT SYMBOLIZATION

Abstract

This chapter discusses symbolic expression of operations and functions, possibility of elementary symbolization of classical mathematics, individual constants and their elimination, and formalization of mathematical theories without primitive equality. To illustrate the expressive possibilities of a symbolic language the chapter takes an example of the elementary axiomatic theory of groups, where elementary means a theory not involving explicitly the notions of integers and sets. The symbols might be enriched so as to include new kinds of operational expressive means, the functional constants. These serve directly to denote the basic operations. The restriction of the means of expression to individual indeterminate and predicate constants does not hinder the formulation of the axiomatic basis of arbitrary elementary binary operations, which are to satisfy certain axioms. Such theories, however, are still relatively poor, unless one has the notions of integers, induction, and—if possible— also the notion of set, in sufficient generality. A convincing argument for the fact that nonlogical means of expression, restricted to individual indeterminate and predicate constants, are already sufficient to carry out the symbolization of the whole of classical mathematics is provided only by the existence of a finitely axiomatizable system of the theory of sets, due to von Neumann and improved by Bernays and Godei. The chapter describes this axiomatic theory.