On ordered pairs

Abstract

Wiener, in 1914, reduced the theory of relations to that of classes by construing relations as classes of ordered pairs and defining the ordered pair in turn on the basis of class theory alone.1 The definition, as improved by Kuratowski,2 identifies the ordered pair x;y with uxyi(ix U iy).In terms of Russell's theory of types, x;y in the above sense is two types higher than x and y. Even when we abandon Russell's theory of fixed types of objects in favor of a theory of stratified formulae,3 there is still significance in saying that ‘x;y’ is of type 2 relative to ‘x’ and ‘y’—meaning that a test of the stratification of any context involves assigning a higher number by 2 to ‘x;y’ than to ‘x’ and ‘y’.