Set—Theoretical Representations of Ordered Pairs and Their Adequacy for the Logic of Relations

Abstract

One of the most significant discoveries of early twentieth century mathematical logic was a workable definition of ‘ordered pair’ totally within set theory. Norbert Wiener, and independently Casimir Kuratowski, are usually credited with this discovery. A definition of ‘ordered pair’ held the key to the precise formulation of the notions of ‘relation’ and ‘function’ — both of which are probably indispensable for an understanding of the foundations of mathematics. The set-theoretic definition of ‘ordered pair’ thus turned out to be a key victory for logicism, providing one admits set theory is logic. The definition also was instrumental in achieving the appearance of ontological economy — since it seemed only sets were needed — although this feature was emphasized only later.