Foreword on Set Theory

Abstract

The building blocks of mathematics are sets, yet no one should say what a set is. An axiomatic treatment of set theory postulates the existence of certain undefined or primitive objects, called sets, together with symbols and axioms governing their use. A graphic analogy can be made with geometry where there are given undefined objects called points, lines, and planes, together with a collection of axioms relating these objects to each other. An axiom of projective geometry is P: If L ₁ and L ₂ are distinct lines, there is one and only one point P on both L ₁ and L ₂. Here the relation “P is on L ₁” is undefined. Another axiom is P*: If P ₁ and P ₂ are distinct points, there is one and only one line L on both P ₁ and P ₂. Any collection of objects which, when properly named as “points” and “lines”, satisfy the axioms of geometry, serve as points and lines, equally as well as any other such collection. It must be possible to replace in all geometric statements the words point, line, plane by table, chair, mug (David Hilbert, quoted by Weyl [44, p. 635]). Nevertheless, from the axioms we soon discover that some objects in a geometry are not “points” and some objects are not “lines”. Analogously in the theory of sets, after we have named our candidates for sets, and listed the undefined symbols and the axioms governing them, we find that not everything in sight is a set. For example, as Cantor observed, there is no such set as “the set of all sets”.