Variable Set Theory

Author’s note. This paper was originally written in 1986 during a brief time when scientific American had published a number of math articles. By the time it was finished and submitted, this window had apparently closed. At any rate this paper was turned down and, as far as we know, no further mathematics appeared. We have recently decided that the paper might have sufficient interest to at least be posted. With light editing, here is the paper as originally written. The concept of set, originated about 100 years ago by Georg Cantor and formalized early in this century by Zermelo and Fraenkel, von Neumann, Gödel and Bernays, is a fundamental tool in modern mathematics. It clarifies difficult points in advanced mathematics and mathematicians routinely define new concepts and theories using the language of sets. The idea of set developed in this way concerns what we will call classical sets. The classical approach is based on a rigid concept of membership which does not always faithfully reflect practice either inside or outside of mathematics. This article describes a new, more general notion of set; variable sets, in which the membership varies with respect to a parameter such as time. The theoretical basis for the concept of variable set is topos theory. Topos theory was developed by A. Grothendieck of the Institut des Hautes Études Scientifiques in France (he is now at the University of Grenoble) in the late 1950’s for use in algebraic geometry; in 1969, working at Dalhousie University in Halifax, Nova Scotia, F. W. Lawvere, now at SUNY at Buffalo, and M. Tierney, now at Rutgers University, generalized the concept of topos and worked out the way topos theory could be used to explicate the notion of variable set. The axioms for topos theory were later simplified by Lawvere and Tierney and by C. J. Mikkelsen, then a student at the University of Aarhus, Denmark. We now turn to two examples which will illustrates various aspects of classical set theory.