Virtual Logic: Cantor's Paradise and the Parable of Frozen Time

Georg Cantor (Cantor, 1941; Dauben, 1990) is well-known to mathematicians as the inventor/discoverer of the arithmetic and ordering of mathematical infinity. Cantor discovered the theory of transfinite numbers, and an infinite hierarchy of ever-larger infinities. To the uninitiated this Cantorian notion of larger and larger infinities must seem prolix and astonishing, given that it is difficult enough to imagine infinity, much less an infinite structure of infinities. Who has not wondered about the vastness of interstellar space, or the possibility of worlds within worlds forever, as we descend into the microworld? It is part of our heritage as observers of ourselves and our universe that we are interested in infinity. Infinity in the sense of unending process and unbounded space is our intuition of life, action and sense of being. In this column we discuss infinity, first from a Cantorian point of view. We prove Cantor's Key Theorem about the power set of a set. Given a set X, the power set P(X) is the set of all subsets of X. Cantor's Theorem states that P(X) is always larger than X, even when X itself is infinite. What can it mean for one infinity to be larger than another? Along with proving his very surprising theorem, Cantor found a way to compare infinities. It is a way that generalizes how we compare finite sets. We will discuss this generalization of size in the first section of the column below. In the world of Cantor, if X is an infinite set, then X < P(X) < P(P(X)) < ... in an unending ascent into higher infinities. Set theory in the classical mode is a theory about forms and structures that are eternally unchanging. When we speak of the set of integers or the set of all sets they are assumed to exist in eternity. When set theory is propelled into the world of Time, a different and equally beautiful structure emerges. This temporal structure is the domain of performative language. Performative language is language whose very utterance commits the speaker to a process and a contract for action. When we speak of all sets or all distinctions, then the very act of speaking creates new sets and new distinctions. The only eternity here is in the moment, and time is an integral part of the structure of our acts and our mathematics.