Introduction

The twentieth century witnessed the development and refinement of the mathematical notion of infinity. Here, of course, I am referring primarily to the development of set theory, which is that area of modern mathematics devoted to the study of infinity. This development raises an obvious question: is there a nonphysical realm of infinity?

As is customary in modern set theory, V denotes the universe of sets. The purpose of this notation is to facilitate the (mathematical) discussion of set theory – it does not presuppose any meaning to the concept of the universe of sets.

The basic properties of V are specified by the ZFC axioms. These axioms allow one to infer the existence of a rich collection of sets, a collection that is complex enough to support all of modern mathematics (and this, according to some, is the only point of the conception of the universe of sets).

I shall assume familiarity with elementary aspects of set theory. The ordinals calibrate V through the definition of the cumulative hierarchy of sets (Zermelo 1930). The relevant definition is given as follows:

Definition 1. Define for each ordinal α a set Vα by induction on α.

1. (1) V0 = ∅.

2. (2) Vα + 1 = ℙ(Vα = { X ∣ X ⊆ Vα}.

3. (3) If β is a limit ordinal, then Vα = ∪{Vβ ∣ β < α}.

There is a much more specific version of the question raised concerning the existence of a nonphysical realm of infinity: is the universe of sets a nonphysical realm?