It depends on what we want to use it to discuss. If we're going to talk about the question of an infinite regress posed by apologists and theologians, then I think we're actually better served by discussing a different concept of infinity than that of Cantor's set theory. The transfinite numbers are wonderfully interesting, to be sure, but transfinite arithmetic is different than common arithmetic, and this can quite easily lead into equivocation and confusion.

To that end, allow me to briefly introduce the Hyperreal numbers.

First, some history. The Natural numbers are so named because that number system was the first to be developed and explored, amongst modern number systems. It involves the whole numbers with which one counts-- 0, 1, 2, 3, 4, et cetera. However, the Natural numbers are incomplete. Think, for example, of the equation

. This fairly simple algebra problem has no solution on the Natural numbers, but we can

*extend* the concept of the Natural numbers to give us numbers which are less than zero-- that is to say, numbers which are negative. This extension gives us the Integers. The Integers, themselves, can be extended in order to solve equations like

. For this, we need the Rational numbers. But the Rational numbers are insufficient to solve something like

, so we can extend once again to create the Real numbers.

Just as the Reals extend the Rationals, and the Rationals extend the Integers, and the Integers extend the Naturals; so too can we create an extension for the Real numbers. This extension will allow us to discuss infinite numbers with consistency, using the exact same arithmetic and algebra with which we are already familiar. We call this extension the Hyperreal numbers.

On the Hyperreal number system, there exist numbers which are described as being

*infinite*. A Hyperreal number,

*K*, is called

*infinite* if and only if, for all

*n* such that

*n* is a Natural number, it is true that

. Any number which is not infinite is called

*finite*. There are also numbers which are called

*infinitesimal*. A Hyperreal number, ε, is

*infinitesimal* if and only if, for all

*r* such that

*r* is a Real number, it is true that

. Note that this means zero is the only Real number which is infinitesimal.

The Hyperreal numbers have some very familiar properties. For example, given an infinite Hyperreal number,

*K*, it will always be true that

. Note that

*K-1*,

*K*, and

*K+1* are all infinite numbers, but they are

*different* infinite numbers. In exactly the same way, if

*K* is positive, it will also be true that

. For a more detailed breakdown of the Hyperreals,

check out this link, or if you really want to get in depth with the subject, pick up Abraham Robinson's book

*Non-Standard Analysis*. There is no better text on the subject-- after all, it was Robinson who discovered, invented, and developed this number system.

The Hyperreals can be added, subtracted, multiplied, divided, exponentiated, logarithmed, and manipulated in exactly the same ways that the Real numbers can be. They give us a very familiar platform with which to discuss infinities, and as such, they are far more intuitive for use in discussions about infinite regress.