A Brief Look at Infinity

Man, that's a big number.

It also seems to be one of the most misused concepts out there. I hear people casually bandy it about quite a lot, using different words of course: forever, eternity, endless, blah blah blah. It kind of bugs me, because I think that misunderstanding the concept leads to some philosophical and/or theological issues. Kind of hard to discuss an afterlife, for example, without acknowledging the infinite. While the nature of the hereafter is up for grabs, the duration doesn't seem to be, but it seems to me you can't really have a meaningful discussion or pondering of the idea without understanding what infinity might really mean.

You know what helps me put infinity into a better conceptual frame? Mathematics. I know, no surprise there. Specifically, number theory, and a discussion of transinfinite numbers, which leads one to the immensely surprising realization that there are orders of infinity, that some infinities are larger than others. An example? OK, now we turn to our math books for a moment. Don't worry, no higher math really necessary at this point.

Take the set of integers; that is, all numbers that can be written without a fractional or decimal equivalent and their negatives-e.g. ...-3, -2, -1, 0, 1, 2, 3...-and so on, going to infinity in both directions. That's a pretty big set, right? Infinite in size, no?

Ah, but what about whole numbers? Whole numbers, in this case, means the set of all numbers that can be expressed in a decimal or fractional form (this covers rational numbers, too, but we don't care about that right now). A quick glance reveals this set is far larger than the integer set, which we already know is infinite, because for every integer, there's a boatload of whole numbers between it. In fact, whole numbers are a transinfinite set, because there is an infinity of numbers between any two points in the set of integers. That's right, you can divide the space between any two integers into infinitely fine gradations. Wow.

Hold on here, buckaroos, because the whole numbers aren't even the largest infinite set. Let's take a look at real numbers, which not only cover rational numbers, which can be expressed in finite decimal or fractional form, but irrational numbers as well. An irrational number can't be expressed exactly in decimal or fractional form, although you can get functional approximations: commonly known irrational numbers include pi and the square root of 2. Take a look at the real numbers, and now you're cooking with gas.

Guess what? There are still larger sets to be found. But, I'm not going to get into Cantor numbers like aleph-null or concepts like imaginary numbers; the point is already made. Thinking about infinity can be dizzying, but anybody who seriously believes in some form of afterlife needs to be considering it at some level, especially those who subscribe to any of the Judeo-Christian flavors. If the first tick of the clock is a million years, as the hard-case nuns in Catholic schools of the pre-Vatican II era used to say, then eternity's a whole lot of time to fill. And you know something? It's a whole lot of time no matter where you might end up, no matter how slowly or quickly you perceive time...and no matter what manner of being you might be. Next time you turn your mind to the hereafter, make sure you factor that into your calculations.