When it comes to math, I think it is best that I assume everyone is really ignorant. So forgive me for going over some really basic ideas about infinity. But I have good reasons for doing this. Some time ago, I wrote, Infinity Is Not a Number. In it, I discuss how the great Christian apologist William Lane Craig used a total misunderstanding of infinity to “prove” that the universe is not eternal. I was pointing out that this brilliant man was ignorant of the fact that infinity is not a number.
I guess I can understand. Mathematicians use infinity (∞) along with a lot of numbers. To understand infinity as a mathematical concept, you really have to study limits. And that isn’t something students get into until college — if then. It’s really when you start to study calculus that you get into limits. And that is done for the opposite reason: to understand the infinitesimal. It’s actually the same thing, but I won’t bore you with it. You either understand what I’m talking about (It’s the reciprocal, stupid!) or you would require a good deal more exposition than I’m willing to expend.
But there are some things about infinity that are intuitive to people of this time and place. For example, if you think of the whole numbers (0, 1, 2, 3…), you know that there are an infinity of them. Regardless of what natural number you can come up with, there is always a number that is one unit larger. Let’s combine the whole numbers with all the negative whole numbers (…, -2, -1, 0, 1, 2, …) that infinity space is twice as large. But that doesn’t mean there are twice as many integers as whole numbers. Both sets contain an infinite number of elements. But they are both “countably infinite.”
So what does it mean that they are countably infinite? It means that you can pick any number in the series and count to it. Consider the whole numbers again (because it is easier). You can count from 0 to 15,000,000 or 15,000,000,000 or whatever. It will just take time. If you have an infinite amount of time, I guess you could count all the whole numbers (and, unintuitively, the integers). (Note: this is one reason immortality is so terrifying: eventually you would get to the point of counting all the whole numbers.)
Now let’s consider the rational numbers from 0 to 1. These are just the fractions. You can tell that they are countably infinite. You just start counting: 0/1, 1/1, 0/2, 1/2, 2/2, 0/3, 1/3, 2/3, 3/3, etc. It doesn’t matter that many of the numbers of duplicates. The denominators go on forever. Thus, the number series is infinite. And we know it is countable because we are counting it. In theory, we can count all of the numbers. The same is true of all the rational numbers — not just the ones between 0 and 1.
So a countable infinity of countable infinities is a countable infinity. You would think that all infinities would thus be countable. But no.
Irrational numbers are numbers that cannot be represented as a fraction of two integers. You know: numbers like π and e. The set of all these numbers does not represent a countable infinity. Indeed, the irrational numbers between any two numbers are not a countable infinity. Think about it. It’s mind blowing.
Rick and Morty
I was thinking about this because I was watching an episode of Rick and Morty, “Rick Potion No 9.” In it, Rick turns his human world into Cronenberg-world. So in order to fix things, Rick goes looking for another reality, “There’s an infinite number of realities, Morty. And in a few dozen of those, I got lucky and turned everything back to normal. I just had to find one of those realities in which we also happened to both die around this time.” (Note: if there were an infinite number of realities, there would be an infinite number of realities where Rich “got luck,” but I’ll leave that as something to consider on your own time.)
The implication of this episode (and others) is that at every instant, there are an infinite number of realities, and each one creates an infinite reality at that point — and on and on and on. I’ve long thought this is what Planck time must represent: the time at which every possible quantum reality is spawned. But that doesn’t much matter. It’s just a thought. But I like Rick’s idea of an infinite universes (“realities”) constantly spawning an infinite number of universes.
Countably Infinite Universes
As outrageous as this notion is, it would still represent a countably infinite number of universes. I find that vaguely comforting. Of course, there’s all that dark matter and dark energy. Maybe that’s where the non-countably infinite universes exist. I don’t really care. I don’t like thinking about things that can’t be counted. Irrational numbers are as strong a proof of the non-existence of God that I can think of.
But let me leave you with a practical thought. As many of you know, I think memorizing multiplication tables is a waste of time. I think children would be better served watching Rick and Morty. They’d have a chance of thinking about math. And it wouldn’t make them hate math. Finally, it might get them to watch Existenz.