Q&A: A Theoretical Question About Infinity
A Theoretical Question About Infinity
Question
Hi Michi, greetings.
I understood that you reject various arguments about the impossibility of an eternal world because it isn’t defined properly to begin with, or because they treat infinity incorrectly {potential and concrete}. I have a thought experiment that’s similar to those arguments, and I’d appreciate it if, assuming it’s wrong, you could explain what’s wrong with it or how it differs from the other arguments.
So here it is: there are two things that can’t both exist at the same time. One is that space is infinite, and the other is that space grows over time. That’s because if it’s infinite, then it already has nowhere left to grow. Just like if space can grow up to some size, say 10 square meters, then the moment it reaches 10 square meters it has nowhere left to grow—and the same applies {in my opinion} to infinity. Meaning, if it can grow up to infinity and it is already there, then it has exhausted its potential for growth, and so it has nowhere left to grow because it is already there.
I’d be happy to hear your criticism and insights. Thank you very much. 🙂
Answer
First of all, I don’t understand your argument. What are you trying to prove? If the world is eternal, meaning it has existed for an infinite amount of time, then it can’t grow? Do you mean grow in time? It can’t advance along the time axis? That isn’t true, but in any case I don’t understand what you’re trying to prove.
As for the claim itself, you’re making a mathematical mistake. Unlike any finite number, infinity can definitely grow. Take, for example, the set of natural numbers (= the positive integers). How many are there? Infinitely many. Now look at the rational numbers (= quotients of natural numbers). How many are there? Also infinitely many. Admittedly, in Cantor’s cardinal theory this is the same infinity (there is a correspondence between the two sets), but there are also infinities that are genuinely larger (such as the real numbers—all the points on the real line—whose number is greater than that of the natural numbers).
Another example. Think about the interval on the real line (0,1). It contains infinitely many points (= representing real numbers). Now I inflate the interval until its length is 2, so now I have the interval (0,2). Both intervals contain the same infinite number of points, but the second is longer than the first.
Search Wikipedia for the entry “Hilbert’s Hotel,” and you can read some amusing things on this topic.
Discussion on Answer
So what does that have to do with the eternity of the world? So it is eternal and doesn’t grow.
There’s a similarity between what I said and the eternity of the world, but that isn’t the issue. What I mean here is that even if the world is not eternal, infinite space still can’t grow. The point of my question isn’t whether the world is eternal or not, but rather the relation to infinity.
Similarities are in the eye of the beholder. In any case, regardless of what this argument is supposed to prove, it is mistaken.
From what I remember, both intervals have cardinality aleph-one, just like the entire real line. In general, by Cantor’s diagonal argument you can show that it is larger than the set of natural numbers, which is aleph-null, but still at least the same size as the real line. And in general, from the assumption that there are no intermediate sizes, you can conclude that it is aleph-one.
First of all, thank you very much.
Second, my argument is only about whether growth of the world is possible if space is infinite.