>>45741967So get this right there are an infinite number of countable numbers, right, like you can just start counting 1, 2, 3, etc. and since you can just keep adding 1 forever this sequence never ends, which means you've just constructed a set of numbers that is infinitely large. But, like, what if you go the other way too, like, you've got 0, 1, 2, ... but then add to that collection all the negative numbers as well. So you've got ..., -2, -1, 0, 1, 2, 3, ... And you might think that this is 2x as many numbers as before, right? But what the fuck does that even mean? Because 2 times infinity is still infinity. In fact, we can even say that these two quantities have the exact same size.
By similar logic, you can say that the set of all real numbers between 0 and 1 contains the same number of numbers as the set of all real numbers between positive and negative infinity. Furthermore, these sets are bigger than 0, 1, 2, ... discussed previously. Specifically, this infinity is infinitely bigger than the previous infinity, which is the only real way you can distinguish infinities from each other.
And you can go even further beyond by constructing still-more exotic sets that are infinitely larger than the set of all reals. If the set of whole numbers is infinity-0, or aleph-0, and the set of all reals is aleph-1, then this would be aleph-2. There's also aleph-3, aleph-4, all the way up to aleph-infinity. Which would be a collection of infinite numbers that's infinitely-infinitely bigger than the set of all infinite countable numbers.