The problem is that infinity is not a number. Infinity plus one is equal to infinity. This is not a trait that any number has. There is no number where you can

The problem is that infinity is not a number. Infinity plus one is equal to infinity. This is not a trait that any number has. There is no number where you can add one and get the same number. The subset of an infinite set can itself be infinite. If you can map bi-directionality (which you can for square numbers and naturals) between an infinite set and its' infinite subset, then they are equivalent in size. Your problem is that you expect infinity (well, one type of infinity, there are ordered infinities [edit: when I said infinities, I meant infinite numbers, which the common infinite is not an infinite number, it's infinity], but that's outside our scope here) to be ordered, which it is not. Numbers are ordered, but infinity is not a number and is not ordered in the numbers.