As I understand it: In ZFC the continuum hypothesis can neither be proven true nor false, or in other words, a new axiom could be added to ZFC that says the continum hypotheis is true or that it is false and it wouldn't contradict any of the other axioms in ZFC assuming that ZFC is consistent.
There are other systems where the continuum hypothesis is false. For example, in the surreal number system, the continuum hypothesis is not held as true and $\omega$, which is defined as the first ordinal greater than the natural numbers, is less than $\omega + 1$.
My question is, if the set of all natural numbers (excluding $0$) contains $\omega$ numbers (or some other transfinite ordinal number of numbers), then wouldn't this imply that the set of all whole numbers (including $0$) contains $\omega + 1$ numbers, and so is larger? Also, wouldn't the entire set of integers contain $\omega + 1 + \omega = 2\omega + 1$ numbers? If this is the case, then how could you justify comparing the cardinality of different infinite sets, what makes the number of natural numbers $\omega$ why is that not the number of integers or the number of even numbers or whatever else, doesn't this conflict with Georg Cantor's intuition for comparing infinite sets by a correspondence between their components? I'm curious how the cardinalities of infinite sets are described in the surreal number system, and other systems where the continuum hypothesis isn't held as true.
Edit: I was reading a book about the surreal numbers and this question cropped up out of a misunderstanding of the distinction between different kinds of infinity. However, I've agreed to not delete the question anyway, in case anyone else runs into the same misunderstanding and needs a response.