How are the cardinalities of infinite sets described in systems where the continuum hypothesis doesn't hold? 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.
 A: Cardinality of sets are indeed something of interest in set theory. And set theory is generally considered as a list of rules for objects which we call "sets", and they come to model our intuition as to what is a collection of mathematical objects (e.g. given a collection of collections, we can take the union of these, etc.)
Cardinality, in specifics, is a way to measure the size of sets. And it is based on bijections between sets, as this is a way that has nice properties.
Cantor proved that the cardinality of the real numbers equals to that of the power set of $\Bbb N$, and that it is uncountable. While he hypothesized that there are no intermediate cardinals between them, he could not prove this. Many years after his death, we learned through collective efforts of Gödel, Cohen and many others, that indeed it is impossible to prove or disprove the continuum hypothesis from the axioms of $\sf ZFC$. 
The surreal numbers, on the other hand, are not a set theoretic system. They are an ordered field, which is maximal, in the sense that every ordered field has an embedding into them. Moreover, it is a complete field, in the same sense that the real numbers are complete. But it is just a field. It is not meant to describe "sets" or "collections", and therefore has no inherent concept of what are the natural numbers (as an object), or their power set, or the real numbers. Moreover, cardinalities are not described using the surreal numbers any more than they are described by the real numbers (and just to be clear, they are not described by the real numbers at all). 
So, the question itself is based on a type mismatch. But let me address the ordinals, which do have some relevance here.
The ordinal $\omega$ is used to model the natural numbers—and it is often regarded as the natural numbers in set theory—and indeed it can be seen as a surreal number, since the ordinals themselves have a canonical embedding into the surreal line. But as far as cardinal arithmetic, $\omega$ and $\aleph_0$, are different objects. For example $\aleph_0+1=\aleph_0$ whereas $\omega+1\neq\omega$. In fact, ordinals also come with their own arithmetic, where $1+\omega=\omega\neq\omega+1$, so that is also something which is different from the surreal numbers and from the cardinal arithmetic. Even if we identify $\aleph_0$ with the ordinal $\omega$.
One could ask, perhaps we can use the ordinals to measure "size" of sets. But this is only true for finite sets. Both $\omega$ and $\omega+1$, which are different ordinals as mentioned above, have the same cardinality. Suddenly, the way that we place objects matters, and cardinality is a measure of size which ignores "extraneous structure" (like order). So what would be the ordinal that you match for $\Bbb Z$ or $\Bbb Q$? Well, the natural choice would be "the smallest one possible". But then we are back to the cardinals, at least in the context of $\sf ZFC$.
A: When you say “the set $X$ has $n$ elements,” what you mean is “the cardinality of the set $X$ is the same as the cardinality denoted by the cardinal number $n$.”
For a natural number $n$, the cardinality denoted by $n$ is the cardinality of $\{1,2,\dots,n\}.$ Furthermore one lets the cardinal number $\aleph_0$ denote the cardinality of the set of natural numbers.
Surreal numbers are not necessarily cardinal numbers and they are certainly not in (natural) bijection. Indeed if we allow $\omega$ to denote the cardinality of $\{1,2,\dots\}$ and $\omega+1$ the cardinality of $\{0,1,2,\dots\},$ we see that the two sets are in bijection ($n\mapsto n-1$) and so the cardinalities are equal. 
The continuum hypothesis is about cardinalities and sets. It says “There exists no set $X$ such that there are injections $\mathbb N\to X$ and $X\to\mathbb R$ but there does not exist surjections $\mathbb N\to X$ and $X\to\mathbb R.$” That is, it says that there is no cardinality between those of $\mathbb N$ and $\mathbb R$. The continuum hypothesis says nothing about the way cardinal numbers may represent cardinalities. It is certainly not true to say that it is independent of our usual number system and false in the surreal numbers. I think the root of this misunderstanding is confusing different kinds of infinity (cardinal, ordinal, and number-system).
