Is it true that $\omega=\{0,1,2,3,\ldots\}$ in ZFC? This is a bit of a philosophical question. According to "Set Theory" by Jech, the set $\omega$ of natural numbers is defined as the least nonzero limit ordinal. After thinking about this definition recently I found no reason why this set should neccessarily be restricted to contain only the numbers obtained by applying the successor operation to zero, what is the intuitive notion of natural number. 
I did not even find a way to formulate the statement that all elements of $\omega$ are of this form. 
To be more technical, is it in fact possible to consistently add an element $x$ to the theory together with axioms that assert $x\neq0,x\neq 1,\ldots$ ?
 A: There are two fine points here.
There is internal interpretation, and external interpretation.


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*Internal interpretation is what happens when we work internally to a theory, or in a more semantic/Platonic way, we work with the entire universe, rather than looking at models from the outside. In the case, $\omega$ is the set of natural numbers. Because we prove, from $\sf ZFC$ that $x\in\omega$ if and only if $x$ is a finite ordinal.

*External interpretation is what happens when we examine a model of a theory, which may or may not agree with the universe of mathematics on some key notions. And as pointed out in the comments, it is consistent that there are models of set theory whose integers include strictly more than what is really an integer.
But the key point to notice here is that if $M$ is a model which has "too many natural numbers", the model does not know about that. When talking about a model $M$, it only regards as "set" the elements of its own universe, not necessarily things we can define, observe and identify from outside of that model. 
So while $M$ may disagree with the "real universe" about what are natural numbers, $M$ is not aware that there is a larger universe, and is not aware of the fact that it includes "false" (or non-standard) integers. And in fact, it might be that the "real universe" is itself a model in a larger universe, and that it has non-standard integers of its own, and so on and so forth. Of course, if you take a Platonist approach, then this has to stop at the actual universe, but if you don't, then you can go on, unveiling layers over layers of larger universes, each with less and less natural numbers than before.
So internally, $\omega$ is exactly the set of natural numbers. Externally, it might not be. In fact this is such an important property that models of set theory which agree with the universe on what $\omega$ is, are called "$\omega$-models". And it is a key property, because agreeing with the universe on $\omega$ means agreeing with the universe about basic arithmetic truths, something which is very fundamental.
But I digress, and this seems like a good point to stop.
To conclude, in $\sf ZFC$, means - to me as a reader - internally to $\sf ZFC$. So the answer is positive. But this question can be phrased slightly different, to suggest the possibility of an external point of view, in which case the answer would be negative.
A: $\omega$ here is an ordinal, so it is supposed to come with an order structure. When you add your element $x$, where do you put it in the order ?
If it is the maximal element, then you get $\omega+1$, which is not a limit ordinal (and therefore does not correspond to the definition). If you put it elsewhere (first or between two elements), then you get $\omega$ again, up to renaming elements. 
The important thing is that an ordinal is just defined by its order structure, the name of elements are not important. In this way, $\omega=\{0,1,2,\dots\}$ is the unique least limit ordinal.
