I know that there are many questions here related to this one, but they are all not really satisfying for the particular detail I want to understand. So I try my best to express what I mean.
That there are several set-theoretic notions that depend on the model of $\mathsf{ZFC}$ we work in is not new to me. For instance, the Löwenheim-Skolem Theorem gives a countable model $U$ of $\mathsf{ZFC}$, which means that its set of real numbers has also to be countable (from a meta perspective, that is, we started with a model $\mathcal M$ of $\mathsf{ZFC}$ which $U$ is an element of and applied the Löwenheim-Skolem Theorem there, and in $\mathcal M$ the set of real numbers of $U$ is countable), but of course it is uncountable as an element of $U$.
If I want to consider questions about absoluteness, I always have to fix a sort of meta set theory in order to even talk about models of set theory. This means, a priori, I have to say "Let $\mathcal M$ be a model of $\mathsf{ZFC}$.". Now, $\mathcal M$ has a set of natural numbers.
If I now want to pass on to questions about absoluteness (and when I say this, I actually mean absoluteness among the elements of $\mathcal M$ that are models of $\mathsf{ZFC}$, because these are now the structures of the language of set theory that I can consider), I have to say "Let $U, V \in \mathcal M$ be models of $\mathsf{ZFC}$." and then I can ask a question like "Is the set of natural numbers of $U$ the same as the set of natural numbers of $V$?", because both sets of natural numbers are elements of $\mathcal M$ and as such I can ask whether they are equal.
Now comes the crucial part: When constructing natural numbers, if I understand it right, we do the following. We know that we have a unique empty set $\emptyset$. Now, following von Neumann, we apply the recursion
$$ z_0 = \emptyset,\\
z_{n+1} = z_n \cup \{z_n\}, $$
where the indices are meta natural numbers, that is, elements of the set of natural numbers of $\mathcal M$. And now comes the point where it seems that I got something not quite right: If $x_i$ resp. $y_i$ denotes the $i$-th natural number of $U$ resp. $V$, where $i$ varies over natural numbers of $\mathcal M$, then it should be easy to see that all three sets of natural numbers are the same, by just noting that the notion of empty set is absolute and the recursion gives the same sets.
So what I wrote down seems to imply that, relative to a meta set theory, that is represented by $\mathcal M$ in my example, the notion of the set natural numbers is absolute. Is this correct?
Thank you for your help!
Best Answer
So, when you say absoluteness to a set theorist, you're asking "If $M$ is a model of set theory and $N$ is a submodel of $N$, do they have the same natural numbers?" and the answer is generally yes in the "well-behaved" cases, but no otherwise. In the case of transitive classes, this is really a question about Levy complexity, in which case, again, the natural numbers are very absolute between two models and will satisfy the same theory.
Now, what you're misunderstanding is that we are not defining the natural numbers in the meta-theory. Not at all. We are defining the natural numbers internally to a model of $\sf ZFC$.
Consider the reverse Skolem paradox. Take a countable model of set theory, $M$, a free ultrafilter on $\omega$, and the ultrapower $N=M^\omega/U$. We can show that this model is not only uncountable, but in fact $\omega^N$, namely the set that $N$ "thinks" to be $\omega$, is uncountable. So there is a model of set theory which has uncountably many finite ordinals.
You could not possibly construct these integers using the meta-theory's integers, since those are countable.
Moreover, when we construct sets "one-by-one" in the meta-theory, then we need to come up with a really good reason as to why the collection of all of these sets is also a set. Usually it is not, because the complexity of the definitions will, normally, tend to increase.
So instead we are defining the natural numbers internally to a model of $\sf ZFC$. Namely, we define the idea of an inductive set, and we posit through the Axiom of Infinity that one exists, and we use that to prove that there is a smallest set which is inductive, and we show that it is the ordinal $\omega$, and that all of its elements satisfy one of many definitions of finiteness, and are therefore worthy of being called "the natural numbers [of that model]".
One thing, though, is that you are correct that we can still define a copy of the meta-theory's integers inside any model of $\sf ZFC$. One step at a time. Yes. However, if the model has non-standard integers, then we will not be able to collect those into a set, and we will simply have to rely on the internal construction instead.