how to express the set of natural numbers in ZFC

I know that it is possible to represent each natural number using the concept of successor. But how do you assign meanings into the representation? I mean, when presented as a set, it's just a set unless we agree on some definition that some form of sets refer to natural numbers. Or is it something impossible at ZFC?

Thanks.

I am turning my comment into an answer. The best way I know to output the set $\{0,1,2,\dots \}$ from the set-theoretical construction goes as follows : $$ 0 = \varnothing, \quad 1 = \{0\}, \quad 2 = \{0, 1\}, \quad 3 = \{0, 1, 2\}, \quad \dots \quad \mathrm{successor}(n) = n \cup \{n\}. $$ In other words, to create the positive integer $n$, you consider the "set that contains the set that contains the set that $\dots$ that contains $\varnothing$" + "the set that contains the set that $\dots$",and so on. Again in other words, the integer $n$ is the union of the sets that contains $\varnothing$ at $i$ levels of deepness, $i$ ranging from $0$ to $n-1$ (All this is only in familiar terms).

This construction can also be used to inductively define addition : $$ n+0 = n, \quad n + 1 \overset{def}= \mathrm{successor}(n),\quad n+2 \overset{def}= \mathrm{successor}(\mathrm{successor}(n)), \quad \dots \quad n+(m+1) \overset{def}= (n+m)+1. $$ Try to understand what I exactly said in the last definition.

You can also define multiplication using this definition : $$ n \cdot 0 \overset{def}= 0, \quad n \cdot (m+1) \overset{def}= (n \cdot m) + n $$ It is an exercise to show that those definitions have all the properties we know over positive integers : associativity, commutativity, etc.

Hope that helps,

In the ZFC set theory everything is a set. There are no non-set elements. Indeed there can be several interpretations of the natural numbers in a given model of ZFC.

Furthermore, if you constructed one model of the natural numbers and used it to generate the complex numbers you can identify the natural numbers as the canonically embedding of the natural numbers there, and of course these would be different sets.

It is common to take the finite von Neumann ordinals as the natural numbers. However even if we take different sets, since the addition and multiplication is not a part of the language of set theory we define them within the model. We can also define those differently (again, depending on the interpretation of the numbers).

The important thing is that the chosen representation will satisfy certain axioms, and have binary operations of addition and multiplication.