How is $\mathbb N$ actually defined? I know perfectly well the Peano axioms, but if they were sufficient for defining $\mathbb N$, there would be no controversy whether $0$ is a member of $\mathbb N$ or not because $\mathbb N$ is isomorphic to a host of sub- and supersets of $\mathbb Z$.
So it seems that mathematicians, when specifying $\mathbb N$, rarely mean the set of natural numbers according to its definition but rather some particular embedding into sets defined by different axiomatic systems.
Doesn't this sort of misstate then the role of the Peano axioms as the defining characteristics of $\mathbb N$?  It seems like most of the time the Peano axioms are not more than a retrofit to a subset of $\mathbb Z$ or other sets defined by a different axiomatic system with one or several axioms incompatible with the Peano axioms.

In the set theory course I took, natural numbers were not defined by Peano's axioms. One can define an inductive set in this manner: $I$ is inductive set if
$$\emptyset\in I\ \wedge\ \forall x\in I\  ((x\cup \{x\})\in I) $$
We say that $n$ is natural number if $n\in I$ for any inductive set $I$. Some examples of natural numbers are:
\begin{align}
0 &= \emptyset\\
1 &= \{\emptyset\}\\
2 &= \{\emptyset,\{\emptyset\}\}\\
&\ \ \vdots\\
n &= \{0,1,\ldots ,n-1\}
\end{align}
But, we still don't know if the set of all natural numbers exists. This is guaranteed by Axiom of infinity:

There exists an inductive set.

and with this we can prove existence of set $\mathbb N = \{n\ |\ n\ \text{is natural}\}$. Also, we can see that $\mathbb N$ is the smallest inductive set in the sense of set inclusion.
