I've seen differing takes on whether or not $0$ is a Natural number so I decided to check it with Peano's 5 axioms.\
The Peano Axioms are...
$\textbf{N1}$ 1 belongs to $\mathbb{N}$ and the rest of the axioms follow.
$\textbf{N2}$ If $n\in\mathbb{N}$, then its successor $n+1\in\mathbb{N}$ which is true for $n=1$ then $1+1=2$ and $2\in\mathbb{N}$
$\textbf{N3}$ $1$ is not the successor of any element is $\mathbb{N}$
$\textbf{N4}$ If $n,m\in\mathbb{N}$ have the same successor, then $n=m$
$\textbf{N5}$ A subset of $\mathbb{N}$ which contains $1$, and which contains $n+1$, whenever it contains $n$, must equal $\mathbb{N}$
which is proven by Mathematical induction.
It seems just as reasonable to me that all of these axioms would still hold if you began by letting $0\in\mathbb{N}$ in axiom $\textbf{N1}$ and the mathematical induction would follow as well. My question is whether or not $0$ is a Natural Number, and if not which axiom does it violate? Unless this is just a matter of tradition and general consensus, but it doesn't seem plausible that a number system created nearly 300 years ago would have such a loosely defined definition with respect to $0$.
