I have the axiom from Peano's axioms:
If $A\subseteq \mathbb{N}$ and $1\in A$ and $m\in A \Rightarrow S(m)\in A$, then $A=\mathbb{N}$.
My book tells me that it secures that there are no more natural numbers than the numbers produced by the below 3 axioms (also from Peano's axioms):
$1\in \mathbb{N}$
For every $n\in\mathbb{N}: 1\neq S(n)$
For every $m,n\in \mathbb{N}:m\neq n\Rightarrow S(m) \neq S(n)$
And I'm not sure why? Is there someone who can explain this?
S(n) is an unary function $S: \mathbb N \rightarrow \mathbb N$. Does this means that $S(n)=n+1$?