Peano axioms: S(1)=1?

Question

I was reading through the Peano axioms here, and a question came up:

Can we define $S(0)=1$, and $S(1)=1$?

It seems to me (at least as it is stated) that it would satisfy all of the axioms listed. And I couldn't find any restrictions on the successor function, which would disallow this.

So would this be a valid Peano arithmetic or am I missing something here?

Thanks!

Answer 1

Your structure fails to satisfy the axiom that the Wikipedia article gives number 7:

7. For all natural numbers $m$ and $n$, $m = n$ if and only if $S(m) = S(n)$. That is, $S$ is an injection.