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?