The Peano axiom of induction for natural numbers says that For any property $P(n)$ , if $P(0)$ holds, and that whenever $P(n)$ holds, $P(n++)$ holds, then $P(n)$ holds for all natural numbers.
Can this be replaced by
"For every natural number $n$ not equal to $0$, there exists another natural number $m$ such that $m++ = n$" ?
What are the problems associated with replacing this?
I am following Terence Tao's book Analysis I.
Let $N=\{-1\}\times\mathbb N\cup\{1\}\times\mathbb Z$. If $(\pm1,m)\in N$, then let $(\pm1,m)++=(\pm1,m+1)$. Then all the Peano axioms (with the induction one replaced by yours) hold (assuming that $0=(-1,0)$), but what we have here is something which is different from the naturals. For instance, induction doesn't hold here.