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.