Let P be some proposition. If we have that $P(0)$ is true and that if $P(n)$ is true, then $P(S(n))$ is true, where $S(n)$ is the successor of natural number $n$. Then we have that $P(n)$ is true for all natural numbers.
To my understanding, we need this axiom to eliminate formulations like $\{0, 0.5, 1, 1.5, 2, \ldots\}$ which otherwise fulfill the peano axioms. That is, the induction axiom forces the natural numbers to all 'stem' from zero. So why don't we just edit the axiom that says 'no element has $0$ as its successor' to be '$0$ is the only element that isn't a successor to another element'.
Are these two formulation equivalent or am I confused? I'm not sure whether this works for $\mathbb{R} \setminus \mathbb{N}^+$, which also otherwise fulfills peano axioms, but I haven't gotten to real numbers yet.