My question is related to this other question. I was asking myself exactly the same questions as OP, and I read carefully each answer he got, but even after spending some hours on it I was still not totally convinced that the induction principle allows to exclude all numbers such as $0.5$, $1.3$, etc., from what we want the natural numbers to be.
I visited the Wikipedia article about Peano axioms, on which the induction axiom is stated in the following form:
If $K$ is a set such that:
$0$ is in $K$, and
for every natural number $n$, $n$ being in $K$ implies that $S(n)$ is in $K$,
then $K$ contains every natural number.
and from this, I tried to show that this really means that the set of natural numbers contains no other element than those expected. (Because it was still not obvious to me.) The problem is that I have no idea if my proof means anything at all, so I would be really happy if someone could check if it makes sense or not.
So here it is:
Let $K$ be a set such that $0$ is in $K$ and such that, for every natural number $n$, $n$ being in $K$ implies that $S(n)$ is in $K$.
Let's suppose, for the sake of contradiction, that the final statement of the axiom is wrong, i.e. let's suppose that there exists a natural number $n$ such that $n \notin K$.
Let $m$ be the smallest element of $\mathbb{N}$ such that $m \notin K$. Because of the first hypothesis of the induction axiom (i.e. "$0$ is in $K$"), $m \ne 0$.
Because $m \in \mathbb{N}$ and $m \ne 0$, $m$ is the successor of another natural number, say $m = p{++}$, with $p \in \mathbb{N}$.
Because $m$ is the smallest element in $\mathbb{N}$ such that $m \notin K$, we have that $p \in K$. But, because of the second hypothesis of the induction axiom (i.e. "$n$ being in $K$ implies $S(n)$ is in $K$"), this means that $p{++}=m$ should be in $K$, which is a contradiction. Therefore, the induction axiom holds and any number that satisfies the hypotheses of this axiom have to be a natural number. $\square$
What bothers me the most in this "proof" (if it is one) is that I use the fact that there is a "smallest" element which satisfies a certain property, while at that point of the book, the author didn't have defined an order over the natural numbers yet... so how can we know if there is a smallest element among them?
Anyway, my question is still the same: does this proof make any sense?
Thanks in advance and sorry for the n-th question about this topic.