Proving that the set of half-integers is a model of the natural numbers using Peano's axioms

I postulate that the following set $\{0,0.5,1,1.5,...\}$ represents the natural numbers. Of course, intuitively, this isn't true. But let me try to show this using Peano's axioms. I'll first define the successor operation to be the following:$$\sigma(n) = n+0.5$$Now, axiom 1 is satisfied: $0\in S$. It is also the case that if $n\in\mathbb{N}$, then $\sigma(n)\in\mathbb{N}$. Thus, axiom 2 is satisfied. It's also true that $\forall n\in\mathbb{N}, \sigma(n)\neq0$. Thus, axiom 3 is satisfied. Even the axiom 4 - injectivity axiom - is true: $\sigma(n)=\sigma(m)\implies n=m$.

Now, onto axiom 5. I've seen two kinds of formulations so far:

1. If $P(n)$ is some property such that $P(0)$ is true, and whenever $P(n)$ is true $P(\sigma(n))$ is true, then $P(n)$ is true for every number
2. For every $S\subseteq\mathbb{N}$ with the property that $0\in S$ and $s\in S\implies \sigma(s) \in S$, $S = \mathbb{N}$

Here are my questions:

1. How are the two formulations of the axiom equivalent?
2. How do you show that the set obeys/disobeys axiom 5 using each of the formulations?
3. If the set obeys axiom 5, isn't that a problem since the set is intuitively not what we mean when we refer to natural numbers?