Minimal Infinite Successor Set forms Peano Structure

Theorem

Let $\cdot^+: \omega \to \omega$ be the mapping assigning to a set its successor set:

$n^+ := n \cup \left\{{n}\right\}$

Let $\varnothing \in \omega$ be the empty set.

Then $\left({\omega, \cdot^+, \varnothing}\right)$ is a Peano structure.

We need to check that all of Peano's axioms hold for $\left({\omega, \cdot^+, \varnothing}\right)$.

Suppose first that for $m, n \in \omega$, we have $m^+ = n^+$.

Since $n \in n^+$ it follows that $n \in m^+$.

Hence, either $n \in m$ or $n = m$.

Similarly, either $m \in n$ or $m = n$.

Now if $n \ne m$, both $m \in n$ and $n \in m$.

Hence it must be that $n = m$, and Axiom $(P3)$ holds.

Next, since $n \in n^+$ for all $n \in \omega$, it follows that $n^+ \ne \varnothing$.

Hence, Axiom $(P4)$ holds as well.

Finally, let $S \subseteq \omega$ satisfy:

$\varnothing \in S$

$\forall n \in S: n^+ \in S$

Then by definition, $S$ is an infinite successor set.

Therefore, by definition of $\omega$ as the minimal infinite successor set:

$\omega \subseteq S$

Consequently $S = \omega$ by the definition of set equality.

Thus Axiom $(P5)$ is seen to hold.

That is, $\left({\omega, \cdot^+, \varnothing}\right)$ is a Peano structure.

$\blacksquare$