Let $ N$ be a set containing an element $ 1$ and $ \sigma: N \to N$ an injective function satisfying the following two properties:

$ \tag 1 1 \notin \sigma(N)$

$ \tag 2 (\forall M \subset N) \;\text{If } [\; 1 \in M \land (\sigma(M) \subset M) \;] \text{ Then } M = N$

We call $ (N, \sigma)$ a Peano system.

The set of all injective functions on $ N$ form a semigroup under composition.

Let $ \mathcal C$ denote the set of all injective functions on $ N$ that commute with $ \sigma$ . It is easy to see that $ \mathcal C$ is a semigroup containing the identity transformation.

Theorem 1: If $ \mu,\nu \in \mathcal C$ and $ \mu(1) = \nu(1)$ then $ \mu = \nu$ .

Proof

Let $ M$ be the set of all elements in $ N$ where the two functions agree. Applying induction with $ \text{(2)}$ , it is immediately evident that $ M = N$ . $ \quad \blacksquare$

Theorem 2: For any $ m \in N$ , there exist a $ \mu \in \mathcal C$ such that $ \mu(1) = m$ .

Proof

Again, simply apply induction. $ \quad \blacksquare$

So $ \mathcal C$ is in bijective correspondence with $ N$ .

Theorem 3: Let $ (N, \sigma)$ and $ (N’, \sigma’)$ be two Peano systems and $ \mathcal C$ and $ \mathcal C’$ the corresponding semigroups. Then there exist one and only one bijective correspondence $ \beta: N \to N’$ satisfying

$ \tag 3 \beta(1) = 1’$

$ \tag 4 \beta \circ \sigma = \sigma’ \circ \beta$

Proof

The function $ \beta$ is defined using recursion. Induction is used to show that $ \beta$ is injective. Induction is used to show that $ \beta$ is surjective. $ \quad \blacksquare$

Using the above an argument can be supplied to prove the following.

Theorem 4: Let $ (N, \sigma)$ and $ (N’, \sigma’)$ be two Peano systems and $ \mathcal C$ and $ \mathcal C’$ the corresponding semigroups. Then the mapping $ \sigma \mapsto \sigma’$ can be extended to an algebraic isomorphism between $ \mathcal C$ and $ \mathcal C’$ .

We reserve the symbol $ \mathbb N$ to denote $ \mathcal C$ and use the symbol $ +$ to denote the binary operation of composition.

Using the axioms of $ ZF$ , the existence of Peano systems is no problem.

I couldn’t find this technique here or on wikipedia, prompting

Is the theory described above coherent?

If it is it would certainly appeal to students who like to see some ‘motion/action’ when studying mathematical constructions, say somebody born to be a functional analyst.

Also, it is interesting to see how $ \mathbb N$ obtained in this way can be viewed in some sense as the ‘dual’ of the Peano system.