Natural Numbers

We write

$$\mathbb{N}=\{1,2,3,\dots\}.$$

The elements of $\mathbb{N}$ are called natural numbers. We regard addition and multiplication on $\mathbb{N}$ as binary operations

$$+ : \mathbb{N}\times\mathbb{N}\to\mathbb{N} \quad\text{and}\quad \cdot : \mathbb{N}\times\mathbb{N}\to\mathbb{N},$$

written in infix form as $(m,n)\mapsto m+n$ and $(m,n)\mapsto mn$. We also regard the successor on $\mathbb{N}$ as a function

$$S:\mathbb{N}\to\mathbb{N}$$

written in infix form as $n\mapsto S(n)$. These data are related by the following recursive identities for all $m,n\in\mathbb{N}$.

1. $S(n)=n+1$.
2. $m+S(n)=S(m+n)$.
3. $m\cdot 1=m$.
4. $m\cdot S(n)=m\cdot n+m$.

We also require that $1$ is not a successor and that the successor map is injective; that is,

$$S(n)\ne 1\quad\text{for every } n\in\mathbb{N},$$

and

$$S(m)=S(n)\implies m=n$$

for all $m,n\in\mathbb{N}$.