The Natural Numbers

The real reason why the elements of $\ensuremath{\mathbb{N} }$ are called `natural numbers' was explained by Kronecker when he stated:

Die ganzen hat der liebe Gott gemacht, alles andere ist Menschenwerk. [God made the whole numbers, all the rest is the work of Man.] (KRONECKER, Jahresber. DMV 2, S. 19)

Definition 1.1     The natural numbers form a set $\ensuremath{\mathbb{N} } ,$ containing a distinguished element 0, called zero, together with a successor function $S : \ensuremath{\mathbb{N} }\to \ensuremath{\mathbb{N} }$ which satisfies the following axioms:

1.

S is injective

2.

$0 \notin S(\ensuremath{\mathbb{N} } )$

3.

If a subset $M \subset \ensuremath{\mathbb{N} }$ contains 0 and is mapped into itself by S then $M=\ensuremath{\mathbb{N} } .$

The existence of the natural numbers depends on the axiom of infinity, [E⁺91], and the uniqueness is gauranteed by the following result [Ded88].

Theorem 1.2 (Dedekind 1888)     Let A be an arbitrary set containing an element $a \in A$ and g a mapping $g: A \to A.$ Then there is a one-to-one mapping $\phi : \ensuremath{\mathbb{N} }\to A$ satisfying $\phi(0)=a$ and making the following diagram commute:

$$\begin{CD} \ensuremath{\mathbb{N} } @ > S >> \ensuremath{\mathbb{N} }\\ @ V \phi VV @ VV \phi V \\ A @ >> g > A \\ \end{CD}$$