Definition:Multiplication/Natural Numbers

Definition
Let $\N$ be the natural numbers.

Multiplication on $\N$ is the basic operation $\times$ everyone is familiar with.

For example:


 * $3 \times 4 = 12$
 * $13 \times 7 = 91$

Every attempt to describe the natural numbers via suitable axioms should reproduce the intuitive behaviour of $\times$.

The same holds for any construction of $\N$ in an ambient theory.

Also defined as

 * $\forall m, n \in \N: m \times n = \begin{cases}

m & : n = 1 \\ \left({m \times r}\right) + m & : n = r + 1 \end{cases}$

or:


 * $\forall m, n \in \N: \begin{cases}

1 \times n & = n \\ s \left({m}\right) \times n & = m \times n + n \end{cases}$