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.

Let $+$ denote addition.

The binary operation $\times$ is recursively defined on $\N$ as follows:

$\forall m, n \in \N: \begin{cases} m \times 0 & = 0 \\ m \times \left({n + 1}\right) & = m \times n + m \end{cases}$

This operation is called multiplication.

Equivalently, multiplication can be defined as:

$\forall m, n \in \N: m \times n := +^n m$

Some treatments of Peano's axioms define the non-successor element (or primal element) to be $1$ and not $0$.

The treatments are similar, but the $1$-based system results in an algebraic structure which has no identity element for addition, and so no zero for multiplication.

Under this $1$-based system, multiplication is consequently defined as follows:

$\forall m, n \in P: \begin{cases} m \times 1 & = m \\ m \times \map s n & = m \times n + n \end{cases}$

or:

$\forall m, n \in P: \begin{cases} 1 \times n & = n \\ \map s m + n & = m \times n + n \end{cases}$