Definition:Addition/Peano Structure

Definition
Let $\left({P, 0, s}\right)$ be a Peano structure.

The binary operation $+$ is defined on $P$ as follows:
 * $\forall m, n \in P: \begin{cases}

m + 0 & = m \\ m + s \left({n}\right) & = s \left({m + n}\right) \end{cases}$

This operation is called addition.

The definition can equivalently be structured:
 * $\forall m, n \in P: \begin{cases}

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

Also defined as

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

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

or:


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

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

Also see

 * Addition in Peano Structure is Well-Defined