Definition:Addition/Peano Structure

Definition
Let $\struct {P, 0, s}$ be a Peano structure.

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

m + 0 & = m \\ m + \map s n & = \map s {m + n} \end{cases}$

This operation is called addition.

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

0 + n & = n \\ \map s m + n & = \map s {m + n} \end{cases}$

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


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

m + 1 & = \map s m \\ m + \map s n & = \map s {m + n} \end{cases}$

or:


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

1 + n & = \map s n \\ \map s m + n & = \map s {m + n} \end{cases}$

Also see

 * Addition in Peano Structure is Well-Defined