Definition:Addition/Peano Structure

Definition

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

The binary operation $+$ is defined on $P$ as follows:

$\quad \forall m, n \in P: \begin{cases} m + 0 & = m \\ m + \map s n & = \map s {m + n} \end{cases}$

This operation is called addition.

The definition can equivalently be structured:

$\quad \forall m, n \in P: \begin{cases} 0 + n & = n \\ \map s m + n & = \map s {m + n} \end{cases}$

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, addition is consequently defined as follows:

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

or:

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