Definition

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

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

Let $\N$ be the natural numbers.

Addition in Peano Structure

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.

Addition in Naturally Ordered Semigroup

Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup.

The operation $\circ$ in $\left({S, \circ, \preceq}\right)$ is called addition.

Addition in Minimal Infinite Successor Set

Let $\omega$ be the minimal infinite successor set.

The binary operation $+$ is defined on $\omega$ as follows:

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

where $m^+$ is the successor set of $m$.

This operation is called addition.

Addition for Natural Numbers in Real Numbers

Let $\left({\R, +, \times, \leq}\right)$ be the field of real numbers.

Let $\N$ be the natural numbers in $\R$.

Then the restriction of $+$ to $\N$ is called addition.