Definition:Addition/Natural Numbers

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Definition

Let $\N$ be the natural numbers.

Addition on $\N$ is the basic operation $+$ everyone is familiar with.

For example:

$2 + 3 = 5$
$47 + 191 = 238$


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.


Addition in Peano Structure

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.


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.


Also see


Sources