# Definition:Addition/Natural Numbers

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## 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 $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.

The operation $\circ$ in $\struct {S, \circ, \preceq}$ is called **addition**.

### Addition in Minimally Inductive Set

Let $\omega$ be the minimally inductive set.

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

- $\forall m, n \in \omega: \begin {cases} m + 0 & = m \\ m + n^+ & = \paren {m + n}^+ \end {cases}$

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

This operation is called **addition**.

### Addition for Natural Numbers in Real Numbers

Let $\struct {\R, +, \times, \le}$ 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

- Results about
**natural number addition**can be found here.

## Sources

- 1974: Murray R. Spiegel:
*Theory and Problems of Advanced Calculus*(SI ed.) ... (previous) ... (next): Chapter $1$: Numbers: Real Numbers: $1$ - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): Chapter $1$: Complex Numbers: The Real Number System: $1$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**add**:**1.**