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:
$\quad \forall m, n \in P: \begin{cases} m + 0 & = m \\ m + \map s n & = \map s {m + n} \end{cases}$
This operation is called addition.
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): add: 1.