Definition:Ordering on Natural Numbers
Informal Definition
Let $\N$ denote the natural numbers.
The ordering on $\N$ is the relation $\le$ everyone is familiar with.
For example, we use it when we say:
- James has $6$ apples, which is more than Mary, who has $4$.
which can be symbolised as:
- $6 \ge 4$
Every attempt to describe the natural numbers via suitable axioms should reproduce the intuitive behaviour of $\le$.
The same holds for any construction of $\N$ in an ambient theory.
Definition
Ordering on Peano Structure
Let $\struct {P, 0, s}$ be a Peano structure.
The ordering of $P$ is the relation $\le$ defined by:
- $\forall m, n \in P: m \le n \iff \exists p \in P: m + p = n$
where $+$ denotes addition in $\struct {P, 0, s}$.
Ordering on Naturally Ordered Semigroup
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.
The relation $\preceq$ in $\struct {S, \circ, \preceq}$ is called the ordering.
Ordering on $1$-Based Natural Numbers
Let $\N_{>0}$ be the axiomatised $1$-based natural numbers.
The strict ordering of $\N_{>0}$, denoted $<$, is defined as follows:
- $\forall a, b \in \N_{>0}: a < b \iff \exists c \in \N_{>0}: a + c = b$
The (weak) ordering of $\N_{>0}$, denoted $\le$, is defined as:
- $\forall a, b \in \N_{>0}: a \le b \iff a = b \lor a < b$
Ordering on Minimally Inductive Set
Let $\omega$ be the minimally inductive set.
The strict ordering of $\omega$ is the relation $<$ defined by:
- $\forall m, n \in \omega: m < n \iff m \in n$
The (weak) ordering of $\omega$ is the relation $\le$ defined by:
- $\forall m, n \in \omega: m \le n \iff m < n \lor m = n$
Ordering on Von Neumann Construction of Natural Numbers
Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction.
The strict ordering of $\omega$ is the relation $<$ defined by:
- $\forall m, n \in \omega: m < n \iff m \subsetneqq n$
The (weak) ordering of $\omega$ is the relation $\le$ defined by:
- $\forall m, n \in \omega: m \le n \iff m \subseteq n$
Ordering on 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 ordering of $\N$ is the restriction of $\le$ to $\N$.
Also defined as
As seen above, not all sources define both the strict ordering $<$ and the weak ordering $\le$ on $\N$.
However, by Reflexive Reduction of Ordering is Strict Ordering and Reflexive Closure of Strict Ordering is Ordering, this is seen to be immaterial.