User:Abcxyz/Sandbox/Real Numbers/Definition:Ordering on Real Numbers

Ordering/Axiomatic Definition
Let $\left({\R, +, \times, \le}\right)$ denote the real numbers, as axiomatically defined as a Dedekind complete totally ordered field.

The ordering on $\R$ is $\le$.

Ordering/Construction from Cauchy Sequences
Let $\R$ denote the set of real numbers, as constructed from Cauchy sequences.

The ordering on $\R$, denoted $\le$, is defined as:
 * $\forall x, y \in \R: x \le y \iff \forall \left\langle{x_n}\right\rangle: \forall \left\langle{y_n}\right\rangle: \left({x = \left[{\!\left[{\left\langle{x_n}\right\rangle}\right]\!}\right]}\right.$ and $\left.{y = \left[{\!\left[{\left\langle{y_n}\right\rangle}\right]\!}\right]}\right) \implies \forall \epsilon \in \Q_{>0}: \exists N \in \N: \forall n \in \N: n > N \implies x_n < y_n + \epsilon$

Ordering/Construction from Dedekind Cuts
Let $\R$ denote the set of real numbers, as constructed from Dedekind cuts.

The ordering on $\R$, denoted $\le$, is defined as the subset relation $\subseteq$.

That is:
 * $\forall \alpha, \beta \in \R: \alpha \le \beta \iff \alpha \subseteq \beta$

Ordering/Real Numbers as Dedekind Completion of Rational Numbers
Let $\left({\R, \le}\right)$ denote the ordered set of real numbers, as defined as the Dedekind completion of the rational numbers.

The ordering on $\R$ is $\le$.