User:Abcxyz/Sandbox/Real Numbers

Axiomatic Definition
The real numbers are a totally ordered field $\left({\R, +, \times, \le}\right)$ such that $\left({\R, \le}\right)$ is Dedekind complete.

The axiom that $\left({\R, \le}\right)$ is Dedekind complete is known as the (Dedekind) completeness axiom.

Construction from Cauchy Sequences
Let $X$ be the set of all rational Cauchy sequences.

Let $\sim$ be the relation on $X$ defined as:
 * $\left\langle{x_n}\right\rangle \sim \left\langle{y_n}\right\rangle \iff \forall \epsilon \in \Q_{>0}: \exists N \in \N: \forall n \in \N: n > N \implies \left\vert{x_n - y_n}\right\vert < \epsilon$

By Equivalence Relation on Cauchy Sequences, $\sim$ is an equivalence relation on $X$.

The set $\R$ of real numbers is defined as the quotient set $X/{\sim}$.

Construction from Dedekind Cuts
The set $\R$ of real numbers is defined as the set of all Dedekind cuts of the totally ordered set $\left({\Q, \le}\right)$ of rational numbers.

Real Numbers as Dedekind Completion of Rational Numbers
Let $\left({\left({\R, \le}\right), \phi}\right)$ be the Dedekind completion of the ordered set $\left({\Q, \le}\right)$ of rational numbers.

Then $\left({\R, \le}\right)$ is the ordered set of real numbers.

Addition/Axiomatic Definition
Let $\left({\R, +, \times, \le}\right)$ denote the totally ordered field of real numbers.

The binary operation $+$ is called addition.

Addition/Construction from Cauchy Sequences
Let $\R$ denote the set of real numbers.

Addition, denoted $+$, is the binary operation on $\R$ defined as:
 * $\left[{\!\left[{\left\langle{x_n}\right\rangle}\right]\!}\right] + \left[{\!\left[{\left\langle{y_n}\right\rangle}\right]\!}\right] = \left[{\!\left[{\left\langle{x_n + y_n}\right\rangle}\right]\!}\right]$

Addition/Construction from Dedekind Cuts
Let $\R$ denote the set of real numbers.

Addition, denoted $+$, is the binary operation on $\R$ defined as:
 * $\alpha + \beta = \left\{{p + q: p \in \alpha, \, q \in \beta}\right\}$

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

Let $\left({\left({\R, \le}\right), \phi}\right)$ be the Dedekind completion of the ordered set $\left({\Q, \le}\right)$ of rational numbers.

Let $\left({\Q, +}\right)$ denote the additive group of rational numbers.

We have that $\left({\Q, +, \le}\right)$ is an Archimedean ordered group.

By this theorem, there exists a unique binary operation $+$ on $\R$ such that:
 * $({1}): \quad \left({\R, +, \le}\right)$ is an ordered group
 * $({2}): \quad \phi$ is a group homomorphism from $\left({\Q, +}\right)$ to $\left({\R, +}\right)$

This binary operation $+$ is called addition.