User:Abcxyz/Sandbox/Real Numbers/Definition:Real Addition

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

The binary operation $+$ is called addition.

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

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, as constructed from Dedekind cuts.

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, as defined as the Dedekind completion of the 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.

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.