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

Addition/Axiomatic Definition
Let $\struct {\R, +, \times, \le}$ 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:
 * $\eqclass {\sequence {x_n} } {} + \eqclass {\sequence {y_n} } {} = \eqclass {\sequence {x_n + y_n} } {}$

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:
 * $\alpha + \beta = \set {p + q: p \in \alpha, \, q \in \beta}$

Addition/Real Numbers as Dedekind Completion of Rational Numbers
Let $\struct {\R, \le}$ denote the ordered set of real numbers, as defined as the Dedekind completion of the rational numbers.

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

Let $\struct {\Q, +}$ denote the additive group of rational numbers.

We have that $\struct {\Q, +, \le}$ is an Archimedean ordered group.

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

This binary operation $+$ is called addition.