Real numbers

• categories: Real Analysis, Definition

$\mathbb{R}$ - set of all Dedekind cuts. Ordered complete field with LUB property and contains $\mathbb{Q}$ as subfield. The only field with thos property.

$x^2=2$ has no solutions in $\mathbb{Q}$. Proved by contradiction ⇒ $\mathbb{Q}$ has holes

Dedekind cut definition :

$\alpha$ is a subset of $\mathbb{Q}$ s.t.:

• Not empty and not $\mathbb{Q}$
• Closed to the left (for every x in $\alpha$ if y<x ⇒ y in $\alpha$ )
• Does not contain the greatest element

Then we define the order and arithmetics using cuts.

Cantor completion definition:

Completing $\mathbb{Q}$ (Complete metric space)