The real numbers: $\mathbb{R}$

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The real numbers

$\mathbb{R}$

Definition

Cantor's construction of the real numbers

The set of real numbers, $\mathbb{R}$, is the quotient space, $\mathscr{C}/\sim$ where:

• $\mathscr{C}$ - the set of all Cauchy sequences in $\mathbb{Q}$ - the quotients
• $\sim$ - the usual equivalence of Cauchy sequences

We further claim:

1. that the familiar operations of addition, multiplication and division are well defined and
2. by associating $x\in\mathcal{Q}$ with the sequence $({ x_n })_{ n = 1 }^{ \infty }\subseteq \mathbb{Q}$ where $\forall n\in\mathbb{N}[x_n:=x]$ we can embed $\mathbb{Q}$ in $\mathbb{R}:=\mathscr{C}/\sim$

Axiomatic construction of the real numbers

Axiomatic construction of the real numbers/Definition

$\mathbb{R}$ is an example of:

• Vector space
• Field ($\implies\ \ldots\implies$ ring)
• Complete metric space ($\implies$ topological space)
  • With the metric of absolute value

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TODO: Flesh out

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Properties

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• The axiom of completeness - a badly named property that isn't really an axiom.

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If $S\subseteq\mathbb{R}$ is a non-empty set of real numbers that has an upper bound then:

• $\text{Sup}(S)$ (the supremum of $S$) exists.