# Real numbers/Completeness/Introduction/Section

Within the rational numbers there are Cauchy sequences which do not converge, like the Heron sequence for the computation of ${\displaystyle {}{\sqrt {5}}}$. One might say that a nonconvergent Cauchy-sequence addresses a gap. Within the real numbers, all these gaps are filled.

## Definition

An ordered field ${\displaystyle {}K}$ is called complete or completely ordered, if every Cauchy sequence in ${\displaystyle {}K}$

converges.

The rational numbers are not complete. We require the completeness for the real numbers as the final axiom.

## Axiom

The real numbers ${\displaystyle {}\mathbb {R} }$ form a complete

Archimedian ordered field.

Now we have gathered together all axioms of the real numbers: the field axioms, the ordering axiom and the completeness axiom. These properties determine the real numbers uniquely, i.e., if there are two models ${\displaystyle {}\mathbb {R} _{1}}$ and ${\displaystyle {}\mathbb {R} _{2}}$, both fulfilling these axioms, then there exists a bijective mapping from ${\displaystyle {}\mathbb {R} _{1}}$ to ${\displaystyle {}\mathbb {R} _{2}}$ which respects all mathematical structures (such a thing is called an "isomorphism“).

The existence of the real numbers is not trivial. We will take the naive viewpoint that the idea of a "continuous number line“ gives the existence. In a strict set based construction, one starts with ${\displaystyle {}\mathbb {Q} }$ and constructs the real numbers as the set of all Cauchy sequences in ${\displaystyle {}\mathbb {Q} }$ with a suitable identification.