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 ${}{\sqrt {5}}$. One might say that a nonconvergent Cauchy-sequence addresses a gap. Within the real numbers, all these gaps are filled.

Definition

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

converges.

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

Axiom

The real numbers ${}\mathbb {R}$ 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 ${}\mathbb {R} _{1}$ and ${}\mathbb {R} _{2}$, both fulfilling these axioms, then there exists a bijective mapping from ${}\mathbb {R} _{1}$ to ${}\mathbb {R} _{2}$ 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 ${}\mathbb {Q}$ and constructs the real numbers as the set of all Cauchy sequences in ${}\mathbb {Q}$ with a suitable identification.