Real numbers/Completeness/Introduction/Section

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

An ordered field is called complete or completely ordered, if every Cauchy sequence in


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

The real numbers 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 and , both fulfilling these axioms, then there exists a bijective mapping from to 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 and constructs the real numbers as the set of all Cauchy sequences in with a suitable identification.