completeness axiom

The completeness axiom is the statement that the real line is solid, with no missing points where a number ought to be. It is what turns the merely arithmetical reals into a continuum on which limits, derivatives, and integrals can live.

There are several equivalent ways to phrase completeness for an ordered field. The most common is the least upper bound property: every nonempty set bounded above has a supremum. Equivalent formulations include that every Cauchy sequence converges, that every bounded monotone sequence converges, that nested closed intervals have a common point, and that every Dedekind cut is realized by a number. Over an Archimedean ordered field these say the same thing.

Completeness is precisely what the rationals lack and the reals supply. It is not a consequence of the field axioms or the order axioms; it must be added on top. Up to a relabeling, the reals are the unique complete ordered field, so completeness, together with being an ordered field, pins them down entirely.

A subtle warning: Cauchy completeness by itself, without the Archimedean property, is weaker than least upper bound completeness. The two coincide only once you also assume the line has no infinitely large or infinitely small elements.