Complete ordered field

Summary

A complete ordered field is an ordered field $F$ in which each nonempty subset that is bounded above has a supremum in $F$. The real numbers form a complete ordered field, by the completeness axiom. Any two complete ordered fields are isomorphic, by a unique isomorphism that preserves the order. Thus "the complete ordered field" is a characterization of $\mathbb{R}$.

Remarks

The ordered field $\mathbb{Q}$ is not complete. Each complete ordered field is Archimedean. Other forms of completeness of $\mathbb{R}$ are equivalent for Archimedean ordered fields, for example the convergence of each Cauchy sequence.

Explanations

L1 human

Many number systems allow addition, multiplication, division and comparison, for example $\mathbb{Q}$ and $\mathbb{R}$. Among them, the real numbers are the only one that has no gaps. So you can define the real numbers by this list of properties, without a construction.