---
tag: 0A9E6X0S3E
kind: concept
name: Complete ordered field
msc: [12J15]
sources:
  - url: https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers
    locator: Forms of completeness
  - url: https://en.wikipedia.org/wiki/Real_number
    locator: '"The complete ordered field"'
---

## Summary

A complete ordered field is an [[ordered field|0P3VCBDEE2]] $F$ in which each nonempty subset that is bounded above has a [[supremum|0R3H5PANRQ]] in $F$.
The [[real numbers|08J2M1JYAG]] form a complete ordered field, by the [[completeness axiom|0AXWET80YW]].
Any two complete ordered fields are isomorphic, by a [[unique isomorphism|0G8G1TH5WK]] 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|0XH79S06J3]].
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.