---
tag: 0G8G1TH5WK
kind: statement
role: theorem
name: Uniqueness of the complete ordered field
msc: [12J15]
uses: [0A9E6X0S3E, 010JCV9J4E]
sources:
  - url: https://en.wikipedia.org/wiki/Real_number
    locator: '"The complete ordered field"'
---

## Statement

Let $F$ and $K$ be [[complete ordered fields|0A9E6X0S3E]].
Then there is exactly one field isomorphism $\varphi \colon F \to K$ that preserves the order, that is, with $\varphi(a) \le \varphi(b)$ whenever $a \le b$.

## Remarks

The proof first maps the rational elements of $F$ to those of $K$. Each other element is the supremum of the rational elements below it, and $\varphi$ maps it to the supremum of their images.
In fact each field isomorphism between two complete ordered fields preserves the order, because the nonnegative elements are exactly the squares.

## Explanations

### L1 human

Two people may build the real numbers in different ways, for example with Dedekind cuts and with Cauchy sequences.
This theorem says that they get the same number system, with only different names for the numbers.