Uniqueness of the complete ordered field

Statement

Let $F$ and $K$ be complete ordered fields. 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.