WEB PAGE UNDER CONSTRUCTION.

This web page is devoted to a very beautiful result that *any two
complete ordered fields are isomorphic*.
In fact, the result is even better than this and the isomorphism is
canonical, in particular is unique
and natural

.

This is a sketch of the idea. Suppose $F,G$ are two complete ordered fields and we are trying to construct an embedding of $F$ in $G$. Then for reasons to do with the ordered field structure, the copy of the rationals in $F$ and the copy of the rationals in $G$ are the same so we must map $F$'s copy of $\mathbb{Q}$ onto $G$'s copy of $\mathbb{Q}$ in the obvious and only way possible.

Now for $x\in F$ there is a monotonic nondecreasing
sequence $\left({a}_{n}\right)$ of rationals converging to $x$
by one of the density theorems.
Since we are forced to send ${a}_{n}$ to the copy
of ${a}_{n}$ in $G$ (as ${a}_{n}\in \mathbb{Q}$)
we must therefore send $x$ to the *limit* of the copy of
the sequence $\left({a}_{n}\right)$ in $G$.
Fortunately (by the completeness axiom in $G$) this sequence does
indeed have a limit so there is some copy of $x$ in $G$
to send $x$ to. This construction will give a
canonical
embedding of
$F$ in $G$, and if $F$ is also complete this
embedding will turn out to be an isomorphism.