WEB PAGE UNDER CONSTRUCTION.
1. Introduction
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 ,
are two complete
ordered fields and
we are trying to construct an embedding
of in . Then for reasons to do with the ordered field
structure, the copy of the rationals in and the copy of the
rationals in are the same so we must map 's copy
of onto 's copy of in the
obvious and only way possible.
Now for
there is a monotonic nondecreasing
sequence (
)
of rationals converging to
by one of the density theorems.
Since we are forced to send
to the copy
of
in (as
)
we must therefore send to the limit of the copy of
the sequence (
)
in .
Fortunately (by the completeness axiom in ) this sequence does
indeed have a limit so there is some copy of in
to send to. This construction will give a
canonical
embedding of
in , and if is also complete this
embedding will turn out to be an isomorphism.