From: apollo@math.berkeley.edu (Apollo Hogan)
Subject: Re: Mapping Ordinals onto Rationals
Date: 4 Aug 2000 20:42:44 GMT
Newsgroups: sci.math
Summary: Aronszajn trees

In article <8mev65$2mc$1@nnrp1.deja.com>,
David C. Ullrich  <david_ullrich@my-deja.com> wrote:
>In article <8mc3i3$10n0@edrn.newsguy.com>,
>  daryl@cogentex.com (Daryl McCullough) wrote:
>> I've heard that every countable ordinal can be mapped into the reals
>> so that if alpha < beta, f(alpha) < f(beta).
>
>    Exactly what you say here is true, but it seems like a
>statement that could easily be misinterpreted. An ordinal
>is in particular a set of ordinals, and it's in this sense
>that yes every countable ordinal can be mapped into the
>reals: if gamma is a countable ordinal then there is a
>mapping from gamma to R such that the above holds for all
>alpha and beta in gamma (ie for all ordinals alpha, beta
>less than gamma).

One of my favorite mathematical constructions uses this fact quite
nicely:  construction an Aronszajn tree in ZFC.

An aronszajn tree is a tree with omega_1 - many levels, each of
countable width and with no branch of length omega_1.  The basic
idea behind this construction is to make each point on the tree
a countable (order-preserving) sequence of rationals.
Then order these sequences by inclusion.  (So S<T iff T is a proper
extension of S.)  Then we can't have any branches of length omega_1,
since such a branch would give us an omega_1 - sequence of rationals,
which is impossible.  But, if we're careful about how we extend branches
at limit stages, we can ensure we have only countable levels and that
we have omega_1 - many levels, since for every alpha<\omega_1, we have
a branch of length at least alpha.

It's a cute proof and I like the fact that it uses something as concrete
as the rationals to build something as esoteric and abstract as an
\omega_1-aronszajn tree...

--Apollo Hogan
(student UC Berkeley)