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Topic: This Week's Finds in Mathematical Physics (Week 236)
Replies: 29   Last Post: Aug 24, 2006 9:00 AM

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 David Madore Posts: 120 Registered: 12/13/04
Re: Order-preserving embeddings of ordinals in the real numbers
Posted: Jul 28, 2006 11:30 AM

John Baez in litteris <ead71m\$etd\$1@news.ks.uiuc.edu> scripsit:
> It's easy to map the ordinal omega^2 into the real numbers
> in a one-to-one and order-preserving way. Here's an artist's
> conception, which uses the second dimension to make things
> easier to see:
>
> http://math.ucr.edu/home/baez/omega_squared.png

Thanks for calling me an artist :-) but I don't think I deserve the
title. I created that image for Wikipedia, see <URL:
http://commons.wikimedia.org/wiki/Image:Omega_squared.png > for a
larger version.

> Which ordinals can we do this for?

If you're asking which ordinals are order-isomorphic to a subset of
the real numbers, the answer is simple (at least, assuming the axiom
of choice): exactly the countable ordinals. First, any well-ordered
subset of the reals is countable because between any element of the
subset and the next ("the next" makes sense, of course, since the set
is well-ordered) there is a rational. Conversely, we can prove by
transfinite induction that any countable ordinal can be embedded in
the reals: it is true of 0, if it is true of alpha it is true of
alpha+1, and if it is true of every ordinal alpha<delta for a limit
ordinal delta, choose an increasing sequence alpha_n leading up to
delta, embed each alpha_n between (n-1)/n and n/(n+1) and put them
together... the details are left to the reader (it may be necessary
to remove some elements since we took the sum of the alpha_n rather
than the limit, but it can also be arranged so that the two coincide).

In fact, every countable ordinal can be embedded in the reals as a
closed set (then the embedding is a homeomorphism from the ordinal
with the order topology to the subset of the reals with the induced
topology), or as a discrete set, but obviously not both (except up to
omega).

[I won't bet on what happens in the absence of Choice, but it wouldn't
at all surprise me it were consistent that some large countable
ordinals can't be embedded in the real line.]

I had produced a graphical representation of epsilon_0, once, but it's
actually entirely uninteresting to look at, it's just a mess.

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