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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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 baez@math.removethis.ucr.andthis.edu Posts: 446 Registered: 12/13/04
Re: Order-preserving embeddings of ordinals in the real numbers
Posted: Jul 29, 2006 12:30 PM

>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 [....]

Thanks! I didn't check to see who made it. The phrase "artist's
conception" was intended as a slight joke, since in pop science
magazines one often reads things like "here is an artist's conception
of romance among australopithecines" adorning pictures that required
a lot of imagination to draw - but this time, it was actually a
mathematically precise picture!

>> 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.

Yay! Great! That's exactly what I was asking.

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

If you still have it around, I would be interested to see it - and maybe
even attach it to week236. I can see why it would be a mess, though.

I suppose drawing it bigger wouldn't help, but it might be fun to
take some large ordinal and draw it in your style on the scale of
this artist's conception of a hydrogen atom:

http://www.phrenopolis.com/perspective/atom/index.html

This may be the world's biggest webpage: it's 18 kilometers wide!
(That's 50 million pixels at 72 pixels per inch.)

I hadn't known my webbrowser could scroll that far. My wrist didn't
even get tired. So, it might be possible to draw omega^omega or
something and have it look interesting, even if epsilon_0 is too big.