
Re: Orderpreserving embeddings of ordinals in the real numbers
Posted:
Jul 29, 2006 12:30 PM


In article <eadai0$g34$1@news.ks.uiuc.edu>, David Madore <david.madore@ens.fr> wrote:
>John Baez in litteris <ead71m$etd$1@news.ks.uiuc.edu> scripsit:
>> It's easy to map the ordinal omega^2 into the real numbers >> in a onetoone and orderpreserving 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 orderisomorphic 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.

