Wikipedia: Surreal numbers/Talk

Could someone please explain what surreal numbers actually are? -- Janet Davis

I'll give it a try. See the new paragraph just below the introduction -- MattBrubeck

I second that! --LMS

This is a huge improvement on what was here when I last looked. I even think I understand it. :-) Thanks! --Janet Davis

Hm. Perhaps I went a bit overboard with my explanation. (I felt that the construction had to be explained to understand the difference between hyperreal numbers and surreal numbers.) Unfortunately I have to get back to work now and the page is not really finished. Perhaps next week. --Jan Hidders

Jan: Excellent work on editing the introduction. It's now much clearer than I left it after my contributions. -- MattBrubeck

Wow, this page is great. Two questions: are the hyperreals embedded in the surreals? How about the ordinals? Ordinal arithmetic is non-commutative, so there must be some problems. --AxelBoldt

And two more questions: what about the topology of the surreals? Also, the first paragraph says they don't form a "class" of numbers, but then later it says they form an ordered field. These don't seem to go together.

Good questions. I don't know enough about hyperreal numbers to say if they can be embedded into the surreal numbers. Studying they hyperreals is still somewhere on my to-do list. There are some very nice resources on Hyperreals and non-standard analysis on-line:

  http://online.sfsu.edu/~brian271/nsa.pdf
  http://www.ugcs.caltech.edu/~shulman/math/nonstandard/node9.html

And if you want to find more the magic word is "ultrafilters" :-) Actually I think we should quickly extend the article on hyperreals because it is #1 in Google right now. :-) I think you are right about the Ordinals; hyperreals and surreals both satisfy the algebraic rules of the reals, so there can be an order homomorphism but not an homomorphism that respects the operators. -- JanHidders

I don't think it's really correct to say that the surreals form an ordered field, as they are a proper class, not a set. There is no largest ordered field - in fact there are hyperreal fields of arbitrarily large cardinality.
Zundark, 2001-08-17

You are of course correct; in a proper treatment one would distinguish between fields which are sets and "big fields" which are classes, and then we could say that the surreals form a big ordered field and every big ordered field embeds in the surreals.

By the way, do you know if the embeddings are unique?

--AxelBoldt

I don't know if they're unique. In fact I didn't even know every big ordered field embeds in the surreals, although I did know this was true for ordinary ordered fields. (But I don't know much about the surreals anyway.)
Zundark, 2001-08-17

The URL http://www.tondering.dk/claus/surreal.html for the "gentle yet thorough introduction" doesn't (currently) seem to work, nor does any obvious modification of it. As for the non-commutativity of ordinal addition, sure, but there is something called the "natural" or "Hessenberg" sum on ordinals which is commutative. (The definition uses the Cantor normal form, and basically just "sorts" the summands.) Maybe this is what extends to Conway's addition?