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 hyperr. numbers to say if they can be embedded into the surr. 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

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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