Talk:Surreal number/Archive 1

Older discussions
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

Embeddings of hyperreals and ordinals
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

Broken external link
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? - Have you tried that lately? It works fine for me. Koyaanis Qatsi, Monday, July 8, 2002

Infinitesimals
Why aren't infinitesimals listed on this page at all? What about * and up?

Yes those numbers should be mentioned, but you should note they are pseudosurreal numbers(or Games).--SurrealWarrior

Star(*) and up are examples of games that are not numbers. Is that what you mean by pseudosurreal numbers? And &epsilon; is an infinitesimal, so that problem is answered. --Dan Hoey 21:01, 26 October 2005 (UTC)

Algebraic closure
What's the algebraic closure of surreal numbers? -- Kaol

There's some theorem telling you that you get the alg. closure of real-closed fields by adjoining the square root of -1 (Artin and another, I recall). Anyway my guess is that it's the 'obvious' complex number analogue, ie as small as it could be.

Charles Matthews 11:12, 25 Oct 2003 (UTC)

Simplicity and cleanliness
Mathematicians have praised the surreal numbers for being simpler, more general, and more cleanly constructed than the more common real number system.


 * Really? Maybe for graduate students and professionals. First of all, I don't understand the comparison, it's apples and oranges. The reals only aim to construct the reals, the surreals are much more ambitious. For dealing with a very abstract notion of "Dedekind cut", comparisons, a system containing lots of other systems, etc. surreals are good. But if you JUST want to get the reals, it's like hitting a fly with frying pan. While I find the subject fascinating and would love to read a more rigorous presentation (i.e. one that explicitly quotes results from set theory, instead of "intuitively" doing things), I would find it difficult to present surreals to say, an undergrad analysis class (at the level of baby Rudin or so). To do them justice (in what supposed to be a rigorous class) would require lots of set theory, a clear presentation of recursive definition, ordinal numbers and arithmetic, and distinction between sets and proper classes. Neither Dedekind cuts nor Cauchy classes require much understand of these.


 * I think the point was that for dedekind cuts, you first need to construct the integers, then construct the rationals as ordered pairs of integers, then use those to construct the reals. In doing so you get a second construction of all the rationals and integers (i.e the rational elements of your new real numbers) and it's a bit ugly. The surreal numbers have the advantage that all the numbers used in the construction are surreal numbers themselves.

Confusing notation
"&#969; + &#969; = { &#969; + S&#969; | } where x + Y = { x + y | y in Y }" looks confusing since "|" seems to mean "left-right cut" when first used and "such that" in the second use. The same is true for the original definitions of addition, negation and multiplication. --Henrygb 10:01, 2 Jun 2004 (UTC)

Maybe { &#969; + S&#969; | } should be changed to [ &#969; + S&#969; | ] (It would have to be changed through out the article).

2&epsilon;
I don't know much about this (yet, I'm learning), but shouldn't
 * 2&epsilon; = { &epsilon; | ..., &epsilon; + 1/16, &epsilon; + 1/8, &epsilon; + 1/4, &epsilon; + 1/2, &epsilon; + 1 }

be
 * 2&epsilon; = { &epsilon; | ..., 1/16, 1/8, 1/4, 1/2, 1 }

...or are the rwo simply equivalent? Roie m 15:27, 9 Jul 2004 (UTC)

Any surreal is unchanged by adding extra right options larger than some existing right option. Since &epsilon;+1 > 1 > &epsilon;+1/2 > 1/2 > &epsilon;+1/4 > 1/4 > ..., both the above forms of 2&epsilon; are unchanged by replacing their right option sets with the union of their right option sets. --Dan Hoey 21:01, 26 October 2005 (UTC)

Tic-tac-toe
Would Tic-tac-toe be an example of such a game? Do surreal numbers have any bearing on it?


 * Two players (named Left and Right)
 * Deterministic (no dice or shuffled cards)
 * No hidden information (such as cards or tiles that a player hides)
 * Players alternate taking turns

We get into trouble on the following:


 * Every game must end in a finite number of moves, even when the players don't alternate, and one player can move multiple times in a row

Players must always alternate, no player may skip taking a turn, but game always has at least 5 and at most 9 moves.


 * As soon as there are no legal moves left for a player, the game ends, and that player loses

Tic-tac-toe can end in a draw; the game otherwise ends by meeting a winning condition, not a losing condition as above, though a player can make two possible wins with one move, thereby guaranteeing his/her win, since opponant cannot block both at once.

Comments?

Good questions. The first objection isn't a problem, since Tic-tac-toe still makes sense if the players didn't alternate moves. And we can deal with the winning condition vs. losing condition issue by adding a rule that it's illegal to play on a board that already has a completed line; this new rule wouldn't affect the classic game, and in the Conway game it would force the non-line-making player to lose. But in order to make it a true Conway game you'd have to change the rules to eliminate draws. 171.64.71.123 05:31, 20 May 2006 (UTC)

a draw could happen in one of two ways:the board is full, in this case neither player can move, so its just the zero game, or boards with 2 winning lines, which can be dealt with by saying that it's legal to play on a board iff the number of lines of your side is ≥ the number of lines the other player has and there is a free space to move on 13:48, 7 April 2007 (UTC)

No Mathematics
This article has nothing to do with mathematics: A basic definition is given by itself and some simple calculations lead to an contradiction. This article has to be deleted!

You can say that again!! This article is a fine example of a Wiki BM (bowel movement). Similar articles are 'proof 0.999... = 1', definition of real numbers, and the list goes on. Wike editors are idiots who don't know shit. 70.110.81.253 00:19, 30 November 2005 (UTC)
 * ''You misspelled 'Wiki', dumbass.


 * Please explain that comment. I don't see any contradiction referred to in the article, and I think it's fairly well-established that surreal numbers are just as consistent as ZFC mathematics.  Prumpf 03:42, 30 Aug 2004 (UTC)

The relation <= ist defined by <=! In addition, this definition is not precisely defined.


 * Yes, that's the nature of a recursive definition. I think that section should be improved to make it more clear what actually happens, but deleting the article certainly doesn't seem justified. Prumpf 10:09, 30 Aug 2004 (UTC)

Ok, you are right. But more and more I think about this definition (and the recursive construction), I find it really not trivial.

Differentiation in Surreals
Shouldn't this page say something about difficulties/advances in differentiation/integration of surreal functions? Anyways to define functions such as $$ln(x), e^x, sin(x), cos(x)$$? I'd also like to include some of my own findings but don't know how to.--SurrealWarrior

The answers to some of the questions on surreal numbers.

Addition and multiplication on surreals do extend the natural sum and product on ordinals, defined via Cantor's normal form. The embedding of the ordinals in the surreals is defined if the following way: Every ordinal $$\alpha$$ can be identified with the set of ordinals strictly smaller than it, namely $$\alpha = \{\beta: \beta < \alpha\}$$. The embbeding $$f$$ is defined inductively by $$ f(\alpha) = \{f(\beta):\beta < \alpha | \}$$. One can readily verify that this embedding is well-defined, preserves the order, and that the sums and products of ordinals are also ordinals. The restriction of surreal addition and multiplication to the ordinals are the natural addition and multiplication (so, if you do not know what natural + and x are, you can define them as the restriction of surreal + and x).

The embedding of an ordered field in the surreal is (in general) not unique: for instance, there are many embeddings of the reals into the surreals. If we avoid the set-theoretic problems, due to the fact that surreal numbers form a proper class, for instance by restricting ourselves to the Grothendieck universe, it is easy to show that the surreal numbers have many automorphisms (as ordered field). This is general model theory: if $$\kappa$$ is the strongly inaccessible cardinal we used to construct the Grothendieck universe, the corresponding set of surreal numbers will be the saturated real closed field of cardinality $$\kappa$$, and any saturated real closed field has plenty of automorphisms.

The functions $$ln(x), e^x$$ have been defined on the surreals, and share many of the properties of the corresponding functions on the reals (to be precise, the surreals are an elementary extension of the reals, in the first order language given by the field operations, plus the $$ln$$ function).

Moreover, every analytic function $$f(x_1,\ldots,x_n)$$, defined on the real poly-interval $$[0,1]^n$$, can be extended to the surreal poly-interval $$[0,1]^n$$, using the fact that every surreal number can be represented canonically as an infinite sum of powers of $$\omega$$ (with real coefficients and surreal exponents). Again, the extended analytic functions share many of the properties of the corresponding functions on the reals.

However, extending the full funtion $$sin(x)$$ on the surreals is problematic, essentially because the 0 set of such function would be an extension of the set of natural numbers, and there is no good candidate for such set (Conway discusses briefly this problem in his book, and shows, for instance, why the set of omnific integers is not a good candidate).

As a source, you can consult the following book:

Gonshor, Harry "An introduction to the theory of surreal numbers." London Mathematical Society Lecture Note Series, 110. Cambridge University Press, Cambridge, 1986. vi+192 pp. ISBN: 0-521-31205-1

and article:

van den Dries, Lou; Ehrlich, Philip "Fields of surreal numbers and exponentiation." Fund. Math. 167 (2001),  no. 2, 173--188. 03C64 (12J99)

--Manta 17:41, 11 Apr 2005 (UTC)

What is the definitions for the functions:$$ln(x)$$&$$e^x$$?--SurrealWarrior 01:46, 13 Apr 2005 (UTC)

If you are really interested, you should read the book of Gonshor, where the topic is treated in detail. However, here is the definition for $$\exp x$$:

Given $$n$$ natural number, let $$[z]_n$$ be the $$n$$-truncation of the Taylor expansion of $$exp z$$ at 0:
 * $$[z]_n := \sum_{i=0}^{n} \frac{z^i}{i!}.$$

If $$ x = \{x^L | x^R\}$$, the recursive definition of $$\exp x$$ is the following:
 * $$\exp x = \{ 0, (\exp x^L )[x  - x^L ]_n, (\exp x^R)[x - x^R]_{2n+1} |

\frac{\exp x^R}{[x^R - x]_n}, \frac{\exp x^L}{[x^L - x]_{2n+1}} \},$$ where $$n$$ varies amomg the natural numbers, and, if $$z < 0$$, then $$[z]_n$$ must be positive.

For the logarithm, you can either define it to be the inverse of the exponential, or use a suitable formula, that I am not willing to write down now.--Manta 10:11, 13 Apr 2005 (UTC)

Thanks!--SurrealWarrior 01:56, 15 Apr 2005 (UTC)

As a further reference about integration of functions on surreal numbers, you may read the following (it is not an introductory text):
 * A. Fornasiero "Integration on Surreal Numbers" (PhD. thesis) http://www.dm.unipi.it/~fornasiero/phd_thesis/thesis_fornasiero_linearized.pdf

--Manta 08:27, 19 Apr 2005 (UTC)

There appers to be a flaw in the comparison rule: say y is a member of Xl (and Xr), then y is indeed less than or equal to a member of Xl, so (x ≤ y) is false under the definition, similarly if x is a member of Yr (and Yl). Numbers that one would reasonably consider to be equal are not comparable. Morosoph 13:49, 27 August 2006 (UTC)

Not a flaw, Xl and Xr must be disjoint, as a consequence of the requirement that no member of Xl can be &ge; any member of Xr. --Vaughan Pratt 00:45, 14 July 2007 (UTC)

Surreal quantization
The construction of the surreals in this article and in Knuth's book is strongly reminiscent of John Baez's nth quantization using category theory. Instead of building numbers from the empty set, he builds quantum states and Hilbert spaces. And it sure is a neat trick: "So, starting with the system with no states and repeatedly applying the second quantization functor, we have gotten to string theory.  It would be crazy to stop now...."

Oh, if I could only establish a connection between string theory and surreal numbers, I'd be the coolest kid on the block. (If anyone else does it, just remember that User:Anville was there first. Heh heh.)  This makes me wish I understood category theory well enough to see if surreals fit in there somehow. Anville 16:29, 6 January 2006 (UTC)

limit ordinals
Are there any surreals whose birthdays are limit ordinals? For example, whose birthday is ω?

For that matter, how do we define Sω? Normally, I would think it would be something like the union of all Sa for a < ω. Except I don't know how to take the union of surreals. For example, what is the union of {0|1} and {1|2}? I suppose there should be some recursive def. I'd appreciate some pointers. -lethe talk 20:57, 7 January 2006 (UTC)


 * Isn't any number which is not "rational" from the surreal perspective born on ω? Any number requiring an infinite number of Dedekind cuts should be born on ω.-CKnapp 3:37, 23 Sept 2006(UTC)


 * From my reading of the article, Sω is indeed that union (remember that Sx is a set of surreal numbers, not a surreal number itself). A real number that is not a dyadic rational is born on Sω+1. --Ihope127 23:12, 13 February 2007 (UTC)


 * That part of the definition seems odd to me. It defines Sα differently for when α is a limit ordinal than when α is not. It seems far more reasonable to define Sα to be the set of all older surreals along with all cuts from from the older surreals, regardless of what type of ordinal α is. This further has the advantage that the birthday of every ordinal is itself. With the definition as given, only finite ordinals have this property; infinite ordinals α have birthday α + 1. 68.103.121.107 17:39, 17 February 2007 (UTC)


 * Well, doing it this way means you can easily state the union of all Sα for α < some limit ordinal. The question then is why S0 is {0} rather than &empty;, as this would make all ordinals α be born on α+1... well, not everything can make sense, now, can it? Then again, there's nothing stopping you from saying Sα is whatever you want, as long as you say what it is. --Ihope127 11:56, 26 February 2007 (UTC)


 * The most appealing (to me) definition would be to say that Sα is the set of all surreals with birthdays strictly less than α, and that a surreal number has birthday α if it can be constructed from members of Sα. This most closely parallels the usual definition of the rank of a set (a set has rank α if it's a subset of Vα, not if it's a member of Vα), and it allows S0 = &empty; as well as having each ordinal be born on its own day.  But if the usage in the article is the most established one (I wouldn't know), we should probably stick with that. 71.121.10.21 (talk) 13:12, 12 January 2008 (UTC)


 * Figure 0 on p.11 of On Numbers and Games exhibits the surreal numbers born on day ω (and not before) as consisting of the dyadic irrationals (those reals that are not dyadic rationals, such as 1/3, e, and &pi;), &plusmn;ω, and all numbers of the form x &plusmn; 1/ω for x a dyadic rational. A bit of a hodge-podge, but no day combined with all prior days produces a field, that's only obtained as all numbers born prior to a sufficiently large day.  That's the day you pop out of your chosen Grothendieck universe, your day of rest.  If you work on the day of rest you'll mess up your field and be condemned to continue on to the next larger Grothendieck universe.  --Vaughan Pratt 23:30, 13 July 2007 (UTC)

Would it be at all possible to define the cardinality of rough sets as a surreal number x={XL|XR} where XL and XR are the cardinalities of the lower and upper approximations respectively?--SurrealWarrior 17:50, 1 February 2006 (UTC)

The "equivalence class" formulation should be dropped
Conway didn't use equivalence classes, and they don't belong in the standard presentation. { 1 | } = { 0, 1 | }, they aren't just "==". If someone wants to present an "equivalence class" formulation, I invite them to add another "alternative formulation" section where it doesn't confuse the issue.

If you want to consider this a non-extensional definition of equality, that's what it is. If you can't cope with that, the extensional view is that numbers and games are abstract objects and {. | . } is a function (or FUNCTION) mapping Ug &times; Ug → Ug. But calling { 1 | } and { 0, 1 | } unequal is broken. I'm planning to fix this unless someone cares to dissuade me.--Dan 03:07, 3 March 2006 (UTC)


 * 1)  But then you have to explain to Janet Davis, at the beginning of this discussion, how "what they are" is a question that can't be answered in any concrete sense.  Abstract object just are, and we can only describe them by what they do.  So give up concreteness or give up extensionality (but shun this second-hand equivalence classism).--Dan 03:07, 3 March 2006 (UTC)

On p.5 of On Numbers and Games Conway calls the pair {L|R} of sets L and R the form (what an algebraist more often calls a term) of the number to distinguish it from the number itself. He says the notion of equality is a defined relation, by which he means it is defined on, and thereby constitutes an equivalence relation on, the forms. In algebra one writes "x+y = y+x" (not the same thing as "x+y" = "y+x") to mean that the terms "x+y" and "y+x" are equivalent by virtue of denoting the same number. As terms, "{ 1 | }" and "{ 0, 1 | }" are distinct, as surreal numbers, { 1 | } and { 0, 1 | } are equal. Conway's "equality is a defined relation" means that the set of surreal numbers is by definition the quotient by this equivalence relation of the set of forms. --Vaughan Pratt 00:23, 14 July 2007 (UTC)

Alternate notation?
I seem to recall reading a Discover magazine article attributing surreal numbers to Martin Kruskal, and there was a notation involving up- and down-pointing arrows. E.g. 1&frac34; would be written &uarr;&uarr;&darr;&uarr;, and &omega; would be &uarr; with a little hat over it, etc. Does anyone know more about this? 67.87.115.207 22:08, 17 April 2006 (UTC)


 * I think I discovered the article you were talking about at http://www.findarticles.com/p/articles/mi_m1511/is_n12_v16/ai_17863372. This article does introduce an arrow notation but it attributes the discovery of Surreal Numbers to Conway.--SurrealWarrior 13:27, 20 May 2006 (UTC)


 * The "arrows" are more of an invention of Conway, Guy, and Berlekamp, in Winning Ways for your Mathematical Plays. If I remember correctly, it also deals specifically with nimbers. An interesting case found when a surreal number a=-a. That could be a lie...

C Knapp 22 Sept. 2006


 * The "sign sequence" concept, see the article, is in fact presented by Conway in the first edition of ONAG; up arrows and down arrows amount to the same thing as pluses and minuses (see this article's section on "sign expansion"). Gonshor used them as the definition of the surreals in his book.  It has nothing whatsoever to do with nimbers, which form a completely different field (with characteristic 2).  Nimbers describe the impartial games; surreal numbers describe the cold partizan games. Mr Death 23:46, 30 September 2007 (UTC)

Confusing expression
In the “to infinity and beyond section” there is the expression:
 * 1/3 = { { a / 2b in Sω | 3a < 2b } | { a / 2b in Sω | 3a > 2b } }

Which confused me at first, because I thought both the left set and right set were single numbers in the form {XL|XR}. Then I realized they were sets, in which case a better expression, to avoid confusion, would be to use the alternative set notation:
 * 1/3 = { { a / 2b in Sω : 3a < 2b } | { a / 2b in Sω : 3a > 2b } }

Is this correct? (I didn’t make the change because I’m not well versed enough in this topic to be sure my understanding is correct) GromXXVII 00:26, 29 October 2007 (UTC)

Inconsistent notation
Does anyone know in which reading (or anywhere else) the notation "Si" came from? (introduced in the Generating surreal numbers using finite induction section). Because it seems to be causing some inconsistencies in the To Infinity and Beyond section: as it says that The birthday of 1/3 is ω+1 yet the birthday of 1/3 is actually ω: but it appears with the Si notation it's in set Sω+1. I find it hard to believe that the notation should yield the index of the set and the birthday to disagree like so. which I find hard to believe and would like to verify/correct. GromXXVII 23:46, 9 November 2007 (UTC)

Okay, so the article defines birthday as The smallest value of a for which a given surreal number appears regarding Sa. This is either inconsistent with the literature, or as I suspect, the indices of the Sa are slightly off: that Sω is actually the set corresponding to all numbers created when ω is created, and not The set of all surreal numbers that are generated in some Si (i a natural number) as stated in the article. GromXXVII 23:56, 9 November 2007 (UTC)

Formulas with squares
On the formulas of the addition and multiplication I see some squares. Must be to convert to LaTeX. Thanks, 77.125.95.201 (talk) 15:00, 17 March 2008 (UTC)


 * You are probably using an older web browser, Internet Explorer 6 or lower. I know there is a tag that lets us get around this, though. It converts to TeX if you're web browser can't handle the notation. Too bad I don't know how to use TeX. -- trlkly 05:23, 26 April 2008 (UTC)


 * I’ve seen the boxes myself, for instance, about half the symbols in Mathematical symbols show up as boxes for me on IE7 on this computer. They all show up correctly on firefox, so there’s probably an update or something for IE7 that could fix it, but if so this computer doesn’t have it.
 * However, the issue he was referring to I believe changed to TeX awhile ago, but never bothered to say anything here. GromXXVII (talk) 19:01, 26 April 2008 (UTC)

Section "To Infinity and Beyond" should be rewritten slightly
The section begins with an attempt to define the birthday of a number. This should use the definition of ordinals in terms of the surreals, i.e. it should not depend on the "usual" ordinals and transfinite induction. An ordinal in Conway's system is a number that can be written as $$\{L|\}$$ (empty set on the right). Then given such an ordinal $$\alpha$$, one defines a set (it can be shown to be a set) $$M_\alpha$$ of numbers such that $$x=\{x^L|x^R\}$$ is in $$M_\alpha$$ if $$x^L, x^R\,\epsilon\,O_\alpha = \cup_{\beta<\alpha}M_\alpha$$. Let $$N_\alpha=M_\alpha\setminus O_\alpha$$. Then we say that $$\alpha$$ is the birthday of $$x$$ if $$x\,\epsilon\,N_\alpha$$ (one can prove every $$x$$ belongs to a unique $$N_\alpha$$). An equivalent "genetic" definition is: $${\rm bday}(x)=\{{\rm bday}(x^L),\,{\rm bday}(x^R)|\}$$. Note this is again an ordinal in Conway's sense so there is no reliance on numbers defined outside the system. JanBielawski (talk) 00:41, 25 June 2008 (UTC)

Recent edits – field/ordered field
One thing to note, as I see that a discussion of relating to fields has been expanded. I think it is important to not give the impression that he class of surreal numbers is a field, as it is not a set. Also to note is that in some of the literature such as ONAG this problem is rectified by calling it a Field, with a capital F, instead. GromXXVII (talk) 21:17, 28 August 2008 (UTC)


 * I'll take a shot at that once I have fixed various errors (including the ones I'm introducing as I go). I'm working without a copy of ONAG handy, trying to structure the exposition so that it stands alone (without too much use of facts from standard analysis or ordinal arithmetic).  Corrections are of course welcome.  Michael K. Edwards (talk) 17:28, 1 September 2008 (UTC)

comparison with Robinson's hyperreals
It would be helpful to have a more meaningful comparison with the hyperreals, e.g. comparison of the logical tools used, comparison of difficulty of construction, etc. Katzmik (talk) 10:56, 1 September 2008 (UTC)


 * I might give that a shot once I've at least corrected the errors that I've introduced. I lack expertise in that area, though, so I'll have to do some reading.  Michael K. Edwards (talk) 17:30, 1 September 2008 (UTC)


 * IMHO, to get too much into hyperreals in this article would be to get substantially off topic; surreals and hyperreals are only tangentially related. They're both ordered superfields of the reals (which naturally means they must include infinitesimals), the hyperreals have to embed into the surreals (although even this is nowhere near obvious), and that's about it.  One works with them using utterly different tools, et cetera et cetera. Mr Death (talk) 01:53, 2 September 2008 (UTC)


 * I think a brief comparison might be helpful. Basically, the hyperreals were designed to be useful for doing analysis, whereas the surreals were designed to be all-encompassing. The hyperreals have the transfer principle, which applies to all propositions in first-order logic, including those involving relations. So for instance you can define a full-fledged sine function on the hyperreals, but not on the surreals.--76.167.77.165 (talk) 04:38, 22 March 2009 (UTC)

Holy length, Batman!
Don't get me wrong, Michael K. Edwards. I did one and a half theses on surreal numbers. But this is Wikipedia, and surreal numbers don't need a 90kB article. What they deserve is a 10-20kB article that grabs people's interest, possibly with subsidiary articles to explain some of the more interesting details. Mr Death (talk) 21:26, 5 September 2008 (UTC)


 * Also, can we please have section headings which look like they belong in an encyclopedia and not lighthearted exposition ("To infinity"/"And beyond")? I added an tag. And why is there so much math jargon and notation?  The only notation should be for the surreal numbers themselves, with the rest of what currently is in jargon and symbols needs to conform to WP:MSM, especially WP:MSM. Neut Nuttinbutter (talk) 21:43, 7 September 2008 (UTC)


 * Given that the transfinite surreals are so important to the topic, as is the method of constructing them, I don't have a problem with that particular title (if you go back in the article's history, it started out as a single section and it worked fine at that point; I agree that the current set of section names has got somewhat out of hand). It seems to me that Michael K. Edwards is trying to turn this article into an online version of (the first half of) ONAG, which it emphatically should not be.  I'm just not sure how to fix it at this point, short of going back a month in time. Mr Death (talk) 04:15, 8 September 2008 (UTC)


 * I have whacked back a lot of overgrowth, trying to leave only the essential steps on the way to the conclusion that $$S_{\omega \uparrow\uparrow *}$$ is closed under the multiplicative inverse. I have also tried to make the tone more encyclopedic.  Perhaps one of you would care to review this, prune more if needed, and remove the tags?  Michael K. Edwards (talk) 18:43, 8 September 2008 (UTC)


 * By the way, I read an article (I'll see if I can dig it up) that showed:
 * $$S_{<n} = \cup_{k < n} S_k$$ is a group iff $$n = \omega^k$$ for some nonnegative integer $$k$$;
 * $$S_{<n}$$ is a ring iff $$n = \omega^{\omega^k}$$;
 * $$S_{<n}$$ is a field iff $$n = \epsilon_k$$.
 * This agrees with the results you have posted, since your $$S_+$$ is exactly $$S_{<\omega}$$ and your $$S_{\omega \uparrow \uparrow *}$$ is exactly $$S_{<\epsilon_0}$$. Might be worth a passing mention. Mr Death (talk) 19:44, 10 September 2008 (UTC)


 * Excellent; I was hoping to find a good reference for those results (and standardize the notation). I've been in touch with Philip Ehrlich, who is sending me a reprint of a paper of his (joint with Lou van den Dries) that provides accurate bounds for the birthdays of x+y, x·y, 2x, etc.  I have some material on the use of surreal exponentiation and Knuth's arrow notation to identify and name the epsilon numbers, starting with $$\epsilon_0 = \omega \uparrow \uparrow \omega$$ and $$\epsilon_1 = (\omega \uparrow \uparrow \omega) \uparrow \uparrow \omega$$; I am not sure whether to put it in a sub-article of this one or to add a new "epsilon numbers" article.  It's a concrete example of what surreal numbers are good for; epsilon numbers are trivial to identify in the surreals because they're the fixed points of base 2 surreal exponentiation x → 2x, which I find much easier to think about than base ω ordinal exponentiation.
 * Is the article still too long after my cuts? If so, what would you cut or move elsewhere?  Michael K. Edwards (talk) 22:37, 10 September 2008 (UTC)


 * I was thinking earlier today that an "epsilon numbers" article would be in order; since you concur, I'll whip one up later on tonight or tomorrow (it will probably need some editing). It should be a sub-article, since epsilon numbers occur in non-surreal-number settings as well.  (Conway notes that there are also non-ordinal epsilon numbers, e.g., $$\epsilon_{-1}$$, and I could get into that too (-:)  There are further generalisations to be made that I haven't gotten around to publishing, and again, Wikipedia is not....
 * By $$\epsilon_{-1}$$, do you mean the simplest surreal less than ω that is a fixed point of x → 2x, namely $$\omega^{\omega^{-\epsilon_0}}$$ (I think)? Michael K. Edwards (talk) 06:08, 11 September 2008 (UTC)
 * $$\epsilon_{-1}$$ is the simplest surreal n less than $$\epsilon_0$$ that satisfies $$n = \omega^n$$. It is not less than ω; in fact, it is greater than $$\omega, \omega^\omega, \omega^{\omega^\omega}, \ldots$$ (as all epsilon numbers must be). It is briefly mentioned in the new article, alongside $$\epsilon_{\frac{1}{2}}$$, the simplest epsilon number between $$\epsilon_0$$ and $$\epsilon_1$$. Mr Death (talk) 10:58, 11 September 2008 (UTC)


 * Personally, I prefer $$\omega$$ "exponentiation" over $$2^x$$ for a variety of reasons, mostly due to predilection and advisor bias I'm sure. A section and/or subarticle about Gonshor's extension of $$\exp$$ and $$\log$$ to the surreal numbers might be in order as well.
 * By the way, if it's the Ehrlich-v.d.D. paper from around 2002, I'll warn you that you're in for some disappointment on those birthday bounds, except for addition!
 * I do think the article is still a bit too long, and I will get around to editing it soon, I promise! Mr Death (talk) 00:48, 11 September 2008 (UTC)


 * If you get a chance to fix my rather confused and circular attempt to define 1/x and 2x without the "recursive options" wheeze used in ONAG (and attributed to Kruskal), that would be an improvement. That, more than the birthday bounds per se, is what I hope to find in v.d.D.-Ehrlich ("Fields of Surreal Numbers and Exponentiation", Fund. Math. 167 (2001), 173-188).  There's an essential connection between the fact that square roots exist (i. e., x → x2 is 1:1 and onto on the positive half of $$S_{<\epsilon_k}$$) and the closed-form definition of 2x for x with non-limit-ordinal birthday, and I'm not quite putting my finger on it.  It's probably done right in Gonshor's book, which seems to have been reprinted recently; perhaps I should order up a copy.
 * Do you have a good reference for the Knuth-arrow-notation forms of the epsilon numbers? In particular, one that handles the transition to $$\omega \uparrow \uparrow \uparrow 3$$ cleanly?  Michael K. Edwards (talk) 01:57, 11 September 2008 (UTC)
 * It seems clear to me now that $$\epsilon_k = \omega \uparrow \uparrow (\omega^{k+1})$$ (for natural number k), $$\epsilon_\omega = \omega \uparrow \uparrow (\omega^\omega)$$, and so on, up to $$\epsilon_{\epsilon_0} = \omega \uparrow \uparrow \uparrow 3$$, and so forth. Easy enough to work out once you realize that $$(x \uparrow \uparrow y) \uparrow \uparrow z = x \uparrow \uparrow (yz)$$ works for surreals as well as natural numbers.  I could be wrong, of course; that's why I'd like to find a reference.  :-)  Preferably one that follows this sequence to the limits of the Knuth arrow notation, and gets into Conway arrow notation, etc.  And it would be nice to identify the sort of set theory that is needed to pursue transfinite recursion this far.  Michael K. Edwards (talk) 04:25, 11 September 2008 (UTC)


 * Houston, we have epsilon numbers! Mr Death (talk) 10:48, 11 September 2008 (UTC)
 * JRSpriggs replaced your new article with a redirect to epsilon nought, so I merged your new material there (along with some additions). Please have a look and see whether you approve.  Michael K. Edwards (talk) 23:04, 11 September 2008 (UTC)

Well if someone else comes along and complains about those quoted section headings, don't say I didn't warn you. But I removed the tag I added, in recognition of Michael's recent work. Neut Nuttinbutter (talk) 08:39, 9 September 2008 (UTC)

Continuum Hypothesis
With $$\omega$$ being a fixpoint of exponentiation, a naive argument would say that the Continuum Hypothesis does not hold for surreal numbers. However, it could be that $$\omega$$ is not $$\aleph_0$$. Can anybody explain what the status really is? -- Zz (talk) 03:19, 11 January 2009 (UTC)
 * $$\omega$$ is a fixpoint of surreal(and ordinal) exponentiation, not cardinal exponentiation. —Preceding unsigned comment added by Kaoru Itou (talk • contribs) 22:16, 25 January 2009 (UTC)

Perfectly correct, and perfectly stupid
>
 * Negation
 * $$\scriptstyle -x = - \{ X_L | X_R \} = \{ -X_R | -X_L \}$$,
 * where $$\scriptstyle -X = \{ -x: x \in X \}$$.

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 * You're using x for two different things in the same statement. It's technically correct, but it's also technically correctly to use squares of slightly-different sizes to represent all of my variables (hyperbole). Why sow unnecessary confusion? --68.161.151.179 (talk) 08:40, 12 March 2009 (UTC)

First-order properties
In the article on infinitesimals, I've been trying to make the treatment more coherent by classifying the systems according to their first-order logical properties. Am I correct in thinking that the surreals have all the same first-order properties as the reals for statements involving the basic ordered-field relations +, ×, and ≤? (This is stronger than the statement that they're an ordered field, e.g., it requires that square roots exist.)--76.167.77.165 (talk) 16:46, 22 March 2009 (UTC)