Talk:Surreal number

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1 Older discussions
2 Embeddings of hyperreals and ordinals
3 Broken external link
4 Infinitesimals
5 Algebraic closure
6 Simplicity and cleanliness
7 Confusing notation
8 2ε
9 Tic-tac-toe
10 No Mathematics
11 Differentiation in Surreals
12 Surreal quantization
13 limit ordinals
14 The "equivalence class" formulation should be dropped
15 Alternate notation?
16 Confusing expression
17 Inconsistent notation
18 Formulas with squares

[edit] 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

[edit] 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

[edit] 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

[edit] 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 ε is an infinitesimal, so that problem is answered. --Dan Hoey 21:01, 26 October 2005 (UTC)

[edit] 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)

[edit] 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.

[edit] Confusing notation

"ω + ω = { ω + Sω | } 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 { ω + Sω | } should be changed to [ ω + Sω | ] (It would have to be changed through out the article).

[edit] 2ε

I don't know much about this (yet, I'm learning), but shouldn't

2ε = { ε | ..., ε + 1/16, ε + 1/8, ε + 1/4, ε + 1/2, ε + 1 }

2ε = { ε | ..., 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 ε+1 > 1 > ε+1/2 > 1/2 > ε+1/4 > 1/4 > ..., both the above forms of 2ε are unchanged by replacing their right option sets with the union of their right option sets. --Dan Hoey 21:01, 26 October 2005 (UTC)

[edit] 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)

[edit] 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.

[edit] Differentiation in Surreals

Shouldn't this page say something about difficulties/advances in differentiation/integration of surreal functions? Anyways to define functions such as $l n (x), e x, s i n (x), c o s (x)$ ? 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 $α$ can be identified with the set of ordinals strictly smaller than it, namely $α = {β:β < α}$ . The embbeding $f$ is defined inductively by $f (α) = {f (β):β < α | }$ . 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 $κ$ 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 $κ$ , and any saturated real closed field has plenty of automorphisms.

The functions $l n (x), e x$ 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 $l n$ 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 $ω$ (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 $s i n (x)$ 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: $l n (x)$ & $e x$ ?--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 $e x p z$ 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 ≥ any member of Xr. --Vaughan Pratt 00:45, 14 July 2007 (UTC)

[edit] 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)

[edit] limit ordinals

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

For that matter, how do we define S_ω? Normally, I would think it would be something like the union of all S_a 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, S_ω is indeed that union (remember that S_x is a set of surreal numbers, not a surreal number itself). A real number that is not a dyadic rational is born on S_ω+1. --Ihope127 23:12, 13 February 2007 (UTC)

That part of the definition seems odd to me. It defines S_α differently for when α is a limit ordinal than when α is not. It seems far more reasonable to define S_α 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 S_α for α < some limit ordinal. The question then is why S₀ is {0} rather than ∅, 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 S_α 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 S_α 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 S_α. This most closely parallels the usual definition of the rank of a set (a set has rank α if it's a subset of V_α, not if it's a member of V_α), and it allows S₀ = ∅ 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 π), ±ω, and all numbers of the form x ± 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)

[edit] 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^[1] and { . | . } is a function (or FUNCTION) mapping Ug × 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)

^ 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)

[edit] 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. 1¾ would be written ↑↑↓↑, and ω would be ↑ 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)

[edit] Confusing expression

In the “to infinity and beyond section” there is the expression:

1/3 = { { a / 2^b in S_ω | 3a < 2^b } | { a / 2^b in S_ω | 3a > 2^b } }

Which confused me at first, because I thought both the left set and right set were single numbers in the form {X_L|X_R}. 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 / 2^b in S_ω : 3a < 2^b } | { a / 2^b in S_ω : 3a > 2^b } }

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)

[edit] Inconsistent notation

Does anyone know in which reading (or anywhere else) the notation "S_i" 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 S_i notation it's in set S_ω+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 S_a. This is either inconsistent with the literature, or as I suspect, the indices of the S_a are slightly off: that S_ω 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 S_i (i a natural number) as stated in the article. GromXXVII 23:56, 9 November 2007 (UTC)

[edit] 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 Wikipedia: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)