Talk:Surreal number/Archive 2

To Infinity... And Beyond
This is clearly a joking reference to the Toy Story quote... Would someone like to change it to something like "Numbers Approaching Infinity" and "Numbers Beyond Infinity"? Asperger, he&#39;ll know. (talk) 05:17, 22 March 2009 (UTC)
 * Why? It's cute the way it is.--76.167.77.165 (talk) 16:43, 22 March 2009 (UTC)


 * I agree: The Wiki on Surreal Numbers is a monster. Certainly an interesting one but still... Stuff like this, being slightly joking while quite perfectly describing what is meant, makes it a tiny bit nicer to read. (On a less useful note it's probably also a bit in the spirit of On Numbers And Games in certain ways...) —Preceding unsigned comment added by 88.116.101.242 (talk) 01:23, 22 January 2011 (UTC)

Is the Birthday property superfluous in the construction?
...keeping in mind that my question does not refer to the usefulness in understanding and describing the construction, but rather whether or not it is an absolute necessity to the construction. If I understand correctly, the following conditions, combined with techniques of induction and transfinite induction, seem sufficient to describe $$\mathbb{S}$$, the proper class, and ordered field (sort of), of surreal numbers. All references to L, R, xL, etc. are, naturally, sets, not proper classes.

$$\forall L \forall R \{ L|R \} \in \mathbb{S} \iff L \subset \mathbb{S} \land R \subset \mathbb{S} \land \neg \exists x_L \in L [\exists x_R \in R (x_R \le x_L)]$$

$$\forall (x=\{ X_L|X_R \} \in \mathbb{S}) \forall (y=\{ Y_L|Y_R \} \in \mathbb{S}) [x \le y \iff \neg \exists x_L \in X_L (y \le x_L) \land \neg \exists y_R \in Y_R (y_R \le x)] $$

$$\forall x \in \mathbb{S} \forall y \in \mathbb{S} (x=y \iff x \le y \land y \le x)$$

$$\forall (x=\{ X_L|X_R \} \in \mathbb{S}) \forall (y=\{ Y_L|Y_R \} \in \mathbb{S}) [x+y = \{ X_L + y, x + Y_L | X_R + y, x + Y_R \}] $$ (using notation from article)

$$\forall (x=\{ X_L|X_R \} \in \mathbb{S}) \forall (y=\{ Y_L|Y_R \} \in \mathbb{S}) [xy = \{ X_L y + x Y_L - X_L Y_L, X_R y + x Y_R - X_R Y_R | X_L y + x Y_R - X_L Y_R, x Y_L + X_R y - X_R Y_L \}]$$

Am I correct in thinking that the Birthday property is, in fact, superfluous? Or is there something subtle that I'm missing? Again, I fully understand, even if the the property is superfluous, why it is such an important one to include in the article. I just want to make sure that my own understanding is not out of whack.

All responses appreciated. --69.91.95.139 (talk) 18:47, 28 March 2009 (UTC)


 * I think you need birthdays to see intuitively why {1|5} = 2 and not {1|5} = 3. --Michael C. Price talk 21:46, 5 April 2010 (UTC)


 * The birthday property is absolutely critical to the definition of Surreals. The first condition you listed should also include the requirement that the contents of L and R must be older than x = {L|R}, or else the comment further down this page about circular reasoning would be correct. Because the Surreals are commonly introduced by construction, as in this article, that requirement is often not explicitly stated. The birthday property (& the equivalent well-founded pre-order "is older than") appears to be just a convenient tack-on to the theory. But without it, the trans-finite inductions and recursive definitions on which the Surreals rely would not be possible. — Preceding unsigned comment added by 68.102.164.99 (talk) 03:12, 14 June 2012 (UTC)


 * No, that's not correct. E.g., { −1 | +1 } is a perfectly good representative of 0, even though −1 and 1 are both younger than 0.  The point that is not clearly explained in the article is that as the construction proceeds, the equivalence classes grow in a well-defined manner.  That is, once any two numbers are constructed (once any two equivalence classes have members), the two numbers retain the same order relationship as the construction proceeds and adds more members to their equivalence classes.  So equivalence classes are never "complete" at any stage in the construction (they continue to acquire new members), but the order and arithmetic relationships between them, once established, don't change.  Similarly, for any set of surreals, the members of the set will all be constructed by some initial stage of the construction; and although the construction proceeds, the relationships between members of that set will not be changed by further stages in the construction. -- Elphion (talk) 16:07, 26 July 2012 (UTC)

Infinitesimals are impossible
The surreal number system is invalid because the existence of infinitesimals leads to an obvious contradiction:

Let x be an infinitesimal.

By definition, $$x\le\frac{1}{n}$$ for all positive integers n.

Lemma: $$x=\frac{1}{q}$$ for some (infinite) positive integer q.
 * Assume that $$x=\frac{p}{q}$$ with $$p>1$$.
 * Since $$p>1$$, $$\frac{p}{q}>\frac{1}{q}$$.
 * Therefore, $$x>\frac{1}{q}$$.
 * This is a contradiction, so $$x=\frac{1}{q}$$.

This means that, if $$x=\frac{1}{q}$$, $$\frac{x}{2}=\frac{1}{q}$$.

By definition, $$x\le\frac{x}{2}$$.

However, since x is positive, this implies that $$1\le\frac{1}{2}$$, an obvious contradiction.

Therefore, infinitesimal numbers are impossible and the surreal number system is invalid. --75.28.53.84 (talk) 15:42, 15 August 2009 (UTC)


 * The contradiction in your proof of the lemma only gives you $$x\le\frac{1}{q}$$, not $$x=\frac{1}{q}$$.


 * Yes something like that is the reason that infinitesimals were originally thought to be impossible until last century. What you are missing is that in modern versions of infinitesimals, there are two types of natural number, with a distinction between ordinary finite and non standard finite numbers. Use F(n) for standard finite then the infinitesimal is smaller than 1/n for any n with F(n). But it can be equal to a rational or real, e.g. can be p/q for some p, q with !F(q). There the F(n) numbers continue endlessly without ever reaching an end so for instance F(n) -> F(n+1) yet there are natural numbers larger than any of those as well in these theories. Robert Walker (talk) 01:42, 26 July 2012 (UTC)


 * I suggest taking a look at $$\mathbb{R}(X)$$, the field of rational functions over the reals. There's a straightforward way of defining an ordering on this field, and resulting ordered field contains infinitesimals. These infinitesimals are just things like $$\frac{1}{X}$$ and $$\frac{X}{X^3-X+1}$$, so it's relatively easy to see what's going on. --Zundark (talk) 18:19, 15 August 2009 (UTC)

Since all of you are so well-informed, could anybody explain to me how to take the sqaure-root of a surreal number, especially of 2, of omega, and of 1/omega? -Pete- —Preceding unsigned comment added by 95.157.2.158 (talk) 15:37, 21 July 2010 (UTC)
 * See the "Powers of ω" section. (Even before reading that section, I guessed that an obvious candidate for a square root of omega would be {1,2,3,...|ω,ω/2,ω/3,ω/4,...} - and, since numbers like 1/3 cannot be created using finite sets, this could be simplified to {1,2,3,...|ω,ω/2,ω/4,ω/8,...}. Likewise, an obvious candidate for a square root of ε is {ε,2ε,3ε,...|1,1/2,1/4,...}.) Square root of 2 is easy - it's a real number, so it's just {rational numbers of the form A/2^B smaller than square root of 2|rational numbers of the form A/2^B bigger than square root of 2}. - Mike Rosoft (talk) 06:09, 9 February 2014 (UTC)

Circular logic! The explanation is bullshit.
Let me condense the complete explanation of construction (up to the point of failure):
 * 1) We assume it is known what sets (lists of things) and pairs (groupings of two associated things) are.
 * 2) There is something called surreal numbers, which is not explained (yet).
 * 3) There are sets of surreal numbers.
 * 4) Forms are pairs of those sets.
 * 5) One part of the <tt>pair</tt> is called <tt>L</tt>. The other one <tt>R</tt>.
 * 6) A <tt>form</tt> is <tt>numeric</tt>, if
 * 7) * <tt>R</tt> and <tt>L</tt> don’t overlap.
 * 8) * All elements in <tt>R</tt> are bigger than all elements in <tt>L</tt>.
 * 9) There is something called <tt>equivalence classes</tt>, which is not explained (yet).
 * 10) *Breathe… Too many new terms* (A pedagogic blunder.)
 * 11) <tt>Numeric forms</tt> are placed is <tt>equivalence classes</tt>. (Apparently, <tt>equivalence classes</tt> can contain <tt>numeric forms</tt>.)
 * 12) Such <tt>equivalence classes</tt> with <tt>numeric forms</tt> in them, are called <tt>surreal numbers</tt> (the ones mentioned in point 1).

For those who have not noticed it by now: This states, that <tt>surreal numbers</tt> are containers for pairs of sets of surreal numbers (themselves!) again. (Containers = <tt>equivalence classes</tt>, pairs = <tt>forms</tt>, sets = R and L.) Which is circular logic, since <tt>surreal numbers</tt> are only explained, based on themselves. To make things worse, the are defined as complex structures with themselves as their base elements. Something that is a kind of oxymoron.

I’m not saying the theory is wrong. I’m just saying, if it is right, the one explaining it did not get it, was drunk, or does not understand basic logic.

Of course, since a human brain is physically unable to store conflicting concepts, it flat-out rejects this, and hence is unable to read the statements following this, because they build on the aforementioned ones.

Funnily, the next thing in the article, is the “equivalence rule”, which states nothing more, than that two <tt>numeric forms</tt> are equal… and here’s the kicker… if, and only if, they are equal!

Man, there are whole religions which are founded on a smaller amount of circular logic, than one can find in this article! ^^

— 188.100.192.146 (talk) 07:06, 19 August 2010 (UTC)


 * A recursive definition is not the same as circular logic.
 * Circular logic says, for example, that A is true because B is true, and B is true because A is true. A recursive definition defines a term in terms of itself.
 * Two examples of a recursive definition:
 * Factorial:
 * 0! = 1.
 * (n+1)! = (n+1)&middot;n!.
 * Positive integral surreal number:
 * 0 = {Ø|Ø}
 * n = {{n-1}|Ø}
 * --Oz1cz (talk) 13:04, 19 August 2010 (UTC)


 * Another way of looking at it might be thus: Even though we don't know what a surreal number is, we can still create a set of surreal numbers, namely the empty set. And based on the empty set, we can continue to build the other numbers.
 * I can, for example, provide you with a set of fnytus. Here it is: Ø. I don't know what a fnytu is, but I know for certain that the empty set contains no fnytus.
 * The number zero was regarded with some suspicion in the middle ages. How can you talk about having nothing of a type? A blank piece of paper contains a drawing of zero fnytus. And zero elephants look exactly like zero fnytus. The empty set provides a similar trick.
 * --Oz1cz (talk) 13:17, 19 August 2010 (UTC)
 * As for the equivalence class part, I think the article is actually wrong, although I also have a strong feeling that you don't know what an equivalence class is (read the Wikipedia article on the subject). Two things can belong to the same equivalence class without being equal. For example, the relation "is parallel to" defines a set of equivalence classes of straight lines; two straight lines belong to the same equivalence class if they are parallel. But that does not make them equal.
 * As for the equivalence class part, I think the article is actually wrong, although I also have a strong feeling that you don't know what an equivalence class is (read the Wikipedia article on the subject). Two things can belong to the same equivalence class without being equal. For example, the relation "is parallel to" defines a set of equivalence classes of straight lines; two straight lines belong to the same equivalence class if they are parallel. But that does not make them equal.


 * The part of the article which I think is actually wrong is this statement: "each such equivalence class is a surreal number". This statement seems to ignore the difference between two identical surreal numbers and to equal surreal numbers. {0,1|} is equal to {1|}, but they are definitely not identical. The equivalence class is formed by the "is equal to" relation, but that does not make the equivalence class itself a surreal number.


 * I think that should be rewritten.
 * --Oz1cz (talk) 19:28, 19 August 2010 (UTC)


 * You're confusing number with representation. Consider rational numbers:  2/3 and 4/6 are equal; in fact they are identical as rational numbers; but they are different representations of the same number. When we exhibit a number, we exhibit a representation, not the totality of representations, so "2/3 equals 4/6" is shorthand for "2/3 and 4/6 are representations of the same number".  Similarly, { | } and {&minus;1|1} are different representations of the same surreal number; as surreal numbers they are identical.  "Identical" and "equal" are synonyms. -- Elphion (talk) 16:44, 31 May 2013 (UTC)


 * I think that the article itself confuses the representations and their equivalence classes. It says: "Surreal numbers are constructed inductively as equivalence classes of pairs of sets of surreal numbers ...", and "A form is a pair of sets of surreal numbers ..." This would seem to imply that the members of the left and right set of the number form {L|R} were the equivalence classes, rather than the number forms themselves. (I don't think that's possible - only sets, not proper classes, can be members of a set, and the equivalence class of, say, 0={|} is at least as big as the class of ordinal numbers.) - Mike Rosoft (talk) 18:32, 9 February 2014 (UTC)


 * At one level you're right, of course: since the numbers are never *finished* (we're always adding more representations), at each stage, we're only working with the numbers (equivalence classes) as built up so far, and with representations of them that have been built so far.  But this is not really a distinction we need to worry about.  In the definition of a representation as two sets of (numbers|representations|whatever), clearly we are technically taking sets of representations; but the broader issue is that the construction allows us to replace any representation appearing in L or R with an equivalent representation without changing the number defined -- i.e., the specific representations used don't matter, and therefore it's useful to think of the members as *numbers*, any of whose representations will work.  This is the flavor of constructions like this:  you retain a willful ambiguity between the numbers and their representations.  That way, it's easier to show a representation like { 1 | 2 }, rather than to having to pick a specific representation for 1 and for 2 -- because it *doesn't matter* which ones you pick. If backed into a corner, you could unwrap all the notation as a mess of representations, but that only obscures what's going on.  Since the construction is well-defined, it's not necessary to be that precise. -- Elphion (talk) 20:33, 9 February 2014 (UTC)


 * This paper takes the opposite position: it specifically defines 0 as {|}, 1 as {0|}, 2 as {1|}, 1.5 as {1|2} etc., and distinguishes the relations "a≡b" (a and b are identical [as sets]) and "a=b" (a and b are equal [represent the same surreal number]). - Mike Rosoft (talk) 23:00, 9 February 2014 (UTC)

Well, that rather illustrates my point. He starts by being very careful to distinguish between representations (which he calls "surreal numbers") and equivalence classes (defined implicitly through the equivalence relation (x is equivalent to y) iff x <= y and y <= x), and explicitly defines 0, 1, 2, etc. to be specific representations ("numbers" in his terminology). But then in Corollary 4, p. 17, he shows that the notion of equivalence class is well-defined w.r.t. the order relation, and says of this result: "It relieves us -- at least for now -- of the tiresome burden of having to be very careful about which representative of 1 [sic] we use, when writing something like { 1 | } or 1 &isin; A." I.e., he is treating 1 as a collective here, meaning all of the things (or any or the things) that are equivalent to the "number" (representation) he called "1". And this was really inevitable because the surreal numbers are not (in common usage) representations, since the surreals are linearly ordered by <=, while the representations are not. -- Elphion (talk) 00:17, 10 February 2014 (UTC)


 * I don't really object to defining surreal numbers as equivalence classes. I guess one author may define it as an equivalence class (with a caveat that an equivalence class is not a set, but a proper class), another as the representation itself (with a caveat that the relations 'a and b are equal' and 'a and b are identical' are distinct). What I do object is saying (or implying) that in a surreal number form (representation) {L|R} the members of sets L and R are equivalence classes, rather than the forms themselves. (One thing is using shorthands, the other thing is being sloppy even with the definition.)
 * If I were to formally define surreal numbers, I would go along these lines:
 * Let α be an ordinal number.
 * Let S be a set of surreal number forms generated in all generations β, β<α.
 * Then the set of surreal number forms generated in generation α is the set of pairs {L|R}, where L and R are subsets of S, and for no l∈L, r∈R: l>=r.
 * I could specifically define that in generation 0 the sole form {|} is generated (L and R are empty sets), but I could also say that it follows from the definition, since there is no ordinal number less than 0.
 * X is a surreal number form, if and only if there exists an ordinal number α such that X is generated in generation α.
 * Forms a and b represent the same surreal number, if and only if a>=b and b>=a.
 * I am not saying that the Wikipedia article needs to be this formal. But I think that in order for the formal definition to be possible, it is necessary that the members of L and R are the number forms, NOT their equivalence classes. Or am I wrong? Does any author say otherwise? - Mike Rosoft (talk) 18:20, 10 February 2014 (UTC)


 * Defining the forms as L and R sets of incipient equivalence classes would also work; but I agree that using L and R sets of forms is technically cleaner. Tøndering's treatment and the approach you outline above both work fine (though Tøndering's terminology is not well chosen).  But at some point it's important to explain the deliberate ambiguity about whether 1, say, is a number or a form, and why the construction makes the ambiguity work.  For the reality is that for most people working with surreals (once the construction is shown to be well-defined) the numbers are the equivalence classes, not the forms -- and use 1 to mean a class or any of its forms, since the distinction makes little difference.  You need to be able to think of the forms as L and R sets of linearly ordered numbers, even if that's an abuse of notation. (See also my remark above about the birthday property.)


 * And that's the way with all such constructions. You have to be careful until you show that the construction is consistent and well-defined, so that you won't get into trouble if you ignore the distinction between object and representation -- and once you get there, you promptly ignore the distinction between object and representation! -- Elphion (talk) 23:42, 10 February 2014 (UTC)
 * I haven't really studied surreal numbers, so I can't continue this debate. I'll just note that even I have been (of course) using shorthands - e.g. in the above section, I have given two equivalent representations of a square root of ω, and the article gives a third (using the formula for ω^0.5): {0, all positive real numbers|ω*(all positive real numbers)}. And likewise, I don't really care which one of the equivalent representations of real numbers I choose, as long as I know that they behave like real numbers - the article gives both a full representation (for 1/3) and a simplified one (for π). - Mike Rosoft (talk) 05:30, 11 February 2014 (UTC)

Surreal Numbers (book) should not redirect here
"Surreal Numbers" is the title of an important work of fiction by computer scientist Donald Knuth. http://www.amazon.com/Surreal-Numbers-Donald-Knuth/dp/0201038129 I'm sure there are other books titled 'surreal numbers' that belong as subsets of the topic, but not that one. Uh, at least, a disambiguation page, no? Redirect isn't appropriate. Sorry if i'm barking up the wrong tree. Best, baxrob (talk) 04:29, 28 November 2010 (UTC)


 * Knuth's book is the one that popularized surreal numbers, and it is mentioned in the body of this article and in the references. Since it does not have its own article (yet), a redirect to this one seems appropriate. — Loadmaster (talk) 17:47, 3 February 2011 (UTC)

"biggest"/"smallest" Surreal Number?
I'm thinking about Surreals since weeks now and stumbled across something that I wonder about...

If $$x=\{X_l | X_r\}$$, $$x$$ is a Surreal Number ($$\in \mathbb{S}$$) and $$X_l$$ and $$X_r$$ are sets of other Surreal Numbers, smaller and bigger than $$x$$, respectively, then one of the allowed sets seems to be the set of all Surreal Numbers $$\mathbb{S}$$. Now, what if I do this?


 * $$x=\{\mathbb{S}|\}$$
 * or
 * $$x=\{|\mathbb{S}\}$$

Wouldn't those be outside the Surreals? While still being inside them at the same time? Is a number like that mathematically meaningful or would that not work? It seems to me that this number would infinitely grow on its own, based on all the Surreals discovered so far. It is always one step ahead, but as it's a proper Surreal, it has to grow a step further to statisfy itself... —Preceding unsigned comment added by 88.116.101.242 (talk) 12:13, 30 January 2011 (UTC)


 * That construction fails because you can't take "the set of all Surreal Numbers" -- it's too big to be a set; it's a proper class. This is pretty much the same reason you can't take "the set of all sets" and use it to form a contradiction.  Joule36e5 (talk) 10:00, 7 February 2011 (UTC)


 * Ok, so that's the big difference between a proper class and a set. I read about that but admitedly didn't read up what makes the difference. I suspected this to be the case, though, after sleeping over it :) —Preceding unsigned comment added by 88.116.101.242 (talk) 19:46, 8 February 2011 (UTC)

Such notations are indeed of interest in the theory of surreal numbers; they are called gaps. Let the class of surreal numbers S=L∪R, and for every l∈L,r∈R: l<r; then {L|R} is a gap [meaning that by definition, there is no surreal number between L and R]. Apart from your examples, some interesting gaps are: (Of course, such notations are not surreal numbers.) - Mike Rosoft (talk) 21:11, 18 February 2014 (UTC)
 * L={l: l is negative, or zero, or a finite positive number [i.e. there exists real number x:x>l]}; R={r: r is a positive infinite number}.
 * L={l: l is negative, or zero, or a positive infinitesimal number [i.e. for every positive real number x: x>l]}; R={r: r is a non-infinitesimal positive number }.

Error in the "order" section
It appears that in the section explaining order, the > signs and the text explanation of what is smaller and what is larger do not match. I have no real understanding of this subject so i don't want to edit the text but it seems pretty obvious that something got confused there. —Preceding unsigned comment added by 87.215.198.122 (talk) 15:13, 2 February 2011 (UTC)


 * Are we talking about this text:
 * Given numeric forms x = { XL | XR } and y = { YL | YR }, x ≤ y if and only if:<ul><li>there is no $$\scriptstyle x_L \in X_L$$ such that y ≤ $$\scriptstyle x_L$$ (every element in the left part of x is smaller than every element in the whole y), and<li>there is no $$\scriptstyle y_R \in Y_R$$ such that $$\scriptstyle y_R$$ ≤ x (every element in the right part of y is bigger than every element in the whole x).</ul>A comparison y ≤ c between a form y and a surreal number c is performed by choosing a form z from the equivalence class c and evaluating y ≤ z; and likewise for c ≤ x and for comparison b ≤ c between two surreal numbers.
 * It does look okay to me. Could you be more specific? —Tobias Bergemann (talk) 13:02, 3 February 2011 (UTC)

Exponentiation
The article uses notation like $$\omega^\omega$$, and also defines $$2^x$$, and there's a function $$e^x$$ defined in the talk archive (and evidently in Gonshor's book). There's no mention of whether there's a general concept of exponentiation for arbitrary (positive base) surreal numbers, and if so, which (if any) of these three are consistent with it. I've convinced myself that there are problems with using any of these three as the basis for general exponentiation, but my argument is original research, and I don't have the relevant books. Joule36e5 (talk) 10:47, 7 February 2011 (UTC)


 * Yes, I just asked about this on MathOverflow and Philip Ehrlich says Gonshor's definition is the one everyone uses. Seems this definition was added by User:Michael K. Edwards.  He's on wikibreak, unfortunately, but I'll post on his talk page anyway... Sniffnoy (talk) 07:21, 21 February 2012 (UTC)

For the record, the passage in question was removed in. Melchoir (talk) 08:40, 23 February 2016 (UTC)


 * Thanks for correcting that. It was an attempt to obtain substantially the same results as Gonshor's strategy without requiring a Taylor series, which is problematic for the "generational" approach to proving the consistency of the arithmetic definitions.  At some point it morphed from a paraphrase/simplification to original research, and ran away with me at about the same time that it became apparent that wikibreak was indicated.  Apologies!  Michael K. Edwards (talk) 22:30, 1 July 2016 (UTC)


 * Oh, hey, wasn't expecting to see you here! Glad to have that confirmed, thanks! Sniffnoy (talk) 14:32, 2 July 2016 (UTC)

Linear continuum?
The introduction claims the surreals form an arithmetic continuum and links to linear continuum. The definition of linear continuum, as stated, requires least upper bounds. Unless I'm missing something, Conway's Simplicity Theorem contradicts the existence of least upper bounds. Is arithmetic continuum meant to refer to something distinct from linear continuum, or is that just a mistake?--Antendren (talk) 02:43, 20 June 2011 (UTC)


 * I think that "arithmetic continuum" is correct for surreal numbers, and that the hyperlink to "linear continuum" is a mistake. http://www.ohio.edu/people/ehrlich/Unification.pdf Com3t (talk) 22:19, 22 November 2016 (UTC)


 * Ehrlich seems to use a definition of linear continuum that doesn't include the least upper bound property; he also seems to use "linear continuum" and "arithmetic continuum" interchangeably. The 'completeness' property that there's a number between any two subsets (if one is > the other) is also different from the order 'completeness' discussed in other articles (which is about LUBs too).


 * The link is wrong in any case and I don't think an article for Ehrlich's concept exists in Wikipedia right now; it's probably not important enough to create one. Also, until somebody adds something to this article about the surreals as a continuum, I don't think it should be mentioned in the abstract at all. I replaced it with 'totally ordered class' which is true though it leaves out the more interesting property. Lhmathies (talk) 15:25, 1 December 2016 (UTC)

Surreals and Transfinites
What is the relationship between surreals and Cantor's transfinites? And why does this article not mention it? Ethan Mitchell (talk) 19:53, 14 August 2011 (UTC)
 * OK, sorry...the surreals contain the transfinite ordinals? It still seems like this point could use expansion. Ethan Mitchell (talk) 19:56, 14 August 2011 (UTC)

Isomorphism of maximal surreals and maximal hyperreals
It would be helpful to include a more detailed discussion of Philip Ehrlich's recent paper. Tkuvho (talk) 16:26, 29 January 2012 (UTC)

Not multiple issues
I don't see anything on the talk page to suggest that the article has multiple issues. The article seems fine to me except for a possible lack of sufficient references. Therefore I've changed { { multiple issues | refimprove } } to { { refimprove } }.--75.83.64.6 (talk) 15:46, 20 January 2013 (UTC)
 * Sorry, I should have caught that when I removed the other flags.DavidHobby (talk) 19:31, 20 January 2013 (UTC)

Order and tone of presentation
This article really could use some editing... it would be nice to remove the "cute" whimsical references to things by Conway or Knuth and focus more on a "dry" presentation of the actual topic of the article, that is, Surreal Numbers themselves. At the very end of the article finally some axioms to guarantee Surreals get listed. Why not put that at the top of the article where it belongs just like every other math article does? (They all start out with a formal definition of their object). As it is now, trying to read this article, it is quite repetitive, and confusing, and there is no reason to spend 80% of the article on PARTICULAR types of constructions of Surreals as opposed to GENERAL conditions for them. Clearly general OUGHT to be the focus. — Preceding unsigned comment added by 71.201.95.139 (talk) 13:05, 31 May 2013 (UTC)


 * I don't entirely agree. The article currently does sprawl a bit, but the order of presentation is not bad.  A good case could be made for separating the details of construction into a separate article (as we do for Construction of the real numbers).  I don't see any particularly "cute" or "whimsical" references from Conway or Knuth.  As for taking a "dry" tone, I categorically disagree:  the vast majority of the mathematics articles in WP are more or less impenetrable; they are not a model we should be promoting.  The "Overview" is the right place to start; the axiomatic approach is not illuminating unless you already have an intuitive notion of what is going on. -- Elphion (talk) 16:30, 31 May 2013 (UTC)

Comparison with hyperreals
Concerning some recent edits: Until a few years ago common wisdom had it that the hyperreals are a proper subfield of the surreals. However, recent research showed this not to be the case. If you are unfamiliar with these developments, I can provide some references (some of them are already cited in the page). Tkuvho (talk) 13:29, 3 November 2013 (UTC)
 * Bajnok's use of the definite article as in "the largest ordered field" is misleading and casts doubt on his reliability. Tkuvho (talk) 14:24, 3 November 2013 (UTC)

Tkuvho-- Sorry I reverted your recent edit. I admit to not reading the article too carefully, but it seems that the same point is made several places, that the maximal class of hyperreals is isomorphic to the maximal class of surreals. I find the phrasing in your version confusing: "both the hyperreal numbers and the surreal numbers are the largest possible ordered field; all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, are subfields of the surreals ;"  The naive question would be "If the hyperreals are equal to the surreals as the largest possible ordered field, then why does it also say they're a subfield?" If the article misleads casual users, it's poorly written. The best quick fix I had was to remove "both the hyperreal numbers and". DavidHobby (talk) 16:25, 3 November 2013 (UTC)
 * I agree that the introduction needs to be rewritten. As I already mentioned, it can no longer be claimed that the surreals are the largest ordered field. Namely, the definite article is inappropriate.  Combined with the other comment, it implies that the hyperreals are necessarily a proper subfield.  This is not the case.  What is the case is that the surreals are one possible way of specifying a maximal field. Tkuvho (talk) 16:44, 3 November 2013 (UTC)

Can I ask a few questions here, Tkuvho? Because what's being said doesn't really make sense to me. First off, as I understand it, the hyperreals, at least by the usual construction, are set-sized, and hence cannot possibly be a largest ordered field. Of course, by the usual construction, it isn't really correct to speak of "the" hyperreals (despite common parlance), since they might not be isomorphic depending on the ultrafilter. So in order for what you're saying to make sense, I would have to infer that A. there is a more abstract notion of a "hyperreal field" that does not depend on the construction, and B. that there exists one of these hyperreal fields that is a proper class and is a largest ordered field. Is this correct? Is this what you are claiming? Because this would make sense -- some hyperreal fields are proper subfields of the surreals, others are not.

Secondly, it's worth pointing out here that what we care about is maximal fields -- not ways of specifying them. If this maximal hyperreal field that you talk about is in fact isomorphic to the surreal numbers, as I would expect, then regardless of whether it is a different construction it is not a different field. I would certainly expect any two maximal fields to be isomorphic (probably by some sort of zig-zag argument), justifying the use of the definite article, but I must admit I haven't actually seen this proved myself. Are you claiming that this maximal hyperreal field is, in fact, not isomorphic to the surreal numbers? Or are you just claiming that is not known to be isomorphic? Or are you just claiming that it's a different construction? If it's just the last of these, it's not particularly notable.

Actually, I should probably say -- there does seem to me to be something misleading in that part of the article at the moment (which I should probably just go and fix). Namely, it's not really correct to say that all ordered fields are subfields of the surreal numbers, but rather that they can be embedded in the surreal numbers, or that they can be realized as subfields of the surreal numbers. Because -- correct me if I'm wrong here -- the embedding is, in general, non-canonical. Obviously it's unique for Q and its algebraic extensions, but in general it's not (the surreal numbers contain infinitely many subfields isomorphic as ordered fields to the subfield Q[&omega;]). Sniffnoy (talk) 02:03, 4 November 2013 (UTC)
 * I believe a lot of things in mathematics are done up to isomorphism, so it's fine to say "X is a subfield of Y" instead of "X is isomorphic to a subfield of Y". If someone feels like making the distinction in the article, they can go ahead and do the edits.  Although writing "isomorphic to" too often might make the article less readable.DavidHobby (talk) 13:07, 4 November 2013 (UTC)


 * A hyperreal field constructed by a limit ultrapower is a proper class field isomorphic (non-canonically, as far as I can tell) to a maximal hyperreal field, as explained in the paper by Ehrlich already referenced in this page. Any use of the definite article in this context should be taken with a grain of salt because a lot depends on the details of the axiomatic background you are using.  For example, do you assume the axiom of global choice, etc.  Tkuvho (talk) 13:11, 4 November 2013 (UTC)


 * And these are isomorphic to the surreals? Or what?  Or is that dependent on the axiomatic background?  Yuck.  For what it's worth, I was implicitly assuming global choice, since I was assuming we were working in NBG (or at least, something NBG-like).  I'll admit, my knowledge of hyperreals pretty small. Sniffnoy (talk) 01:44, 5 November 2013 (UTC)


 * Well, you need to be careful with that. It's fine to talk up to isomorphism when you're just considering things in and of themselves (like I do above with the surreals and hyperreals).  You need to be more careful when you're talking about e.g. subobjects.  To say "X is a subobject of Y" rather than "X is isomorphic to a subobject of Y" implies that the embedding is canonical somehow.  So it's definitely OK to say "Q is a subfield of the surreals," and it might be fine to say "R is a subfield of the surreals" since I think there's a nice construction (it's just everything with birthday at most &omega; except for &plusmn;&omega;, right?), but I don't think it's OK to say "Any ordered field is a subfield of the surreals" since that implies that the general construction is canonical, which I don't think it is.  Sniffnoy (talk) 01:44, 5 November 2013 (UTC)


 * I agree. Furthermore in the absence of global choice, there is no proof that the limit ultrapower of hyperreals is isomorphic to the surreals, so it is not quite correct to say that every ordered field has an isomorphic copy in the surreals.  This is simply not known.  The page should be edited accordingly. Tkuvho (talk) 17:55, 12 November 2013 (UTC)


 * Hm, OK. I feel like global choice is kind of a default assumption though (since it's in NBG), so while certainly this should be mentioned as you say, I don't think it's a problem if, say, it's written with the point of view that "normally this is what happens, but if you don't assume global choice then it might not" rather than "normally it's hard to say what happens, but if we assume global choice than we can." Sniffnoy (talk) 22:40, 12 November 2013 (UTC)


 * According to our pages on both axiom of global choice and NBG, global choice is not a default assumption by any means. Therefore the claim that the surreals contain all ordered fields is misleading, since this does not apply to the limit ultrapower of the hyperreals unless one makes additional assumptions. Tkuvho (talk) 14:20, 13 November 2013 (UTC)


 * But global choice is a consequence of limitation of size, which is one of the axioms of NBG; and NBG is basically the usual way of extending ZFC to allow proper classes. Am I missing something here? Sniffnoy (talk) 22:55, 13 November 2013 (UTC)


 * At NBG I find the following comment: Limitation of Size cannot be found in Mendelson (1997) NBG. In its place, we find the usual axiom of choice for sets, and the following form of the axiom schema of replacement: if the class F is a function whose domain is a set, the range of F is also a set. Tkuvho (talk) 13:35, 14 November 2013 (UTC)


 * I'm not sure how to respond to that. OK, so this one particular book doesn't include it?  It still is a part of standard NBG.  And, well, my experience as a mathematician is that (outside of set theory and logic, at least, where things may be more variable, and more specification in advance is needed) NBG is the default setting for working with proper classes unless something else is specified.  I don't think the lone example of Mendelson really changes that, but I'm not sure how I could really cite something for that. Sniffnoy (talk) 01:50, 15 November 2013 (UTC)
 * At NBG we find: Mendelson, Elliott, (1997), An Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall. ISBN 0-412-80830-7. Pp. 225–86 contain the classic textbook treatment of NBG, showing how it does what we expect of set theory, by grounding relations, order theory, ordinal numbers, transfinite numbers, etc. (emphasis mine) Tkuvho (talk) 08:25, 15 November 2013 (UTC)
 * At google scholar mendelson's book is cited 2486 times. Tkuvho (talk) 08:28, 15 November 2013 (UTC)
 * Tkuvho, do you have any evidence that people ever work with the surreals without assuming global choice? JoshuaZ (talk) 15:28, 18 November 2013 (UTC)

Intrusive Editorial Opinion
Somebody has introduced, at the top of the article, the opinion '''This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (April 2012)'''

in a big colourful box. 1.) I disagree. I think the footnoting of Knuth's book alone would be sufficient citation, and as it is the explication is self-evidently all the citation anyone could wish for.

2.) Even if this opinion had any chance of being even faintly respectable, what business does it have intruding in full loudness and colour at the top? 2.a.) While we all value the good work done by Wiki-world's contributors and editors, I think they sometimes get a wee bit full of themselves -- and I think this intrusion is a good example of that.

DavidLJ (talk) 14:58, 19 November 2013 (UTC)


 * The "big colourful box" is simply the WP machinery that is automatically produced for the "needs more references" tag -- it's not the tagger trying to be flamboyant. It was designed to be hard to miss so that both users and editors would be alerted to the judgment that the article does not do a good job of indicating where the information is coming from.  Usually this is not a claim that the information is wrong or biased, just that the sources aren't clearly spelled out.  One can disagree with that judgment, but the tag means that someone did not find the article transparent that way, so it's worth considering if improvements can be made.  I did not tag the article, but I agree with the tagger that the sources are not clearly stated.  The WP ideal is that each section, at least, is supported by a source.  For this article that's probably overkill, but a section near the top discussing where this stuff comes from would not be amiss. -- Elphion (talk) 16:30, 19 November 2013 (UTC)

Non-reliable source referenced.
According to WP:Circular, Wikipedia is not a reliable source. This article should instead reference the source that Von Neumann–Bernays–Gödel set theory used to source the same information. Blackbombchu (talk) 04:03, 2 December 2013 (UTC)

Disputed information
I think the sentence the surreal numbers are the largest possible ordered field is quite likely wrong. I think saying it's the largest possible ordered field is another way of saying no class that's an ordered field can be larger than that class. It's tempting to say it is the largest one because a class can't contain a proper class or anything that's defined in terms of one. Therefore, you can't fill in the holes of the surreal number system defining the new numbers in those holes in terms of the surreal number system because those numbers would be defined in terms of the surreal number system which is a proper class and a class can't contain anything defined in terms of a proper class. However, maybe one can find a larger ordered field by defining another 2 operations and another ordering relation on the surreal number system just like one can find a well ordered class that's larger than the class of all ordinal numbers by taking an ordering relation that's the same as the standard one except that the ordinal number 0 exceeds all other ordinal numbers using that relation. Furthermore, anything that can be proven in Von Neumann–Bernays–Gödel set theory can be proven in ZFC but I don't think the statement There is no larger totally ordered class that follows the conditions of a field can be expressed in ZFC because there is no class of all classes to quantify over and if you allow quantification over objects with arbitrary properties, you could get a paradoxical statement like All statements that don't universally quantify over themselves are statements in ZFC. Blackbombchu (talk) 13:49, 22 May 2014 (UTC)
 * It has been proven mathematically that every ordered field whose collection of elements forms a class in Von Neumann–Bernays–Gödel set theory is a subfield of the surreals . Do you have a concrete reason for objecting to this statement? Because what you write above looks like vaguely formulated intuition rather than mathematics, and that's a dangerous thing to rely on in this sort of area. —David Eppstein (talk) 03:59, 25 May 2014 (UTC)
 * I think when people say an ordered field is larger than another ordered field, they don't just mean there exists an injection from the latter to the former and not vice versa but mean that in addition to that, such an injection preserves the operations and the relation that define them as an ordered field. An ordered field is not a class but a combination of a class and operations and a relation on it so a field can be larger than another field without its class being larger than the other class, but the article still says the surreal number system is a proper class instead of saying it's ring operations are defined on a proper class. I think the article meant to say that assuming the axiom of choice, all ordered fields are isomorphic to a subfield of the surreal number system, not that they're equal to one. Is there a proper subfield of the surreal number system that's isomorphic to the surreal number system which explains it being the largest ordered field, unlike the well-ordered class of all ordinal numbers which can be made into a larger well-ordered class by changing the relation? Blackbombchu (talk) 05:05, 25 May 2014 (UTC)
 * You are making basic true-but-trivial statements as if they added something new to the discussion, as if they contradicted anything that anyone else was saying when they don't, and also not reading what David actually said. You are also ignoring conventions of mathematical language in your statements and how you read other's statements.  First off, obviously we are just not talking about cardinality; we are talking about ordered fields, it would be silly to just talk about cardinality, as if that were what were relevant.  Yes, when people say "A is larger than B", they don't just mean that there is some set-theoretic injection of B into A (and possibly excluding the reverse), they mean there is an embedding, an injection that preserves the relevant structures.  (And possibly excluding the reverse.)  (Note, by the way, that in mathematics, we frequently just use "injection" to mean "embedding", i.e. structure-preserving injection, because if it didn't preserve structure, it wouldn't be worth talking about!)  This was never in question.  David explicitly said, and any source or textbook on the matter will tell you, that any ordered field embeds into the surreals as an ordered field -- not just as a set or as a field, but as an ordered field.  The thing you are trying to question is simply settled mathematics; you are managing to suggest it is not only by misreading the things that state it.  Also, saying that the surreals "have their operations defined on a class" instead of that they "are a class" is pedantry beyond pedantry.  Apparently you've learned enough algebra to understand that it's the operations that are important, not the underlying set; but not enough algebra to understand that we usually elide this and refer to the structure as the underlying set when this is unambiguous, because it's easier.  I'm not sure how this happened.  If this is really your objection, you'll have to object to nearly every algebra-related article on this website.  Generally, anyone who also knows enough to understand what's meant by the statement "the surreals are the largest ordered field" should not be confused by the statement "the surreals are a proper class".  Your distinction between isomorphism and equality is also irrelevant here.  Perhaps David should have said "is isomorphic to" rather than "is", because the isomorphism might not be canonical, but this is not relevant to anything.  Demanding actual equality is completely silly in context; actual equality is both impossible and irrelevant -- which, again, you ought to know if you've learned enough algebra to raise the objection in the first place.  The only real question is isomorphism vs. canonical isomorphism -- but none of this is relevant to the original point!  Also, we don't need to explicitly say "assuming the axiom of choice", as it's a standard assumption; it's the C in ZFC, after all.  (Although, as Tkuvho has pointed out above, we actually need the axiom of global choice -- which is also a standard assumption; if you're working with these kinds of things, generally it's in NBG.)
 * As to your actual concrete question about whether the surreals have subfields isomorphic (as ordered fields) to the whole thing, my answer to that is "I think the answer is yes, but you should probably consult an actual source". (The answer has to be yes for the surreals really to be "the" largest ordered field, as I've discussed below; otherwise calling it that is a misnomer, but the fix would not be what you are suggesting.)  But really, you just seem out of your depth here.  You have one or two good points but mostly you are raising irrelevant, trivial, or pedantic objections, and it's a real pain to deal with the latter to get to the former.  I think you should learn the relevant subject matter a bit better before continuing this. Sniffnoy (talk) 15:14, 25 May 2014 (UTC)
 * Blackbombchu, you appear to be a bit confused about several issues, especially regarding proper classes.
 * Firstly, any statement purely about sets that can be proven in NBG can be proven in ZFC; this is what we mean when we say NBG is a "conservative extension" of ZFC. Obviously NBG can make statements about proper classes that cannot even be talked about in ZFC; otherwise, there would be no point in having NBG in the first place.
 * Secondly, your statement "a class can't contain a proper class or anything that's defined in terms of one" is a bit confused. Regarding the first part, no class can have a proper class as an element, but of course a class can have a proper class as a subclass, which is what is relevant here.  Regarding the second part, it is not the case that "a class can't contain anything that's defined in terms of one".  Rather, it is just the case that NBG's axiom of comprehension not guarantee the existence of a class in such a case; it might exist anyway.  For instance, let us take the following property of a set S: "S is an element of all classes C that satisfy the following three properties: 1. the null set is in C; 2. for any x in C, x&cup;{x} is in C; 3. C is closed under union".  Now, I may want to consider the class of all sets S satisfying this property, but, as you point out, because I have quantified over classes in the definition of this property, NBG's axiom of comprehension does not guarantee its existence.  Nonetheless, this class exists anyway, because it is precisely the class of all Von Neumann ordinals, which can also be defined without quantifying over proper classes.  Thus, if you try to make a definition that quantifies oper proper classes, the result might exist; it just isn't guaranteed to.
 * Thirdly, it's not a question whether you can extend the surreals -- you can. For instance, if S denotes the surreals, one can easily make S(x) into an ordered field (with of course the caveat that the result will be a proper class).  Nonetheless, this doesn't necessarily invalidate the statement that "The surreals are the largest ordered field".
 * Let's ask the question -- what do people mean when they say S is the largest ordered field? I can think of at least two possible answers:
 * 1. All (actual, set-sized) ordered fields embed in S. This statement is true.
 * 2. All (actual, set-sized) ordered fields embed in S, and it is (up to isomorphism) the unique ordered field with this property. I'm less than clear on the truth of this statement -- but note that being able to extend S does not immediately invalidate it!  Extensions of a field can be isomorphic to the original field; if every ordered extension of S is in fact isomorphic to S as an ordered field, then the existence of extensions does not invalidate this.
 * Of course, if only (1) and not (2) is true, then it is certainly a misnomer to say that the surreals are "the" largest ordered field, when there can be others as well. Nonetheless, this appears to be standard.  I suspect this means (2) is true, but if not, well, we can deal with that some other way.  (For instance, "All ordered fields embed in the surreals, so the surreals are often referred to as 'the largest ordered field', although there are other (class-sized) ordered fields with this property'.)  Still, I don't actually know the answer to that question.  I'd consult an expert -- or a textbook.
 * Really, as a whole, your comment seems ill-informed. Your comment has the form "This page states the following theorem.  However, maybe there's some way the theorem could be false, possibly along the following lines?  Maybe the people who wrote this page just didn't know what they were talking about?"  And, sure, some real junk ends up on Wikipedia, and many claimed theorems are false.  That said, you should really do a bit more research before marking something disputed like this -- you aren't disputing it with "This is false" or even "This is not proven", you're disputing it with "Here's a general idea of why maybe it might be false according to me".  The statement you want to dispute has a source listed; perhaps you could look it (or some other source on the matter) up.  You are seeing the statement on Wikipedia, but Wikipedia didn't pull it from nowhere, and you are giving the mathematicians behind this theorem very little credit -- saying that, sure, they say they've proved this theorem, but it doesn't seem right to me, so even though I have neither a source nor a proof of my own to the contrary, it's in dispute!  And as I've pointed out above, you seem to be pretty confused about some basic matters regarding both the question, and proper classes in general; you don't seem to have "done your homework", as they say.  I think you should make sure you have a better idea what you're talking about before marking something disputed. Sniffnoy (talk) 04:55, 25 May 2014 (UTC)


 * Sniffnoy: It is true that the term "largest" is misleading. The standard term for an object in a category in which any other object of the category embeds is "universal", while maximal often refers to having no strict upper bound, and thus here, to having no proper extension. (for instance the real numbers form a maximal archimedean ordered group/field, hahn series fields form maximal but non universal in general valued fields of given residue field and value group)
 * Here No is just "a universal" ordered field (in NBG, even for ordered Fields), since as you pointed out, one can form a proper non isomorphic extension No(X) of No (where X is taken to be above every surreal number), which is universal for it is an extension of No, larger for it is a proper extension, and non isomorphic to it for it has countable cofinality. (alternatively, it admits no square root of X for instance, but the former argument also works when considering a real closure of No(X)). Vincent Bagayoko (talk) 22:38, 13 December 2017 (UTC)


 * That's a good point! Somehow I didn't notice No(X) isn't isomorphic to No even though it clearly isn't.  That really does seem to knock down the claim that it's the "largest ordered field" in any meaningful sense. Sniffnoy (talk) 06:52, 14 December 2017 (UTC)


 * How does this work? If No is a universal ordered field then either there should be an embedding of No(X) in No, or No(X) isn't a Field, or I'm misunderstanding something else. Lhmathies (talk) 10:10, 15 December 2017 (UTC)


 * Thanks for the change Sniffnoy. This mistake is enduring because there is a sketch of a proof that No is the unique universal ordered Field in some edition of ONAG.Vincent Bagayoko (talk) 12:01, 15 December 2017 (UTC)


 * Lhmathies: No(X) does embed in No but not isomorphically (not even cofinaly).Vincent Bagayoko (talk) 12:01, 15 December 2017 (UTC)


 * I think I see how to do that... but is it possible to extend the exponential function to No(X) or might No (still) be the largest elementary extension of the real exponential field?Lhmathies (talk) 14:12, 15 December 2017 (UTC)


 * There is no exponential satisfying exp(x) > x on No(X) because otherwise exp(X^2) would be greater than any exp(nX) and thus than any X^n. I don't think No is maximal among elementary extensions of the real exponential field but I am not sure (it is not easy to find an extension small enough for it to exist as a class and big enough for it to be stable under exponentiation). At least it isn't in any Ackermann set theory which allows to define the Cauchy-completion of No which is a non isomorphic elementary extension of the ordered exponential field structure of No. However, this result cannot be transferred to NBG so the question remains open. Even if my intuition is true, No might be the unique universal elementary extension of the real exponential field (but is it universal at all?). I don't think I can say much more on those problems which seem difficult.Vincent Bagayoko (talk) 18:29, 15 December 2017 (UTC)


 * In the "Gaps and Continuity" section of the main article I cited a paper showing that the Dedekind completion of 𝐍𝐨 is Cauchy-complete as well -- is that mis-stated or just not valid in NBG? (If not in NBG it should be noted). As far as I can see, exp cannot be extended to 𝐍𝐨𝕯 because by monotonicity we would have to have exp(∞) = ∞. Lhmathies (talk) 08:21, 16 December 2017 (UTC)


 * What they call the Dedekind completion here is a special type of Dedekind completion where cuts are non-empty, thus excluding the final cut and the initial cut, so there is no relative infinite. This is not to be confused with the Cauchy completion of a an ordered field, whereby only a certain type of cuts is allowed, so that the resulting linear order can be equipped with extensions of the field operations. The Cauchy completion is another dense linear extension, which is maximal as a dense ordered field extension, and which embeds naturally as a linear order in the special Dedekind completion of the underlying order structure. The special Dedekind extension cannot said to be Cauchy-complete since there is no adequate notion of Cauchy sequence in it. As a sidenote, what Conway defines as a gap is not necessarly a cut: for instance Conway defines (]-∞;0],]0;+∞[) as a gap (denoted 1 / +∞ I believe), but this is not a Dedekind cut / section, and it would be infinitesimally close to 0 with respect to No (or rather it would be its successor) if the order of inclusion was considered.
 * In Ackermann set theory, you can see that No does not surject onto its Cauchy completion, by a diagonal argument similar to Cantor's using normal form instead of dyadic expansion. This contradicts the axiom of limited size in NBG, which proves that this Cauchy completion does not exist in NBG. As for defining the exponential, the details should be worked out but I think the following works:
 * Notice that exp is Cauchy-continuous (it is locally uniformly continuous) on No, so it extends uniquely to the Cauchy completion by exp(lim u) := lim exp o u. This is the same for every restricted analytic function, and we can see that the extensions satisfy the axioms of the complete theory Th_an of the real field with restricted analytic functions given in, while one can see that the extension of exp satisfies the axioms of the axioms given in the same paper which turn Th_an into an axiomatisation of the theory of real exponential field with restricted analytic functions. So in particular, the thus equipped Cauchy completion of No is an elementary extension of the real exponential field, which is strictly larger than No.
 * If I recall well, the authors of the paper you cite were careful not to speak of the Cauchy completion as a class in NBG. You can still talk about it in NBG - just like one may want to consider the class of all sets as a big set -; it is then a device to use set-like (here class-like) folklore rather than formulas in NBG. Vincent Bagayoko (talk) 12:13, 16 December 2017 (UTC)
 * So there are 'traditional' Dedekind cuts where the right set (only) may have a minimum which then represents that minimum in the completion, so to speak, and there are Conway gaps where either side may have an extremum which is different from the gap in the completion -- so the main article as it stands is not strictly correct. Not to speak of the 'Dedekind representations' that Rubinstein-Salzedo and Swaminathan use where the representation of a number simply omits it from both sides.
 * If I read them correctly, a Cauchy sequence either approaches a number or a gap of the form ∑α∈𝐍𝐨rαωaα where the aα are cofinal in 𝐍𝐨- (their type Ia) -- and the truncations of such a form constitute a Cauchy sequence that approaches it, so the collection of such forms is exactly the Cauchy completion of 𝐍𝐨 if it fits in your set theory. (And the diagonal argument is obvious from this).
 * Sorry for harping on, I'm a bit out of my depth here but I want to make sure the article is not wrong. Lhmathies(talk) 11:00, 19 December 2017 (UTC)
 * No problem Lhmathies. Yes, except that the exponent sequence rather needs to be coinitial in No because here omega is infinite. From there it is not trivial to justify that the class of bounded normal forms plus those coinitial normal forms is a field, but it is. As for the article, I think it is enough to say that Conway considers gaps which do not necessarily correspond to cuts.Vincent Bagayoko (talk) 15:11, 19 December 2017 (UTC)
 * Wait, if we want to talk about Cauchy-completeness here, surely sequences are not sufficient? Shouldn't we be using filters?  Or are we just deciding by definition to stick just to sequences here? Sniffnoy (talk) 01:09, 20 December 2017 (UTC)
 * I meant generalized sequences. Just like some topologies are sequential in that open sets can be defined in terms of the convergence of regular sequences, the natural uniform structure on an ordered field F (and the order topology it induces) is cof(F)-sequential (where cof(F) denotes the cofinality of F) in that entourages and neighborhoods can be defined in terms of cof(F)-sequences. For such uniform structures, the notion of Cauchy filter can be replaced by that of a Cauchy sequence (and indeed every uniform notion has a sequential equivalent), so Cauchy-completeness and sequential Cauchy-completeness coïncide, and a completion can be given by considering the ring of Cauchy cof(F)-sequences in the field and taking the quotient by the ideal of sequences which tend to zero. Here with surreals, Ord plays the role of cof(F), and a similar diagonal argument can be applied to the quotient in terms of Cauchy sequences to see that it is not a class in NBG. Vincent Bagayoko (talk) 08:13, 20 December 2017 (UTC)
 * Oh interesting, didn't know that. Thanks! Sniffnoy (talk) 09:07, 20 December 2017 (UTC)

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

I think the conditional "and that player loses" is not necessary, but am new to the subject. Can someone review this? 63.99.16.220 (talk) 23:38, 8 January 2015 (UTC)


 * It should be kept. Otherwise there's nothing specifying which player wins.  Of course, the convention that you lose if you can't move is standard in combinatorial game theory, but since this part of the article is exactly explaining the basics of combinatorial game theory, it would not make sense to assume that readers would be familiar with that convention.  (The condition where you win if you can't move is also studied; it's known as "Misère".)  Sniffnoy (talk) 20:53, 10 January 2015 (UTC)

Infinity
In the section currently titled "... And Beyond", this definition is given: x − Y = { x − y | y in Y } Unfortunately, this seems to be ambiguous as to whether the right hand side is a surreal number or set builder notation. After rereading it several times I'm still not sure what is intended. A few words should be added to clarify this.

Also I think the two sections titled "To Infinity ..." and "... And Beyond" should be renamed. The titles should directly describe the contents of the section, not be cute allusions to an irrelevant movie. Mnudelman (talk) 17:41, 29 June 2015 (UTC)


 * It seems to be prety clearly set-builder notation to me; it makes sense as set-builder notation, and doesn't make any sense as a surreal number. If you really think it's confusing, you could change the bar to a colon. Sniffnoy (talk) 18:54, 29 June 2015 (UTC)


 * I have done so. The notation { X | Y } is used dozens of times in the article as notation for a surreal number; I think it's too prone to misinterpretation to suddenly use the same notation with a different meaning.  The string { x − y | y in Y } is not really syntactically incorrect as a surreal: "x-y" is a number and "y in Y" could be a somewhat roundabout way of saying "the set Y".  Mnudelman (talk) 19:18, 29 June 2015 (UTC)


 * An IP editor recently renamed the sections, and I mistakenly thought the IP introduced the "To infinity ..." section heading and reverted. Now that I see what happened, I reverted myself, so the section in question is now titled Transfinite induction. --JBL (talk) 19:13, 29 June 2015 (UTC)


 * Sorry, the IP editor was me. I failed to login.  Mnudelman (talk) 19:21, 29 June 2015 (UTC)


 * Not a problem. I've also added some explanatory text with your edit.  --JBL (talk) 19:24, 29 June 2015 (UTC)

Deemphasize the Conway construction?
Would it be better to start the article off with either the 'map from an ordinal to {+, -}' approach or the binary simplicity hierarchy one? Conway's { | } may have been the first concrete construction, but that doesn't mean it's the most pedagogical. And while it's neat to start with the empty set and carry off a simultaneous, transfinitely inductive, mutually recursive definition of forms, numbers, ordering and equality, that's not the important thing about the surreals in the larger picture. -- Lhmathies (talk) 17:44, 5 May 2016 (UTC)
 * The "map from an ordinal to {-1,+1}" approach is mentioned later, and has the disadvantage that it does not immediately make sense. With Conway's approach, one can read the "|" as "number midway between" or something, and the surreals make some sense even if one doesn't delve into the higher mathematics.  I guess the question is, who is the article written for? DavidHobby (talk) 18:01, 5 May 2016 (UTC)
 * Well, does the Conway construction make sense if you don't have the 'mathematical maturity' to follow the mutual recursion? Maybe it does, if you just go with the flow -- but if you look through the talk archive it's clear that many people get caught up in the seeming paradoxes. Maybe the solution is to point out at the start that there are more direct but less intuitive ways of doing it, and you don't have to prove that the Conway construction works by itself, it falls out of the other approaches too. My real goal is to get rid of some of the very basic manipulations that fill up the first half of the article and still only get to ordinal 2ω -- start with a recap of von Neumann ordinals, and then 'by the same sort of argument, we construct ... . This can be made rigorous following Conway, or using the axiomatic approach, ...' Make sense? -- Lhmathies (talk) 21:12, 5 May 2016 (UTC)
 * That's not the approach I took in the Overview, which I wrote precisely because people were getting too hung up in the details without seeing the basic idea. I think the Conway construction is a good intuitive introduction if you don't bog down in the weeds.  By constrast, an axiomatic approach is almost totally unenlightening right from the start without a lot of mathematical maturity. -- Elphion (talk) 21:21, 5 May 2016 (UTC)
 * The overview helps. I guess what bothers me is the abrupt change of pace in the middle, going from S* in one paragraph to transfinite ordinals, set theory and epsilon numbers in the next (which I wrote, but the original next one wasn't much better). But to match the pace I'd have had to write a textbook... On the other hand I'd like to put in some bits about how constructions like multiplication are arrived at and how inverse and square root can be done the same way, but again matching the pace would make it too long... -- Lhmathies (talk) 05:50, 6 May 2016 (UTC)

Note that Conway wrote the following statements (with which I agree) in an epilog to "On Numbers in Games" more than two decades after its initial publication.  Most of the authors who have written about them have chosen to define surreal numbers as just their sign-sequences. This has the great advantage of making equality be just identity rather than be an inductively defined relation, and also of giving a clear mental picture from the start. However, I think it has also probably impeded further progress. Let me explain why. The greatest delight, and at the same time, the greatest mystery, of the Surreal numbers is the amazing way a few simple "genetic" definitions magically create a richly structured Universe out of nothing. Technically this involves in particular the facts that each surreal number is repeatedly redefined, and that the functions the definitions produce are independent of form. Surely real progress will only come when we understand the deep reasons why these particular definitions work that way rather than avoid the issue completely right from the start. (I left my copy of the book at work, so I paraphrase from memory. Com3t (talk) 22:30, 22 November 2016 (UTC))
 * The Conway construction is interesting in itself because of the recursion from nothing and how operations turn out to respect equality, but I'm not sure everybody agrees about the deep mathematical significance -- whereas the surreal number field independent of that construction seems to be gaining in importance as the maximal extension of the reals with exponentiation. Alling's formulation was earlier, though it's hard to know if the surreals would have the position they have now without Conway's book. Lhmathies (talk) 15:49, 1 December 2016 (UTC)

Gaps
http://discovermagazine.com/1995/dec/infinityplusonea599 "For one thing, unlike the real number line, which has no holes in it, the surreal number line is riddled with gaps. There's a gaping hole between the finite numbers and the positive infinites, for example. (That particular hole is what people often mean when they talk about infinity.) There's another gap between the infinitesimals and the positive real numbers. Theres another that separates zero from every positive number."

http://web.archive.org/web/20070615123345/http://www.valdostamuseum.org/hamsmith/surreal.html "But the far greater scope of the surreals also spans enormous numbers of 'holes,' which riddle the realm of the surreals. One, for example, pops up at the frontier between the finite and infinite numbers, while another divides the finite from the infinitesimally small numbers.

https://books.google.com/books?id=tXiVo8qA5PQC&pg=PA38&dq=%22on+numbers+and+games%22+%22collection+of+all+gaps%22&hl=en&sa=X&ved=0ahUKEwjVzOv7or3QAhXJwFQKHXXjCkkQ6AEIHTAA#v=onepage&q=%22on%20numbers%20and%20games%22%20%22collection%20of%20all%20gaps%22&f=false "On Numbers and Games", by Conway, chapter "The Structure of the General Surreal Number" discusses the fact that there are very many gaps in the surreal numbers.

1) I think that a section mentioning the gaps should be added to the main Wikipedia article on Surreal numbers, because these gaps are an important feature of the surreal number line.  https://arxiv.org/pdf/1307.7392.pdf "Analysis on surreal numbers," by Simon Rubinstein-Salzedo AND Ashvin Swaminathan mentions surreal numbers are not cauchy-complete.

https://books.google.com/books?id=tXiVo8qA5PQC&pg=PA212&dq=%22on+numbers+and+games%22+%22games+in+the+gaps%22&hl=en&sa=X&ved=0ahUKEwiRuN79qL3QAhVS4mMKHbNIBG4Q6AEIIzAB#v=onepage&q=%22on%20numbers%20and%20games%22%20%22games%20in%20the%20gaps%22&f=false Ibid, Conway, chapter "The long and the short and the small", subsection "Games in the Gaps" mentions that there are games in some gaps with the property that the sum of any two games in the type of gap referenced again lie within the gap.

Since games can have values strictly between surreal numbers (that is, a surreal number is either less than or greater than, but not equal to the value of a game), supports the importance that surreal numbers have gaps.

2) I think that the opening line of the article on surreal numbers in Wikipedia, "In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number." should be replaced, because I feel that this emphasizes "artihmetic continuum" over a more fundamental characteristic of the surreal numbers. 

Note that I do not disagree with the statement with which the article begins; I just think it could be improved. If it is kept, perhaps it may benefit from a link to the following paper. https://web.archive.org/web/20090205010450/http://oak.cats.ohiou.edu/~ehrlich/Peirce.pdf

P.S. If anybody wants to edit the Wikipedia markup of my entry here in the talk section, because you can do a better job and don't like how I did it, then please, just go ahead and fix it up, and feel free to delete this line. (Apologies for not being expert and relying on the friendliness of others to fixup the markup--I felt it more important to communicate the gist than get caught up in syntax. Mea culpa.)

Com3t (talk) 21:49, 22 November 2016 (UTC)
 * I agree that a short section on (non)-continuity and gaps would be natural to have, but I never got around to looking for references. I'm reading the Analysis paper now to see if I can extract something suitable.Lhmathies (talk) 21:58, 1 December 2016 (UTC)

Pseudosurreal numbers
Can we please have a paragraph on these in this page? I mean, I understand why they'd be on moth.stackexchange due to the programming language aspects of them (i.e., Knuth), but there really should be a wikipage on them here. — Preceding unsigned comment added by Kaliratra (talk • contribs) 04:46, 20 March 2017 (UTC)
 * There is a section on games already, linking to combinatorial game theory. If you have a reference for the term pseudo-surreal number being used to mean the same thing, you can add a mention to the article. Lhmathies (talk) 09:38, 21 March 2017 (UTC)