Talk:Surreal number

Norman Larrabee Alling, his WP entry

I thought it would be a good idea to link to his WP page from the introduction, but it turns out it doesn't exist. I woiuld think he should be notable enough... There's a biography online at http://www-history.mcs.st-andrews.ac.uk/Biographies/Alling.html and one of his colleagues published his Collected Papers in 1998: P Ribenboim (ed.), Collected papers of Norman Alling (Queen's University, Kingston, ON, 1998) -- MathSciNet review 19 13 629 912 according to an online reference -- which even has a "Vita Norman Alling" starting on page vii -- but there is no online version or structured reference that I can find, so there's about 4 lines of WP-suitable material to be gained from the online biography. It's a bit thin, all in all -- can anyone help? One last google gave me a hit on Philip Ehrlich's work page: Collected Papers of Norman Alling, edited by Paulo Ribenboim, Queen’s Papers in Pure and Applied Mathematics, Volume 107, 1998, Kingston, Ontario, Canada. Should be good enough, though an ISBN would be nice, and I'd like a peek at that Vita. — Preceding unsigned comment added by Lhmathies (talk • contribs) 23:01, 18 February 2018 (UTC)

Now I'm confused, the MacTutor page above says NLA "was" a mathematician which would imply he was no longer with us, but I cannot find a death date anywhere, not even on the reference at https://mathematics.library.cornell.edu/works/A which seems up to date in this regard, judging by other entries. Since the rules for biographies of living persons are stricter, it would be nice to have this confirmed... Lhmathies (talk) 04:23, 23 February 2018 (UTC)

Controversy

There should be a section here, to preserve neutrality, noting that surreal numbers are non-existent, in the finitist form of mathematics.

That doesn't make a lot of sense. The fraction of math articles that could have such a disclaimer is huge. Nothing special about the surreals there. (I wouldn't call it a "controversy" either. Some formulations of mathematics just don't include it.) Sniffnoy (talk) 18:53, 10 May 2018 (UTC)

It is clear that the surreals are intended to be developped in an infinitary set theory so there is no need to specify that. Note that although they form an infinite class, they can still be defined in finitary mathematics.Vincent Bagayoko (talk) 11:13, 13 May 2018 (UTC)

Definition of y^R and y^L

It seems that there is no definition of the expressions y^R and y^L present in the equation for division

My reading is that they're the same as y_R and y_L in the previous paragraph. But I'm not confident enough to make the change. Help?DavidHobby (talk) 02:45, 7 May 2019 (UTC)

True, this is inconsistent notation and I changed it.Lhmathies (talk) 13:40, 8 May 2019 (UTC)

When is the birthday number / older-than relation needed?

This is mostly idle musings, a.k.a. original research -- I would have put them under this previous Talk page entry, but it has been archived.

I think the answer to that question is that you can construct the ordered class of surreal numbers without using the older-than relation, but in the process you will necessarily define it.

On the other hand, to get from forms to numbers and thus to prove existence and consistency of arithmetic operations and define a field structure, the existence of the relation is crucial.

Sketch: Given a set S of totally ordered sets, totally ordered by inclusion (order homomorphies), you can create a 'completion' by taking the union U and adding special forms { L | R }, L < R, L ∪ R = U -- there is an obvious extension of the ordering and you can extend S with the completion to get a new totally ordered set of totally ordered sets. Now start with the empty set of sets and apply transfinite induction to get the surreals as the union of all the sets in the union of the sets of sets (where the sets of numbers are the same as the generations in the Conway construction). The step of taking the union is only strictly necessary for the steps corresponding to limit ordinals, including 0; in other cases S has a maximal element.

Note that the induction step only needs a total order on the set of sets. The well-ordering is a result of using (transfinite) induction. And instead of the 'special forms' you can just take the set of upwards (or downwards) closed sets -- each 'new' number corresponds to exactly one closed set, there are no equivalence classes involved and no double or parallel induction happening. Lhmathies (talk) 13:24, 10 January 2020 (UTC)

Actually this came out a bit misleading -- you do need a total order relation on S to make the induction step work and that is in effect an older-than relation, but the order type of S is not important so you might not be able to express it as a number. — Preceding unsigned comment added by Lhmathies (talk • contribs) 14:29, 13 January 2020 (UTC)

Mistake in article image

Where the value of a third is shown in the diagram is in fact where two thirds should be. The tree branches right, left, right, left, r, l, r, l etc. for 2/3rds and right, left, left, right, left, right left, r, l, r, l etc. for 1/3rd. --82.30.170.253 (talk) 02:51, 20 April 2020 (UTC)

Yup, that's an error alright. I'd suggest commenting on the page for the image, or perhaps directly messaging the person who made it. (You could also try editing it yourself, obviously, but that may be difficult; I tried it myself and didn't get very far...) Sniffnoy (talk) 17:13, 20 April 2020 (UTC)

I agree. It is an error. You beat me to it. I tried to comment on the page for the image, but could not find it. The answer is 2/3 as per a video from John Conway himself. — Preceding unsigned comment added by 2001:16B8:170A:4700:E942:4EC6:B9E3:1C31 (talk) 21:35, 1 June 2020 (UTC)

Does Conway need to be defended?

One of the most recent edits has "remove conflicting naming claims" as the edit summary -- but the old text did say that the name was first used by Conway. It also said that equivalent treatments had been developed earlier, and that is gone too... so what was the real reason to change it? (The previous formulation was mine from a few years ago, full disclosure).

It looks like Conway has to be the first to ever come up with anything that looks like surreal numbers -- confer the emphasis in "History of the concept" on Alling's prior treatment not being as useful, which may be true but possibly not worth so many words. Is this a cult? Lhmathies (talk) 08:59, 1 June 2020 (UTC)

Error in epsilon definition?

Am I misunderstanding or should

{ 0 | 1, ⁠1/2⁠, ⁠1/4⁠, ⁠1/8⁠, ... } = ε

actually be

{ 1, ⁠1/2⁠, ⁠1/4⁠, ⁠1/8⁠, ... | 0 } = ε

? In the former version, as it currently is in the article, 0 | 1 would be 1/2 and the subsequent infinite series of fractions seems misplaced. Kufat (talk) 02:03, 13 August 2022 (UTC)

Order does not matter in sets. The order used in the article is more convenient in that it describes an infinite sequence of simpler numbers that are greater than ε. The latter is not a surreal number because 0 is not less than 1. –LaundryPizza03 (dc̄) 03:16, 13 August 2022 (UTC)

Exponentiation

Should the article contain a paragraph on exponentiation of surreal numbers, and perhaps even a hint of tetration, pentation, and other hyperoperations? —Quantling (talk | contribs) 15:45, 7 November 2022 (UTC)

Did you see Surreal number#Exponentiation (a subsection under the treatment of the Surreal number#Exponential function)? This is a general definition for any two surreal numbers -- but it seems to me that hyperoperations are only defined for integer arguments (because you need to count down to zero) and I'm at a loss as to any possible generalizations. (Of course you could allow infinite numbers as the first argument, but I'm not sure if anything more interesting happens than with integers).

Exponentiation with an exponent that is a dyadic fraction (of which the integers are a subset) doesn't even need the log and exp functions, but that is not treated separately.

Lhmathies (talk) 13:41, 21 February 2023 (UTC)

Surcomplex numbers

One passage of the article reads as follows:

"A surcomplex number is a number of the form a+bi, where a and b are surreal numbers and i is the square root of −1. The surcomplex numbers form an algebraically closed field (except for being a proper class), isomorphic to the algebraic closure of the field generated by extending the rational numbers by a proper class of algebraically independent transcendental elements."

I find a great similarity between a) "(except for being a proper class)" and b) "Other than that, Mrs. Lincoln, what did you think of the play?".

My point is that there is a tremendous, indescribably large distinction between

a) an algebraically closed field that is a set

and

b) one that is not a set, that cannot be a set.

User:2601:200:c082:2ea0:48d7:7772:5ebd:f634 00:04, 29 March 2023‎

Sure. But to my naive perception, this wanders off into model theory and requires posing questions like "is there a model of the surreals in the Grothendieck universe or maybe in NBG?" or maybe this is a question about "how does the Yoneda lemma work out in this case?" or maybe the question is "how does the transfer principle work out in this case?" -- and I have no clue about this. But since the transfer principle is mostly about transferring over statements written in first-order logic, maybe something goes wonky higher logics, ascending up the sigma-pi hierarchy? Who knows? On the other hand, it is my understanding that there is one and only one model of the complex numbers. Since the paragraph closes with the statement: Up to field isomorphism, this fact characterizes the field of surcomplex numbers within any fixed set theory.[6]: Th.27 and so that feels very cut-n-dried and unambiguous. "Any fixed set theory" would seem to include ZFC and New Foundations and etc. I tried to rephrase your statement as a menu of plausible questions. Rephrasing statements as questions allows them to get answered. I am not sure what you might have been thinking of. 67.198.37.16 (talk) 20:51, 30 April 2023 (UTC)

"Esoteric Pun" Parenthetical.

In "Gaps and Continuity" there's a part describing gap "On" and its relation to "No". Then, in parenthesis it is described as an "Esoteric Pun", followed by a colon (indicating a link between ideas), followed by some more text.

I have a few issues with this section that I will do my best to loosely describe, and I'm hoping that either my sentiments will be reflected and something will be done, or that justification will be given and I will understand the choices that have been made in the writing of the section. In no particular order:

1. The word "Esoteric" links to "Western Esotericism". Forgive my ignorance, but I see no way in which that is actually relevant here. I believe it is a different use of the word "Esoteric", possibly an informal (or even wrong) one.
2. Even if it is somehow relevant, the portion after the colon following "Esoteric Pun" fails to explain how it is related to esotericism at all, or just how it is "esoteric".
3. It also fails to elaborate on the pun. There is the obvious observation that "No" and "On" are English inverses of one another, but it isn't mentioned or clued into at all, in favor of simply elaborating on the mathematical relationship between the two. If this does actually serve to explain the pun a bit more, I'm just not seeing how.
4. Even without the issues of the first sentence, I'm not sure what function the rest serves as a parenthetical. It's fairly long and, at least to me, just feels poorly written. I think that if there's something important that needs saying in there, it should be given it's own paragraph, rather than being stuffed into parentheses.

[Conclusive paragraph which sums up my thoughts] TheJonyMyster (talk) 03:09, 19 May 2023 (UTC)

@TheJonyMyster: Fixed it. Esoteric is one of the types of mathematical joke as explained in the lead of that article. So the first link was just a red herring and I've removed it. Though of course since the book includes its own creation myth, it's possible that there is some esoteric content, should a third-party source report on it... Skyerise (talk) 12:38, 19 May 2023 (UTC)

sup and inf

Do the surreals have the least-upper-bound property? That is, does every non-empty set of surreals with an upper bound have a least upper bound (sup) in the surreals? Whether yes or no, mention of this in the article would be nice. I assume the answer would be the same for a greatest-lower-bound property (inf), and mention of that too would be nice. Thank you —Quantling (talk | contribs) 13:13, 7 September 2023 (UTC)

This question is answered in the negative in the section "Gaps and continuity". –LaundryPizza03 (dc̄) 13:17, 16 September 2023 (UTC)

Dead Link

This link in the section Further reading is dead: http://www-cs-faculty.stanford.edu/~knuth/sn.html Hubert1965 (talk) 14:39, 3 November 2023 (UTC)

@Hubert1965: Thanks, I've fixed it. Skyerise (talk) 14:43, 3 November 2023 (UTC)

magnitudes and absolutes

... infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number.

The word positive was removed, and restored with "This is necessary; an infinitisimal is not smaller than -1." But it is smaller in absolute value, unless I missed a memo.

It is not smaller than the real number zero, though. —Tamfang (talk) 05:23, 3 May 2024 (UTC)

I think the sentence should be reworded. "X is smaller in absolute value than Y" is unclear and could be interpreted as "the absolute value of X is smaller than Y", which is how I initially read it. Lumping infinites and infinitesimals together in one phrase also makes the sentence a little awkward. Perhaps ... but also infinite numbers, whose absolute value is larger than any real number, and infinitesimal numbers, whose absolute value is larger than zero but smaller than any positive real number. It's a little wordy but is less ambiguous than the current sentence. CodeTalker (talk) 07:06, 3 May 2024 (UTC)

Games & Applications to combinatorial game theory—merge sections?

It seems like these sections repeat some information and are in general about a similar topic. Should they be merged? Theanswertolifetheuniverseandeverything (talk) 12:44, 17 June 2024 (UTC)

Replacing an ambiguous conclusion to a paragraph

A paragraph in Surreal number#Induction_rule closes with "These labels will also be justified by the rules for surreal addition and multiplication below". I am concerned about an ambiguity: When I reached the aforementioned "below", I expected to see a definition of addition that establishes that the labels were correct (i.e., a derivation or proof.) Instead, I found a circular definition of addition that requires a a priori knowledge of the labels. On my first few readings, I thought that I had skipped over (or forgotten) a step. I am thinking of a replacement conclusion to the paragraph with something like "These labels will used to define addition and multiplication below in a way that permits any given surreal number to be described by a bewildering array number of different forms ( such as $\{2{\tfrac {1}{2}}|4{\tfrac {1}{2}}\}=\{2|\}=3\neq 3{\tfrac {1}{2}}.)\,$" While this sentence clarifies the true nature of the temporary assignment of labels, it is a heavy sentence. Ordinarily I would place some or all of it in a footnote. But Wikipedia is not too keen on footnotes. For that reason I decided not to "Be Bold" on this edit. Guy vandegrift (talk) 16:32, 18 June 2024 (UTC)

It would suffice to emend that to "by many different forms". -- Elphion (talk) 03:51, 19 June 2024 (UTC)

Is this a typo?

At the time of writing, Surreal_number#Transfinite_induction contains the statement:

""The surreal number ω − 1 is not an ordinal; the ordinal ω is not the successor of any ordinal. This is a surreal number with birthday ω+1, which is labeled ω− 1 on the basis that it coincides with the sum of ω = { 1, 2, 3, 4, ... } and −1 = { | 0 }"

It seems to me that "ω = { 1, 2, 3, 4, ... }" should be replaced by "ω = { 1, 2, 3, 4, ... | }". Am I right??? Guy vandegrift (talk) 22:42, 16 September 2024 (UTC)

Yes. I made an edit for this and another edit for another typo in that section. —Quantling (talk | contribs) 01:29, 17 September 2024 (UTC)

Potentially problematic images

Illustrating x ≤ y

Illustrating 0 ≤ 3

I mostly work on Wikiversity, but have also done lots of images on other wikis. Those experiences taught me that sometimes it is necessary to consult with an article's active editors, especially with images that contain too many innovations. One innovation in these images is the left to right ranking of the sets and surreal numbers:

$$X_{L}\leq ^{*}x\leq ^{*}X_{R}\leq ^{*}Y_{L}\leq ^{*}y\leq ^{*}Y_{R}$$ By placing the x and y above the left and right sets, the reader can view it in terms of this ordering. The "problematic" aspect of this is the use of an asterisk to denote inequalities between sets of numbers. Tøndering does something similar, using things like $a\not \geq A$ (where a is a number and A is a set.)

As somewhat of a novice to set theory, I find $\leq ^{*}$ much easier on the eye and brain (compared with entities such as $a\not \geq A$ or $a\neg \geq A$.) For one thing, I like to see the "wide" part of < next to the larger element. Also, the asterisk warns the novice that "inequalites" between sets and numbers are tricky. It notifies readers of the complexities of comparing numbers with sets, but in a way that encourages them to ignore the details on a first reading.

Also, I attempted to use a chatbot to find a convention for describing inequalities between numbers and sets of numbers. Gemini was unable to find anything. See v:Chatbot_math/Gemini/Ranking_sets_of_numbers --- Guy vandegrift --(talk) 21:48, 22 November 2024 (UTC)

Expression formatting

There's rather a lot of inconsistency in the maths formatting in this article. The first few subheadings use {{math}}, {{mvar}}, or <math display>, which produce serif text in slightly larger type size; several portions use {{mset}} or no formatting (apart from html superscripting / subscripting), both of which appear unformatted.

Is this intentional? I'd go through and add or alter the relevant templates, but I wanted to check in to verify that this is a result of inaction rather than choice. Folly Mox (talk) 13:42, 11 December 2024 (UTC)

At the risk of stating the obvious: {{mset}} is for putting braces around an expression, such as showing elements in a set: ω = {0, 1, 2, ...}. Usually it should be inside {{math}} markup. {{mvar}} merely provides a shorthand for {{math}} markup that would have been entirely in italics: {{mvar|xy}} is the same as {{math|''xy''}}. In the other camp, {{tmath}} formats like <math>, though with different syntax inside. <math display=block> is for math that gets its own display block and <math> is for math that is inline within a sentence.

So the question that is most relevant for me is whether we should be using {{math}} and its accomplices, which produce unicode characters to be displayed, vs. <math> and its accomplices, which produce image renderings of the math. The general rule is that if the article generally uses <math> for inline math then it should do that everywhere. If it generally uses {{math}} then it should do so everywhere, except where that renders abysmally, in which case substitute in an occasional <math>. Regardless, generally use <math display=block> for displayed blocks of math.

In articles that are 50/50, I tend to go with {{math}}. I like that it is easier to cut and paste subsets of the unicode characters than it is to cut and paste subsets of an image. —Quantling (talk | contribs) 14:32, 11 December 2024 (UTC)

Thank you for stating the obvious: I wasn't directly comparing the templates to the rendered page at any point, nor did I look at the template documentation. I'll check out the options in depth after work (or forget entirely) and attempt to harmonise. Folly Mox (talk) 19:02, 11 December 2024 (UTC)