Talk:Ordinal number

I am no expert, to be sure, on ordinals but can we please finally dump these attempts at graphical representations? the 'spiral' and 'matchstick' images -- they add nothing to the explanation and as far as I can see are not even discussed in the article text. 71.139.124.132 (talk) 13:20, 30 December 2022 (UTC)

The images are intended to illustrate the structure of (initial segments of) the ordinal numbers. The spiral form is just a hack to avoid an extreme aspect ratio. Do you have suggestion about alternatives? - Jochen Burghardt (talk) 19:07, 30 December 2022 (UTC)

The matchsticks are good. They clearly show how ω² can be embedded into the reals.

The spiral could be improved by defining clearly what normal function corresponds to one turn of the spiral. If I could, I would change it so that that function is α -> ω·(1 + α). However, I do not know how to edit the image. JRSpriggs (talk) 00:35, 31 December 2022 (UTC)

It seems that the Svg file can be edited with an ordinary text editor, so I can offer to change the labels according to your suggestions. However, I'm afraid I didn't understand them. It seems that after n turns, we are at ωⁿ; but I'm not sure whether your α corresponds to my n. - Jochen Burghardt (talk) 13:12, 31 December 2022 (UTC)

The outermost ring of the spiral should begin with 0 at the 12:00 high position and continue with the natural numbers coming closer together until the first turn is completed. At that point it should meet ω which should be directly under 0. The spoke beginning with 0 would continue with ω, ω², ω³, etc. until it reaches the center which would be ω^ω. The spoke beginning with 1 would continue with ω·2, ω²·2 ω³·2, etc.. Between these, new spokes should begin in the second turn, ω+1 would continue with ω²+ω. Clearly, most of these numbers would have to be replaced with just tick-marks and as you proceed around the spiral, and most of those would have to be omitted altogether. JRSpriggs (talk) 00:05, 1 January 2023 (UTC)

I started to redesign the picture according to your suggestion; it will take a few days to get the limit ordinals arranged. - Jochen Burghardt (talk) 17:19, 23 January 2023 (UTC)

It took me longer than expected, but now I came up with a first version that meets the above requirements (I hope), see File:Omega-exp-omega-normal svg.svg and File:Omega-exp-omega-normal.pdf. Comments are appreciated. - Jochen Burghardt (talk) 18:14, 8 February 2023 (UTC)

Thank you for your effort. Two problems:

1. Use ωn instead of nω. 2ω = ω < ω + ω = ω2 .

2. Please use darker colors. Yellow on white is virtually illegible (to me at least). JRSpriggs (talk) 01:00, 9 February 2023 (UTC)

I would use ${\displaystyle \omega \cdot 2}$ rather than ${\displaystyle \omega 2}$; the latter is unusual in my experience and the 2 looks like it could be a misplaced superscript or subscript. I agree about the yellow; that's pretty hard to see. --Trovatore (talk) 01:44, 9 February 2023 (UTC)

 Done. As for 1.: oops, sorry! As for 2.: I now darkened the colors from RGB (100%,0%,0%) for red (similar for green, blue) and (80%,80%,0%) for yellow (similar for cyan, magenta) to (80%,0%,0%) and (50%,50%,0%). I find the result still hard to read. On the other hand, I'd like to keep the "rainbow" effect which makes it easier to recognize which label belongs to which spiral turn. Moreover, inspired by the current image File:Omega-exp-omega-labeled.svg, I'd added some fade-out effect, e.g. at ${\displaystyle \omega ,\omega +1,...,\omega +4}$; should I turn that off? - Jochen Burghardt (talk) 12:08, 9 February 2023 (UTC)

Yes, I agree that File:Omega-exp-omega-normal.pdf should replace what is in the article. Thanks again. JRSpriggs (talk) 19:46, 9 February 2023 (UTC)

I think you put the wrong link. We don't want a PDF as an image (not sure that even works). Probably you meant File:Omega-exp-omega-labeled.svg. --Trovatore (talk) 01:52, 10 February 2023 (UTC)

No, Trovatore. That is not it. That is an old incorrect version. JRSpriggs (talk) 14:48, 10 February 2023 (UTC)

@Jochen Burghardt After recently looking at the Ordinal number article, I see that you changed the black and white picture to a color one (March of last year). Is that really an improvement? The old version seemed a lot easier to look at. The new layout is good, but the lack of contrast with the washed-out tones of the colors makes the whole thing hard to follow and understand what is going on. Any thoughts? PatrickR2 (talk) 04:49, 15 May 2024 (UTC)

Yes, please turn off the fade-out effect. PatrickR2 (talk) 04:55, 15 May 2024 (UTC)

I've uploaded a b/w version at File:Omega-exp-omega-normal-bw svg.svg for comparison purposes. The colors are intended to ease distinguishing the different turns of the spiral and to ease recognition of which label belongs to which spoke.

I also have uploaded a version using the (darker) spoke colors for label coloring, too, see File:Omega-exp-omega-normal-dark svg.svg; it has a reduced fade-out effect; I' prefer this one. - Jochen Burghardt (talk) 07:51, 19 May 2024 (UTC)

I agree that https://upload.wikimedia.org/wikipedia/commons/a/af/Omega-exp-omega-normal-dark_svg.svg is the best of the three. The colors make it easier to distinguish the turns, and it's also easier to read without the fade-out effect. PatrickR2 (talk) 18:18, 22 May 2024 (UTC)

i dont know why, but (for me at least) the colors make it harder to process where the spiral ends/starts; i think it's due to their colors mixing into each other like a gradient or because most triangles are kind-of the same size when compared to their neighbors, or simply because i got accostumed to the old monochrome image.

would it look better if each spiral turn was just one solid color? maybe an alternating one like:

red ${\displaystyle 0,\cdots ,\omega }$

blue ${\displaystyle \omega ,\cdots ,\omega ^{2}}$

red ${\displaystyle \omega ^{2},\cdots ,\omega ^{3}}$

or just the regular "cycling through the rainbow" one.

it would(?) help distinguish to where each spiral starts/ends, im not sure though

also, why is the most-outer spiral ${\displaystyle 0,\cdots ,\omega }$ almost invisible? i almost didnt notice it (im srry in advance if this sentence sounds rude, idk how else to word it) It is I! ALonelyPhoenix !!!!!!!! (talk) 17:45, 26 May 2025 (UTC)

Everyone is free to make their own picture suggestion, so we can discuss it here. The TeX code for the current one is available for experimenting with different color schemes. - Jochen Burghardt (talk) 14:42, 27 May 2025 (UTC)

The article currently claims that the ${\displaystyle \infty }$ symbol is sometimes used for the class of all ordinals. Is this true? What textbook does that? Thank you —Quantling (talk | contribs) 13:57, 7 October 2025 (UTC)

I am confused by the proof. Suppose T = ω = {0, 1, 2, ...}. Suppose D = {0}. Suppose S = {1, 2, 3, ...}. Even though S is a proper subset of T, the ordinal that S is isomorphic to is T = ω, which is not an element of T nor is it equal to inf(D) = 0. What am I missing? —Quantling (talk | contribs) 02:21, 11 December 2025 (UTC)

S is not a (von Neumann) ordinal. That said, I don't think this is a particularly important fact to include. --Trovatore (talk) 02:41, 11 December 2025 (UTC)

Apparently, the lemma is used to prove trichotomy of ${\displaystyle \in }$, which, in turn, is used in several other proofs (if I remember correctly). - Jochen Burghardt (talk) 09:49, 11 December 2025 (UTC)

I responded to Quantling above without actually having looked at the text referred to. I decided I'd better look it up and make sure I hadn't said something silly.

When I did I was bit shocked; it seems to have changed a lot since I last checked. Specifically there are a bunch of elementary proofs. We don't normally do proofs in Wikipedia, except for sketches of especially notable ones (say Banach–Tarski or Gödel incompleteness), because that's not normally what people come to an encyclopedia for (cf. "Wikipedia is not a textook"). These proofs in particular would normally not be spelled out even in a textbook; they would be relegated to the exercises.

I think this section should be sharply trimmed. First, the von Neumann version of ordinals is an implementation detail; it isn't really that conceptually important to the notion of ordinal number. And even if it were, see my paragraph above. --Trovatore (talk) 06:38, 11 December 2025 (UTC)

It has now been trimmed.Tosiaki! (talk) 09:55, 11 December 2025 (UTC)

Not enough. The proofs should be removed altogether. I'll try to get to it later. --Trovatore (talk) 18:27, 11 December 2025 (UTC)

Looking around, "We don't normally do proofs in Wikipedia, except for sketches of especially notable ones" doesn't seem quite true, there are many articles with proofs, such as Heine–Borel theorem etc. So whether to trim the proofs doesn't seem to come down to simply "they shouldn't be there."Tosiaki! (talk) 21:18, 11 December 2025 (UTC)

There may be other proofs that should be removed. I think it's completely clear that we should not give details of elementary proofs that would be exercises even in textbooks. That's just way outside the bounds of what an encyclopedia is for. --Trovatore (talk) 22:17, 11 December 2025 (UTC)

No, it doesn't seem so clear cut, proofs in the text do serve a purpose. Also, it's been trimmed further so that it's no longer given as a proof, but rather as a discussion, so it doesn't look like it really needs any further trimming.Tosiaki! (talk) 23:27, 11 December 2025 (UTC)

No, there's still too much. I'm going to trim it. --Trovatore (talk) 02:15, 12 December 2025 (UTC)

This doesn't seem to be something commonly agreed upon, so there's no consensus on this decision.--Tosiaki! (talk) 04:17, 12 December 2025 (UTC)

I would say there was never consensus to add it. Anyway we should get more opinions here. --Trovatore (talk) 06:04, 12 December 2025 (UTC)

I have dropped a note at WT:WPM. --Trovatore (talk) 06:20, 12 December 2025 (UTC)

@Tosiaki! There is actually a consensus on that, even if some people don't follow it. It's explained at WP:NOTTEXTBOOK. I have not looked in detail all the changes you added. But the main thing is that wikipedia is not meant to "teach". It is meant to be a distillation of useful information, backed by sources that interested readers can follow to dig deeper into the subject. In particular, it is not meant to teach someone about a particular subject, and it is not meant to give proofs. Learning about a subject for the first time is better done through textbooks or sites like Math Stackexchange, where people can ask question and discuss the topic. PatrickR2 (talk) 06:46, 12 December 2025 (UTC)

The proofs have since been reduced to a single sentence explaining the general rationale rather than giving out the whole detail. This should be okay in any case.Tosiaki! (talk) 07:24, 12 December 2025 (UTC)

That looks much cleaner. Thanks. --Trovatore (talk) 07:31, 12 December 2025 (UTC)

"First, the von Neumann version of ordinals is an implementation detail..." I would like to disagree on just this point. The von Neumann ordinals capture one of the defining aspects of ordinals, that we always get the next ordinal by counting up the previous ones. (The other defining aspect is that they are equivalence classes of well-orders with respect to isomorphism, but I would imagine that is even more abstract.) It also gives a representation that uniquely represents every ordinal in ZF, which is not an easy task to achieve without using the same idea. In fact the whole reason I am feeling the urge to talk about this is these sentences in the lead section:

The axiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one is isomorphic to an initial segment of the other. So ordinal numbers exist and are essentially unique."

I do not know how the second sentence could be interpreted so that it follows from the first sentence, but I would like to read it as "counting a (well-ordered) set will always give a ordinal number that uniquely represents its order type", in which case it follows from the von Neumann representation. Bbbbbbbbba (talk) 12:03, 16 January 2026 (UTC)

I'd be careful of the word "counting"; to me it is more related to cardinality than ordinals. —Quantling (talk | contribs) 19:39, 16 January 2026 (UTC)

@Bbbbbbbbba: The "implementation detail" is exactly how you "get the next ordinal by counting up the previous ones". In the von Neumann coding, you get it in arguably the simplest possible way; you just take the set of all the previous ones. But while very convenient, this is not essential to the idea of an ordinal. Cantor, for example, did not particularly trouble himself with how you represent the next ordinal as a set. (In fact I somewhat doubt that he thought of ordinals as being sets.) --Trovatore (talk) 21:54, 16 January 2026 (UTC)

While plausible, I have not read anything Cantor wrote to speculate on how he viewed the idea of ordinals. And I would assume that was why we had the Burali-Forti paradox! I would assume that it matters to a significant portion of readers that the ordinals could be defined in a paradox-free way, which was why someone put "ordinal numbers exist and are essentially unique" there in the lead section. It sounds reassuring even if currently it is unclear what it actually means. Bbbbbbbbba (talk) 00:46, 17 January 2026 (UTC)

Jourdain's very readable translation of Cantor's Beiträge is in the public domain and available online here. If you prefer having a book in your hands, Dover has it for cheap. Really more people should read it; it would be a good antidote to the narrative about there being something inherently contradictory about "naive set theory". --Trovatore (talk) 02:29, 17 January 2026 (UTC)

I am reading the book right now, and I am at around this sentence on page 57:

Cantor called it the second principle of generation of real integers, and defined it more closely as follows: If there is defined any definite succession of real integers, of which there is no greatest, on the basis of this second principle a new number is created, which is defined as the next greater number to them all.

I wonder if, historically, this "succession" should be understood as "(countable) sequence" or "(possibly well-ordered) set". I guess the footnote on page 59 implies that it could be uncountable? Bbbbbbbbba (talk) 05:19, 18 January 2026 (UTC)

I also wonder what you disagree about "the narrative about there being something inherently contradictory about 'naive set theory'". It is a mathematical fact that the system consisting of axiom of extensionality and unrestricted comprehension is contradictory, right? Bbbbbbbbba (talk) 05:23, 18 January 2026 (UTC)

I mean that there's nothing in the Beiträge that suggests unrestricted comprehension. The sets Cantor described are not extensions of properties; they're collections of preexisting objects. Following that intuition doesn't lead to unrestricted comprehension (there's no "set of all sets" because it would have to be there before you could include it) but to something like the von Neumann hierarchy.

It's true, of course, that a naive set theory that considers sets to be extensions of arbitrary properties is inconsistent.

(If you want to delve deeper into this there's a considerable history to the discussion. The points I'm making are essentially echoing Hao Wang. On the other hand María Frápolli asserts different moments in Cantorian thought and claims that there's an earlier strand that is vulnerable to the antinomies, but which Cantor cleaned up later; she may very well be right; I haven't looked into the entire history. My concern is that we not fall uncritically into the formalist line that the cure for the antinomies was restricting to a particular formalism.) --Trovatore (talk) 05:39, 18 January 2026 (UTC)

I jumped ahead and I see what you mean. Cantor was quite cautious in using the collection of "the second number class" (all countably infinite ordinals) as a set, only doing so after he has spent a lot of time studying well-orders of countable sets. Either Burali-Forti or Russell (the Burali-Forti paradox page does not make it clear what Burali's original theorem was), on the other hand, seems to have stepped over the line by considering "the series of all ordinal numbers". So the verdict is that, there probably is some principles that mathematicians intuitively follow when constructing new sets, but if it is not stated explicitly, it would be easy to violate, especially when other people start working upon your results. Bbbbbbbbba (talk) 06:22, 18 January 2026 (UTC)

I think I get your point somewhat: The decimal representation is not essential to the idea of natural numbers, but we do not shy away from using it when taking about natural numbers either. Of course we also do not want to spend a lot of space proving representation-specific facts, and I agree with you on that point. Bbbbbbbbba (talk) 01:00, 17 January 2026 (UTC)

And indeed, the article Natural number makes no mention of notation. My only problem with the revised section is that it does not clearly relate to the Cantor normal form, i.e. relating ${\displaystyle \mathbb {N} }$ to ${\displaystyle \omega }$ as seen in the rest of the article. –LaundryPizza03 (dc̄) 05:41, 17 January 2026 (UTC)

I am actually considering the position of this section in the whole article. I understand that we may take equivalence classes between well-orders as the "true" definition of ordinals, but the von Neumann ordinals are more concrete. They also give explicit examples of well-orders of every order type, and these examples are arguably more "natural" than e.g. the ordering on natural numbers that puts all even numbers before odd numbers. In fact the current first non-lead section "Ordinals extend the natural numbers" is already using some von Neumann ordinal language. Bbbbbbbbba (talk) 04:04, 18 January 2026 (UTC)

I wouldn't call equivalence classes the "true definition". That's still a coding, just a different one. It seems to me that the ordinals are intuitively understandable without having to represent them as sets at all; these "naive ordinals" are the ones we should be trying to get across in the early sections. --Trovatore (talk) 04:45, 18 January 2026 (UTC)

Not that equivalence classes are sets in ZF, but I get the idea. At the most "naive" level, ordinals are just labels that we assign to things in a sequence, such that we never run out of labels (of course with the caveat that proper classes like "all ordinals, but with 0 moved to the end" are not eligible sequences) and never have an ambiguity as to which label to use next. Bbbbbbbbba (talk) 09:27, 18 January 2026 (UTC)

Would that imply that we may actually want to explicitly use the phrase "transfinite sequence" earlier? The section itself maybe remains where it is now, for formal definitions. Bbbbbbbbba (talk) 12:43, 18 January 2026 (UTC)

I added back some sentences explaining the motivation for each statement. Do you find this OK? Bbbbbbbbba (talk) 11:58, 22 January 2026 (UTC)

Currently this page is tagged {{More citations needed}} and {{More footnotes}}. However from a glance I cannot tell which part of this page was deemed in need of citations. From my understanding this is basically a textbook topic and thus does not need that many references to verify. Bbbbbbbbba (talk) 15:33, 15 January 2026 (UTC)

To be safe, I'd suggest scattering additional footnotes across paragraphs that don't have one. It is likely that everything in the article can be cited to one of the listed references, so {{More citations needed}} can be dropped. –LaundryPizza03 (dc̄) 02:54, 22 January 2026 (UTC)

In the references:

{{Citation |last=Jech |first=Thomas |title=Set Theory |date=2013 |url=https://books.google.com/books?id=GHjmCAAAQBAJ |edition=2nd |publisher=Springer |isbn=978-3-662-22400-7 |ref={{harvid|Jech|2003}} }}

I understand that the book has many editions, but should not we use a consistent year in the reference and in the {{sfn}}s? Bbbbbbbbba (talk) 12:40, 21 January 2026 (UTC)

2003 is correct. The ISBN was also for the 1997 edition, so that has been corrected as well. –LaundryPizza03 (dc̄) 02:52, 22 January 2026 (UTC)