Talk:Ordinal arithmetic

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When it comes to natural addition, the article says

We can also define the natural sum of α and β inductively (by simultaneous induction on α and β) as the smallest ordinal greater than the natural sum of α and γ for all γ < β and of γ and β for all γ < α.

It seems to me that a similar definition would work for the ordinary addition as well, though the induction is only on β, and a base case appears to be required:

When the right addend β = 0, ordinary addition gives α + 0 = α for any α. For β > 0, the value of α + β is the smallest ordinal greater than the sum of α and γ for all γ < β.

That is, assuming I'm right, the article's current definition that separately describes cases for β when it is a successor ordinal vs. a limit ordinal is making a distinction that need not be made. We could avoid making that senseless distinction. Should we change the article? —Quantling (talk | contribs) 18:31, 13 June 2024 (UTC)[reply]

I edited boldly to give both the approaches. —Quantling (talk | contribs) 13:44, 17 June 2024 (UTC)[reply]

... or so says the article. But is it? Is ${\textstyle \alpha \otimes \beta }$ equal to the smallest ordinal that is both greater or equal to ${\textstyle (\gamma \otimes \beta )\oplus \beta }$ for all ${\textstyle \gamma <\alpha }$ and is greater or equal to ${\textstyle (\alpha \otimes \delta )\oplus \alpha }$ for all ${\textstyle \delta <\beta }$? That is, $${\displaystyle \alpha \otimes \beta =\sup \left(\sup \lbrace (\gamma \otimes \beta )\oplus \beta :\gamma <\alpha \rbrace ,\sup \lbrace (\alpha \otimes \delta )\oplus \alpha :\delta <\beta \rbrace \right)\,.}$$ The base cases, ${\textstyle 0\otimes 0}$, ${\textstyle 0\otimes \beta }$, and ${\textstyle \alpha \otimes 0}$, are handled in the above if we assume that sup(∅) = 0, or we could define them explicitly as equal to 0. —Quantling (talk | contribs) 19:00, 14 June 2024 (UTC)[reply]

I am going to edit boldly. If you disagree please also comment in this discussion. —Quantling (talk | contribs) 21:17, 24 June 2024 (UTC)[reply]

I think we should not go on at too much length about the so-called "natural" operations, which are not very important and are not what would normally be understood as "ordinal arithmetic". --Trovatore (talk) 00:32, 25 June 2024 (UTC)[reply]

Could you please provide a proof or citation for this? I don’t see it immediately Scott (talk) 18:41, 28 August 2024 (UTC)[reply]

For natural addition, I am finding Theorem 2.4, Statement 1 of this, which appears to support what is in the Wikipedia article. For natural multiplication, Statement 2 of that theorem is not what is currently in the Wikipedia article. The formula from that source gives that α ⊗ β is equal to the smallest ordinal x such that for all α′ < α and all β′ < β, it is the case that x ⊕ (α′ ⊗ β′) is strictly greater than (α ⊗ β′) ⊕ (α′ ⊗ β). We could switch to that formula ... at least until we find a source that verifies the current formula. —Quantling (talk | contribs) 20:04, 28 August 2024 (UTC)[reply]

Fixed with a citation! Scott (talk) 23:56, 28 August 2024 (UTC)[reply]

@Sctfn thank you for fixing the transfinite recursion for natural multiplication. However, I think the latest change to natural addition is wrong. It needs to be "strictly greater than" rather than "greater than or equal", try an example such as 2 + 2. The cited source (Altman) writes sup′ with a prime, which I suspect is the author's way of indicating "strictly greater than". —Quantling (talk | contribs) 13:18, 29 August 2024 (UTC)[reply]

Fair enough. I misunderstood the paper Scott (talk) 16:19, 29 August 2024 (UTC)[reply]

@Unseemly Levity et al. The way I see it, there are multiple ways to define addition on ordinals. The second is "natural addition" and third is "nim addition". What do we call the first? I am thinking it should be called "ordinary" because it isn't either of the other two. However, the article currently calls it "ordinal". I find that "ordinal" could be used to describe all three. So, what do you think about "ordinary" or "standard" instead? —Quantling (talk | contribs) 02:23, 16 November 2025 (UTC)[reply]

It's the standard addition (and multiplication) on the ordinals, what you mean when you say addition (multiplication) without further qualification. The "natural" and "nim" versions need to be called out specially. Honestly they're not very important. --Trovatore (talk) 02:37, 16 November 2025 (UTC)[reply]

I disagree both in terms of following standard terminology and in terms of good exposition. "Ordinary" is not widely used for this, and "ordinal" is. It's also confusing to use "ordinary" for this, because this same article also uses "ordinal" to mean integer arithmetic in contrast to ordinal arithmetic (a much more normal way to use that term), so it is confusing to use the same word to mean ordinal arithmetic in contrast to natural or nim arithmetic. Unseemly Levity (talk) 04:42, 16 November 2025 (UTC)[reply]

Are there any applications of this type of math? -- Beland (talk) 02:25, 15 March 2026 (UTC)[reply]

See Talk:Veblen function#What are they good for?. JRSpriggs (talk) 16:54, 15 March 2026 (UTC)[reply]

Would it be accurate to say, then, that they are used to solve the halting problem for specific programs or algorithms? Do you have a citation that documents this? -- Beland (talk) 16:57, 16 March 2026 (UTC)[reply]

Are you really asking about applications of ordinal arithmetic, or is your question more about what ordinals are good for? Ordinals are sort of the "backbone" of modern set theory. You can't understand set theory without understanding ordinals. Part of understanding them is knowing how they behave under certain obvious operations. --Trovatore (talk) 17:57, 16 March 2026 (UTC)[reply]

This article is about ordinal arithmetic, so it should say what ordinal arithmetic, specifically, is used for. Also saying what ordinal numbers are used for would probably be helpful, if not necessary, as background.

It's definitely helpful and interesting to know what other types of math are built on this type of math, and that should be added to the article, but in asking my question I was thinking about real-world applications. -- Beland (talk) 18:07, 16 March 2026 (UTC)[reply]

I am not aware of any direct applications to the physical world. --Trovatore (talk) 19:19, 16 March 2026 (UTC)[reply]

I agree that it would make sense to spend a few words in the opening paragraph (maybe even the opening sentence) briefly recapping what an ordinal number is. --Trovatore (talk) 19:22, 16 March 2026 (UTC)[reply]

It's very practical. For instance, if we're both farmers and if you have ω² sheep and I have ω3 + 7 sheep and all of your sheep are less than all of my sheep then, together, we have ω² + ω3 + 7 sheep. —Quantling (talk | contribs) 14:28, 16 March 2026 (UTC)[reply]

To @Beland:: I am not talking about trying to tell whether some arbitrary given process will halt. I am talking about using ordinals to modify the design of new processes to ensure that they halt without unduely restricting what they can do by putting them on a clock. To this end, the destructive arithmetic operations (e.g. subtraction, division, roots, logarithms) are not helpful, since googolplex minus 1 divided by two, and so forth will not get you to zero in a realistic time. Only the constructive operations of addition, multiplication, exponentiation, etc. are useful. Removing them or replacing them with combinations of smaller such operations will be effective. JRSpriggs (talk) 12:08, 18 March 2026 (UTC)[reply]

Ah, OK. Can this be documented with references? -- Beland (talk) 16:11, 18 March 2026 (UTC)[reply]

Wikipedia articles need to be accessible to a general audience. I went to MIT and I'm confused at the end of the first sentence and lost by the end of the second. Some suggestions:

• For about an hour, I was confused because I didn't know there was a difference between ordinal numbers (which it seems are infinite sets?) and ordinal numerals (1st, 2nd, 3rd). It would be helpful to clarify in the first sentence that this math only takes place on infinite sets, if that's correct.
• Add non-technical material, such as applications (discussed in previous talk section) and history of the field.
• It's unclear why addition, multiplication, and exponentiation are the "usual" operations. Subtraction and division are more common in everyday arithmetic than exponentiation.
• It would be helpful to explain in words the difference (and why we need both?) between addition and natural addition, and multiplication and natural multiplication. I see there are sections discussing both, but without more background, they are mathematical gobbledygook.
• The "Cantor normal form" section is difficult to understand because it seems to be expressed in abstract mathematical terms. It would be helpful to have a concrete example, like how would I write the ordinal number equivalent of "3rd" or something simple like that?

-- Beland (talk) 18:03, 16 March 2026 (UTC)[reply]

These are actually fair points. I think we can address them without the use of that template.

The thing is that most of these really should be addressed at ordinal number rather than here. This article is sort of an appendix to that one, covering some important but fairly dull aspects of that topic.

Addition and multiplication would be well-served by a couple of simple images, I think. Exponentiation is harder.

The "natural" operations are not very important. I don't think it would be that much of a loss to remove them, but they probably are "notable", so I'm not sure exactly what to do with the content. It is a bit unfortunate that a reader can read this article and be confused about "why we need both" as you say.

"Subtraction" and "division" don't have any completely standard meaning on the ordinals. I'm not sure exactly how to phrase that, for a couple of reasons. First, it's always hard to find references for something not being standard. Second, there actually is a kind of important sort-of-subtraction operation; it's just that it's not usually called out explicitly. It's just implicitly used from time to time that, for ${\displaystyle \alpha <\beta }$, there's a unique ${\displaystyle \gamma }$ such that ${\displaystyle \alpha +\gamma =\beta }$. --Trovatore (talk) 18:17, 16 March 2026 (UTC)[reply]

If "there is no standard definition" is a fair summary of reliable sources, I think it's reasonable to just write that even if no source says so explicitly. To document that I'd cite sources that give differing definitions, or sources that talk about other operations but not these, if you're saying they aren't widely used, or something. -- Beland (talk) 18:20, 16 March 2026 (UTC)[reply]

@Trovatore: You removed {{too technical}} from the article, but the problem identified by this tag has not been resolved. The purpose of the tag is to recruit assistance from other editors and readers who might not yet be editors, but who might have the specialized knowledge required to fix the problem. Some editors also browse categories listing articles that have problems, and pick up articles on topics they enjoy editing. -- Beland (talk) 03:32, 17 March 2026 (UTC)[reply]

If you want to make articles on which readability has been raised as an issue visible in categories, you can put the template on the talk page. In my opinion this template should appear only on talk pages. It's of extremely limited value to readers (as opposed to editors).

It's your opinion that the article is too technical. We do not have to satisfy you personally that it is not too technical just to avoid having that template appear on the page.

That said, I agree that there are issues that should be addressed, but I don't agree that the template adds anything to the discussion. --Trovatore (talk) 03:51, 17 March 2026 (UTC)[reply]

That's not how this template is intended to be or actually is used; it's on over 3,400 articles, not on talk pages. The place to debate the use of the template is on its talk page or a request for deletion. -- Beland (talk) 05:12, 17 March 2026 (UTC)[reply]

Be that as it may, I don't agree that this article is too technical. Yes, you have identified some problems with the writing. But the article is pretty basic set theory for the most part.

The biggest problem you've flagged is that it's not reasonable to try to read the article until you know what an ordinal is. We don't want to import the whole discussion of that from ordinal number. On the other hand we don't really want to merge this entire article there, because it's too long and detailed on a subject that isn't extremely important to that topic. In my opinion it could be cut down fairly drastically without really losing too much but even so there's probably more content than fits comfortably at the other article.

I'll see if I can come up with a sentence or two to add to the opening para. --Trovatore (talk) 19:04, 17 March 2026 (UTC)[reply]

I have requested a third opinion on the tag question. -- Beland (talk) 19:21, 17 March 2026 (UTC)[reply]

Without providing a "formal" 3rd opinion: it's clearly appropriate to leave a maintenance tag on a page if the issue hasn't been addressed yet. WP:CLEANUPTAG tells us that the purpose of maintenance tags is "to foster improvement of the encyclopedia by alerting editors to changes that need to be made. Cleanup tags are meant to be temporary notices that lead to an effort to fix the problem."

But, that's clearly not the crux of the disagreement here. WP:ONEDOWN suggests "A general technique for increasing understandability is to consider the typical level where the topic is studied and write the article for readers who are at the previous level." I would expect ordinal arithmetic to be discussed briefly in an undergraduate set theory course (or maybe a real analysis course?), so we should hope that the article and especially the lede is comprehensible to a reader with secondary or minimal post-secondary maths exposure.

Overall, I think it's pretty close, to be honest. I think it would be reasonable to maintain the tag until a slightly longer lede is written or a Background section added in order to provide more context. I do think Trovatore shouldn't have reverted the tag a second time; no change has been made to the lede, and Beland added significant comments to the Talk page as you requested! On balance, I would restore the tag until the lede can be copy-edited. Cheers, Suriname0 (talk) 21:02, 19 March 2026 (UTC)[reply]

Tag restored; thanks for your time considering this. -- Beland (talk) 21:04, 19 March 2026 (UTC)[reply]

This article is awfully technical, but I wouldn't call it {{too technical}}.

WP:TECHNICAL says that Wikipedia articles should be written for the widest possible general audience. It's probably possible to write a less technical introduction to the subject that would be more accessible. The problem is that I don't know if there are reliable sources that would support it. Ordinal arithmetic is rarely discussed in detail outside of set theory textbooks, and those textbooks don't have anything more accessible than what's currently in the article.

Sure, it would be nice to talk about the history of the subject, or its applications, or explain why we don't have subtraction and division. It'd be nice to explain why ordinal arithmetic and cardinal arithmetic are the same for natural numbers. However, I'm not familiar with any sources that discuss any of this. Trovatore, do you know such sources?

As for subtraction and division specifically, the problem is that unless we have a source noting their absence, this feels like original research. We'd be saying something like "unlike arithmetic of natural numbers, arithmetic of ordinal numbers has no operations of subtraction and division", which is synthesis.

Also, as Trovatore already mentioned, an article about the arithmetic of ordinal numbers can't reasonably be made accessible to people who are unfamiliar with the concept of ordinal numbers. I know that the similarity between "ordinal number" and "ordinal numeral" is really confusing (especially considering that they're also similar concepts), but I don't know what we can do about it. We can't say "hey, note that ordinal numbers aren't the same as ordinal numerals" in every single article where one of these subjects is mentioned.

This article also has other issues that affect readability. Parts of it are written like a {{textbook}}. The definitions that use transfinite induction have unnecessarily technical notation (S(β) instead of β+1, union instead of supremum). The nonstandard operations shouldn't be in the lead, and the "natural operations" should probably be in their own article (or, if they're insufficiently notable, deleted outright). Cantor normal form should also ideally be in its own article. There should be more examples of specific arithmetic calculations.

In short, this is a technical article and it has much room for improvement, but I don't think it can be made less technical. Streded (talk) 06:13, 28 March 2026 (UTC)[reply]

Checking a few textbooks should be enough to say something like "standard textbooks for ordinal arithmetic do not define the operations of subtraction and division" without synthesis.

I am finding sources for the history of the field, though you may need to look at the history of related concepts. See for example: [1]

I don't see any reason why we can't give a one-sentence definition of ordinal numbers in this article, and doing that substantially increases its accessibility. If doing that in every article that talks about ordinal numbers likewise improves them, then we should do it.

I'm looking at Special:Whatlinkshere/ordinal number, and it looks like there are less than 1,300 articles that do. Most of those I think are just due to nav templates. Spot-checking to see how many would need a thumbnail definition reveals a problem that I think actually requires us to define the term. "Ordinal number" is actually used as a synonym for ordinal numerals like "1st", "second", etc. This explains why I was confused by the terminology. I'm pretty sure I was not aware that there are two different concepts before reading this article, much less that the same term is used for both of them.

Many articles are I think linking to the wrong article, like Ordinal date and ISO 8601. Some articles like Natural number use the term "ordinal number" to mean both the simple number and the concept in set theory, and link to different articles on each use. This is horribly confusing!

We could probably use an audit of all incoming links and textual references. One thing that might help is moving this page to Ordinal number (set theory) and making Ordinal number a disambiguation page. That would force incoming links to consciously choose the correct link, now and in the future, while still getting readers coming from the search function to the right places.

I can do a database scan to get a complete list of affected articles if that would be helpful, but you can quickly see problems by skimming references in the results from this search.

-- Beland (talk) 07:13, 28 March 2026 (UTC)[reply]

As it turns out, one of the textbooks currently referenced in the article (Cardinal and Ordinal Numbers, pages 278–279) does talk about the sort-of-subtraction that Trovatore mentioned, and actually calls it "the difference ${\displaystyle \alpha -\beta }$" in the case that ${\displaystyle \alpha >\beta }$. We should probably mention that.

Beland, that's a useful reference. For example, it mentions that arithmetic of ordinals and of cardinals are the same for finite numbers. It's good to know that such sources exist.

As for ordinal numbers vs. ordinal numerals... I'm now realizing why this is horribly confusing, and why the links in some articles seem to point to the wrong target. The simple concept of "ordinal number" that you're familiar with (that is, finite ordinal position) is actually covered by neither article. Ordinal number says that Ordinal numeral covers it, and vice versa. The one place where it seems to be covered is Natural number § Position in a sequence, and it is covered poorly there.

I think our next course of action is to determine where the concepts of "natural number as a cardinal" and "natural number as an ordinal" should be covered, since the current answer appears to be "nowhere". I should probably open a discussion about this in WT:WPM, but in all honesty, this looks like a lot of effort and I don't feel like doing it. Ugh. Streded (talk) 10:24, 28 March 2026 (UTC)[reply]

@Trovatore: You wrote "Addition and multiplication would be well-served by a couple of simple images". What kind of images did you have in mind? The current ${\displaystyle \omega \cdot 2}$ image can also be used to illustrate addition ${\displaystyle \omega +\omega }$ (maybe illustrations of ${\displaystyle \omega +5}$ and ${\displaystyle 5+\omega }$ are helpful also), and I could extend it to a picture of ${\displaystyle \omega \cdot \omega }$, but I don't have any ideas beyond that. Suggestions are welcome. - Jochen Burghardt (talk) 08:33, 17 March 2026 (UTC)[reply]

I had in mind sort of "generic" images, one for addition showing an ordinal before another, and one for multiplication showing a rectangle where maybe you read it in row-major order. I hadn't noticed when I wrote that that there were already images sort of along those lines already. But they are kind of buried. We might want to put images higher up. --Trovatore (talk) 20:37, 17 March 2026 (UTC)[reply]

Response to third opinion request:

Just an FYI that I've closed this request. It was stale, and even if you were to refile, it currently appears that there are more than two editors involved in any case. If there's still a dispute on this matter, I'd advise you to either start a new thread specifically regarding the dispute in question, or pursue other forms of dispute resolution. Cheers! DonIago (talk) 23:46, 28 March 2026 (UTC)[reply]

Ah, thanks. I guess the editor who already supplied a third opinion forgot to close it. -- Beland (talk) 01:38, 29 March 2026 (UTC)[reply]

FYI, I intentionally didn't close it because I'm not a regular WP:3O user, so I wanted to leave it open for more experienced users to handle the process properly. @Doniago: thanks for clerking. Suriname0 (talk) 16:58, 29 March 2026 (UTC)[reply]

Since addition is not commutative, one must distinguish between subtraction from the left and subtraction from the right. Subtraction from the left could be defined by:

${\displaystyle \alpha -_{L}\beta =\inf\{\gamma :\beta +\gamma \geq \alpha \}.}$

Subtraction from the right could be defined by:

${\displaystyle \alpha -_{R}\beta =\inf\{\gamma :\gamma +\beta \geq \alpha \}.}$

If β ≥ 1, division from the left could be defined by:

${\displaystyle \alpha \div _{L}\beta =\inf\{\gamma :\beta \cdot \gamma \geq \alpha \}.}$

If β ≥ 1, division from the right could be defined by:

${\displaystyle \alpha \div _{R}\beta =\inf\{\gamma :\gamma \cdot \beta \geq \alpha \}.}$

If β ≥ 2, logarithm (from the left) could be defined by:

${\displaystyle \log _{\beta }\alpha =\inf\{\gamma :\beta ^{\gamma }\geq \alpha \}.}$

If β ≥ 1, root (from the right) could be defined by:

${\displaystyle {\sqrt[{\beta }]{\alpha }}=\inf\{\gamma :\gamma ^{\beta }\geq \alpha \}.}$

OK? - JRSpriggs (talk) 00:09, 29 March 2026 (UTC)[reply]

These operations lack many of the desirable properties of the operations on the real numbers. In particular, they are not inverses of the corresponding constructive operations. One could choose to limit the domain to the case when you have equality to α. Or we could come at the value from the other side:

Subtraction from the left could be defined by:

${\displaystyle \alpha -_{L}\beta =\sup\{\gamma :\beta +\gamma \leq \alpha \}.}$

Subtraction from the right could be defined by:

${\displaystyle \alpha -_{R}\beta =\sup\{\gamma :\gamma +\beta \leq \alpha \}.}$

If β ≥ 1, division from the left could be defined by:

${\displaystyle \alpha \div _{L}\beta =\sup\{\gamma :\beta \cdot \gamma \leq \alpha \}.}$

If β ≥ 1, division from the right could be defined by:

${\displaystyle \alpha \div _{R}\beta =\sup\{\gamma :\gamma \cdot \beta \leq \alpha \}.}$

If β ≥ 2, logarithm (from the left) could be defined by:

${\displaystyle \log _{\beta }\alpha =\sup\{\gamma :\beta ^{\gamma }\leq \alpha \}.}$

If β ≥ 1, root (from the right) could be defined by:

${\displaystyle {\sqrt[{\beta }]{\alpha }}=\sup\{\gamma :\gamma ^{\beta }\leq \alpha \}.}$

But the supremum may fail to be a member of the set, so the sum, product, or power may still end up being larger than α as well as smaller. One could choose to exclude from the domain the cases when the set is empty rather than taking the supremum of the empty set to be zero.

OK? - JRSpriggs (talk) 14:02, 29 March 2026 (UTC)[reply]