Talk:Ordinal arithmetic

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)

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

... 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)

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

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)

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

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)

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

@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)

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

@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)

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)

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)

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

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

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)

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)

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)

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

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)

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)

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)

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)

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)

@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)

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)

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)

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)

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

@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)

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)