Talk:Ordinal arithmetic

Creation

I created this article by extracting the section "Arithmetic of ordinals" from the article "Ordinal number". JRSpriggs 09:34, 15 March 2006 (UTC)

Warning about notation

I inserted the following sentence that was removed

"In practice, unless it is clear from context that an expression such as $2^{\omega }$ is intended to denote ordinal exponentiation, it is read to mean the corresponding cardinal exponentiation (in this case, $2^{\aleph _{0}}$)."

This was motivated by a discussion here. What is the objection? Is the concern about the other possible meanings of 2^ω? CMummert · talk 13:50, 31 March 2007 (UTC)

I just disagree with your statement (especially in an article on ORDINAL arithmetic) that one should presume that 2^ω really means $2^{\aleph _{0}}$. I always presume that it means what it says, i.e. the ORDINAL exponential. JRSpriggs 06:40, 1 April 2007 (UTC)

In my field, 2^ω is used either to denote $2^{\aleph _{0}}$ or to denote the set of functions from 2 to ω; the number of times I have seen it used to indicate ordinal exponentiation could be counted on my fingers. That's why the "unless it is clear from context" phrase. CMummert · talk 12:23, 1 April 2007 (UTC)

I think you meant functions from ω to 2. Of course, context will make it clear in almost all cases. But if other aspects of the context do not make it clear, then the remaining part of the context, i.e. whether the symbols being used are symbols for ordinals or for cardinals, should be allowed to decide.

I noticed that you started to remove the italics from the Greek letters. I am now aware that not using italic Greek letters is the rule here. But when I worked on this article over a year ago, I thought that it was the other way around. So most of the Greek letters are italicized. Do you think that we should bother to remove all the italics from Greek letters in this article and also in ordinal number and large countable ordinal? JRSpriggs 06:27, 2 April 2007 (UTC)

No, I don't think its worth it. I had second thoughts after editing it that I shouldn't have removed the italics, since the entire article uses them. I'll put them back, so the article is self-consistent. CMummert · talk 11:50, 2 April 2007 (UTC)

Is this related to the notation ^ω2, which I've seen in regards to functions (mappings) of sets? — Loadmaster 21:30, 11 June 2007 (UTC)

Yes. Some people use ^ω2 and some use 2^ω for the set of functions. The cardinality of that set is $2^{\aleph _{0}}$. If you restrict the functions to those with finite support, then its order type is the ordinal 2^ω. JRSpriggs 05:18, 12 June 2007 (UTC)

Cantor normal form

The article says:

Every ordinal number α can be uniquely written as $\omega ^{\beta _{1}}c_{1}+\omega ^{\beta _{2}}c_{2}+\cdots +\omega ^{\beta _{k}}c_{k}$ where k is a natural number, $c_{1},c_{2},\ldots ,c_{k}$ are positive integers, and $\beta _{1}>\beta _{2}>\ldots >\beta _{k}$ are ordinal numbers

and also

$\omega ^{\beta }c+\omega ^{\beta '}c'$ is $\omega ^{\beta '}c'$ if $\beta '>\beta$

Does that not imply that

$\omega ^{\beta _{1}}c_{1}+\omega ^{\beta _{2}}c_{2}+\cdots +\omega ^{\beta _{k}}c_{k}=\omega ^{\beta _{1}}c_{1}$

 ? --Michael C. Price ^talk 12:05, 5 October 2007 (UTC)

The first equation is for ordinals, but the second is for cardinals. For example ω + ω is not ω in the sense of ordinal addition. — Carl (CBM · talk) 13:23, 5 October 2007 (UTC)

If that is the case the article needs a major check over, since the 2nd equation reads in full:

The Cantor normal form also allows us to compute sums and products of ordinals: to compute the sum, for example, one needs merely know that $\omega ^{\beta }c+\omega ^{\beta '}c'$ is $\omega ^{\beta '}c'$ if $\beta '>\beta$

which specifies "ordinal" not "cardinal". And $\omega$ is the symbol for an ordinal. --Michael C. Price ^talk 13:29, 5 October 2007 (UTC)

There is also the commutativity issue; I misunderstood where your confusion was and didn't read far enough in the article. Ordinal addition is not commutative. For example $\omega +\omega \cdot \omega$ is ω followed by ω copies of ω, which is the same as just $\omega \cdot \omega$, so the sum collapses. But if you put the sum in the other order, $\omega \cdot \omega +\omega$ then it does not collapse. In the article text, the Cantor normal form is set up so that the exponents decrease from left to right, so that the sum does not collapse. When you add two ordinals in Cantor normal form, you can simplify that sum by collapsing binary sums with the first term of the second ordinal. — Carl (CBM · talk) 13:55, 5 October 2007 (UTC)

Ah! I overlooked the commutativity issue. I shall reread it. Thanks. --Michael C. Price ^talk 14:16, 5 October 2007 (UTC)

Another question. I'm okay with:

$\alpha n=\omega ^{\beta _{1}}c_{1}n+\omega ^{\beta _{2}}c_{2}+\cdots +\omega ^{\beta _{k}}c_{k}$, if n is a non-zero natural number.

but doesn't that imply

$\alpha \omega =\omega ^{\beta _{1}+1}+\omega ^{\beta _{2}}c_{2}+\cdots +\omega ^{\beta _{k}}c_{k}$.

rather than just

$\alpha \omega =\omega ^{\beta _{1}+1}$.

? i.e. what is the mechanism that cancels the extra RHS terms in the limit as n -> ω ? (Does n have to be in the final term in the sum for this limit substitution to work?) --Michael C. Price ^talk 11:55, 10 October 2007 (UTC)

So $\alpha \omega =\sup\{\alpha n:n<\omega \}$. Since $\alpha n<\omega ^{\beta _{1}+1}$ for each n, this means $\alpha \omega \leq \omega ^{\beta _{1}+1}$. But also $\omega ^{\beta _{1}+1}=\omega ^{\beta _{1}}\omega =\sup\{\omega ^{\beta _{1}}n:n<\omega \}$. Since $\alpha n\geq \omega ^{\beta _{1}}n$ for all $n<\omega$, this means that $\alpha \omega \geq \omega ^{\beta _{1}+1}$ (take the sup of both sides). Since both inequalities hold, $\alpha \omega =\omega ^{\beta _{1}+1}$. I'm not sure if there is a good way to visualize the infinite product, except to think of the fact that any time there is a copy of $\omega ^{\beta +1}$ in the product it will absorb all copies of $\omega ^{\gamma }$ for $\gamma <\beta$ that appear to its left. In the product $\alpha \omega$ everything in the infinite product is followed by at least one copy of $\omega ^{\beta _{1}}$, so everything gets absorbed. — Carl (CBM · talk) 14:05, 10 October 2007 (UTC)

Thanks, that was most helpful. --Michael C. Price ^talk 09:42, 11 October 2007 (UTC)

Natural operations

I've added a "disputed" banner to this material. The definition of the natural sum, as the least upper bound of the set of ordinals with a certain property doesn't even make sense, because that set is unbounded.

The section is also unreferenced. Perhaps if someone is convinced that it can be salvaged, the "in need of attention from an expert" banner would be best, but in the meantime readers need to be warned of the inaccuracies. 67.150.246.120 (talk) 13:39, 9 December 2008 (UTC)

The definition makes perfect sense, the set is trivially bounded by $|S\cup T|^{+}$. — Emil J. 13:50, 9 December 2008 (UTC)

Aha, now I got it: the wording "well-orders which extend this partial order" is actually ambiguous. I'll clarify it. — Emil J. 13:58, 9 December 2008 (UTC)

Thanks, I guess that interpretation of "extension" should have occurred to me as a possibility. 67.150.252.101 (talk) 09:39, 10 December 2008 (UTC)

Tetration

Is there any reason not to continue the sequence of ordinal addition, multiplication, and exponentiation also with ordinal tetration? Indeed, it seems to me that the recursive definition of Knuth's up-arrow notation would work also in the ordinal case.

Is it then furthermore the case that $\varepsilon _{0}=\omega \uparrow \uparrow \omega$? I suppose it would have to be, since $\omega ^{\omega \uparrow \uparrow \omega }=\omega \uparrow \uparrow (1+\omega )=\omega \uparrow \uparrow \omega$. 90.230.192.94 (talk) 12:16, 15 January 2009 (UTC)

See ε₀ where tetration of ordinals is discussed. I have seen very little written on this subject; perhaps because it becomes essentially trivial. If one wants to describe large countable ordinals, there are better ways to do so. JRSpriggs (talk) 07:59, 16 January 2009 (UTC)

Exponential — ordering of P=B^E

However, if we try to use any effectively defined ordering on this set [the set of all functions from one infinite well-ordered set to another infinite well-ordered set], we find it is not well-ordered.

Sligocki said "All sets may be well ordered". This is true (if the axiom of choice holds) but irrelevant since it disregards the words "effectively defined".
EmilJ said "This makes no sense. Effective is a synonym for computable, and a computable ordering has by definition countable domain.". This is false. For example, the set of all functions from E to B has an effective total ordering defined as follows. Suppose f and g are distinct functions from E to B. Find the smallest element e of E where f(e)≠g(e). Then let f<g just in case f(e)<g(e). This example ordering differs from the definition of the exponential in the article in two ways: (1) it is not restricted to functions with finite support; and (2) the ordering on the exponent E enters into the definition in a reversed way. JRSpriggs (talk) 22:05, 4 November 2009 (UTC)

Hi JRSpriggs, sorry if my comment seemed adversarial, just worried about fuzzy wordings, what are effective orderings. Even if this can be precisely defined, it seemed a bit arbitrary. To me the clearest reason for defining exponentiation this way is so that it behaves well with limits. i.e. so that $\alpha ^{\lim \beta _{n}}=\lim \alpha ^{\beta _{n}}$ if I understand this correctly. Cheers, — sligocki (talk) 22:46, 4 November 2009 (UTC)

Define your notion of effective, JRSpriggs. Normally in mathematics, effective is a synonym of algorithmically computable, and the latter applies to subsets or partial functions on ω (or something equivalent, like finite binary strings). You cannot have an effective relation on ^ωω. If you rely on another definition of effective which allows uncountable domains, you have to state it properly. — Emil J. 11:04, 5 November 2009 (UTC)

In descriptive set theory, effective is generally synonymous with lightface; that is, not requiring an arbitrary real-number parameter. This is coextensive with computable only at the very bottom of the hierarchy. --Trovatore (talk) 11:10, 5 November 2009 (UTC)

And what is the thing you are talking about that does not require an arbitrary real-number parameter? I mean, I can understand what is, say, a lightface $\Sigma _{2}^{1}$ ordering, but what is a "lightface ordering"? Anyway, this should be properly explained in the article, if it is the intended meaning. Somehow I don't think that readers struggling with the definition of ordinal exponentiation can be expected to be familiar with descriptive set theory. — Emil J. 12:00, 5 November 2009 (UTC)

Now that I thought about it: whatever the actual result is, why is it lightface rather than boldface? I find it hard to believe that a single real parameter would help to define a well-ordering of all reals; it's not as if the well-order is continuous or something like that so that it could be encoded by a real. — Emil J. 12:58, 5 November 2009 (UTC)

I thought "effective" here was supposed to mean "definable in the language of set theory without parameters". I don't think this is what most people would mean by "effective". Of course there are definable orderings under V=L, but I was ignoring that. If this is not what was intended, then I agree the text is not clear enough. — Carl (CBM · talk) 17:15, 5 November 2009 (UTC)

(There being no well-order of the reals definable in the language of set theory with real parameters is also consistent with ZFC.) I think that the discussion above sufficiently demonstrates that even if this is what was intended, the text is not clear enough. — Emil J. 16:26, 7 November 2009 (UTC)

This is exactly what I'm talking about. This whole discussion is way over the level of the article (I don't even know what lightface vs. boldface means). Rather than making and proving some high level theorem about well-ordering the continuum, I think that this article should stick to simpler arguments. For example, the ordering that User:JRSpriggs suggested is the naively logical one to pick, so we could say that and then show that it is not well-ordered. Alternatively, I still think that it is worth pointing out that we would also like a definition that is the limit of lower terms. Cheers, — sligocki (talk) 17:05, 5 November 2009 (UTC)

As for Sligocki's suggested wording, it amounts to making continuity of the mapping from the exponent to the power a defining characteristic, unlike the case for addition and multiplication of well-ordered sets where it is a consequence. Since his suggestion results in a conflict with the definition of exponential for cardinal numbers, I feel that a stronger justification is needed. Essentially the proper justification is that there is no other way to do it which relies on the orderings rather than attributes of the model of set theory being used. Unfortunately, I do not know how to reword the sentence to make it clearer to novices as EmilJ desires. JRSpriggs (talk) 16:23, 9 November 2009 (UTC)

It's not clear to anybody, not just novices. Above you have three logicians in the discussion, and three different answers to what "effective" might mean in the context. Note that we still do not know which (if any) of them is the one you want, since you carefully avoid to tell us. — Emil J. 16:41, 9 November 2009 (UTC)

OK, originally it was just intuitive for me, but since you press the point I will try to be precise. It should be absolute, that is, Δ⁰₁ in the language of set theory. In other words, there should be two provably (in ZF) equivalent forms of the definition: one with a single unbounded universal quantifier followed by as many bounded quantifiers (of either kind) as needed; the other with a single unbounded existential quantifier followed by as many bounded quantifiers as needed. And there should be no free variables (parameters) other than the base, exponent, and power. The result being that the power would not change when one passes to a transitive super-model or a transitive sub-model which contains the base and the exponent. To ensure that the power does not depend on any peculiarities of the base and exponent other than their orderings, we require that they be given as Von Neumann ordinal numbers while the power is just the set of functions from the exponent to the base together with an ordering on it. JRSpriggs (talk) 03:42, 19 November 2009 (UTC)

(sandbox) tables are cool

Property	Description or counter-example
Identity	α + 0 = 0 + α = α
Associativity	(α + β) + γ = α + (β + γ)
Strict increase in the right argument	α < β implies γ + α < γ + β
Strict increase in the left argument	0 < 1 but 0 + ω = 1 + ω
Non-strict increase in the left argument	α < β implies α + γ ≤ β + γ
Left-cancellation	α + β = α + γ implies β = γ
Right-cancellation	1 + ω = 0 + ω = ω but 1 ≠ 0

> Any chance to see here some official work on the property of some ordinals like integration and derivation ? — Preceding unsigned comment added by StefanoMaruelli (talk • contribs) 18:27, 28 November 2016 (UTC)

Left-cancellation only works if α>β, and strict increase in the right argument only when γ>β. 80.98.179.160 (talk) 16:14, 4 January 2018 (UTC)

To 80.98.179.160: You are mistaken. JRSpriggs (talk) 19:44, 4 January 2018 (UTC)

Axiomatic definitions?

Can axiomatic definitions for addition, etc be added to the article? I believe that they'd help clarify matters. For example, Conway, in "On Numbers and Games" provides very nice, clean axiomatic definitions for order, addition, etc. which can then be used to give fairly easy proofs that e.g. 0<1 and that a < a+1 and that a<a.2 etc. (something I can't do with this article). However, his system is not isomorphic to the ordinals, and so I think it would be interesting to compare these two, to see how they compare. (OK, I guess this is two requests: 1) provide axioms, and 2) compare to Conway's system).

Well, maybe a third request: Conway explicitly defines addition to be commutative, but none-the-less is able to construct a system of infinite numbers that look a lot like the usual ordinals given here, although there are important differences: besides commutativity, he is able to define ω -1 and prove that ω-1< ω etc. but is unable to construct 2^ω. So the third request here is: explain why Conway's system isn't as popular/widely used. Is it due to in-grained tradition, or due to underlying shortcomings? linas (talk) 17:08, 26 November 2011 (UTC)

I don't know of any definitions for ordinal arithmetic that are as "neat" as the ones Conway comes up with for arithmetic on his "numbers". It would be interesting to know if they exist. Not particularly important from my POV, more of a curiosity, but interesting.

As to "shortcomings", I wouldn't use that word. There's nothing wrong with Conway's numbers, but they just aren't the right tool for the job, if the job is one for which ordinals are currently used. Most obviously, Conway's numbers are not wellordered, whereas in some sense the most important fact about the ordinals (or even in some sense the only fact about them) is that they are wellordered. --Trovatore (talk) 20:32, 26 November 2011 (UTC)

Large Countable Ordinals

Just added a short section describing these as I thought it needed to be mentioned, with link to the main page about them. Robert Walker (talk) 09:28, 28 July 2012 (UTC)

Ordinal Exponentiation

I've had a go at making this section more concise. Also seemed necessary to put a motivating paragraph a the start of this section as for newbies to ordinal exponentiation it can seem unintuitive and hard to understand why it is done this way. I'm sure the section can do with more work, it's a tricky subject to explain clearly. Robert Walker (talk) 10:18, 28 July 2012 (UTC)

"The so-called "natural" arithmetical operations retain commutativity at the expense of continuity."

Is this supposed to means something other than what I think it means? Because ordinal arithmetic is not commutative. — Preceding unsigned comment added by 109.149.188.194 (talk) 17:36, 9 January 2013 (UTC)

The "natural" operations ARE commutative, like arithmetic on natural numbers. JRSpriggs (talk) 08:20, 10 January 2013 (UTC)

I do somewhat object to the "value-laden" wording chosen in this section. We are told that these operations have the "advantage" of associativity and commutativity, at the "expense" of continuity. Why are associativity and commutativity an advantage? Why is losing continuity an expense? This is not really explained.

The usual operations are not defined with an aim to make them continuous, especially, and the fact that they are not commutative is not felt as a loss — it's just what happens to be true of them. They arise naturally in contexts where ordinals are applied; their properties are discovered, not chosen. --Trovatore (talk) 09:13, 10 January 2013 (UTC)

Both the ordinary and "natural" operations of addition and multiplication are associative. So that is not a difference.

To Trovatore: Surely it must be obvious, even to you, that having additional properties is advantageous for someone doing arithmetic calculations. JRSpriggs (talk) 16:35, 10 January 2013 (UTC)

I agree with JRSprigg's brother comment. Furthermore, I disagree with your comment that the ordinary operations are not defined with an aim to make them continuous. That depends on what definition you use! If you define them in terms of a new order on the disjoint union or the Cartesian product, then yes, they are not defined so as to make them continuous. But if one uses the recursive definition, then they certainly are defined so as to make them continuous (in the second operand, anyway). And indeed, without this motivation, why would one define ordinal multiplication to use reverse lexicographic order, where the second coordinate is compared first? It is because this way it matches up with the recursive definition. And this is leaving out exponentiation, which is most naturally defined recursively; ordinal exponentiation isn't a very useful operation on other order types, if I'm not mistaken. Sniffnoy (talk) 18:33, 10 January 2013 (UTC)

Oh come on. The definitions are not some arbitrary thing, "let's see what we can do with ordinals that will have nice properties". These are operations that arise naturally in applications of ordinals. They have the properties that they have. Granted, continuity is intimately connected with the way they arise, but it wasn't like people said, "hey, let's see what we can define that would be continuous". --Trovatore (talk) 21:31, 10 January 2013 (UTC)

Once again, I think you are failing to take the recursive definition seriously. The recursive definition isn't building in continuity just because continuity would be nice; it's building in continuity because doing so the obvious way to take the usual Peano arithmetic recursions for addition, multiplication, and exponentiation, and extend them to a setting in which some inputs can only be expressed as limits of previous inputs, rather than successors of previous inputs. Again -- both the recursive and the direct definitions are commonly used definitions; neither is particularly complicated. You seem to be speaking as if the direct definition were the one true definition. To someone who prefers the recursive definition, continuity is indeed built-in. But I'm getting the idea this is besides the point -- well, I'll address that in a cousin comment. Sniffnoy (talk) 04:59, 17 January 2013 (UTC)

OK, I have to disagree with the statement "the fact that they are not commutative is not felt as a loss". We refer to operations as "addition" and "multiplication" when they have properties reminiscent of the archetypical examples of these. As such, these names carry with them certain expectations. The more properties lost, the worse the fit. Now I don't think any mathematician would think that the name "multiplication" carries the expectation of commutativity; but I would say that the name "addition" does carry the expectation of commutativity. And calling one operation "addition" and another "multiplication" in the same context carries the expectation of distributivity on both sides. Not only does losing properties mean that it is harder to prove things, or harder to work with, it means more mistakes you're going to make when you accidentally use a property that the operation doesn't have, because it was named in a way that suggested it does. I mean, OK -- I guess I can't really argue against the statement "the fact that they are not commutative is not felt as a loss" without, like, poll data or something, but it seems just an absurd statement I don't know what to make of it, really. It honestly is not a point of view I'd heard expressed before. (I think may have heard it said that not having commutativity of multiplication is not a loss, in certain contexts. But multiplication isn't expected as much to be commutative anyway.) Anyway, I think I'm straying from the point once again. The point is that you are objecting to the "value-laden" wording. But I think these are such common mathematical values that it makes sense to present things from that point of view, and maybe even makes things a little clearer. Sniffnoy (talk) 04:59, 17 January 2013 (UTC)

Well, OK, we just flat disagree on this point. To me it makes no more sense to call it "bad" that the operations are not commutative, than it does to call it "bad" that 2+2 is not 17. There is no possible world in which things are any different, so bad compared to what? As for the so-called "natural" operations, well, they just don't do what is asked of the ordinary operations, so you can't compare with them. I feel very strongly that the wording should be changed to eliminate reference to advantages and costs. --Trovatore (talk) 05:12, 17 January 2013 (UTC)

Yes, that's perhpas not worded the most clearly -- it seems strange to say that they "retain" commutativity, when the ordinary operations do not have it. I think what the writer intended is the idea that these retain commutativity from addition and multiplication of whole numbers, rather than the ordinary operations on ordinals. Sniffnoy (talk) 18:33, 10 January 2013 (UTC)

I think that the ordinary operations are actually more natural than the "natural" operations. As Trovatore indicated, we are forced to define the ordinary operations as we do because that is what is needed for most applications. However, the "natural" sum is used in some systems of ordinal notations including Takeuti's ordinal diagrams. In that case, the commutivity and absence of absorption (e.g. 1+ω=ω, but 1#ω≠ω) appear to be the motivations for using "natural" addition. JRSpriggs (talk) 03:46, 11 January 2013 (UTC)

Of course they're more natural -- that's why they're the ordinary operations! Sniffnoy (talk) 04:59, 17 January 2013 (UTC)

As someone who came to this article to learn the basics of ordinal arithmetic, I was extremely confused by this language in the intro that seemed to say that the operations were commutative, only to read in both the addition and multiplication section that they are *not* commutative. I have rephrased the intro to (hopefully) be clearer, but as I mentioned, I'm not an expert so if something I added isn't correct, of course feel free to change it--but I do hope there isn't simply a revert, because the intro really did throw me off. biggins (talk) 07:30, 8 January 2019 (UTC)

I renew my objection to "at the expense of".

Another problem is that the text seems to give the two sets of operations equal weight. That's completely inappropriate. The "natural" operations are almost never used; the "ordinary" operations just are ordinal arithmetic, plain and simple. The "natural" operations need to be downgraded to a side note somewhere. --Trovatore (talk) 19:07, 8 January 2019 (UTC)

Strongly disagree with "almost never". I (studying well partial orders) use them routinely, and they're useful for getting upper bounds even when you're dealing just with well orders (as frequently there's a WPO that will map onto it). (Although I notice that the text currently doesn't mention the theory of WPOs at all... likely I should add in a note about that -- probably I should replace the current order-theoretic definitions with a more general statement about how they apply there.) And of course they're the operations used in the surreals. Giving them equal weight in the intro might be a mistake, and one could reduce that there, but note that the actual text does *not* give them equal weight, discussing the natural operations only in one short section. There's no need to downgrade this useful information into a sidenote! Anyway, likely I should do some rewriting of that section... it's kind of a mess at the moment. Replace the complicated definitions with a note on WPOs, and also toss in a quick note on how one applies these to the theory of well orders, not just well partial orders. Sniffnoy (talk) 18:50, 9 January 2019 (UTC)

Conway operations

It appears that "Conway sum" and "Conway product" are used for operations on knots, and on "nimbers", not for the "natural" sum and product of ordinal numbers. At least I can't find a reference. I suppose I could ask Conway, himself, if he has heard of the term. (Quoting him would nominally be a violation of WP:SPS, but I've never known him to falsely claim credit for something.) — Arthur Rubin (talk) 03:22, 7 March 2015 (UTC)

Interesting....the paper in the Notre Dame Journal of Mathematics does seem to have "Conway sum" to be the same as the "natural sum", but "Conway product" is a right-continuous function with the same relationship to Conway sum as regular product has to regular sum. A completely different operation. — Arthur Rubin (talk) 09:40, 7 March 2015 (UTC)

Now that's truly odd! The multiplication you describe was considered earlier by Ernst Jacobsthal, for what it's worth; I've seen it called "Jacobsthal's natural multiplication" (or just "Jacobsthal multiplication", but I think it might just be me that calls it that :P ). What paper is this that you're referring to? (Apologies if this has gotten a little off topic. I agree that should probably be removed, and I'd go and just do it, but I want to see this reference first. :) ) Sniffnoy (talk) 12:47, 7 March 2015 (UTC)

Examples:

• Combinatorial Game Theory by Aaron N. Siegel https://books.google.com/books?id=VUVrAAAAQBAJ&pg=PA415&lpg=PA415&dq=%22conway+multiplication%22&source=bl&ots=c8mL-eUOCf&sig=6JHy2M6s2aoDq-aN5MqvvhT4Loo&hl=en&sa=X&ei=ZBH7VJTeJsnToASkqICQAQ&ved=0CEwQ6AEwBQ#v=onepage&q=%22conway%20multiplication%22&f=false

  Uses it for multiplication on the surreal numbers, which happens to be the same as "natual multiplication" on the ordinals treated as surreal numbers.

• A note on Conway multiplication of ordinals, John Hickman, Notre Dame J. Formal Logic, Volume 24, Number 1 (1983), 143-145.

  This is the paper I described above; "Conway addition" is natural addition, but "Conway multiplication" is continuous in the second argument, satisfying $\alpha \times (\beta +1)=(\alpha \times \beta )+\alpha ,$, where all the "+" are Conway/natural addition, although it doesn't matter for the first "+".

• OEIS A160679

  Only on the natural numbers, but this one uses "Nim (or Conway) multiplication"; clearly the operation of multiplication on nimbers.

• https://eprint.iacr.org/2007/157.pdf

  Same as above, for use in cryptography

I don't really see a principle use; there are so many other "X-Conway multiplication" or "Conway" "product"s, that a Google search is not very productive. — Arthur Rubin (talk) 15:21, 7 March 2015 (UTC)

Good point. I've gone ahead and edited out the reference to "Conway operations". The reason I asked about the Hickman paper, btw, had nothing to do with the article; it was just that I have an interest in Jacobsthal's multiplication, and was interested to see another paper discussing it that I didn't know about. :) Seems it's because Conway independently rediscovered the operation, not knowing it had been studied earlier by Jacobsthal, and neither Hickman's nor Gonshor's paper credits Jacobsthal. Good to know! So thank you. Sniffnoy (talk) 02:58, 8 March 2015 (UTC)

Moreover the relation α+β = β+α is an equivalence relation on the set of nonzero ordinals, and all the equivalence classes are countable infinite.

I removed the above sentence from the text. The equivalence class of a finite ordinal is finite. Indeed, if we take math>n=\alpha + \beta</math> both α and β will be finte and β+α = n. Also, I don't see how it cannot be uncountable. Wisapi (talk) 16:34, 21 January 2016 (UTC)

The equivalence classes are countably infinite, not their elements. For each ordinal, there are aleph_0 others equivalent to it; it's not making any claims about the sizes of the ordinals involved. There's no mistake. (That said, that bit could use some rephrasing.) Sniffnoy (talk) 16:56, 21 January 2016 (UTC)

Citation needed for claim in Ordinal Exponentiation?

(Sorry if I'm not doing this correctly.)

R.e.b edited the page "Ordinal Arithmetic" on July 20, 2014 (compare versions), adding this statement:

Jacobsthal showed that the only solutions of αβ = βα with α≤β are given by α=β, or α=2 β=4, or α is any limit ordinal and β=εα where ε is an ε-number larger than α.), adding the following statement:

Can anyone provide a reference for this statement? I've looked at two Jacobsthal papers on ordinals (the ones that seem to be cited a lot and the only ones I could find), namely
'Ueber den Aufbau der transfiniten Zahlen' (Mathematische Annalen 66) and
'Zur Arithmetik der transfiniten Zahlen' (Mathematische Annalen 67)
but cannot find said result in those.

Thanks
Feel free to email me at username@gmail.com Georg.anegg (talk) 11:49, 13 October 2017 (UTC)

Nevermind, I found it. The proof is in

Ernst Jacobsthal, 'Vertauschbarkeit transfiniter Zahlen' (Mathematische Annalen 64, 1907) FULL ARTICLE

Georg.anegg (talk) 16:30, 16 October 2017 (UTC)

"Warning: [...]" in a Wikipedia article?

Near the bottom of § Exponentiation, "Warning: Ordinal exponentiation [...]" just doesn't seem encyclopedic, because it seems solely aimed at "hobbyists" thinking about ordinal arithmetic (for lack of a better thing to do) than at anyone else at all. I would be bold and edit it out but I'm not sure how to replace it. — Preceding unsigned comment added by Nameless6144 (talk • contribs) 00:10, 12 November 2018 (UTC)

Leave it alone. Anyone reading this article needs to know that there is a possibility of confusion. JRSpriggs (talk) 21:53, 12 November 2018 (UTC)

I kind of agree that the tone is wrong for an encyclopedia. I think it's reasonable to make the point, but without the word "warning", and it seems like it might go better in an explanatory footnote. --Trovatore (talk) 23:25, 12 November 2018 (UTC)

I have tried to make the tone somewhat more "encyclopediac". Please check that I didn't make a mess of it. – Tea2min (talk) 09:31, 13 November 2018 (UTC)

What makes this potentially confusing is that for ordinals α and β:

$\vert \alpha +\beta \vert =\vert \alpha \vert +\vert \beta \vert$

$\vert \alpha \cdot \beta \vert =\vert \alpha \vert \cdot \vert \beta \vert$, but

$\vert \alpha ^{\beta }\vert <{\vert \alpha \vert }^{\vert \beta \vert }$

unless α < 2 or β < ω. JRSpriggs (talk) 20:55, 13 November 2018 (UTC)

Zero^Zero

With real number arithmetic, 0⁰ is often left as undefined because of trouble like $\lim _{x\rightarrow 0+}x^{0}\neq \lim _{x\rightarrow 0+}0^{x}$. Is there a similar conundrum with ordinal arithmetic? Whether the answer is yes or no, I think an explanation of this case would be a nice addition to the article. —Quantling (talk | contribs) 21:58, 24 July 2022 (UTC)

Continuity is not an issue because ordinal arithmetic is not continuous except at limit ordinals of which 0 is not one.

How many functions are there from the empty set to the empty set (with or without finite support)? One, namely the empty function. So 0⁰= 1. Full stop. JRSpriggs (talk) 15:11, 25 July 2022 (UTC)

Thank you. —Quantling (talk | contribs) 16:40, 25 July 2022 (UTC)

Sources for ordinal arithmetic definitions

An editor thinks that these need citations. If you have one, please supply them here (or edit the article):

1. Addition of ordinals recursive step: $\alpha +\beta =\bigcup _{\delta <\beta }((\alpha +\delta )+1)$ where "+ 1" denotes the successor of an ordinal, regardless of whether β > 0 is a limit ordinal.
2. Multiplication of ordinals recursive step: $\alpha \cdot \beta =\bigcup _{\delta <\beta }((\alpha \cdot \delta )+\alpha )$, regardless of whether β > 0 is a limit ordinal.
3. Exponentiation of ordinals recursive step: $\alpha ^{\beta }=\bigcup _{\delta <\beta }(\alpha ^{\delta }\cdot \alpha )$, regardless of whether β > 0 is a limit ordinal.

Thank you —Quantling (talk | contribs) 17:12, 26 July 2022 (UTC)

Note that 1 and 3 fail when β equals zero. --Trovatore (talk) 17:29, 26 July 2022 (UTC)

@Trovatore: Agreed; each of these is the recursive step for β > 0, not the base step for β = 0. The indication of β > 0 is part of the above text, but your note tells me that that needs to be clarified. —Quantling (talk | contribs) 18:11, 26 July 2022 (UTC)

I mean, besides a source, I think what's missing is a reason to care about these formulations. Basically it looks like a clever hack to unify the successor and limit cases, but it doesn't seem to be conceptually enlightening in any way I can see. If a lot of sources presented them, that would be a good reason to do likewise, but if you're struggling to find just one, that could be because they're not generally seen as that interesting. --Trovatore (talk) 19:30, 26 July 2022 (UTC)

Perhaps it is a tomaTOE vs. toMAHto thing. The current text with distinct cases for successor and limit ordinals makes me wonder what the two cases have in common that makes them both represent the same concept. When it is presented as a single case, it is clearer (to me) that they represent the same concept. (Note that keeping the two-cases approach is also important (to me) because they show the "practical" way to compute in these two cases.) It's been decades since I took this material in school and I don't have any sources handy. My hope is that pretty much anyone with a textbook will find support for the combined case; I don't expect this to be a struggle to find a rare source. —Quantling (talk | contribs)

To me it looks like what you've done is taken the successor step and wrapped it in some boilerplate that cleverly falls away at successor ordinals but takes suprema at limits. I don't see that as "making them represent the same concept". The successor step is still the meat of the operation, without all the $\bigcup$ padding around it, and what happens at limits is trivial, as is usually the case with transfinite recursion (though there are exceptions).

As for it being in "pretty much any textbook", well, I don't recall ever seeing it. I would be interested to find out if you find it in any prominent textbook. --Trovatore (talk) 17:24, 27 July 2022 (UTC)

Yes, the successor ordinal recursion doesn't need the union and the limit ordinal doesn't need the "+ 1", "+ α", or "· α" -- well, other than that the union has to exclude δ = 0 because "· α" was dropped. These are useful simplifications of the combined case and definitely should be retained for the article.

For this article, we could define the successor recursion (as it currently is) and also indicate that the recursion definition for a limit ordinal (as it currently is) follows from our desire that the function be continuous (on that right-side variable) and satisfy some montonicity criterion (which isn't quite normality because of the behavior on some very small values of α for some of these operations). The article doesn't say that we are doing that -- though we could change that.

To be clear, I don't have access to the textbooks so it is unlikely that *I* will find the combined expression in any prominent textbook; I am hoping that someone with a prominent textbook handy will come forward. —Quantling (talk | contribs) 18:10, 27 July 2022 (UTC)

Hmm, well, I checked Kunen and I don't see it there. I have a copy of Jech lying around somewhere; maybe I'll take a look when I find it. --Trovatore (talk) 07:32, 28 July 2022 (UTC)

I checked Jech Set theory 3e Ch 2 and it is the standard transfinite recursion definition. Also various web sources. I think there is unlikely to be a source with the combined formulation, as transfinite recursion itself is split into successor and limit cases. Mathnerd314159 (talk) 17:57, 1 August 2022 (UTC)

While I'm sure the combined formulations come from somewhere, I think it does need a source to show utility/convenience. It is conceptually better in my view to show successor and limit cases as separate, because fundamentally, there really are three different types of ordinals: 0, successors, and limits. Without the limits, you just have finite recursion/induction; with them, you have transfinite recursion/induction.

Induction/recursion with multiple (but finitely many) recursive cases, or base cases, is not unusual: structural inductions of this form occur regularly in type theory/functional programming or, equivalently, in formal logic. — Bilorv (talk) 21:58, 4 August 2022 (UTC)

Please see Talk:Ordinal arithmetic § Simplifying the definition of addition below. In particular, the approach that I advocate for above, which combines the successor ordinal and the limit ordinal cases, allows a short definition for (ordinary ordinal) addition

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.

which is quite similar to the short definition for natural addition already present in the article. (Please put replies in that section rather than here.) —Quantling (talk | contribs) 18:18, 14 June 2024 (UTC)

Dead external link to calculator

The external link ordCalc ordinal calculator at the bottom of the page leads to a 404 page, likely meaning that the calculator has been removed. I've tried looking for a new ordinal calculator myself but I couldn't find any. Can anyone update the link with a new calculator? Many thanks. KittySaya (talk) 18:50, 5 December 2022 (UTC)

@KittySaya:: Archive.org has archived it at https://web.archive.org/web/20210605160128/http://mtnmath.com/ord/ so can you see if the info provided there helps you? (pending a more permanent solution with a live alternative). --𝕁𝕄𝔽 (talk) 20:09, 5 December 2022 (UTC)

Commented out what now?

Hi Jochen Burghardt. I wonder if you could say a little more about this edit. The edit summary says that it's "comment[ing] out a wrong claim", but as far as I can tell the edit doesn't comment out any claim, but rather adds a commented-out counterexample to a claim that nobody made. --Trovatore (talk) 22:18, 22 January 2023 (UTC)

The comment runs until the very end of the paragraph. Maybe you overlooked this due to the "---" between counter-example and wrong claim? - Jochen Burghardt (talk) 10:53, 23 January 2023 (UTC)

Oh, I see. Yes, when I looked at the diff, I thought that triple-hyphen was ending the comment. --Trovatore (talk) 20:39, 23 January 2023 (UTC)

Maybe you have sufficient experties to fix the wrong claim I commented out? I guess it can be fixed (i.e. a simple condition for commutativity can be given), but I don't know how. - Jochen Burghardt (talk) 16:21, 24 January 2023 (UTC)

Hmm, it's a cute little problem; might be fun to think about. But I doubt we really need to give such a condition in the article. --Trovatore (talk) 21:47, 24 January 2023 (UTC)

Very good thinking with $(\omega .2+5)+(\omega +5)=(\omega +5)+(\omega .2+5)$. You can write $\omega .2+5=5.(\omega .2+1)$, but not with the natural number as the right term in the multiplication. I wonder if maybe $\alpha ,\beta ,\gamma$ should be restricted to limit ordinals. In regards to whether addition commuting is an equivalence, reflexivity and symmetry are trivial but I don't see a trivial proof or counterexample to transitivity. — Bilorv (talk) 12:05, 28 January 2023 (UTC)

Concise representation of natural sum, as restricted to ordinals

Regarding natural (surreal) arithmetic on ordinals, the article currently 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. There is also an inductive definition of the natural product (by mutual induction), but it is somewhat tedious to write down and we shall not do so (see the article on surreal numbers for the definition in that context, which, however, uses surreal subtraction, something that obviously cannot be defined on ordinals).

Does that mean we can write the following definitions for natural addition and natural multiplication, using the supremum function sup:

α ⊕ β = sup({(γ ⊕ β) + 1 | γ < α} ∪ {(α ⊕ γ) + 1 | γ < β})

α ⊗ β = sup({(γ ⊗ β) ⊕ β | γ < α} ∪ {(α ⊗ γ) ⊕ α | γ < β})

regardless of whether or not α and β are limit ordinals, countable ordinals, etc? That is, is this true, WP:CALC, and sufficiently notable — or do you have a citation? —Quantling (talk | contribs) 21:09, 25 October 2023 (UTC)

Are there similar formulations for the (presumably non-commutative) natural exponentiation, natural tetration, etc.? —Quantling (talk | contribs) 13:34, 26 October 2023 (UTC)

Comma in "if ... then" statements

@Trovatore: et al: My English teacher taught me that a comma is used when "then" is implied but not explicit, but not merely for readability. (As in "If a = 1 then we will get that a + 1 = 2." or "If a = 1, we will get that a + 1 = 2." are good, but not "If a = 1, then we will get that a + 1 = 2.") But, of course, my personal experience don't mean squat; we'll need a manual of style to get anywhere with this. Do you happen to have a handy reference that backs your approach? I do find web pages that support your approach, but none that are particularly authoritative either way. —Quantling (talk | contribs) 20:09, 14 December 2023 (UTC)

This page says that a comma can be used before "then" if it is replacing an implied "and". (As in "She filled in the last square in Sunday's puzzle, then yawned.") I agree with that. But I don't think that is what you are arguing. —Quantling (talk | contribs) 20:15, 14 December 2023 (UTC)

This page claims that one should use a comma if one would put a pause there. Are these sentences where you would pause before "then"? —Quantling (talk | contribs) 20:23, 14 December 2023 (UTC)

There are examples where a clause is set off with commas immediately before "then", such as the clause "or even if B" in "If A, or even if B, then C". —Quantling (talk | contribs) 20:27, 14 December 2023 (UTC)

I think the "pause" explanation is closest. To be fair, I wouldn't pause very long at these spots, but yes, there would be a change in rhythm to help the listener understand that we're entering a separate branch of the parse tree, and that change can be transcribed by a comma. --Trovatore (talk) 20:29, 14 December 2023 (UTC)

To me, the use of the word "then" is itself the marker that we are entering a separate branch of the parse tree. (Which I guess is why I need a comma when the "then" is not explicit.) Or, roughly in the same vein, if someone were to pause near the "then" for additional emphasis, might it be just as likely to be after the "then" as before it? —Quantling (talk | contribs) 20:42, 14 December 2023 (UTC)

My view is that one of the main purposes of punctuation is to help the reader chunk the sentence into semantic units before reading the actual words, so requiring them to notice the word "then" does not fully accomplish that.

I think the pause would more likely come before "then". In addition to helping with parsing, it also gives the listener a little time to process the previous semantic unit. --Trovatore (talk) 20:57, 14 December 2023 (UTC)

If there are no more editors with opinions on this then ... I guess we are at a stalemate. Let's do it your way. Best —Quantling (talk | contribs) 22:01, 14 December 2023 (UTC)

Simplifying the definition of addition

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)

An inductive definition of the natural product is somewhat tedious to write down

... 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, $$\alpha \otimes \beta =\sup \left(\sup \lbrace (\gamma \otimes \beta )\oplus \beta :\gamma <\alpha \rbrace ,\sup \lbrace (\alpha \otimes \delta )\oplus \alpha :\delta <\beta \rbrace \right)\,.$$ 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)