Talk:Ordinal arithmetic

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

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

text: 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)
replies: text: 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)
thread_title: Natural operations

message: 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)
replies: text: 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)
thread_title:  Tetration 

message: :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)
replies: text: 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)

text: 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)
replies: text: 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)
replies: text: 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^1_2 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)

text: 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)
replies: text: 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)
replies: text: (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)

text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Exponential &mdash; ordering of ''P''=''B''<sup>''E''</sup> 

message: 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)
replies: text: 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)
replies: text: To 80.98.179.160: You are mistaken. JRSpriggs (talk) 19:44, 4 January 2018 (UTC)
thread_title:  (sandbox) tables are cool 

message: 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)
thread_title:  Large Countable Ordinals 

message: 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)
thread_title:  Ordinal Exponentiation 

message: 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)
replies: text: 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)
thread_title:  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.  

message: 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)
replies: text:  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)
thread_title:  Dead external link to calculator 

message: Hi .  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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)

text: 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)
thread_title:  Commented out what now? 

message: 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 :
:
:
regardless of whether or not  and  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)
replies: text: Are there similar formulations for the (presumably non-commutative) natural exponentiation, natural tetration, etc.? —Quantling (talk | contribs) 13:34, 26 October 2023 (UTC)
thread_title:  Concise representation of natural sum, as restricted to ordinals