Talk:Ordinal number/Archive 3

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GA Re-Review and In-line citations

Members of the Wikipedia:WikiProject Good articles are in the process of doing a re-review of current Good Article listings to ensure compliance with the standards of the Good Article Criteria. (Discussion of the changes and re-review can be found here). A significant change to the GA criteria is the mandatory use of some sort of in-line citation (In accordance to WP:CITE) to be used in order for an article to pass the verification and reference criteria. Currently this article does not include in-line citations. It is recommended that the article's editors take a look at the inclusion of in-line citations as well as how the article stacks up against the rest of the Good Article criteria. GA reviewers will give you at least a week's time from the date of this notice to work on the in-line citations before doing a full re-review and deciding if the article still merits being considered a Good Article or would need to be de-listed. If you have any questions, please don't hesitate to contact us on the Good Article project talk page or you may contact me personally. On behalf of the Good Articles Project, I want to thank you for all the time and effort that you have put into working on this article and improving the overall quality of the Wikipedia project. Agne 05:48, 26 September 2006 (UTC)

I'm not a math geek. Personally, I looked up Ordinal to make sure I was using it correctly to distinquish from Nominal, Interval and Ratio level data. I did not find this entry helpful in this regard. 216.158.5.4 14:27, 7 August 2007 (UTC)

Division undefinable?

From the article:

One can define addition, multiplication, and exponentiation on ordinals, but not subtraction or division.

This seems to be contradicted by the MathWorld article on ordinals[1], which gives as an example:

$\omega +...+\omega \over r$

where r is a real number. How is this to be explained? Simões (^talk/_contribs) 05:28, 25 November 2006 (UTC)

My first reaction was, in general, don't be surprised if you see nonsense on MathWorld. That's maybe a little unfair; their actual common sin is more promoting neologisms as though they were standard usage, which I guess isn't quite as bad as actual false statements.

In this case, though, there's nothing wrong with what they actually wrote. You need to look at it a little closer. That isn't a fraction sign, and r is presumably not an (arbitrary) real number. --Trovatore 05:57, 25 November 2006 (UTC)

What you saw there is NOT an underline indicating division, it is a brace indicating that there are r copies of ω being added together to get ω·r . JRSpriggs 09:07, 25 November 2006 (UTC)

Actually ordinal subtraction x−y is well defined (delete the first y elements of x, returning 0 when x ≤ y). However you don't have x−y = x−z implies y = z in the case y < x, unlike the situation with the natural numbers. For example ω−1 = ω−2 (= ω). You do however have x−x = 0. --Vaughan Pratt (talk) 10:18, 15 June 2008 (UTC)

See also the discussion titled Subtraction lower on this page.  --Lambiam 19:46, 15 June 2008 (UTC)

Intro too technical

The article needs a nontechnical section at the top of the introduction. Currently, it immediately dives into the mathematics. Pcu123456789 04:39, 27 January 2007 (UTC)

I agree that it wouldn't hurt to add some intuitive motivation. However I don't agree with the {{confusing}} tag. It's a mathematics article, after all, so the complaint about "diving into mathematics" is a bit odd. The tag should be reserved for articles whose logical structure is unclear, not those that need to be made more accessible. Discuss the latter issue on the talk page, not the article page. --Trovatore 05:23, 27 January 2007 (UTC)

It's a nice question how to introduce the ordinal concept. Just now I rewrote part of the introduction with the goal of at least making it more precise, if not more intuitive, on the theory that those likely ever to really grasp the concept should find it easier when the definitions are self-contained. The hardest concept might be that of order type. For pedagogical completeness in explaining order type one should mention the negative integers and the rationals between 0 and 1 to make the point that there exist linear orders that aren't well orders, but this bit of the explanation should be delegated to the order type article since otherwise the pedagogically optimal presentation of the ordinals would almost surely be too long to serve as a Wikipedia-style introduction. The premise of my attempt at an introduction is that it should at least mention the main facts about the ordinals. --Vaughan Pratt (talk) 09:33, 15 June 2008 (UTC)

I added grammatical spacing to the finite ordinals, which caused the list of transfinite ordinals to split on my screen. Perhaps it should be converted to <math> notation?

Also, ε₀ = ω^ε₀ doesn't quite look like $\epsilon _{0}=\omega ^{\epsilon _{0}}$ to me. — Arthur Rubin (talk) 12:40, 15 June 2008 (UTC)

Fixes subsequent to my rewrite greatly appreciated. Does it matter whether the list of transfinites splits? I deleted the spaces just now but by all means put them or equivalent back. My only question is to User:JRSpriggs: does "each finite set can be well-ordered and" matter? (Since an ordinal is the order type of a well-foundedordered set, if there did exist such a thing as a finite set that couldn't be well-ordered, how would that change anything?) --Vaughan Pratt (talk) 09:24, 17 June 2008 (UTC)

To Vaughan: The sentence in question is "The finite ordinals (and the finite cardinals) are the natural numbers: 0, 1, 2, …, since each finite set can be well-ordered and any two total orderings of a finite set are order isomorphic." with the part I added underlined. Your version explains why there is at most one finite ordinal per natural number. My addition explains why there is at least one finite ordinal per natural number. Neither explanation is strictly necessary, but we are here to be helpful. JRSpriggs (talk) 21:20, 17 June 2008 (UTC)

The usual construction of 7 is as 6∪{6}, which totally orders 7 by induction. That every finite set can be totally ordered is true but irrelevant, and moreover either circular (if you use the definition of finite set as one in bijection with a natural number) or nontrivial (if you use some other definition such as Dedekind-infinite set). That any two total (and hence well-) orderings of a finite set are order isomorphic is on the other hand a crucial distinction between finite and infinite sets when distinguishing ω from ω+1. --Vaughan Pratt (talk) 21:43, 19 June 2008 (UTC)

(Incidentally, thanks, JRSpriggs, for changing ω to {ω} in the bit I added about topology---I had the sign reversed in my mind, closed instead of open, ω is the infinite subsets of ω are the only nonclosed subsets of ω+1.) --Vaughan Pratt (talk) 00:38, 22 June 2008 (UTC)

(unindent) To Vaughan Pratt: It is difficult to argue about this because the dispute is not over what is true or what is provable, but over what is obvious. Let me just point out that the article on finite sets assumes that the natural numbers are defined independently of and prior to set theory. Thus 7 is not identified with 6∪{6} until after the finite ordinals have been seen to be equivalent to the natural numbers. And any argument from mathematical induction ("which totally orders 7 by induction") cannot be considered obvious to most readers. JRSpriggs (talk) 11:46, 22 June 2008 (UTC)

Agreed. However it is not obvious to me why the existence of specific well-ordered finite sets, namely the finite ordinals, should follow from the well-orderability of all finite sets. In general arguing from a universal to an existential usually involves a tacit existential, in this case the existence of a finite set of cardinality n for every natural number n. Natural ways of constructing such a set tend to well-order it without needing to appeal to the well-ordering of every isomorphic copy of it, and moreover a natural way to prove that any finite set can be well-ordered is via its isomorphism with the set of that cardinality whose existence has been previously established. To me an argument that reverses that arrow seems less natural. But that's just me, and I can appreciate that other organizations of what rests on what might justifiably reverse that arrow (not that I have one in mind in this case). --Vaughan Pratt (talk) 15:12, 23 June 2008 (UTC)

In any case, I think the somewhat fine point may be lost on the very reader who might need the help – assuming the same somewhat clueless reader feels a compelling urge to understand why the statement is necessarily true.  --Lambiam 23:59, 29 June 2008 (UTC)

Common use meaning added

'Ordinal number' or 'ordinal' is a concept that is understood not only by small children, but even by rhesus monkeys !!! So I understand the frustration of Pcu123456789 and respectfully disagree with user Trovatore who says that "It's a mathematics article" : No, it is totally wrong to assume user typing common english word like 'ordinal' into Wikipedia has a PhD in math and wants to read only about set theory. I also disagree that all languages that have ordinal numbers are 'English' so I removed otheruses tag to Names of numbers in English and created a linguistic stub instead. I have seen other examples in Wikipedia, where small explanatory sections like this were used instead of full disambig page so I hope that is OK Warbola 01:18, 10 April 2007 (UTC)

Well, this article is and always has been about mathematics. If you can find references discussing a non-mathematical meaning other than just the linguistic thing about words like "seventeenth" (I'm curious how one might do an experiment showing that Rhesus monkeys distinguish ordinals from cardinals), then by all means, write it up and put a disambiguation link to it. But please don't muddle the issue in this article. --Trovatore 02:02, 10 April 2007 (UTC)

OK, I understood your 'mathematics' meaning 'hard core set theory'. I you are willing to include things like simple counting of numbers to 'mathematics' then I am with you and apologize for misunderstanding. About ordinal numerical abilities of rhesus monkeys, [2] should make things very clear. Warbola 04:17, 10 April 2007 (UTC)

No, I'm not willing to include those. I'm saying "ordinal number" has a specific meaning to mathematicians, and that meaning is what this article is about, and I would like it to stay that way. Whether this article is correctly named is a separate issue -- you can see in the archive of this talk page that I suggested it should be at a title like ordinal number (mathematics) or ordinal (mathematics). (By the way, from a brief glance at the abstract, what the monkeys seem to have understood is what mathematicians would call a linear order.) --Trovatore 04:26, 10 April 2007 (UTC)

You 'have been over this issue many times', yet are not willing to fix the problem ? Why ? In Google, just "Ordinal number" "database" alone gives two times more hits than "ordinal number" "set theory", so other meanings clearly deserve short explanation or disambiguation. If you do not like my version of the clarification, why not make your own? And what was wrong with link to Ordinal numbers (linguistics) stub other than it was written by "wrong" person ? Warbola 12:59, 10 April 2007 (UTC)

I'm not exactly sure whom you're arguing with, but I don't think it's me. I'm quite willing to have articles on other meanings, and disambiguations to them. --Trovatore 18:40, 10 April 2007 (UTC)

I do not think I have anything to argue about and yes, Trovatore, my last comment was not for you. Somebody restored my 'otheruses' to linguistics and I am happy with that. Perhaps there should be also otheruses toOrdinal number (databases). Warbola 23:39, 10 April 2007 (UTC)

On a related issue, on individual number pages such as 100 (number), "ordinal" wikilinks to this article when it clearly needs to link to the article about linguistics. This wouldn't usually be a problem, but there are literally hundreds of these pages, and all the ones I've looked at need modifying. In this case, not even an ordinal disambig page would be appropriate, as a straight link would obviously be preferred. This confusion is futher augmented by the fact that the linguistic page is at present no more than a stub. Also confusingly linked is the cardinal link on the number pages. Any ideas of how to proceed? Poojean 10:42, 12 April 2007 (UTC)

In my opinion, articles like 100 (number) are not worth the disk space they occupy. In any case, it is the responsibility of the editors who maintain those articles to get the links from them correct, not our responsibility to fulfill their mistaken expectations. JRSpriggs 02:18, 14 April 2007 (UTC)

I modified the "Common use meaning" slightly not because it was wrong, but only because I didn't immediately see the connection between readin', 'riting & 'rithmatic. Numbers are math not language! (yes, there are a million things wrong with that thought. lol) Reference materials are not only about containing the "data" but also about being able to find the data we're intending to look for. For example, how could I ever discover the meaning of Phthongometer if I didn't know (or forgot) that it started with a PH and the search options were too narrowly confined. Anyway I stole this modified definition from Japanese counter word: Ordinal numbers which for the first time in 44 years made sense to me. (I'm adding a copy of this to cardinal discussion page too) J-puppy (talk) 22:05, 26 March 2009 (UTC)

Why distinguish

The article says this:

While in the finite world these two concepts coincide, when dealing with infinite sets one has to distinguish between the two.

I then read quite a bit more of the article trying to figure out why "one has to distinguish between the two" but I couldn't figure it out. Can anyone clarify? --Doradus 16:23, 12 June 2007 (UTC)

Because both ordinals ω and ω + 1 have the same cardinality but have distinct order types. The finite cardinalities are the only ones that have a unique possible well-ordering. — Carl (CBM · talk) 16:47, 12 June 2007 (UTC)

Ok, thanks. I think I get it, but not well enough to incorporate into the article. Do you think you could do that? --Doradus 12:06, 16 June 2007 (UTC)

Thanks for your addition. I'm not sure I'm 100% crystal clear, but it helps. --Doradus 03:38, 25 June 2007 (UTC)

GA status

This article is very nice, but even though the citation requirements for GA have been relaxed, this article doesn't meet them at the moment (see WP:WIAGA and WP:SCG). For instance, it states that ordinals were introduced by Cantor in 1897, but does not give a reference. Can someone go through the article and fix it? Geometry guy 21:24, 20 September 2007 (UTC)

Transfinite induction

The article says:

Any property which passes from the set of ordinals smaller than a given ordinal α to α itself, is true of all ordinals.

Is it ok to put it like this? Isn't it necessary that there be a 'basis'? Arthena^(talk) 21:19, 31 October 2007 (UTC)

No, it is not necessary to have a separate "basis". Consider what happens if we take α to be 0. The set of ordinals smaller than 0 is the empty set which vacuously satisfies the condition that all of its members have the property in question. Consequently, 0 has the property. So all members of {0} have the property. But {0} is the set of ordinals less than 1, thus 1 has the property. So all members of {0, 1} have the property. Thus 2 has the property. And so on. See also Transfinite induction. JRSpriggs 03:30, 1 November 2007 (UTC)

Shouldn't it actually read "... is true of all ordinals greater than $\alpha$?" After all, you can't prove inductively a property works for ordinals less than the anchor. Eebster the Great (talk) 02:20, 15 May 2008 (UTC)

To Eebster: No. Are you paying attention? Try reading the article again. JRSpriggs (talk) 04:08, 15 May 2008 (UTC)

Sorry, I missed "set of ordinals smaller" and thought it said "ordinal smaller." So yeah, obviously if the property's already taken to satisfy all ordinals smaller ordinals, and passes to the ordinal itself, it satisfies them all. I jus misinterpreted $\alpha$. 24.165.184.37 (talk) 00:17, 16 May 2008 (UTC)

clarification request

I'd like to request that the intro section (that describes ω, ω+1, ω·2, etc.) describe more explicitly (based on the axioms of set theory) why these ordinals exist and aren't all the same. It's a bit counterintuitive to have a countable ordinal like ω² different from ω, since that means there's a bijection f  between ω² and $\mathbb {N}$ which means (since ω is an element of ω²) that f(ω) "should" be some specific natural number, which clearly doesn't fly. Also, schoolchildren are taught Hilbert's paradox of the Grand Hotel which says that all countable infinities (of course that really means cardinals) are the same. So I think the explanation should begin with finite ordinals 0={}, 1={0}, 2={1,0}, etc.; then get ω by invoking the axiom of infinity explicitly; then use the axiom of union to get the supremum of an ordinal (I edited the article to mention this but I hope it's correct) giving ω+1, ω+2, etc.; getting ω·2 and beyond needs the axiom of replacement. After some head scratching I think I understand the idea of this construction, but I haven't worked out the details well enough yet (I'll keep trying since it seems like a good exercise) to put them into the article, so it would be great if someone else could. Right now what's there comes across like "start with 1,2,3,..., keep going til you get to infinity, then stop for a moment and then keep going" which is bogus. The point to be made is that these infinite sets are logical consequences of the axioms. 75.62.4.229 (talk) 05:45, 25 November 2007 (UTC)

You seem to be confused; and it is hard to see where to start to correct you.

Yes, there are bijections between the elements of ω² and the elements of ω. So they have the same cardinality. However, no such bijection is order preserving. Which is why they are different ordinals.

The elements of ω² have the form ω·n + k. One bijection maps that element to <n, k> using Pairing function#Cantor pairing function. JRSpriggs (talk) 09:01, 25 November 2007 (UTC)

Thanks, that explains the issue pretty well, that the article's presentation of iterating the successor operation out to ω and then continuing the iteration is missing some steps, and evokes a picture of an order-preserving bijection which is of course silly. Really, ω is more like a brand new atomic symbol created by invocation of an axiom put there just for the purpose, and you get ω·2 by invoking another axiom and sup λ (for arbitrary ordinal λ) with yet another axiom. This never made sense to me until I actually tried to piece it together from the axioms. These completed infinities aren't there made from an iterative construction, they're there because they're postulated. It's less like "iterate the successor function infinitely many times" and more like "iterate til you get tired, then invoke a completion axiom to create a limit ordinal". As mentioned I think I just sort of get it now, but IMHO the presentation would benefit from being explicit about the axiomatic justification for saying these limit ordinals actually exist. Btw, another oddity: we have $\omega ^{\varepsilon _{0}}=\varepsilon _{0}$ and $\varepsilon _{0}+1>\varepsilon _{0},$ and therefore $\omega ^{\varepsilon _{0}}<\varepsilon _{0}+1.$ Is that correct? It's a little surprising. 75.62.4.229 (talk) 09:24, 26 November 2007 (UTC)

It's correct that $\omega ^{\varepsilon _{0}}<\varepsilon _{0}+1.$, yes, surprising or not. I don't much agree with the rest of your comments, though. The best way to think about it really is "iterate the successor function infinitely many times, and then do it some more". The axiomatics are secondary (remember for example that when Cantor first described the ordinal numbers, he wasn't using any axiomatic system at all). --Trovatore (talk) 03:31, 27 November 2007 (UTC)

Indeed, no formal set theory is needed. The set of real numbers $\{1-1/(n+2)\mid n\in \mathbf {N} \}$ has order type ω. Add in the number 1.5 and the set now has order type ω+1. It's easy to keep going, adding more elements, until you get to ω + ω. Then you add in 2.5 to get ω + ω + 1, and so on. — Carl (CBM · talk) 13:54, 27 November 2007 (UTC)

Nice!! Incidently, now that this page has a bit of attention on it, is there any chance that someone could add a few cites to meet the scientific citation guidelines? This is both for the benefit of readers and for preserving the GA status. Geometry guy 19:48, 27 November 2007 (UTC)

Thanks for suggesting Cantor. The SEP article about early set theory gives the principle that Cantor used that's missing from this article (section 2) and that I wouldn't have presumed:

Cantor defined them [transfinite ordinals] by means of two “generating principles”: the first (1) yields the successor a+1 for any given number a, while the second (2) stipulates that there is a number b which follows immediately after any given sequence of numbers without a last element. Thus, after all the finite numbers comes, by (2), the first transfinite number, ω (read: omega); and this is followed by ω+1, ω+2, …, ω+ω = ω·2, …, ω·n, ω·n +1, …,ω2, ω2+1, …, ωω, … and so on and on. Whenever a sequence without last element appears, one can go on and, so to say, jump to a higher stage by (2).

I think without stating principle 2 explicitly, there's not really an argument that ω exists. It's like describing a "biggest natural number". I can see with principle 2 that you can get ω+ω and so forth, and upwards in a tree-like fashion to ordinals like $\varepsilon _{0}$. Is that consistent? Who cares, everyone knows that naive set theory of Cantor's era is inconsistent regardless ;-). What I don't see is how to get uncountable ordinals out of this. The whole concept of iteration connotes countability, as does the word "ordinal" ("first, second, third..."). I know the diagonal proof that $\mathbb {R}$ is uncountable, but what reason is there to think there is an ordinal that large, i.e. that they don't run out long before they get to even the first uncountable cardinal (which depending on CH might or might not be as big as $\mathbb {R}$)? For a reader just trying to grok the basics of this subject (that would be me) these important points seem to be missing so I wish they could be filled in. 75.62.4.229 (talk) 10:43, 28 November 2007 (UTC)

Why is ω₁ < ∞? Because using the axiom of powerset one can show that every set has a Hartogs number, and thus that every cardinal number has a successor cardinal. In particular, aleph_null has aleph_one as its successor cardinal. Since ω₁ exists as a set, it is not a proper class like ∞. JRSpriggs (talk) 02:27, 29 November 2007 (UTC)

The point I don't see is why there's an ordinal corresponding to $\aleph _{1}$, and even more why there's one corresponding to ${\mathfrak {c}}$ (the cardinality of $\mathbb {R}$). That is, you can repeat the ordinal iteration construction for as long as you want, but why does that ever reach an uncountable instead of staying countable no matter how far you go? Also according to a good historical article I just found [3], Cantor was never able to prove that ${\mathfrak {c}}$ was an aleph (that required Zermelo's introduction of AC). I'll see if I can contribute a fix this weekend if nobody else does by then, but given my weak understanding of the subject, I'm likely to get something wrong and leave the rest of you to clean it up ;). 75.62.4.229 (talk) 09:52, 29 November 2007 (UTC)

I think that I see your question now. The answer is the axiom of replacement. That article has a detailed explanation of the role of the axiom in constructing large ordinals in ZFC. But, as Trovatore has also pointed out, this is independent of the construction of such ordinals in unformalized mathematics. — Carl (CBM · talk) 13:54, 29 November 2007 (UTC)

I think 75 might be looking for a more "philosophical" answer. Here's how I look at it: it's clear that if all countable ordinals (in the informal sense that's hopefully clear) can be gathered together into a completed whole, then that completed whole is itself wellordered, and its length is an ordinal, and it's not a countable ordinal, because otherwise you could stick another point at the end, and get another countable ordinal larger than it, and by assumption it's larger than all countable ordinals.

So the question is, why can all countable ordinals be taken as a completed whole? We know that it's not possible to have a completed totality of all sets, or even all ordinals, so why all countable ordinals?

But the thing is, for sets, or for ordinals in general, we know a precise reason there can be no such completed totality. The claim that such a completed totality exists is a falsifiable conjecture in Popper's sense -- in the strongest possible way, given that it's already been falsified. Whereas for the case of all countable ordinals, the hypothesis that there's a completed totality of them is a potentially falsifiable proposition that has not been falsified. And if it's true, we want to know' that it's true; we don't want to limit our view just to countable ordinals, if there are bigger ones out there. --Trovatore (talk) 02:24, 30 November 2007 (UTC)

section break

Trovatore, thanks, I actually was looking for a mathematical answer since I don't think I can understand the philosophy without seeing the mechanics. It looks like we're primarily discussing Cantor's unformalized theory, and ZFC; I won't worry about other systems like Principia. To recap the questions I had, and where I currently am in terms of having answers:

Q. How do we know there's an infinite ordinal ω, when there's no biggest natural number?

A. In Cantor's system, this relies on principle 2 as mentioned in the SEP article but which is missing from the Wikipedia article, so IMO the Wikipedia article has a gap. Principle 2 basically says that whenever you see a sequence going "..." you can plop a limit ordinal at the end of the dots, though the exact formation rules aren't totally clear to me once you get to ordinals like $\varepsilon _{0\varepsilon _{0\varepsilon _{0_{\ldots }}}}$. In ZFC, it's there because of the axiom of infinity.

Q. Considering that exponentiation is not commutative (x^y != y*x) and not associative (x^y)^z != x^(y^z), then, do limit x^y or limit (x^y)^z as x, y, z approaches ω exist?

A. My personal answer is no — since, if a limits exists, then it must be the same from all approach-directions [it is emphasized that convergence of a potentially infinite collection of mathematical objects depends on their order so that only sequence-terms (but not set-elements which are not ordered) converge or have a limit]. I resolutely believe that there is no ordinal number greater than or equal to ω^ω. Transfinite ordinal numbers are useful for indexing a transfinite sequence [that is, a noncommutative union of sequences (but not sets whose elements have no order) where the first sequence has a potentially infinite terms]. An example of a transfinite sequence is the right-sequence-union of a convergent potentially infinite sequence with its limit — say, <3, 3.1, 3.14, 3.141, 3.1415, 3.14159, …, π>. Other examples of transfinite sequences are the sequence-union of countable parallel or nested enumerations of potentially infinite mathematical objects — for example, taking into consideration the appropriate respective name-listing rate (say, 1 name per 5 seconds), birth rate (say, 10 persons per second), death rate (say, 2 persons per second), and transient rate (say, 20 persons per second) for each country and its states, provinces, counties, cities, and municipalities, the listing of all living inhabitants of the planet Earth is a theoretivally infinite human task and would require transfinite ordinal numbers to index the list of names (this last example provides a simple natural manifestation of transfinite sequences and ordinal numbers which belie their “abstract-only” Platonist-view existence). [BfnCawaling@Yahoo.com] BenCawaling (talk) 07:45, 17 December 2008 (UTC)

Q. What about further limit ordinals like ω+ω, ω², etc.?

A. in Cantor's system again use principle 2. In ZFC, this needs the replacement axiom.

Q. Is there an uncountable ordinal ω₁? I.e. how do we know that iterating the successor and limit operations indefinitely don't result in more and more countable ordinals forever?

A. I think Cantor had some way to get uncountable ordinals but I don't see exactly how. Trovatore gives a philosophical argument that there should be such an ordinal but I don't see how to get it explicitly from Cantor's rules as described in SEP. Cantor's paper is translated online so maybe I'll try to read it. In ZFC, this is again done with the replacement axiom, need to figure out the specifics.

Q. There's one supremely interesting uncountable set, the real line $\mathbb {R}$ whose cardinality is called ${\mathfrak {c}}$. Is there an ordinal the size of ${\mathfrak {c}}$ and if yes, what is its value?

A. That ordinal (${\mathfrak {c}}$ itself under the von Neumann construction) exists iff $\mathbb {R}$ can be well-ordered. Cantor spent years trying to establish that there was such a wellordering, without success, so in Cantor's system we have to say this is an unanswered question. It took Zermelo's axiomitization of set theory along with introduction of the axiom of choice to create a theory in which $\mathbb {R}$ can be well-ordered, and that's how it's done in ZFC today. As for the value of ${\mathfrak {c}}$, that's the continuum hypothesis. Cantor wanted to prove that ${\mathfrak {c}}=\aleph _{1}$ but was unable to prove that ${\mathfrak {c}}$ even appeared anywhere in his aleph hierarchy. ZFC can prove that there's an ordinal κ with ${\mathfrak {c}}=\aleph _{\kappa }$, but κ can be any ordinal (Cohen's independence proof).

I probably have some of the above wrong (along with some known missing), and the above wording is not in presentable form for the article, but I think the questions are pretty basic and that the article ought to treat them one way or the other, so I'll try to write something once I've made a bit more progress in understanding, unless you guys really think it's inappropriate. I appreciate very much your all taking the time to answer this stuff from me and I hope it can lead to making the article better. 75.62.4.229 (talk) 10:52, 30 November 2007 (UTC)

If and when someone fixes the "proof" in Hartogs number, it will show that there is a least ordinal which cannot be mapped one-to-one into a given set (such as ω). That is the proof that ω₁ exists. ${\mathfrak {c}}\neq \aleph _{0}$ and ${\mathfrak {c}}\neq \aleph _{\lambda }$ when λ has cofinality ω. JRSpriggs 20:23, 30 November 2007 (UTC)

history section

I think the article should have a history section. This is a good source to start with. I may try to add something. 75.62.4.229 (talk) 09:54, 29 November 2007 (UTC)

Scope of the graphical matchstick

Unless I am mistaken, a graphic similar to the one in the article can also be created for $\omega ^{\omega }$, $\omega ^{\omega ^{\omega }}$ and any integer number of iterations thereof. --Dan Polansky (talk) 13:25, 31 January 2008 (UTC)

Indeed, or even larger. Any countable ordinal (i.e., any ordinal $\alpha <\omega _{1}$) is the order type of a well-ordered subset of the reals (which can be chosen either closed or discrete, but not both). So it is in principle possible to give such a "matchstick" representation of some extremely large ordinals. Unfortunately, it is useless, because the representations in question get too crowded to make any sense of. Already I had a hard time fiddling with the exponential constants to get a satisfactory $\omega ^{2}$ (I drew the picture in the article), I never managed to build a good $\omega ^{\omega }$, let alone $\varepsilon _{0}$. You can see an attempt here for $\omega ^{\omega }$ and an explanation of it, but it's just worthless. --Gro-Tsen (talk) 17:09, 1 February 2008 (UTC)

Maybe we could use colour and/or the second dimension? --99.234.59.230 (talk) 06:09, 4 February 2008 (UTC)

Really, you can embed $\omega +1$ into R so that it is discrete? I'm quite certain you cannot.24.58.63.18 (talk) 12:50, 19 March 2009 (UTC)

What do you mean by discrete - no limit points in the image or no limit points in R? — Carl (CBM · talk) 12:54, 19 March 2009 (UTC)

To clarify: a discrete subset of the reals means discrete with the induced topology, so, no accumulation points in the subset. "No accumulation points in the reals" is equivalent to closed and discrete (proof: the implication ⇒ is obvious, and to prove ⇐, assume there is a limit in the reals, that limit must be in the subset because it is closed, contradicting discreteness). No subset of the reals which is closed and discrete has order type $\omega +1$ (proof: the points corresponding to elements $\omega$ would be bounded by the point corresponding to the element $\omega$ itself, so, being an increasing sequence bounded above they would have a limit, so the subset is not closed-and-discrete as remarked previously). However, any countable ordinal is the order type of a subset of the reals which is closed or discrete (again, discrete means no accumulation points in the subset): if you want $\omega +1$ for example, take the set of $-1/n$ for positive integers $n$ and add either 0 if you want a closed subset, or 1 if you want a discrete one — but you can't have both at once. I hope I've made this excruciatingly clear now. --Gro-Tsen (talk) 14:53, 19 March 2009 (UTC)

Subtraction

The statement "One can define addition, multiplication, and exponentiation on ordinals, but not subtraction or division" is nonsense. Of course one can define subtraction and division, one only has to be careful about the order of things, as neither addition nor multiplication is commutative.

• Subtraction: for any ordinals α ≤ β there exists a unique ordinal β − α such that α + (β − α) = β.
• Division with remainder: for any ordinals α > 0, and β, there is a unique pair of ordinals β / α and β % α (for a lack of a better notation) such that β % α < α, and β = α·(β / α) + (β % α).
• One can also define a sort of "logarithm with remainder": if α > 1, and β > 0, there are unique ordinals log_αβ, γ, δ such that 0 < γ < α, δ < log_αβ, and β = α^log_αβ·γ + δ. — EJ (talk) 14:45, 11 April 2008 (UTC)

What keeps you from correcting this by removing the false part?  --Lambiam 00:21, 12 April 2008 (UTC)

To EJ: The point is that those operations do not have all the properties normally expected for subtraction, division, logarithm, etc.; in particular they are not closed, i.e. do not give values for all inputs. If you look at the article on Ordinal arithmetic, you will see that the decompositions to which you referred are mentioned. The main thing is that those ordinal operations are not used by many people for the simple reason that they are not useful. JRSpriggs (talk) 06:49, 12 April 2008 (UTC)

To Lambiam: my uncertainty whether it would be agreed upon.

To JRSpriggs: what exactly do you mean by "not closed"? The operations are (uniquely) defined under the same restrictions as the usual operations on natural numbers. The context of the statement suggests the reading "addition, multiplication, and exponentiation of natural numbers can be extended to ordinals, but subtraction and division cannot". If that's not intended, it should be stated explicitly. Also, "not being used by many people" is quite different from "cannot be defined". — EJ (talk) 09:23, 13 April 2008 (UTC)

I'm not an expert in the area. I do not know whether ordinal subtraction (if definable as claimed) is of sufficient importance to mention it, but if it is, mentioning it does require a citation of a reliable source. Conversely, the statement that subtraction etcetera is not definable (as claimed in the article) also requires a citation. A middle ground is not to mention subtraction at all.  --Lambiam 19:14, 23 April 2008 (UTC)

To Lambiam: Good point. Following the middle way, I took out the sentence, "Subtraction and division with remainder of natural numbers can also be extended to ordinals, but it is rarely used.". It was wrong anyway because the remainder (when it exists) is not constrained to be a natural number. JRSpriggs (talk) 05:44, 24 April 2008 (UTC)

I don't understand what you are talking about, the sentence did not say that the ordinal remainder is constrained to be a natural number. Anyway, omitting the sentence is fine with me, as long as the misleading claim of nonexistence of subtraction and division is not there either. — EJ (talk) 12:55, 24 April 2008 (UTC)

In a hypothetical subtraction on ordinals, what ordinal would correspond to ω - ω₁? — Carl (CBM · talk) 22:11, 24 April 2008 (UTC)

Just like subtraction of natural numbers, ordinal subtraction is only defined for β ≥ α. And ω₁ − ω = ω₁ (because ω + ω₁ = ω₁). — EJ (talk) 09:59, 25 April 2008 (UTC)

Right; there is no subtraction operation defied on the natural numbers, for the same reason. There is a partial subtraction function, but that isn't a binary operation. At least that's the way a significant number of people would look at it; my guess is that this was what the person who originally edited the article was thinking. — Carl (CBM · talk) 12:08, 25 April 2008 (UTC)

(Undent) I can't help the distinctive feeling of going in circles in this discussion. The point, once again, is that whatever the original editor was thinking, it does not look that way to a casual reader, because the intro paragraph at least twice mentions the idea that ordinal numbers are an extension of the finite, natural numbers. In such a context, saying that addition etc can be defined, but subtraction can't, sounds like saying that something special about these infinite thingy numbers precludes them from being subtracted, whereas in fact they behave exactly the same as natural numbers in this respect (the only restriction on the subtraction operation being β ≥ α), just like for addition. So, while the statement may be formally correct, it is terribly misleading. It should not be there unless it includes a proper discussion of why and where the subtraction is not total. — EJ (talk) 12:27, 25 April 2008 (UTC)

I agree with the conclusion there. Sorry for prolonging a discussion that was settling down; I just wanted to make sure the opinion that subtraction isn't defined on the natural numbers, either, was represented here. I think that not mentioning (truncated) subtraction is the most natural choice, as it is essentially undiscussed in set theory texts. — Carl (CBM · talk) 13:03, 25 April 2008 (UTC)

But there must be some properties in common with subtraction on the integers to justify calling it subtraction. I imagine the reason why it is not defined (not that it couldn't be defined) is that a) negation is not defined for ordinals in general, hence even though ω -1 +1 might be parsed as ω + (-1+1), (which gives a different value than (ω-1)+1, which would have to be ω+1 by your proposed definition), for infinite ordinals negation is not even defined. And b) addition follows from the idea of concatenation, which is possible to perform given 2 ordinals, but it is meaningless to "unconcatenate" ω by 1. 24.58.63.18 (talk) 03:07, 29 January 2009 (UTC)

Relying on the axiom of regularity

The section Definitions states that

the axiom of regularity (foundation) is used in showing that these ordinals are well ordered by containment (subset)

and very similarly it says that

Such a set S is automatically well-ordered with respect to set containment. This relies on the axiom of well foundation: every nonempty set B has an element b which is disjoint from B.

My knowledge of set theory isn't really good, but if I remember correctly, these statements are false: the axiom of regularity is not actually needed at all for the theory of ordinals, and while it may simplify some proofs as shown in that section, those proofs can actually be done from the rest of the ZFC axioms. Could someone who is more competent in this area confirm this? – b_jonas 13:17, 16 May 2008 (UTC)

The theory of ordinals can be developed without regularity just fine, but one has to define ordinals as transitive sets well-ordered by set membership. The definition given in the article, which refers to just total order, does not work without the axiom of regularity. So, the caveat in the article is justified, but it really just concerns details of the definition, you do not need the axiom of regularity once you get past such issues. — EJ (talk) 14:10, 16 May 2008 (UTC)

As EJ says, the definitions and proofs are much simpler, if one is allowed to use the axiom of foundation. The definitions given in the article are inadequate, if the axiom is not used.

However, as far as I am concerned, if one does not have at least the axiom of extensionality and the axiom of regularity, one is not talking about sets. (If you do not have a home plate, it is just not baseball.) JRSpriggs (talk) 21:18, 16 May 2008 (UTC)

Well, a lot of people do work in non-well-founded set theory, so unlike extensionality, it is not very wise to rely on the foundation axiom without a good reason. IMO it would not hurt to start with the general definition which is adequate without foundation, and then give the other possible definitions as alternative equivalent characterizations.

Also, now that I come to think about it, the article attributes the definition given to von Neumann, but that does not sound right. As far as I now, von Neumann's definition of ordinals predates the axiom of foundation by a couple of years, hence he probably stated the definition in the way suitable for nonwellfounded theories. So that's another reason to use the latter. — EJ (talk) 13:16, 19 May 2008 (UTC)

Aren't both introduced in some preliminary form in two articles found in From Frege to Gödel: the axiom of foundation by Skolem ("Some remarks on axiomatized set theory" from 1922), and the von Neumann definition of ordinals by von Neumann ("On the introduction of transfinite numbers" from 1923)? I don't have access to the book and can't check myself now if this is indeed the case.  --Lambiam 00:36, 22 May 2008 (UTC)

I have the book here, and in Skolem's paper you refer to it looks like replacement is being proposed, not foundation. Von Neumann's paper does refer to augmenting Zermelo's system with replacement, but does not seem to assume foundation. --Unzerlegbarkeit (talk) 01:32, 22 May 2008 (UTC)

Thanks for all the replies so far. – b_jonas 21:22, 19 May 2008 (UTC)

Rename proposal

At Talk:Ordinal number (linguistics) there is a proposal to rename the present article, so that Ordinal number (linguistics) can be moved here.

(A notification about this fact was removed with the inappropriate editing summary "rv trolling", apparently in response to an equally inappropriate editing summary by the editor referred to. It was clearly not trolling, and I believe technically it didn't violate WP:CANVAS, either. However, the very non-neutral tone of the original notification was likely to cause disruption, so I decided to replace it rather than restore it. I very much hope that's OK. I encourage everybody to stay calm and constructive.) --Hans Adler (talk) 08:36, 17 June 2008 (UTC)

JPG-GR (talk · contribs) closed this proposal as no consensus. I assume that this means the move will not occur. Fortunately. JRSpriggs (talk) 05:30, 3 July 2008 (UTC)