Talk:Ordinal number/Archive 2

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Old stuff about the symbol representing multiplication

The definition of multiplication is not the conventional one, and will probably cause confusion.

The definition given in the article was Cantor's in 1883. He changed it in 1887 -- see Michael D. Potter, "Sets : an Introduction", Clarendon press 1990, p 120.

The point is that one wants a^(b+c) = a^b >< a^c. a >< b should be read "a, b times" (as Potter suggests.)

Fixed. AxelBoldt 16:32, 2 Feb 2004 (UTC)

What is an ordinal set?

The definition of set ordinality (John von Neumann) relies on "set containment". This does not seem to be defined anywhere.

"set containment" is the same as the subset relationship. So for example, the set {1,3} is contained in {1,2,3}. AxelBoldt 16:32, 2 Feb 2004 (UTC)

Perhaps some math person could provide an explination.

It might help answer the question: Is {1,2} an ordinal set?

No, of course! As nought isn't a member of its, it isn't transitive, then it isn't ordinal.

Question: Doesn't S = {0,{0,1,2,...},1,{1,2,3,...},2,{2,3,4,...},...} meet von Neumann's definition as stated? Every element of S is a subset of S, and every two distinct elements are related by containment in one direction. This S does not seem well-ordered to me because it contains a descending infinite sequence. Maybe von Neumann's definition should state the S is well-ordered by set containment, not merely totally ordered. DRLB 18:33, 29 March 2006 (UTC)

No, there is no containment (in either direction), nor inclusion, between 1={0} and {2,3,4,...}. --Gro-Tsen 01:25, 30 March 2006 (UTC)

uncountable ordinals

I wonder whether in ZF there is a proof the class of all countable ordinals is a set. If no, why every text about such a matter takes it trivial that there are some uncountable ordinals, therefore a least one, that it must be less or equal (depending on Continuum Hypothesis) than the power set of naturals; neither greater than nor uncomparable to it?

Possibly because they are working in ZFC. Without the axiom of choice, not every set can be well-ordered, and I expect you fall flat on your face before you get to $\omega ^{\omega }$. (I notice the article doesn't really attempt to define ordinal exponentiation.) -Dan 19:29, 29 August 2005 (UTC)

There are several distinct issues here. Without AC you can still prove that there is a set of all the countable ordinals (you do have to use the axiom of replacement, though). However you can't prove that it has cardinality comparable to the cardinality of the continuum.

As for ω^ω, it depends on what you mean. If you're talking about ordinal exponentiation, as you do above, then ω^ω is a rather small countable ordinal; you definitely don't need AC to prove its existence. (You need Replacement to prove that it exists as a von Neumann ordinal, but that's just a detail of coding--without Replacement you can come up with some alternative coding of ordinals and still use ω^ω for just about anything you'd want to. --Trovatore 19:57, 30 August 2005 (UTC)

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Doesn't

Ordinals which don't have an immediate predecessor can always be written as a limit of smaller ordinals

contradict

no sequence of elements in ω1 has the element ω1 as its limit

? --SirJective 11:58, 4 Dec 2003 (UTC)

Both of these statements are true and (therefore) there is no contradiction. The crucial word in the second statement is sequence. The first statement would be false if we inserted "as a limit of a sequence of smaller ordinals". A sequence here is understood as a countable collection of things. I'll clarify this in the article. AxelBoldt 16:32, 2 Feb 2004 (UTC)

Thank you. The explanation now given in the article (together with the article net) makes this point clear to me, an I will soon change the german article to reflect this. --SirJective 23:17, 2 Feb 2004 (UTC)

---

I certainly don't object to having the von Neumann definition of ordinal in this article; but it would be useful to have it preceded by a more naive- axiomatic. Somethinh along the lines of the following (written in TeX)

axiomatic description. Note that the collection of ordinal numbers do not form a set.

$\alpha$ is an {\em ordinal} is written $\opr{On}(\alpha)$. In addition, there is a binary predicate $\sqsubset$ defined for pairs of ordinals.

\begin{enumerate} \item \label{linear-ordering-property}$\sqsubset$ is a strict linear ordering on the ordinals. This means $\sqsubset$ is transitive: $\alpha \sqsubset \beta$ and $\beta \sqsubset \gamma$ implies $\alpha \sqsubset \gamma$, linear: for every $\alpha, \beta$ one of $\alpha \sqsubset \beta$, $\alpha = \beta$ or $\beta \sqsubset \alpha$ holds and irreflexive: $\alpha \not\sqsubset \alpha$. \item $\sqsubset$ is well-founded. For any non-empty set $A$ of ordinals there is an $a \in A$ such that for all $x \in A$ either $x = a$ or $a \sqsubset x$. We refer to such an element $a$ as a least element of $A$. \item Given an ordinal $\alpha$, there is a set whose members are precisely the ordinals $\beta$ such that $\beta \sqsubset \alpha$. \item There is no set whose members are all the ordinal numbers. \end{enumerate} Write $\alpha \sqsubseteq \beta$ iff $\alpha \sqsubset \beta$ or $\alpha = \beta$. Let $\opr{On}_\alpha=\{\beta: \beta \sqsubset \alpha\}$. By (3) above, $\opr{On}_\alpha$ is a set.

Clearly, least elements of non-empty sets of ordinals are unique.

An {\em initial ordinal segment} is a {\em set} $V$ all of whose members are ordinals and such that for all ordinals $x,y$, if $x \sqsubset y$ and $y \in V$ then $x \in V$. For example, for any ordinal $\alpha$, $\opr{On}_\alpha$ is an initial segment. $\emptyset$ is an initial segment. \begin{prop} \label{recursion-theorem} Suppose $\mathcal{U}$ is an initial ordinal segment, $A$ an arbitrary set, $\mathrm{I}_{\mathcal{U}}$ the set of all functions $g$ with values in $A$ such that $\domain(g)$ is an initial ordinal subsegment of $\mathcal{U}$ with $\domain(g) \neq \mathcal{U}$ and $\mathcal{E}:\mathrm{I}_{\mathcal{U}} \rightarrow A$ an arbitrary function. Then there is a unique function $f: \mathcal{U} \rightarrow A$ such that \begin{equation} \label{recursive-def-prop} f(x)=\mathcal{E}(f | \opr{On}_x) \mbox{ for every $x \in \mathcal{U}$} \end{equation} \end{prop}

CSTAR 22:42, 19 May 2004 (UTC)

Cantor normal form

Does Cantor normal form apply to uncountable ordinals? The article states that it applies to all ordinals > 0. If so, what is the normal form of ω₁? -- Fropuff 01:31, 2005 Mar 8 (UTC)

It's $\omega _{1}=\omega ^{\omega _{1}}$. May seem strange, but it is true. 157.181.80.93 17:29, 31 May 2005 (UTC)

Are you sure? Surely $\omega ^{\omega _{1}}>\omega ^{\omega }\geq \omega _{1}>\omega$, no? -Dan 19:29, 29 August 2005 (UTC)

No. The second inequality above fails. The problem is that the notation ω^ω is ambiguous. It can mean:

1. The set of all functions from ω into ω
2. The cardinality of that set (cardinal exponentiation), or
3. The limit of the sequence ω, ω², ω³, etc (ordinal exponentiation).

In this context the meaning we're interested in is number 3 above, and that's a countable ordinal. --Trovatore 20:08, 30 August 2005 (UTC)

I'll be damned. There's a major difference between $\omega ^{\omega }$ and $\aleph _{0}^{\aleph _{0}}$, isn't there. Thank you. I need to go away and get my head around this. -Dan 23:47, 30 August 2005 (UTC)

For some reason $\omega ^{\omega }$ makes me happy. Anyway, I've added exponentiation to the article. Hopefully I got it right. -Dan 03:30, 31 August 2005 (UTC)

Is The Set N of Natural Numbers Well Defined?

Excerpt from the article: 'In set theory, the natural numbers are commonly constructed as sets, such that each natural number is the set of all smaller natural numbers:

0 = {} (empty set) 1 = {0} = { {} } 2 = {0,1} = { {}, { {} } } 3 = {0,1,2} = {{}, { {} }, { {}, { {} } }} 4 = {0,1,2,3} = { {}, { {} }, { {}, { {} } }, {{}, { {} }, { {}, { {} } }} } etc.'

Sets are enclosed in braces which are inherently balanced and corresponding to each right brace is a corresponding left brace and vice versa. Now consider the set N of all finite ordinals.It must have an infinite number of braces. To the outermost left bracket must correspond a outermost right bracket. Prior to this outermost right bracket must also be a right bracket. Between this right bracket and the corresponding left bracket must be an infinite element, which can only be the set N itself.So, is the axiom of infinity consistent with the axiom of foundation?

--Apoorv1 08:56, 28 November 2005 (UTC)

Good question! One answer: the axioms of set theory do not make reference to brace notation at all! There is no assumption that all sets can be written out in this way. To write out infinite sets, you need an infinite number of braces and commas, which presupposes some theory of infinite strings. (Finite strings, no matter how absurdly huge, are assumed to be well-understood.)

But this answer has a "cowardly" quality to it. Even if brace notation is not part of the foundation of set theory, surely we can come up with some brace notation consistent with the set theory we just built!!

Another answer: "prior to this outermost right bracket must be a right bracket" is a logical error. The outermost right bracket is a "limit bracket", just as omega is a limit ordinal. There is an infinite sequence of brackets and commas before it, but no single immediate predecessor. Naturally I prefer this answer. -Dan 15:42, 28 November 2005 (UTC)

There are two points that you have raised.You stated 'There is an infinite sequence of brackets and commas before it, but no single immediate predecessor.'I would submit that the outermost right bracket is preceded by an infinite number of right brackets only. For, the symbol prior to it cannot be a comma. Nor can it be a left bracket, for than the string would represent two sets (one of them the null set), placed side by side.So, the symbol would have to be a right bracket; in fact, each ordinal ends with a seq. of right brackets and N would, if it could be so represented, end with an infinity of such right brackets.

You also stated that the axioms of set theory do not make a reference to the brace notation at all.True.However, I would submit that the conclusions should be independent of the representation selected.

Please also consider this.The ordinals are defined by the recursive relation k=(k-1)U{k-1}.w (omega) is a limit ordinal defined as the union of 'all its predecessors'.There is an element of circularity here,as the definition of w depends on our being able to identify all its predecessors.In particular, what if w is one of its own predecessors?

In any case,for any recursion k=f(k-1),the limit, if it exists, is given by the condition of stationarity, namely w=f(w). In the present case, this limit would be given by w=wU{w}, so that w belongs to w? --Apoorv1 10:33, 29 November 2005 (UTC)

I agree that my first answer was somewhat of a cop-out, but I stand by my second answer: it ends with a single right bracket, with no symbol immediately before it. It is exactly the same way as there is no ordinal omega-minus-one.

There is also a problem with many commonly used definitions e.g. in the article, omega is introduced as "the first transfinite ordinal" -- they are impredicative, and notable mathematicians have criticised them as such, and I agree with that criticism. (I'd hope nobody actually tries to define it as the fixed point of f(x) = x U {x} though.)

But most mathematicians do accept that sort of thing, and this article doesn't seem like the right place to get into all of that. Especially because there are other definitions possible, and it really doesn't affect most of the article, which is quite constructive!

We might try to remove a bit of classicist bias, perhaps expand on "other definitions", but I'd leave it mostly alone. -Dan 16:07, 29 November 2005 (UTC)

It is rather interesting to speculate what different forms an infinite string of braces may take. If we have a single bracket to the right with nothing immediately before it,it would not ocupy any finite position in the seq. of right brackets. Then, we would need a similar left bracket to keep the braces balanced. Perhaps, it would look like {....{{{}}}}....} or {{},{{}},...{.....}. On the other hand, if we take it as {{{{{....}}}}},then each left brace has a finite position, and each right brace, when counted from the right, again has a finite position .In all cases, it would appear that we still are faced with an infinite element within the set. --Apoorv1 05:01, 1 December 2005 (UTC)

I do like your "that's nice Dan, but what form would it take" approach! Sure, I can think of concrete representations that don't fall under the cases you give above. Like this one! (Maybe we should continue this discussion there.) -Dan 20:33, 1 December 2005 (UTC)

Each finite ordinal k ends with (k+1) right braces. The ordinal w, therefore has a subseq. },}},}}}... of right braces, all occupying finite positions, counting from the left.It ends in (atleast) one limiting right brace not in any finite position, counting from the left.(This is the brace of step 2 in position 6 in your representation.) The question is whether the seq. },}},}}}.. has a limit }}}}..(w braces) or ends with just a single brace. Just as the seq. 0.1,0.11,0.111,...has the non-terminating 0.111... as its limit, I would think that the seq. },}}, }}}...of braces would have a limiting member }}}}..(w braces) rather than just a single brace. --Apoorv1 11:12, 5 December 2005 (UTC)

Hmm! Well, okay, I'll run with your analogy. My response is on this page. 15:17, 5 December 2005 (UTC)

Although the outermost bracket is written in step 2 at position 6 in your representation, it is a limiting bracket only in the natural order as we count the right brackets from the left to right.So, although it is written out in step 2,in the ordering in which it is a limiting bracket, its position corresponds to w(omega). The bracket at position 6 can be shifted to position 5 (or any position between 5 and 6), without changing its meaning. So, the expression written out by you actually is the same as one with a limiting bracket in position 5.

The position 5 is the fixed point for the algorithm by which the successive right brackets are placed.Just as 0.1111...., or 1/9 is the fixed point for the recursion X(n+1)=0.1(Xn+1),and is therefore its own predecesor in terms of the recursion, so also one could say that the predecessor (bracket) of the bracket at position 5 is a bracket (or an infinite number of them)at position 5 itself.

In other words, a situation of no immediate predecessor is indistinguishable from a situation where an infinite number of predecessors are placed at position 5 itself.(These brackets could be placed in any position beyond 5 without any change in meaning)

--Apoorv1 11:54, 15 December 2005 (UTC)

Yes, I chose position 6 to emphasize that, although it is most natural (harrr harrr...) to think of it as "position omega", it is not necessary for it to be a "fix point" or "limit bracket" in the sense you mean. The steps I describe never get to the point where all the finite ordinals are written! Nevertheless, any finite ordinal is written out in a finite number of steps, and only finite ordinals are written within the two outermost braces. That sort of thing is good enough for many purposes, for instance, ordinal arithmetic. That is, there are well-defined ways of taking two open-ended descriptions (or one open-ended description and a finite ordinal) and producing a third which describes (in an open-ended way) their sum, product, or (as I recently learned, see above in the talk page) one raised to the power of the other. -Dan 16:53, 16 December 2005 (UTC)

The problem seems to arise from a flawed extrapolation of intuition from finite numbers. It is not possible to enumerate symbols from right to left in the object you described. This can be formalised by creating an ordered set of symbols for each ordinal which corresponds to the way one would write down the set as brackets for a finite set. This set with its order reversed would not be well-founded for an infinite ordinal, like $\omega \;$. Perhaps it's clearer if one imagines writing down all the integers, as one symbol each, in order (with decreasing sized writing presumably), then write $\omega \;$ after it. What is the symbol to the left of the $\omega \;$ ?Elroch 22:19, 14 March 2006 (UTC)

The usual way to generate an infinite number of symbols for the numbers is to use strings created from a finite alphabet.The natural way is to write numbers as 0, S0,SS0,. . . or as 1 ,11, 111, 1111,... w (omega) is then represented by ...SSS0 or by 11111....In both these cases, we would have Sw=w,that is ,w is its own successor (or predecessor).See also http://mathforum.org/kb/message.jspa?messageID=3808877&tstart=0 and http://mathforum.org/kb/message.jspa?messageID=4165416&tstart=0 and[title]--Apoorv1 05:24, 18 March 2006 (UTC)

ordinal and cardinal numbers

If ordinal number 10 is a sequence. And the number 1 is a series. Explain the number 101. I've had a few critics break the meaning down but it didn't make sense.

Mr. Nelson

"ordinal data type"

A section was just added about computer ordinal data type. Is this appropriate here, and is is really true that decimal numbers are not ordinal, given that on a computer, there is in fact a minimum increment? -Dan 23:49, 4 February 2006 (UTC)

I don't like the section on ordinal data type. It seems, at best, irrelevant to this article. And yes, real numbers are not ordinals. The numbers that a computer stores are not reals, but rather floating point numbers. Those are not ordinals either. Ordinals are totally ordered under inclusion and containment. -lethe ^{talk +} 00:28, 5 February 2006 (UTC)

Other meanings

Ordinal is also used for the numbering of regents, I can´t find that article anywhere./Johan Jönsson 19:04, 6 February 2006 (UTC)

Article: Monarchical ordinal./Johan Jönsson 10:33, 10 September 2006 (UTC)

Cleanup

I did some cleanup on this article, removing unrelated stuff to other pages. Somebody knowing set theory needs to work more on the intro though,to integrate the two meanings for ordinal a bit. Any volonteers? Oleg Alexandrov (talk) 05:01, 8 February 2006 (UTC)

I would prefer not to introduce it with the Von Neumann definition (or even the older "class of order-isomorphic sets" definition), because an ordinal number is not really a set! This has been on my todo list forever. -Dan 19:30, 8 February 2006 (UTC)

An interesting pairing function

By transfinite induction on ∞·α+β, we can define ξ(α,β) = the smallest ordinal γ such that α < γ and β < γ and γ is not the value of ξ for any smaller α or for the same α with a smaller β.

ξ is defined for all pairs of ordinals and is one-to-one. It always gives values larger than its arguments and its range is all ordinals other than 0 and the epsilon numbers.

ξ(0,β)=β+1. ξ(1+α,β)=(ω^(ω^α))·(β+k) for k = 0 or 1 or 2 depending on special situations: k=2 if α is an epsilon number and β is finite. Otherwise, k=1 if β is a multiple of (ω^(ω^(α+1))) plus a finite number. Otherwise, k=0.

ξ(α,β)<ξ(γ,δ) if and only if either (α=γ and β<δ) or (α<γ and β<ξ(γ,δ)) or (α>γ and ξ(α,β)≤δ).

Some examples: 1=ξ(0,0), 2=ξ(0,1), and generally α+1 = ξ(0,α). ω=ξ(1,0), ω·2=ξ(1,1). Generally, ξ(1,α+1)=ξ(1,α)+ω. ξ(2,0)=ω^ω. ξ($\epsilon _{0}$,0)=$\epsilon _{0}$·2.

Using this pairing function on ordinals and a pairing function on natural numbers, one can construct an explicit bijection between ω and $\epsilon _{0}$.

Last revised JRSpriggs 08:06, 13 March 2006 (UTC)

Let's not go overboard with details...

I realize that my edits to the article (starting 2006-02-11) have triggered a gain of interest, which is a good thing, but now I believe we should not go overboard with details: for instance, I think section 6 is full enough as it is, and if more is to be added, the article needs to be split, with the basic stuff (and a summary of more) in the main article and perhaps a new article on "countable ordinals" or "recursive ordinals" or "construction of ordinals" or some such thing. I mean, my motivation for describing the Veblen hierarchy was to give a feeling that (even recursive) ordinals go "very far", something which I think is best achieved by describing the Feferman-Schütte ordinal and perhaps the Bachmann-Howard ordinal: now it's certainly a good thing to say more about the Veblen hierarchy on Wikipedia, but perhaps no longer on this article. On a related note, I think it would be better if the contributor using IP's in the 66.44.0.0/36 range (who made a substantial number of changes in the past few weeks) created an account, it would help others recognize that the edits in question are not coming from a wild source. --Gro-Tsen 14:49, 3 March 2006 (UTC)

I was the user with the IP 66.44.whatever who was doing many edits recently on the ordinal number page. -- JRSpriggs

Suggest split

OK, this article is too long, now. It makes editing cumbersome and it is probably tedious for readers also. I suggest making the arithmetic of ordinals section into a separate article, and something similar should probably be done about some “large” ordinals (but I really don't know what the article should be). We can leave an “executive summary” of arithmetic operations in this article, but all the gory details should go elsewhere. Comments? Opposition, anyone? --Gro-Tsen 23:22, 6 March 2006 (UTC)

Very much agree. I also thought that this article is too big to be maintainable. Oleg Alexandrov (talk) 03:59, 7 March 2006 (UTC)

Moving "large ordinals" to another page is reasonable. But no executive summary could do justice to "ordinal arithmetic". -- JRSpriggs on March 6, 2006

To Oleg: I cannot find help on how to move a section into a new article in order to implement this. JRSpriggs 05:05, 11 March 2006 (UTC)

I don't think there's anything better than cut'n'paste, unfortunately. (You have to make sure your cut'n'paste will preserve even bizarre Unicode characters, however.) But let's wait to see if some wikiexpert can confirm this. (Speaking of which, I think you're supposed to indent your comments one level down every time you reply on a comment page, using semicolons at the beginning of the paragraph.) --Gro-Tsen 19:00, 11 March 2006 (UTC)

As long as you mention the source article in the edit summary when you paste into a new article, the terms of the copyright are considered respected, and you can copy paste the text. See Wikipedia:How to break up a page. -lethe ^{talk +} 20:09, 11 March 2006 (UTC)

Yeah, right. A cut and paste is all one can do. Oleg Alexandrov (talk) 16:33, 12 March 2006 (UTC)

The split is a good idea. The name "large ordinals" doesn't really do it for me, though. Maybe something like hierarchy of countable ordinals? I don't really love that either, but it might be a little better. Other suggestions? --Trovatore 18:19, 12 March 2006 (UTC)

• JA: "Extraordinals". Jon Awbrey 18:22, 12 March 2006 (UTC)

OK. I just did the split. I put the section on "Some "large" ordinals" into the new article large ordinals. JRSpriggs 08:26, 13 March 2006 (UTC)

I put up a {{main|large ordinals}} template link to this new article and I wrote a little summary (not good at all, though, so feel free to improve it in any way) in the original article. Now I still think we should do the same for the section on ordinal arithmetic. --Gro-Tsen 14:16, 13 March 2006 (UTC)

I just split out the section "Arithmetic of ordinals" to a new article "Ordinal arithmetic". JRSpriggs 09:40, 15 March 2006 (UTC)

Are you sure the {{see details|...}} template is more appropriate than {{main|...}}? I've seen {{main|...}} far more often, I think. Could some experienced wikipedian tell us which is recommended? --Gro-Tsen 18:11, 15 March 2006 (UTC)

My impression is that "see details" is used by a broader article to point a more narrowly focused article which is subordinate to it and that "main" is used by the subordinate article to point back at the broader article. Since "ordinal number" is the broader topic compared to "ordinal arithmetic" or "large countable ordinals", "ordinal number" should use "see details" to point at them and they should use "main" to point back at it. Someone please correct me, if I am wrong. JRSpriggs 07:29, 17 March 2006 (UTC)

Executive summary?

The executive summary works as a way to map out the territory in a minute, but is there any way this can be formatted so it looks more in place in an encyclopaedia article? Elroch 19:10, 14 March 2006 (UTC)

I removed the indenting and added subsection headers. Does that fix what you were worried about? JRSpriggs 03:37, 16 March 2006 (UTC)

I don't understand this section at all, and I consider myself reasonably well trained in the foundations of mathematics (A level grade C, over 90% in "Logic and Foundations of Mathematics" university module aimed at first year computer science undergrads). Can we just cut it? I don't see that it has any place in the article if it's this confusing. Hairy Dude 03:35, 17 March 2006 (UTC)

It should be moved into the lead section per WP:LEAD, and there's no need for a section called "Introduction". I agree also that it's a bit too terse and "choppy", but it's along the right lines for a summary-style lead section. Also many of the subsections of the article are duplicative of other articles; I don't think it's really necessary to repeat in detail the content of wellordering, club set, cofinality, and transfinite induction.

To Trovatore: I wrote the executive summary very quickly. I felt that an introduction should "Tell them what you are going to tell them!" and the old introduction did not do that. If you can make it better, please do so. However, the section on "confinality" has some information in it specific to ordinals which I did not see in the other article, so please do not delete it. JRSpriggs 07:37, 17 March 2006 (UTC)

OK. I fixed up the introduction myself. How does it look now? JRSpriggs 06:56, 18 March 2006 (UTC)

Understandability for beginners

One things bugs me now: the section "Ordinals extend the natural numbers" is written so as to be understandable with no knowledge of mathematics (and I think this is important and it should be kept that way: an ordinal is something one can form an intuitive image of without being a mathematician, and I believe it is worth it), but the summary at the beginning of the article is certainly sufficient to deter any such person from reading the rest. Any idea of how we could point out the fact that at least parts of the article should be fully accessible, without deviating from the Wikipedia style rules? --Gro-Tsen 15:54, 18 March 2006 (UTC)

I agree that that is a problem, but I have no idea currently of how to fix it. JRSpriggs 06:39, 20 March 2006 (UTC)

Wow, the introductory paragraph is a mess. It defines ordinals vaguely and lists a bunch of random properties. It should define ordinals precisely and either talk about their domain of use or give a simple conceptual model. Luqui 12:18, 28 March 2006 (UTC)

Which introductory paragraph do you mean, Luqui? The "Ordinals extend the natural numbers" section, or the paragraph which summarizes what is to come later? --Gro-Tsen 12:27, 28 March 2006 (UTC)

The introduction reads oddly. It gives a terse definition of an ordinal number, then seems to announce that this article is actually going to be about transfinite ordinals. Shouldn't that treatment be in a separate article, linked from here, headed transfinite articles? Next thing is the articles starts some definitions related to well-orderings. This is a very mathematical approach, setting down some facts which will then be picked up later in the paragraph to make a conclusion: a mathematician will follow it but a non-mathematician will surely just be bamboozled. "Well-ordering is total ordering with transfinite induction" -- Bam! that's lost every non-mathmo right at the start. Mooncow 23:34, 10 October 2006 (UTC)

Also, the style doesn't feel like wikipedia style -- all that "we" business sounds like a maths research paper. My suggestion: move all the Cantor stuff into "arithmetic of ordinals", move all the transfinite stuff into "transfinite ordinals" or "transfinite induction", and keep in this article just some high-level outlines of the different ways to define and derive ordinals along with some discussion of how they're used and arise, including links to the transfinite stuff, Cantor, von Neumann, Godel, ordinal arithmetic, etc etc. And the terms "well-ordering", "transfinite" and "normal form" should NOT appear in the executive summary! Mooncow 23:38, 10 October 2006 (UTC)

Definition of ordinals by $\subset$ vs. definition of ordinals by $\in$

I have compared your article with some works of Godel and he defines an ordinal, as a set such that every element of it is a subset of it, and it is totally ordered by the relation $\in$. This approach differs from the one in this article where you require that an ordinal is totaly ordered by $\subset$- could you point me in the direction of the article where von Neumann defines ordinals?

Sorry, I do not have that reference. However, the important point here is that there are two different but equivalent ways of talking about Total orderings. You can use either "≤" or "<". Here, we are using "≤" which corresponds to $\subseteq$. Apparently, in your reference, Gödel must have been using "<" which corresponds to $\subsetneq$ (no equality) which for ordinals is the same as $\in$. JRSpriggs 02:11, 10 August 2006 (UTC)

Gödel defines ordinals as sets totally ordered by $\in$ and such that every element of it is a subset of it. Proving later on that, as you said, in these sets $\in$ is the same as $\subsetneq$. I don not know if, starting with sets totally ordered by $\subseteq$ (such that every element of it is a subset of it) you do not allow some other sets to be ordinals. Finding a minimal element in a Gödel's ordinal is trivial - this is the element guaranteed by the Axiom of Regularity to exist in every set. How do you find a minimal element in an ordinal with your definition? It has to exist since it is a well ordered set.

Personally, I would rather start by defining an ordinal as a transitive set of transitive sets and then prove that it has all the desired properties. Part of that would be proving that it is well-ordered by $\in$. But I do not feel comfortable trashing the current definition which claims to have a reference, which I do not have for mine. JRSpriggs 10:15, 22 August 2006 (UTC)

I would like to see some reference, since I cannot see that the definition in here is correct. I can prove all the facts concerning ordinals for Gödel's definition. For the definition given in here I cannot do that. Either this definition is incorrect, or there is a proof I cannot find. If anyone could show me the proof or show me the reference to the place with a proof it would be great. If not, the definition is incorrect and has to be changed.

The current definition says "A set S is an ordinal if and only if S is totally ordered with respect to set containment and every element of S is also a subset of S.".
Since every element is a subset, S is a transitive set. Suppose that $x\in y\in z\in S$, then $y\in S$ and $x\in S$ also. What about $x\in z$? Either $y\subseteq z$ or $z\subseteq y$ by totality. But the latter leads to a contradiction because it implies that $y\in z\subseteq y$ and thus $y\in y$ which is inconsistent with the axiom of regularity applied to {y}. So we get $x\in y\subseteq z$ and hence $x\in z$. Thus any element z of S must be transitive, that is, S is a transitive set of transitive sets. So S is an ordinal by the definition which I have chosen to use personally. Do you need more proof? JRSpriggs 05:01, 23 August 2006 (UTC)

Thanks. I was looking for the other implication, as it is a bit more complicated, but it works. Thanks for your time.

Anyone know how to show that an ordinal defined as a transitive set of transitive sets is also totally ordered by containment? It's been driving me nuts - I cannot sleep at night. All I need to show is that a transitive proper subset of an ordinal is an element of the ordinal and I can do the rest. As a side benefit, it will show that total ordering by containment coincides implies total ordering by elementality. It seems easy to show that a transitive set totally ordered by elementality is also totally ordered by containment. And easy to show that a transitive set totally ordered by containment contains only transitive elements. But how to show that a transitive set which contains only transitive elements is totally ordered by elemantlity? Hard to believe that such a weak definition does the job. 64.42.233.61 18:09, 21 September 2006 (UTC)

Proof

The article on total ordering says "... transitive (a < b and b < c implies a < c) ... trichotomous (i.e., exactly one of a < b, b < a and a = b is true). We can work the other way and start by choosing < as a transitive trichotomous binary relation; then if we define a ≤ b to mean a < b or a = b then ≤ can be shown to be a total order.".

Define "ordinal" to mean a transitive set of transitive sets. Then an ordinal is a transitive set of ordinals. I need to show that the elements of an ordinal are transitive and trichotomous with respect to $\in$. Suppose x, y, z are elements of S which is an ordinal. If $x\in y\land y\in z$, then $x\in z$ because z is a transitive set. Let us call an ordinal S "universally trichotomous" if and only if whenever T is an ordinal $S\in T\lor T\in S\lor S=T.$ Let us call an ordinal S "good" if and only if it and all its elements are universally trichotomous. Consequently any good ordinal S is totally ordered and indeed well-ordered (using the axiom of regularity). Also notice that any element of a good ordinal is good. So the class of good ordinals is transitive.

Suppose S is an ordinal and all of its elements are good. I want to show that S is also good. All the elements of S are universally trichotomous. So all that remains to be shown is that S itself is universally trichotomous. Let T be any ordinal. I want to show that $S\in T\lor T\in S\lor S=T$. Suppose that $T\notin S\land S\neq T$ and try to show $S\in T$. Let $Z=T\setminus S\!.$ If Z were empty, then T would be a subset of S. Since it is not S, there must be an element of S not in T. The least such element would have to be equal to T, but that would contradict the fact that T is not an element of S. Thus Z is not empty. Using the axiom of regularity let z be an element of Z which is disjoint from Z. Then z is an element of T and thus a subset of T, but being disjoint from Z it must be a subset of S also. Consider any w in S, $w\in z\lor w=z\lor z\in w$ by the goodness and thus universal trichotomousness of w. If $w=z\lor z\in w$, then z is in S contradicting the choice of z as in Z and thus not in S. Thus $w\in z$ and hence S is a subset of z. So by the axiom of extensionality, S is z and thus in T.

So now we know that an ordinal S whose elements are all good is universally trichotomous and thus good. Consider any ordinal. If all its elements are good, then it is good. Otherwise, it must have a non-good element. By the axiom of regularity applied to the non-good elements of S, there must be a non-good element of S all of whose elements are good. But that element must be good because all its elements are good, and this contradicts its choice as a non-good element of S. So S has no non-good elements and S itself must be good. So all ordinals are good. Thus the class of all ordinals is the class of good ordinals, and that class is transitive and trichotomous. So the class of all ordinals is totally ordered by $\in$. It can be shown that any non-empty subclass of the ordinals has a least element. So the class of ordinals is well-ordered. JRSpriggs 07:05, 22 September 2006 (UTC)

the phrase "the least such element would have to be equal to T" seems to assume exactly the point I have been stuck on. Though your proof gives me an alternative approach: Define an ordinal to be complete if and only if every transitive proper subset is also an element. Then show that any incomplete ordinal contains an incomplete element. Next apply the axiom of regularity to the incomplete elements of an ordinal (this is the point I missed) to arrive at a contradiction which then shows that there are no incomplete elements in an ordinal. And finally any ordinal is complete. From there given any two ordinals, consider the intersection which is a transitive subset of each ordinal. By the axiom of regularity again, this intersection cannot be an element of both ordinals and so it must equal the ordinal that it is not an element of. This makes the "smaller" ordinal a transitive subset of the other and so either it is an element or is equal to it. This induces a total ordering on ordinals by elementality. Combined with your earlier statements this shows the equivalence of Godel, Von Neumann, and the modern definitions. 64.42.233.61 16:05, 22 September 2006 (UTC)

Justification of "the least such element would have to be equal to T" where "such" refers to elements of S which are not in T. Let $U=S\setminus T$. In this case, we are presuming that U is non-empty. So use the axiom of regularity to choose an element u of U which is disjoint from U. Then u is a subset of the intersection of S and T which intersection is T itself (remember that T is a subset of S in this case). Consider any element t of T. It is an element of S, so it is universally trichotomous. Thus $t\in u\lor u\in t\lor t=u$. If $u\in t\lor t=u$, then u would be an element of T contradicting the choice of u. So t is an element of u, and thus T is a subset of u. Since T and u are both subsets of the other, they must be equal by the axiom of extentionality. This is what was to be proved. And it implies that T is an element of S which would make S universally trichotomous and thus good. JRSpriggs 02:53, 24 September 2006 (UTC)

Should the article really be called transfinite ordinal numbers

It seems to me that this article is miss-named, and transfinite ordinal numbers might be a better title, leaving Ordinal to be a more basic article covering the finite case, basically the same as How to name numbers in English#Ordinal numbers. The problem I see is that some people are likely to come across this looking for something much more basic. --Salix alba (talk) 09:45, 3 September 2006 (UTC)

"Transfinite ordinal number" is an old name which is not really used any more. Everyone just calls them "ordinals". So I disagree. Especially since VERY MANY articles point to this article by the current name. If you want to have another article with a name like "ordinal number (naming)" and a disambiguation link, that would be OK. JRSpriggs 10:03, 3 September 2006 (UTC)

Just looking through the what links here we have

• Danish language
• HyperCard
• Array
• The Sand Reckoner
• Chomp
• Floor numbering
• Differences between Norwegian Bokmål and Standard Danish
• Exit 0
• Year zero
• List of Pretty Sammy minor characters
• Roman numerals
• 0 (number) and many other numbers
• Rastafari movement - many general articles
• Serial number
• Addition
• Ordinal notation

all refer to mainly the finite case. For such a well linked article I feel is is very important to have a basic introduction, which explains in simple terms the concept of order. As such I think if fails GA criteria 1a) it has compelling prose, and is readily comprehensible to non-specialist readers; As such I'm now listing it on Wikipedia:Good articles/Review. --Salix alba (talk) 11:20, 3 September 2006 (UTC)

I agree with Salix alba. At the very least, the first section of this article must address ordinal numbers as they are known to everyone but mathematicians. If the two sections become long enough that they no longer fit in one article, it will be necessary to move to summary style and ask what's a subtopic of what. (By the way, though, you might be surprised by the content at Addition!) Melchoir 03:12, 4 September 2006 (UTC)

Notice that even among the articles which Salix alba selected, Chomp, Addition, and Ordinal notation refer to the set theoretic concept and not to mere number names. This article is already too long, which is why a mere link to another article is appropriate. JRSpriggs 03:20, 4 September 2006 (UTC)

From Addition, it says "Addition of sets.
One extraordinary generalization of the addition of natural numbers is the addition of ordinal numbers. Unlike most addition operations, ordinal addition is not commutative. However, passing to the "smaller" class of cardinal numbers, we recover a commutative operation. Cardinal addition is closely related to the disjoint union of two sets. ..." (emphasis added). JRSpriggs 03:25, 4 September 2006 (UTC)

Yeah, I put that at the bottom of the article because I don't expect the average reader to appreciate it. The original, still-commonly-understood meaning of "ordinal number" is finite. Just because mathematicians have taken the concept and generalized the hell out of it doesn't mean we should push that meaning aside. And you can't know how much longer this article will be if it addresses the common meaning, because it hasn't been attempted yet. Melchoir 03:52, 4 September 2006 (UTC)

There is already another article, referenced by this one, which covers common naming (as I already said). I did not say that this article is almost too long; I said that it is already too long. If any change is made, it should be to make it shorter. Before you-all started this crisis, I was thinking about moving the section on order topology into the article by that name just to make room for simple edits on this article. JRSpriggs 05:20, 4 September 2006 (UTC)

The article is properly named. It's the content that needs (minor) revision. It immediately restricts:

• "Here, we describe the mathematical meaning of transfinite ordinal numbers."

Why? Yes, the smaller numbers are fairly trivial. That doesn't mean ignore them, it means take a paragraph or two and say what little there is to say. Then get on with the serious mathematics that motivated the invention in the first place. --KSmrq^T 05:42, 4 September 2006 (UTC)

But naming conventions ("first", "second", "third", etc.) are a completely different subject. Having a disambiguation pointer to the appropriate article (which is already there) should be enough! JRSpriggs 07:16, 4 September 2006 (UTC)

I'm not so sure that everyday ordinal numbers are simply a naming convention, nor that they're a completely different subject. Names of numbers in English is certainly one possible interpretation, but the number of references (0) makes me somehow doubt that such a shallow, simplistic treatment speaks for everyone. I'm more inclined to believe that a trip to the math section of your local research library on education will reveal quite enough conceptual material for a section here, if not a whole new article. In lieu of such a trip, please try to keep an open mind. Melchoir 08:11, 4 September 2006 (UTC)

Was I not clear? I'm agreeing with you that the article should not be moved to "transfinite ordinal number". Nor is this the place to talk about the linguistic difference between "one, two, three" and "first, second, third", except to note that the distinction naturally occurs outside of mathematics. But we should still take a moment to talk about simpler ordinals in mathematics. --KSmrq^T 18:19, 4 September 2006 (UTC)

My preference would be for this article to be called Ordinal number (set theory). Firstly, I agree that transfinite ordinal number would be a bad name for the article because, as noted, that terminology isn't used much except to set historical context. (It's fine as a redirect.) However, the disambiguation currently present isn't adequate. Everyday ordinal numbers are not just a naming convention. There is a conceptual issue in "school mathematics" education about ordinal number versus cardinal number (and sometimes versus nominal number or serial number), and when ordinary folks are looking up "ordinal number", that's what they're trying to find. What is needed is a separate article describing this; Names of numbers in English#Ordinal numbers doesn't do the job. Once such an article is written (and I am not sufficiently knowledgeable in mathematics education to do it properly), then my preference would be for Ordinal number itself to be a disambiguation page. This would take care of the concern that many articles link here. Michael Kinyon 12:29, 4 September 2006 (UTC)

Incidentally, while I agree that this article is long-ish, it is certainly not too long. The idea, though, of moving some of the topological details to Order topology is a good one. Michael Kinyon 12:33, 4 September 2006 (UTC)

How about creating a page called Ordinal which would cover the more basic usage, I think the term has a long history which predates modern set theory, so could make for an intering and useful article. Yes I agree transfinite ordinal number is not a good name, it been a which since I've studied the subject. I quite like Ordinal number (set theory). Moving the page should not be too problematic as Wikipedia:AutoWikiBrowser is good a perfoming mass page relinking.

What could go in the page, a discussion on ordinal vrs cardinal, a discussion on the concept of ordering, introducing readers to the concept of transitive relations, and a link here. --Salix alba (talk) 17:04, 4 September 2006 (UTC)

To KSmrq: Simplier ordinals are natural numbers. It already makes that clear by saying "Ordinals are an extension of the natural numbers different from integers and from cardinals.". My position is that no change should be made in this article on account of "simplier ordinals". There seem to be at least two types of change being advocated here: (1) moving this article and putting something else in its place which would be disasterous for the many articles on set theory which point to it; or (2) cluttering this article up with material which is not relevant. JRSpriggs 07:33, 5 September 2006 (UTC)

If I am overruled and we have to rename this article, I would prefer Ordinal (set theory). The word "number" is not much used anymore and it makes the name longer than necessary. The reason I say that this article is too long is that I experience problems with the editor function when I try to edit the entire article (or introductory part) rather than a section. JRSpriggs 07:38, 5 September 2006 (UTC)

Furthermore, messing with this article would adversely impact the entire Category:Ordinal numbers for which this is the lead article. Notice that their names are almost the same. And if something else is put in its place, this category would become more vulnerable to pollution by articles about dreck like numbering the floors of a building. JRSpriggs 07:58, 5 September 2006 (UTC)

Yes Ordinal (set theory) sounds good to me. --Salix alba (talk) 10:19, 5 September 2006 (UTC)

• This article should be left at Ordinal number for a "number" of reasons, most of which have been stated by User:JRSpriggs. Perhaps an article Ordinals could be created for the grammatical term, with an appropriate {{For}} included in this article. There seem to be multiple topics we're dealing with here, though....

• Cardinal number (set theory) at Cardinal number
• Cardinal number (grammar) primarily at names of numbers in English, I believe)
• Ordinal number (set theory) now at Ordinal number
• Ordinal number (grammar), hidden in names of numbers in English.

The finite cardinals and ordinal, in the set theoretical sense, are the same, and possibly could be in a separate article finite number or finite number (set theory). If this article were to be renamed (which I'm opposed to) Ordinal (set theory), then Cardinal number should also be renamed to cardinal (set theory). — Arthur Rubin | (talk) 19:28, 5 September 2006 (UTC)

I'm afraid I have to be a traitor to my class here—I don't think we can justify the main article at ordinal or ordinal number being the set theorist's concept. Those terms have a common meaning which is at least somewhat encyclopedic (though less so than the set-theoretic one, of course), so really they should be disambiguation pages. Since ordinals are of use to mathematicians other than set theorists, I'd suggest ordinal (mathematics) as the appropriate location for the current article. --Trovatore 19:34, 5 September 2006 (UTC)

Good grief. Of course this article should not be renamed! Maybe this book will be an antidote to all this silliness:

Halmos, Paul (1974). Naive Set Theory. Springer. ISBN 0-387-900092-6. (reprint of 1960 classic)

See chapter 19 ("Ordinal Numbers"), and consider the big picture. ---CH 21:20, 6 September 2006 (UTC)

Well, as I see it, the big picture is that lots of people use the term "ordinal number" to mean something that's only tangentially connected with the Cantorian concept. Moreover, while this populist meaning is not nearly as interesting as the set-theoretic one, it's not completely unencyclopedic either, and nor can we count on the set-theoretic one as being the overwhelmingly primary target of searches. Therefore ordinal number should probably be a disambig page. --Trovatore 21:23, 6 September 2006 (UTC)

I think the problem is that everyone here is already familiar with that book, so it's not ignorance of that material that is causing the disagreement. I have no opinion as to the "correct" way to resolve this, but I do admit that there is a valid concern for disambiguation here. --C S (Talk) 06:21, 8 September 2006 (UTC)

I just want to emphasize that the disambiguation is already present in the very first sentence which reads "Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc., whereas a cardinal number says "how many there are": one, two, three, four, etc. (See How to name numbers.)". JRSpriggs 07:25, 8 September 2006 (UTC)

Oh, of course, it's not that there's anything unclear about the actual text; that's not really the point. The concern is rather that someone looking for an article on ordinal numbers in the more prosaic sense, won't find it, and someone linking the expression ordinal number in another article, intending the lay meaning, will have made an inappropriate link.

I guess I would agree that there's not much point in making ordinal number into a disambiguation page until such time as an article on the non-mathematician's concept is actually written. Someone should probably do that, as dull an article as it would admittedly be. --Trovatore 15:26, 8 September 2006 (UTC)

Okay, but at least ordinal should still be a disambiguation, no? 192.75.48.150 18:54, 8 September 2006 (UTC)

Yes! I think a plainer article is badly needed. I have the maths to follow this article, but I linked to it from "ranking" and, gosh, you get a short sharp shock when you arrive at a scholarly treatment of the set theoretic definitions and aspects of mathematical ordinals! This article is at the end of links from a whole stack of places for which it is wholely unsuitable, great though much of the article's content is. We need to stop the mathematical bigotry and realise that treating a subject from a specialised standpoint should be qualified as a specialised standpoint, not as the general article. Mooncow 23:49, 10 October 2006 (UTC)

NB, in the long run, "there are a lot of links to it" is not a reason to resist rename/restructuring. It is an argument for accepting shoddy quality in order to save work. If there are a lot of links to it, that's a good reason to think through what we do before going for it willy-nilly, that's all. Mooncow 23:49, 10 October 2006 (UTC)

Ideas for the other article

Just some quick thoughts as to what the non-mathematician's article might include:

• Discussion of when the various endings (st, rd, th) get used
  • Comparison with other languages
• The status of "zeroth"
• The status of nth, (n+1)-th versus (n+1)-st
• "Fractional ordinals" (Duck Dodgers in the 25 1/2th Century)

Everything I can think of to say about the topic seems to be broadly characterizable as linguistics, so the title I'd suggest is ordinal number (linguistics).

• Ranking strategies (1st, 2nd, 3rd, joint 4th, 1224 vs 1334 vs 1234 rankings, etc) Mooncow 23:49, 10 October 2006 (UTC)

Two articles or one

Are people generally agreed on whether we should have two articles, irrespective of the actual names? One sugestions to names could be ordinal number (finite) and ordinal number (infinite). As for the consequences of moving the page I now have AWB access so it should make a mass renaming easier.--Salix alba (talk) 00:27, 11 October 2006 (UTC)

I don't think there's much point in an article on finite ordinals as a mathematical concept. They aren't different extensionally from finite cardinals, and really I don't think they're particularly different intensionally either. As far as I can see the only real interest in them is linguistic; that is, what sort of words are used. --Trovatore 01:18, 11 October 2006 (UTC)

How about having a new article on finite ordinals named Ordinal number (mathematics education), explaining how the New Math of the 1960's tricked a generation of students into believing that there was a significant difference between ordinal and cardinal numbers in the finite case. -- Four Dog Night 04:33, 11 October 2006 (UTC)

Finite ordinals are natural numbers and there is already a separate article on that subject. And there is also a link to it in the sentence "Ordinals are an extension of the natural numbers different from integers and from cardinals." which appears in the lead of this article. JRSpriggs 10:10, 11 October 2006 (UTC)

Apparently, the trickery is still going on, and getting worse. There are now cardinal, ordinal, and nominal numbers (see [1]). Does the discussion of nominal numbers belong in the Natural number article as well? -- Four Dog Night 21:30, 11 October 2006 (UTC)

I'm now thinking it may be best to treat finitie ordinals and cardinals in the same article, explaining the similarities and differences. I've also been reading up on my mathematical eductation litrature. It seems like counting, ordering and assigning cardinals are distinct educational stages in a childs development of mathematics. --Salix alba (talk) 23:29, 11 October 2006 (UTC)

Question on topology

I wonder, why "Any ordinal is, of course, an open subset of any further ordinal"? For example take the Set [0, Ω] where Ω is the first uncountable ordinal. How do you construct {ω} as an open subset, where ω is an infinite countable ordinal? 84.168.77.237 23:39, 8 September 2006 (UTC)

{ω} = [ω] is not open. What the article is saying is that ω = [0, ω) is open in Ω or any other ordinal larger than ω. Remember that in an order topology an open ray

$(-\infty ,b)=\{x\mid x<b\}\!$

is an open set, but this is [0, b). If we let b = ω, then we get the desired set. JRSpriggs 06:35, 9 September 2006 (UTC)

Move section "Topology and ordinals" to the article "Order topology"?

If no one raises an objection in the next few days, I intend to move the section Ordinal number#Topology and ordinals from this article into the article Order topology. My objective is to make this article shorter (and that one longer). JRSpriggs 05:27, 25 September 2006 (UTC)

Move completed. JRSpriggs 08:40, 30 September 2006 (UTC)