Talk:Ordinal number/Archive 2

message: 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)
thread_title: Old stuff about the symbol representing multiplication

message: 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)
replies: text: 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)

text: 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)
thread_title:  What is an ordinal set? 

message: 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)
replies: text: 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)
thread_title:  "ordinal data type" 

message: Ordinal is also used for the numbering of regents, I can´t find that article anywhere./Johan Jönsson 19:04, 6 February 2006 (UTC)
replies: text: Article: Monarchical ordinal./Johan Jönsson 10:33, 10 September 2006 (UTC)
thread_title: Other meanings

message: 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)
replies: text: 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)
thread_title:  Cleanup 

message: 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)
thread_title:  An interesting pairing function 

message: 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)
thread_title:  Let's not go overboard with details... 

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

text: 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)
replies: text: 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)

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

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

text: Are you sure the  template is more appropriate than ?  I've seen  far more often, I think.  Could some experienced wikipedian tell us which is recommended? --Gro-Tsen 18:11, 15 March 2006 (UTC)
thread_title:  Suggest split 

message: 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)
replies: text: 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)

text: 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)
replies: text: 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)

text: 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)
thread_title:  Understandability for beginners 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title: Proof

message: 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)
thread_title: Ideas for the other article

message: 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)
replies: text: 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)

text: 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)

text: 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)
thread_title: Two articles or one

message: 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)
replies: text: Move completed. JRSpriggs 08:40, 30 September 2006 (UTC)
thread_title:  Move section "Topology and ordinals" to the article "Order topology"?