Talk:Ordinal number/Archive 2

291:\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. 2872:" 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. 298:

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}

1448:" 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 Knowledge style rules? -- 2194:

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.

664:.'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. 31: 1497:

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.

735:

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. --

716:

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. --

1513:

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!

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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 Knowledge, but perhaps no longer on

3163:"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. 1384:, 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 1370:, 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. 3930:

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.

3477:

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.

3457:

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

1407:

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

1198:

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

757:

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

4038:

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

3939:

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.

3929:

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

3500:

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

1512:

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

734:

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

2215:

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

2193:

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.

1838:'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. 1331:

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"

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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

802:

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

753:

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

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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

4095:

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

3841:

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

3581:

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

3359:

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

3291:

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

715:

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

1496:

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

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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

629:

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.

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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

3806:

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

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exponentiation, as you do above, then 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

615:

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

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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

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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

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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

671:

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

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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.

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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.

3595:. 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. 616:

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?

1280:

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.

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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.

3349:. 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). 761:

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)

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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.

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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

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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

925:

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)

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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.

3422:

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.

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I wonder, why "Any ordinal is, of course, an open subset of any further ordinal"? For example take the Set where Ω is the first uncountable ordinal. How do you construct {ω} as an open subset, where ω is an infinite countable ordinal?

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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

1468:

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.

1017: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. 705:

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

1781:'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. 281:

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)

1160:(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? -- 686:

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

985:

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?

2837:

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

1792:

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

2219:

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

675:

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?

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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

667:

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.

1317:

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? --

264:. 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. 1125:

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

3436:

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!

1011:

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.

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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. --

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

371: 4192: 4053:, 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. -- 3208: 1208:

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

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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)

555: 3867:; 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 3563:

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. --

2502: 696:, 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.) 1569:

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

1332:

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.

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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.

1906: 1014: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. 3067: 2927: 2737: 1106: 1076: 1049: 272:

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. --

2036: 2010: 1346:

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?

3548:

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

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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

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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

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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

609:

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:

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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....

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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

910:

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.

97:

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.

3325:

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.

2825:

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

829:. 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 1185:

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

711:

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)

3530:

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

3509:), 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; 3360:

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.

288:$ \alpha$ is an {\em ordinal} is written $ \opr{On}(\alpha)$ . In addition, there is a binary predicate $ \sqsubset$ defined for pairs of ordinals. 692:

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

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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

207:

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

3381:

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

3373:

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

3223: 1613:

Sorry, I do not have that reference. However, the important point here is that there are two different but equivalent ways of talking about

1081:

Using this pairing function on ordinals and a pairing function on natural numbers, one can construct an explicit bijection between ω and

4127:{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 4015:

Are people generally agreed on whether we should have two articles, irrespective of the actual names? One sugestions to names could be

1525: 3263: 147:

infinite sequence. Maybe von Neumann's definition should state the S is well-ordered by set containment, not merely totally ordered.

3105:

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

1024:

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(

746: 727: 1152:

OK, this article is too long, now. It makes editing cumbersome and it is probably tedious for readers also. I suggest making the

1777:(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 3992:

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

3777: 1209: 1813:. But I do not feel comfortable trashing the current definition which claims to have a reference, which I do not have for mine. 3453:

I'm not so sure that everyday ordinal numbers are simply a naming convention, nor that they're a completely different subject.

3271: 2858:. It can be shown that any non-empty subclass of the ordinals has a least element. So the class of ordinals is well-ordered. 660:

There are two points that you have raised.You stated 'There is an infinite sequence of brackets and commas before it, but no

119:

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

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Apparently, the trickery is still going on, and getting worse. There are now cardinal, ordinal, and nominal numbers (see

3699:

The finite cardinals and ordinal, in the set theoretical sense, are the same, and possibly could be in a separate article

81: 76: 71: 59: 260:

Both of these statements are true and (therefore) there is no contradiction. The crucial word in the second statement is

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either, and nor can we count on the set-theoretic one as being the overwhelmingly primary target of searches. Therefore

3765:

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

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

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

329: 1239: 4136: 3146: 204:. (I notice the article doesn't really attempt to define ordinal exponentiation.) -Dan 19:29, 29 August 2005 (UTC) 38: 3458:

material for a section here, if not a whole new article. In lieu of such a trip, please try to keep an open mind.

3557: 1226: 1175: 991: 3704: 3149:. The problem I see is that some people are likely to come across this looking for something much more basic. -- 4020: 3605: 2335: 314:

Does Cantor normal form apply to uncountable ordinals? The article states that it applies to all ordinals : -->

4023:. As for the consequences of moving the page I now have AWB access so it should make a mass renaming easier.-- 2984: 2650: 2419: 1238:

The split is a good idea. The name "large ordinals" doesn't really do it for me, though. Maybe something like

594:. Anyway, I've added exponentiation to the article. Hopefully I got it right. -Dan 03:30, 31 August 2005 (UTC) 521: 4016: 2794: 1589:. This approach differs from the one in this article where you require that an ordinal is totaly ordered by 3720: 2469: 211:, though). However you can't prove that it has cardinality comparable to the cardinality of the continuum. 3728: 3712: 3630: 3592: 2389:"good" if and only if it and all its elements are universally trichotomous. Consequently any good ordinal 1199:

level down every time you reply on a comment page, using semicolons at the beginning of the paragraph.) --

930:

I don't like the section on ordinal data type. It seems, at best, irrelevant to this article. And yes,

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and such that every element of it is a subset of it. Proving later on that, as you said, in these sets

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

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itself to be a disambiguation page. This would take care of the concern that many articles link here.

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In this context the meaning we're interested in is number 3 above, and that's a countable ordinal. --

208: 3740:

I'm afraid I have to be a traitor to my class here—I don't think we can justify the main article at

1188:

To Oleg: I cannot find help on how to move a section into a new article in order to implement this.

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I removed the indenting and added subsection headers. Does that fix what you were worried about?

967: 938:. Those are not ordinals either. Ordinals are totally ordered under inclusion and containment. - 157:

No, there is no containment (in either direction), nor inclusion, between 1={0} and {2,3,4,...}. --

47: 17: 1084: 1054: 1027: 897: 766: 736: 717: 677: 620: 94:

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

3501:"school mathematics" education about ordinal number versus cardinal number (and sometimes versus 3239: 2015: 1989: 1740: 1644: 2178:

is an ordinal by the definition which I have chosen to use personally. Do you need more proof?

1760: 1620: 3774: 3724: 3193: 1020: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: --> 854: 832: 810: 1592: 1529: 1298:

I just split out the section "Arithmetic of ordinals" to a new article "Ordinal arithmetic".

1263:

OK. I just did the split. I put the section on "Some "large" ordinals" into the new article

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the bracket at position 5 is a bracket (or an infinite number of them)at position 5 itself.

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

4097: 4024: 3716: 3672: 3634: 3564: 3275: 3203: 3178: 3150: 1371: 1253: 635:

But this answer has a "cowardly" quality to it. Even if brace notation is not part of the

273: 254: 2841: 2223: 1796: 1720: 1700: 1664: 1609:- could you point me in the direction of the article where von Neumann defines ordinals? 1572: 1549: 1242:? I don't really love that either, but it might be a little better. Other suggestions? -- 4240: 4230: 4226: 4222: 4207: 4128: 4082: 4069: 4064: 4040: 3883: 3879: 3868: 3843: 3812: 3808: 3754: 3745: 3685: 3657: 3653: 3613: 3596: 3583: 3531: 3518: 3514: 3502: 3437: 3386: 3382: 3350: 3345:

One extraordinary generalization of the addition of natural numbers is the addition of

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

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section into a separate article, and something similar should probably be done about

1136: 730:! (Maybe we should continue this discussion there.) -Dan 20:33, 1 December 2005 (UTC) 693: 649: 158: 4243: 4233: 4210: 4121: 4105: 4089: 4072: 4057: 4043: 4032: 4004: 3944: 3934: 3915: 3886: 3871:

in another article, intending the lay meaning, will have made an inappropriate link.

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

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

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But we should still take a moment to talk about simpler ordinals in mathematics. --

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Ranking strategies (1st, 2nd, 3rd, joint 4th, 1224 vs 1334 vs 1234 rankings, etc)

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into the article by that name just to make room for simple edits on this article.

803:

down the set as brackets for a finite set. This set with its order reversed would

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If no one raises an objection in the next few days, I intend to move the section

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is not a good name, it been a which since I've studied the subject. I quite like

3268:

it has compelling prose, and is readily comprehensible to non-specialist readers;

1170:

Very much agree. I also thought that this article is too big to be maintainable.

630:(Finite strings, no matter how absurdly huge, are assumed to be well-understood.) 3408: 3292:

subtopic of what. (By the way, though, you might be surprised by the content at

2212: 1617:. You can use either "≤" or "<". Here, we are using "≤" which corresponds to 1614: 1347: 934:

are not ordinals. The numbers that a computer stores are not reals, but rather

874: 561:

I need to go away and get my head around this. -Dan 23:47, 30 August 2005 (UTC)

46:

If you wish to start a new discussion or revive an old one, please do so on the

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doesn't do the job. Once such an article is written (and I am not sufficiently

1458:

I agree that that is a problem, but I have no idea currently of how to fix it.

1408:

ordinals which I did not see in the other article, so please do not delete it.

956:

is also used for the numbering of regents, I can´t find that article anywhere./

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in mathematics education to do it properly), then my preference would be for

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by totality. But the latter leads to a contradiction because it implies that

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coding of ordinals and still use ω for just about anything you'd want to. --

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to be a more basic article covering the finite case, basically the same as

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Furthermore, messing with this article would adversely impact the entire

3340: 3318: 3293: 3250: 1835: 1778: 1694: 1638: 1389: 652:. There is an infinite sequence of brackets and commas before it, but no 591: 1856:

is totally ordered with respect to set containment and every element of

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Oh, of course, it's not that there's anything unclear about the actual

3741: 3545: 3142: 4229:. My objective is to make this article shorter (and that one longer). 2408:

are universally trichotomous. So all that remains to be shown is that

1483:" section, or the paragraph which summarizes what is to come later? -- 4217:

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

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there are": one, two, three, four, etc. (See How to name numbers.)".

3591:

If I am overruled and we have to rename this article, I would prefer

3213: 127:

relationship. So for example, the set {1,3} is contained in {1,2,3}.

124: 2400:

is an ordinal and all of its elements are good. I want to show that

656:. Naturally I prefer this answer. -Dan 15:42, 28 November 2005 (UTC) 2885:

Justification of "the least such element would have to be equal to

894:

http://mathforum.org/kb/message.jspa?messageID=4165416&tstart=0

890:

http://mathforum.org/kb/message.jspa?messageID=3808877&tstart=0

3314: 3198: 3188: 2933:

is non-empty. So use the axiom of regularity to choose an element

294:

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

3612:

by articles about dreck like numbering the floors of a building.

1425:

OK. I fixed up the introduction myself. How does it look now?

471:

The limit of the sequence ω, ω, ω, etc (ordinal exponentiation).

214:

As for ω, it depends on what you mean. If you're talking about

3313:

Notice that even among the articles which Salix alba selected,

888:

Sw=w,that is ,w is its own successor (or predecessor).See also

788:

finite ordinal is written out in a finite number of steps, and

745:

Hmm! Well, okay, I'll run with your analogy. My response is on

639:

of set theory, surely we can come up with some brace notation

25: 3958:

Discussion of when the various endings (st, rd, th) get used

3656:

for a "number" of reasons, most of which have been stated by

2868:

the phrase "the least such element would have to be equal to

248:

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

138:

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

1757:. I don not know if, starting with sets totally ordered by 4081:). Does the discussion of nominal numbers belong in the 468:

The cardinality of that set (cardinal exponentiation), or

102:< b should be read "a, b times" (as Potter suggests.) 4049:

How about having a new article on finite ordinals named

3209:

Differences between Norwegian Bokmål and Standard Danish

3753:

as the appropriate location for the current article. --

3402:

that needs (minor) revision. It immediately restricts:

2697:

by the goodness and thus universal trichotomousness of

1479:

Which introductory paragraph do you mean, Luqui? The "

173:

naturals; neither greater than nor uncomparable to it?

135:

Perhaps some math person could provide an explination.

672:

particular, what if w is one of its own predecessors?

90:

Old stuff about the symbol representing multiplication

4139: 3492:

My preference would be for this article to be called

3037: 2987: 2903: 2844: 2765: 2707: 2653: 2536: 2510: 2472: 2422: 2338: 2300: 2262: 2226: 2142: 2110: 2076: 2044: 2018: 1992: 1966: 1940: 1914: 1876: 1799: 1763: 1743: 1723: 1703: 1667: 1647: 1623: 1595: 1575: 1552: 1532: 1087: 1057: 1030: 857: 835: 813: 570: 524: 497: 391: 332: 183: 3137:

It seems to me that this article is miss-named, and

3556:. Moving the page should not be too problematic as 2591:. The least such element would have to be equal to 2328:"universally trichotomous" if and only if whenever 4186: 3711:article were to be renamed (which I'm opposed to) 3061: 3023: 2921: 2850: 2777: 2731: 2689: 2559: 2522: 2496: 2458: 2377: 2312: 2286: 2232: 2154: 2128: 2088: 2062: 2030: 2004: 1978: 1952: 1926: 1900: 1805: 1769: 1749: 1729: 1709: 1673: 1653: 1629: 1601: 1581: 1558: 1538: 1100: 1070: 1043: 865: 843: 821: 583: 549: 510: 443: 365: 196: 1641:must have been using "<" which corresponds to 366:{\displaystyle \omega _{1}=\omega ^{\omega _{1}}} 3173:Just looking through the what links here we have 2607:is not empty. Using the axiom of regularity let 1661:(no equality) which for ordinals is the same as 1221:Yeah, right. A cut and paste is all one can do. 1120: 851:after it. What is the symbol to the left of the 4187:{\displaystyle (-\infty ,b)=\{x\mid x<b\}\!} 4182: 2552: 784:the finite ordinals are written! Nevertheless, 100:The point is that one wants a^(b+c) = a^b : --> 3147:How to name numbers in English#Ordinal numbers 807:be well-founded for an infinite ordinal, like 1380:It should be moved into the lead section per 605:Is The Set N of Natural Numbers Well Defined? 8: 4179: 4161: 3878:agree that there's not much point in making 2324:is a transitive set. Let us call an ordinal 1697:defines ordinals as sets totally ordered by 3511:Names of numbers in English#Ordinal numbers 2174:is a transitive set of transitive sets. So 2981:, so it is universally trichotomous. Thus 2595:, but that would contradict the fact that 2378:{\displaystyle S\in T\lor T\in S\lor S=T.} 4138: 3036: 3024:{\displaystyle t\in u\lor u\in t\lor t=u} 2986: 2902: 2843: 2764: 2706: 2690:{\displaystyle w\in z\lor w=z\lor z\in w} 2652: 2535: 2509: 2471: 2459:{\displaystyle S\in T\lor T\in S\lor S=T} 2421: 2337: 2299: 2261: 2225: 2141: 2109: 2075: 2043: 2017: 1991: 1965: 1939: 1913: 1875: 1798: 1762: 1742: 1722: 1702: 1666: 1646: 1622: 1594: 1574: 1551: 1531: 1092: 1086: 1062: 1056: 1035: 1029: 856: 834: 812: 575: 569: 550:{\displaystyle \aleph _{0}^{\aleph _{0}}} 539: 534: 529: 523: 502: 496: 444:{\displaystyle \omega ^{\omega _{1}}: --> 429: 416: 401: 396: 390: 355: 350: 337: 331: 188: 182: 3560:is good a perfoming mass page relinking. 3398:The article is properly named. It's the 3117:universally trichotomous and thus good. 2412:itself is universally trichotomous. Let 861: 839: 817: 3677:Cardinal number (grammar) primarily at 2913: 2546: 455:, no? -Dan 19:29, 29 August 2005 (UTC) 4051:Ordinal number (mathematics education) 3911:should still be a disambiguation, no? 3811:should probably be a disambig page. -- 2929:. In this case, we are presuming that 2821:, there must be a non-good element of 2497:{\displaystyle T\notin S\land S\neq T} 1121:Let's not go overboard with details... 465:The set of all functions from ω into ω 452:\omega ^{\omega }\geq \omega _{1}: --> 445:\omega ^{\omega }\geq \omega _{1}: --> 387:\omega ^{\omega }\geq \omega _{1}: --> 315:0. If so, what is the normal form of ω 44:Do not edit the contents of this page. 4096:childs development of mathematics. -- 3690:Ordinal number (grammar), hidden in 3224:List of Pretty Sammy minor characters 2889:" where "such" refers to elements of 1444:One things bugs me now: the section " 123:"set containment" is the same as the 7: 4223:Ordinal number#Topology and ordinals 3986:Duck Dodgers in the 25 1/2th Century 3796:), and consider the big picture. --- 3130:Should the article really be called 2969:in this case). Consider any element 2416:be any ordinal. I want to show that 373:. May seem strange, but it is true. 4225:from this article into the article 4039:is, what sort of words are used. -- 3684:Ordinal number (set theory) now at 2949:is a subset of the intersection of 1848:The current definition says "A set 1481:Ordinals extend the natural numbers 1446:Ordinals extend the natural numbers 643:with the set theory we just built!! 4206:= ω, then we get the desired set. 4146: 2404:is also good. All the elements of 2287:{\displaystyle x\in y\land y\in z} 1870:is a transitive set. Suppose that 536: 526: 24: 3544:How about creating a page called 3141:might be a better title, leaving 2560:{\displaystyle Z=T\setminus S\!.} 2129:{\displaystyle x\in y\subseteq z} 2063:{\displaystyle y\in z\subseteq y} 1866:Since every element is a subset, 1637:. Apparently, in your reference, 584:{\displaystyle \omega ^{\omega }} 511:{\displaystyle \omega ^{\omega }} 197:{\displaystyle \omega ^{\omega }} 4198:is an open set, but this is [0, 3671:Cardinal number (set theory) at 2216:be shown to be a total order.". 1901:{\displaystyle x\in y\in z\in S} 1366:I don't understand this section 1210:Knowledge:How to break up a page 29: 3961:Comparison with other languages 2812:So now we know that an ordinal 2096:which is inconsistent with the 1440:Understandability for beginners 1240:hierarchy of countable ordinals 1007:An interesting pairing function 4244:08:40, 30 September 2006 (UTC) 4234:05:27, 25 September 2006 (UTC) 4155: 4140: 3272:Knowledge:Good articles/Review 3270:As such I'm now listing it on 3122:02:53, 24 September 2006 (UTC) 3062:{\displaystyle u\in t\lor t=u} 2922:{\displaystyle U=S\setminus T} 2877:16:05, 22 September 2006 (UTC) 2863:07:05, 22 September 2006 (UTC) 2732:{\displaystyle w=z\lor z\in w} 2583:, there must be an element of 2199:18:09, 21 September 2006 (UTC) 1546:vs. definition of ordinals by 975:10:33, 10 September 2006 (UTC) 749:. 15:17, 5 December 2005 (UTC) 1: 4211:06:35, 9 September 2006 (UTC) 4122:23:39, 8 September 2006 (UTC) 3916:18:54, 8 September 2006 (UTC) 3887:15:26, 8 September 2006 (UTC) 3847:07:25, 8 September 2006 (UTC) 3831:06:21, 8 September 2006 (UTC) 3816:21:23, 6 September 2006 (UTC) 3801:21:20, 6 September 2006 (UTC) 3758:19:34, 5 September 2006 (UTC) 3732:19:28, 5 September 2006 (UTC) 3643:10:19, 5 September 2006 (UTC) 3617:07:58, 5 September 2006 (UTC) 3600:07:38, 5 September 2006 (UTC) 3587:07:33, 5 September 2006 (UTC) 3573:17:04, 4 September 2006 (UTC) 3539:12:33, 4 September 2006 (UTC) 3526:12:29, 4 September 2006 (UTC) 3483:18:19, 4 September 2006 (UTC) 3463:08:11, 4 September 2006 (UTC) 3441:07:16, 4 September 2006 (UTC) 3428:05:42, 4 September 2006 (UTC) 3390:05:20, 4 September 2006 (UTC) 3365:03:52, 4 September 2006 (UTC) 3354:03:25, 4 September 2006 (UTC) 3330:03:20, 4 September 2006 (UTC) 3301:03:12, 4 September 2006 (UTC) 3284:11:20, 3 September 2006 (UTC) 3168:10:03, 3 September 2006 (UTC) 3159:09:45, 3 September 2006 (UTC) 2833:has no non-good elements and 2170:must be transitive, that is, 1852:is an ordinal if and only if 1101:{\displaystyle \epsilon _{0}} 1071:{\displaystyle \epsilon _{0}} 1044:{\displaystyle \epsilon _{0}} 770:11:54, 15 December 2005 (UTC) 681:10:33, 29 November 2005 (UTC) 624:08:56, 28 November 2005 (UTC) 4106:23:29, 11 October 2006 (UTC) 4090:21:30, 11 October 2006 (UTC) 4073:10:10, 11 October 2006 (UTC) 4058:04:33, 11 October 2006 (UTC) 4044:01:18, 11 October 2006 (UTC) 4033:00:27, 11 October 2006 (UTC) 4005:23:49, 10 October 2006 (UTC) 3994:ordinal number (linguistics) 3945:23:49, 10 October 2006 (UTC) 3935:23:49, 10 October 2006 (UTC) 3377:too long; I said that it is 3343:, it says "Addition of sets. 3077:contradicting the choice of 2747:contradicting the choice of 2031:{\displaystyle z\subseteq y} 2005:{\displaystyle y\subseteq z} 1518:23:38, 10 October 2006 (UTC) 1502:23:34, 10 October 2006 (UTC) 996:05:01, 8 February 2006 (UTC) 961:19:04, 6 February 2006 (UTC) 943:00:28, 5 February 2006 (UTC) 906:ordinal and cardinal numbers 740:11:12, 5 December 2005 (UTC) 721:05:01, 1 December 2005 (UTC) 662:single immediate predecessor 654:single immediate predecessor 3950:Ideas for the other article 3692:names of numbers in English 3679:names of numbers in English 3554:Ordinal number (set theory) 3494:Ordinal number (set theory) 3455:Names of numbers in English 3413:transfinite ordinal numbers 3139:transfinite ordinal numbers 3132:transfinite ordinal numbers 2183:05:01, 23 August 2006 (UTC) 1818:10:15, 22 August 2006 (UTC) 1750:{\displaystyle \subsetneq } 1686:02:11, 10 August 2006 (UTC) 1654:{\displaystyle \subsetneq } 1526:Definition of ordinals by 483:20:08, 30 August 2005 (UTC) 224:19:57, 30 August 2005 (UTC) 4260: 3789:(reprint of 1960 classic) 3719:should also be renamed to 3705:finite number (set theory) 3652:article should be left at 3550:transfinite ordinal number 3498:transfinite ordinal number 2631:, but being disjoint from 1770:{\displaystyle \subseteq } 1630:{\displaystyle \subseteq } 1488:12:27, 28 March 2006 (UTC) 1474:12:18, 28 March 2006 (UTC) 1463:06:39, 20 March 2006 (UTC) 1453:15:54, 18 March 2006 (UTC) 1430:06:56, 18 March 2006 (UTC) 1413:07:37, 17 March 2006 (UTC) 1375:03:35, 17 March 2006 (UTC) 1361:03:37, 16 March 2006 (UTC) 1351:19:10, 14 March 2006 (UTC) 1337:07:29, 17 March 2006 (UTC) 1322:18:11, 15 March 2006 (UTC) 1303:09:40, 15 March 2006 (UTC) 1286:14:16, 13 March 2006 (UTC) 1272:08:26, 13 March 2006 (UTC) 1257:18:22, 12 March 2006 (UTC) 1247:18:19, 12 March 2006 (UTC) 1231:16:33, 12 March 2006 (UTC) 1217:20:09, 11 March 2006 (UTC) 1204:19:00, 11 March 2006 (UTC) 1193:05:05, 11 March 2006 (UTC) 1116:08:06, 13 March 2006 (UTC) 901:05:24, 18 March 2006 (UTC) 878:22:19, 14 March 2006 (UTC) 487:I'll be damned. There's a 162:01:25, 30 March 2006 (UTC) 152:18:33, 29 March 2006 (UTC) 4021:ordinal number (infinite) 3558:Knowledge:AutoWikiBrowser 1180:03:59, 7 March 2006 (UTC) 1165:23:22, 6 March 2006 (UTC) 1140:14:49, 3 March 2006 (UTC) 866:{\displaystyle \omega \;} 844:{\displaystyle \omega \;} 822:{\displaystyle \omega \;} 3606:Category:Ordinal numbers 3496:. Firstly, I agree that 2256:which is an ordinal. If 1602:{\displaystyle \subset } 1539:{\displaystyle \subset } 701:But most mathematicians 378:17:29, 31 May 2005 (UTC) 323:01:31, 2005 Mar 8 (UTC) 305:22:42, 19 May 2004 (UTC) 268:16:32, 2 Feb 2004 (UTC) 257:11:58, 4 Dec 2003 (UTC) 4017:ordinal number (finite) 3984:"Fractional ordinals" ( 3407:"Here, we describe the 3242:- many general articles 3073:would be an element of 2941:which is disjoint from 2795:axiom of extensionality 2635:it must be a subset of 2615:which is disjoint from 2385:Let us call an ordinal 780:get to the point where 276:23:17, 2 Feb 2004 (UTC) 131:16:32, 2 Feb 2004 (UTC) 115:What is an ordinal set? 110:16:32, 2 Feb 2004 (UTC) 4188: 3966:The status of "zeroth" 3660:. Perhaps an article 3236:and many other numbers 3063: 3025: 2977:. It is an element of 2961:itself (remember that 2957:which intersection is 2923: 2852: 2779: 2778:{\displaystyle w\in z} 2733: 2691: 2561: 2524: 2523:{\displaystyle S\in T} 2498: 2460: 2379: 2314: 2313:{\displaystyle x\in z} 2288: 2234: 2156: 2155:{\displaystyle x\in z} 2130: 2090: 2089:{\displaystyle y\in y} 2064: 2032: 2006: 1980: 1979:{\displaystyle x\in z} 1954: 1953:{\displaystyle x\in S} 1928: 1927:{\displaystyle y\in S} 1902: 1807: 1771: 1751: 1731: 1711: 1675: 1655: 1631: 1603: 1583: 1560: 1540: 1252:JA: "Extraordinals". 1154:arithmetic of ordinals 1102: 1072: 1045: 936:floating point numbers 867: 845: 823: 585: 551: 512: 447: 367: 198: 4189: 3781:Parameter error in {{ 3769:Halmos, Paul (1974). 3751:ordinal (mathematics) 3721:cardinal (set theory) 3633:sounds good to me. -- 3064: 3026: 2924: 2853: 2780: 2734: 2692: 2627:and thus a subset of 2599:is not an element of 2575:would be a subset of 2562: 2525: 2499: 2461: 2380: 2315: 2289: 2235: 2157: 2131: 2091: 2065: 2033: 2007: 1981: 1955: 1929: 1903: 1808: 1772: 1752: 1732: 1712: 1676: 1656: 1632: 1604: 1584: 1561: 1541: 1398:transfinite induction 1158:some “large” ordinals 1103: 1073: 1046: 868: 846: 824: 586: 552: 513: 448: 383:Are you sure? Surely 368: 199: 42:of past discussions. 4137: 4112:Question on topology 4085:article as well? -- 4063:Finite ordinals are 3713:Ordinal (set theory) 3631:Ordinal (set theory) 3593:Ordinal (set theory) 3035: 2985: 2901: 2851:{\displaystyle \in } 2842: 2763: 2705: 2651: 2534: 2508: 2470: 2420: 2336: 2298: 2260: 2233:{\displaystyle \in } 2224: 2140: 2108: 2074: 2042: 2016: 1990: 1964: 1938: 1912: 1874: 1860:is also a subset of 1806:{\displaystyle \in } 1797: 1761: 1741: 1730:{\displaystyle \in } 1721: 1710:{\displaystyle \in } 1701: 1674:{\displaystyle \in } 1665: 1645: 1621: 1593: 1582:{\displaystyle \in } 1573: 1559:{\displaystyle \in } 1550: 1530: 1085: 1055: 1028: 855: 833: 811: 568: 522: 495: 389: 330: 209:axiom of replacement 181: 168:uncountable ordinals 4011:Two articles or one 3907:Okay, but at least 2639:also. Consider any 2162:. Thus any element 2098:axiom of regularity 968:Monarchical ordinal 921:"ordinal data type" 546: 491:difference between 239:of smaller ordinals 18:Talk:Ordinal number 4184: 4183: 3240:Rastafari movement 3059: 3021: 2919: 2848: 2775: 2729: 2687: 2579:. Since it is not 2557: 2553: 2520: 2494: 2456: 2375: 2310: 2284: 2230: 2152: 2126: 2086: 2060: 2028: 2002: 1976: 1950: 1924: 1898: 1803: 1767: 1747: 1727: 1707: 1671: 1651: 1627: 1599: 1579: 1556: 1536: 1342:Executive summary? 1098: 1068: 1041: 863: 862: 841: 840: 819: 818: 581: 547: 525: 508: 441: 363: 310:Cantor normal form 194: 101:< a^c. a : --> 3794:"Ordinal Numbers" 3194:The Sand Reckoner 3113:which would make 3109:is an element of 3085:is an element of 2893:which are not in 2623:is an element of 2611:be an element of 2571:were empty, then 1960:also. What about 1021:γ and ξ(α,β)≤δ). 87: 86: 54: 53: 48:current talk page 4251: 4239:Move completed. 4193: 4191: 4190: 4185: 3792:See chapter 19 ( 3788: 3786: 3771:Naive Set Theory 3323:Ordinal notation 3256:Ordinal notation 3068: 3066: 3065: 3060: 3030: 3028: 3027: 3022: 2928: 2926: 2925: 2920: 2857: 2855: 2854: 2849: 2784: 2782: 2781: 2776: 2755:and thus not in 2738: 2736: 2735: 2730: 2696: 2694: 2693: 2688: 2566: 2564: 2563: 2558: 2529: 2527: 2526: 2521: 2504:and try to show 2503: 2501: 2500: 2495: 2465: 2463: 2462: 2457: 2384: 2382: 2381: 2376: 2319: 2317: 2316: 2311: 2293: 2291: 2290: 2285: 2252:are elements of 2239: 2237: 2236: 2231: 2161: 2159: 2158: 2153: 2135: 2133: 2132: 2127: 2095: 2093: 2092: 2087: 2069: 2067: 2066: 2061: 2037: 2035: 2034: 2029: 2011: 2009: 2008: 2003: 1985: 1983: 1982: 1977: 1959: 1957: 1956: 1951: 1933: 1931: 1930: 1925: 1907: 1905: 1904: 1899: 1812: 1810: 1809: 1804: 1776: 1774: 1773: 1768: 1756: 1754: 1753: 1748: 1736: 1734: 1733: 1728: 1716: 1714: 1713: 1708: 1680: 1678: 1677: 1672: 1660: 1658: 1657: 1652: 1636: 1634: 1633: 1628: 1608: 1606: 1605: 1600: 1588: 1586: 1585: 1580: 1565: 1563: 1562: 1557: 1545: 1543: 1542: 1537: 1107: 1105: 1104: 1099: 1097: 1096: 1077: 1075: 1074: 1069: 1067: 1066: 1050: 1048: 1047: 1042: 1040: 1039: 872: 870: 869: 864: 850: 848: 847: 842: 828: 826: 825: 820: 687:omega-minus-one. 590: 588: 587: 582: 580: 579: 564:For some reason 556: 554: 553: 548: 545: 544: 543: 533: 517: 515: 514: 509: 507: 506: 454: 453:\omega }" /: --> 450: 449: 442: 434: 433: 421: 420: 408: 407: 406: 405: 372: 370: 369: 364: 362: 361: 360: 359: 342: 341: 203: 201: 200: 195: 193: 192: 68: 56: 55: 33: 32: 26: 4259: 4258: 4254: 4253: 4252: 4250: 4249: 4248: 4219: 4135: 4134: 4114: 4065:natural numbers 4013: 3977:+1)-th versus ( 3952: 3780: 3768: 3717:Cardinal number 3673:Cardinal number 3534:is a good one. 3347:ordinal numbers 3204:Floor numbering 3179:Danish language 3135: 3093:is a subset of 3033: 3032: 2983: 2982: 2965:is a subset of 2899: 2898: 2840: 2839: 2789:is a subset of 2761: 2760: 2703: 2702: 2649: 2648: 2532: 2531: 2506: 2505: 2468: 2467: 2466:. Suppose that 2418: 2417: 2334: 2333: 2296: 2295: 2258: 2257: 2222: 2221: 2211:The article on 2209: 2138: 2137: 2106: 2105: 2072: 2071: 2040: 2039: 2014: 2013: 1988: 1987: 1962: 1961: 1936: 1935: 1910: 1909: 1872: 1871: 1795: 1794: 1759: 1758: 1739: 1738: 1737:is the same as 1719: 1718: 1699: 1698: 1663: 1662: 1643: 1642: 1619: 1618: 1615:Total orderings 1591: 1590: 1571: 1570: 1567: 1548: 1547: 1528: 1527: 1442: 1344: 1223:Oleg Alexandrov 1172:Oleg Alexandrov 1150: 1123: 1088: 1083: 1082: 1058: 1053: 1052: 1031: 1026: 1025: 1009: 988:Oleg Alexandrov 983: 951: 923: 908: 853: 852: 831: 830: 809: 808: 607: 571: 566: 565: 557:, isn't there. 535: 520: 519: 498: 493: 492: 425: 412: 397: 392: 385: 384: 351: 346: 333: 328: 327: 318: 312: 184: 179: 178: 170: 117: 92: 64: 30: 22: 21: 20: 12: 11: 5: 4257: 4255: 4247: 4246: 4227:Order topology 4218: 4215: 4214: 4213: 4196: 4195: 4194: 4181: 4178: 4175: 4172: 4169: 4166: 4163: 4160: 4157: 4154: 4151: 4148: 4145: 4142: 4129:order topology 4113: 4110: 4109: 4108: 4087:Four Dog Night 4083:Natural number 4076: 4075: 4055:Four Dog Night 4047: 4046: 4012: 4009: 4008: 4007: 3990: 3989: 3982: 3969:The status of 3967: 3964: 3963: 3962: 3951: 3948: 3927: 3926: 3925: 3924: 3923: 3922: 3921: 3920: 3919: 3918: 3896: 3895: 3894: 3893: 3892: 3891: 3890: 3889: 3880:ordinal number 3872: 3869:ordinal number 3854: 3853: 3852: 3851: 3850: 3849: 3834: 3833: 3819: 3818: 3809:ordinal number 3763: 3762: 3761: 3760: 3746:ordinal number 3735: 3734: 3697: 3696: 3695: 3688: 3686:Ordinal number 3682: 3675: 3666: 3665: 3658:User:JRSpriggs 3654:Ordinal number 3646: 3645: 3626: 3625: 3624: 3623: 3622: 3621: 3620: 3619: 3576: 3575: 3561: 3536:Michael Kinyon 3532:Order topology 3523:Michael Kinyon 3519:Ordinal number 3503:nominal number 3490: 3489: 3488: 3487: 3486: 3485: 3470: 3469: 3468: 3467: 3466: 3465: 3446: 3445: 3444: 3443: 3431: 3430: 3420: 3419: 3418: 3417: 3416: 3395: 3394: 3393: 3392: 3383:order topology 3368: 3367: 3344: 3337: 3336: 3335: 3334: 3333: 3332: 3306: 3305: 3304: 3303: 3259: 3258: 3253: 3248: 3243: 3237: 3231: 3229:Roman numerals 3226: 3221: 3216: 3211: 3206: 3201: 3196: 3191: 3186: 3181: 3175: 3174: 3134: 3128: 3127: 3126: 3125: 3124: 3058: 3055: 3052: 3049: 3046: 3043: 3040: 3020: 3017: 3014: 3011: 3008: 3005: 3002: 2999: 2996: 2993: 2990: 2918: 2915: 2912: 2909: 2906: 2880: 2879: 2847: 2774: 2771: 2768: 2728: 2725: 2722: 2719: 2716: 2713: 2710: 2686: 2683: 2680: 2677: 2674: 2671: 2668: 2665: 2662: 2659: 2656: 2556: 2551: 2548: 2545: 2542: 2539: 2519: 2516: 2513: 2493: 2490: 2487: 2484: 2481: 2478: 2475: 2455: 2452: 2449: 2446: 2443: 2440: 2437: 2434: 2431: 2428: 2425: 2374: 2371: 2368: 2365: 2362: 2359: 2356: 2353: 2350: 2347: 2344: 2341: 2332:is an ordinal 2309: 2306: 2303: 2283: 2280: 2277: 2274: 2271: 2268: 2265: 2229: 2213:total ordering 2208: 2205: 2204: 2203: 2202: 2201: 2151: 2148: 2145: 2125: 2122: 2119: 2116: 2113: 2085: 2082: 2079: 2059: 2056: 2053: 2050: 2047: 2027: 2024: 2021: 2001: 1998: 1995: 1975: 1972: 1969: 1949: 1946: 1943: 1923: 1920: 1917: 1897: 1894: 1891: 1888: 1885: 1882: 1879: 1865: 1846: 1845: 1844: 1843: 1842: 1841: 1840: 1839: 1825: 1824: 1823: 1822: 1821: 1820: 1802: 1785: 1784: 1783: 1782: 1766: 1746: 1726: 1706: 1689: 1688: 1670: 1650: 1626: 1598: 1578: 1566: 1555: 1535: 1524: 1523: 1522: 1521: 1520: 1507: 1506: 1505: 1504: 1491: 1490: 1466: 1465: 1441: 1438: 1437: 1436: 1435: 1434: 1433: 1432: 1418: 1417: 1416: 1415: 1402: 1401: 1364: 1363: 1343: 1340: 1329: 1328: 1327: 1326: 1325: 1324: 1310: 1309: 1308: 1307: 1306: 1305: 1291: 1290: 1289: 1288: 1275: 1274: 1265:large ordinals 1260: 1259: 1236: 1235: 1234: 1233: 1219: 1183: 1182: 1149: 1146: 1122: 1119: 1111:Last revised 1095: 1091: 1065: 1061: 1038: 1034: 1008: 1005: 1004: 1003: 982: 979: 978: 977: 950: 949:Other meanings 947: 946: 945: 922: 919: 916: 913: 907: 904: 885: 884: 883: 882: 881: 880: 860: 838: 816: 795: 794: 751: 750: 732: 731: 713: 712: 708: 707: 698: 697: 689: 688: 658: 657: 645: 644: 632: 631: 606: 603: 602: 601: 600: 599: 598: 597: 596: 595: 592:makes me happy 578: 574: 542: 538: 532: 528: 505: 501: 473: 472: 469: 466: 462: 461: 460: 459: 440: 437: 432: 428: 424: 419: 415: 411: 404: 400: 395: 388:\omega }": --> 358: 354: 349: 345: 340: 336: 316: 311: 308: 279: 278: 277: 251: 250: 241: 240: 230: 229: 228: 227: 226: 212: 191: 187: 169: 166: 165: 164: 144: 143: 133: 132: 116: 113: 112: 111: 91: 88: 85: 84: 79: 74: 69: 62: 52: 51: 34: 23: 15: 14: 13: 10: 9: 6: 4: 3: 2: 4256: 4245: 4242: 4238: 4237: 4236: 4235: 4232: 4228: 4224: 4216: 4212: 4209: 4205: 4202:). If we let 4201: 4197: 4176: 4173: 4170: 4167: 4164: 4158: 4152: 4149: 4143: 4133: 4132: 4130: 4126: 4125: 4124: 4123: 4120: 4119:84.168.77.237 4111: 4107: 4103: 4099: 4094: 4093: 4092: 4091: 4088: 4084: 4080: 4074: 4071: 4066: 4062: 4061: 4060: 4059: 4056: 4052: 4045: 4042: 4037: 4036: 4035: 4034: 4030: 4026: 4022: 4018: 4010: 4006: 4003: 3999: 3998: 3997: 3995: 3987: 3983: 3980: 3976: 3972: 3968: 3965: 3960: 3959: 3957: 3956: 3955: 3949: 3947: 3946: 3943: 3937: 3936: 3933: 3917: 3914: 3913:192.75.48.150 3910: 3906: 3905: 3904: 3903: 3902: 3901: 3900: 3899: 3898: 3897: 3888: 3885: 3881: 3877: 3873: 3870: 3866: 3862: 3861: 3860: 3859: 3858: 3857: 3856: 3855: 3848: 3845: 3840: 3839: 3838: 3837: 3836: 3835: 3832: 3829: 3826: 3821: 3820: 3817: 3814: 3810: 3805: 3804: 3803: 3802: 3799: 3795: 3790: 3784: 3779: 3776: 3772: 3766: 3759: 3756: 3752: 3747: 3743: 3739: 3738: 3737: 3736: 3733: 3730: 3726: 3722: 3718: 3714: 3710: 3706: 3702: 3701:finite number 3698: 3693: 3689: 3687: 3683: 3680: 3676: 3674: 3670: 3669: 3668: 3667: 3663: 3659: 3655: 3651: 3648: 3647: 3644: 3640: 3636: 3632: 3628: 3627: 3618: 3615: 3611: 3607: 3603: 3602: 3601: 3598: 3594: 3590: 3589: 3588: 3585: 3580: 3579: 3578: 3577: 3574: 3570: 3566: 3562: 3559: 3555: 3551: 3547: 3543: 3542: 3541: 3540: 3537: 3533: 3528: 3527: 3524: 3520: 3516: 3515:knowledgeable 3512: 3508: 3507:serial number 3504: 3499: 3495: 3484: 3481: 3476: 3475: 3474: 3473: 3472: 3471: 3464: 3461: 3456: 3452: 3451: 3450: 3449: 3448: 3447: 3442: 3439: 3435: 3434: 3433: 3432: 3429: 3426: 3421: 3414: 3410: 3406: 3405: 3404: 3403: 3401: 3397: 3396: 3391: 3388: 3384: 3380: 3376: 3372: 3371: 3370: 3369: 3366: 3363: 3358: 3357: 3356: 3355: 3352: 3348: 3342: 3331: 3328: 3324: 3320: 3316: 3312: 3311: 3310: 3309: 3308: 3307: 3302: 3299: 3295: 3290: 3289: 3288: 3287: 3286: 3285: 3281: 3277: 3273: 3269: 3265: 3257: 3254: 3252: 3249: 3247: 3246:Serial number 3244: 3241: 3238: 3235: 3232: 3230: 3227: 3225: 3222: 3220: 3217: 3215: 3212: 3210: 3207: 3205: 3202: 3200: 3197: 3195: 3192: 3190: 3187: 3185: 3182: 3180: 3177: 3176: 3172: 3171: 3170: 3169: 3166: 3161: 3160: 3156: 3152: 3148: 3144: 3140: 3133: 3129: 3123: 3120: 3116: 3112: 3108: 3104: 3100: 3096: 3092: 3088: 3084: 3080: 3076: 3072: 3056: 3053: 3050: 3047: 3044: 3041: 3038: 3018: 3015: 3012: 3009: 3006: 3003: 3000: 2997: 2994: 2991: 2988: 2980: 2976: 2972: 2968: 2964: 2960: 2956: 2952: 2948: 2944: 2940: 2936: 2932: 2916: 2910: 2907: 2904: 2896: 2892: 2888: 2884: 2883: 2882: 2881: 2878: 2875: 2871: 2867: 2866: 2865: 2864: 2861: 2845: 2836: 2832: 2828: 2824: 2820: 2815: 2810: 2808: 2804: 2800: 2796: 2792: 2788: 2772: 2769: 2766: 2758: 2754: 2750: 2746: 2742: 2726: 2723: 2720: 2717: 2714: 2711: 2708: 2700: 2684: 2681: 2678: 2675: 2672: 2669: 2666: 2663: 2660: 2657: 2654: 2646: 2642: 2638: 2634: 2630: 2626: 2622: 2618: 2614: 2610: 2606: 2602: 2598: 2594: 2590: 2586: 2582: 2578: 2574: 2570: 2554: 2549: 2543: 2540: 2537: 2517: 2514: 2511: 2491: 2488: 2485: 2482: 2479: 2476: 2473: 2453: 2450: 2447: 2444: 2441: 2438: 2435: 2432: 2429: 2426: 2423: 2415: 2411: 2407: 2403: 2399: 2394: 2392: 2388: 2372: 2369: 2366: 2363: 2360: 2357: 2354: 2351: 2348: 2345: 2342: 2339: 2331: 2327: 2323: 2307: 2304: 2301: 2281: 2278: 2275: 2272: 2269: 2266: 2263: 2255: 2251: 2247: 2243: 2227: 2217: 2214: 2206: 2200: 2197: 2192: 2191: 2190: 2189: 2188: 2185: 2184: 2181: 2177: 2173: 2169: 2165: 2149: 2146: 2143: 2123: 2120: 2117: 2114: 2111: 2104:}. So we get 2103: 2099: 2083: 2080: 2077: 2057: 2054: 2051: 2048: 2045: 2025: 2022: 2019: 1999: 1996: 1993: 1973: 1970: 1967: 1947: 1944: 1941: 1921: 1918: 1915: 1895: 1892: 1889: 1886: 1883: 1880: 1877: 1869: 1863: 1859: 1855: 1851: 1837: 1833: 1832: 1831: 1830: 1829: 1828: 1827: 1826: 1819: 1816: 1800: 1791: 1790: 1789: 1788: 1787: 1786: 1780: 1764: 1744: 1724: 1704: 1696: 1693: 1692: 1691: 1690: 1687: 1684: 1668: 1648: 1640: 1624: 1616: 1612: 1611: 1610: 1596: 1576: 1553: 1533: 1519: 1516: 1511: 1510: 1509: 1508: 1503: 1500: 1495: 1494: 1493: 1492: 1489: 1486: 1482: 1478: 1477: 1476: 1475: 1472: 1464: 1461: 1457: 1456: 1455: 1454: 1451: 1447: 1439: 1431: 1428: 1424: 1423: 1422: 1421: 1420: 1419: 1414: 1411: 1406: 1405: 1404: 1403: 1399: 1395: 1391: 1387: 1383: 1379: 1378: 1377: 1376: 1373: 1369: 1362: 1359: 1355: 1354: 1353: 1352: 1349: 1341: 1339: 1338: 1335: 1323: 1320: 1316: 1315: 1314: 1313: 1312: 1311: 1304: 1301: 1297: 1296: 1295: 1294: 1293: 1292: 1287: 1284: 1279: 1278: 1277: 1276: 1273: 1270: 1266: 1262: 1261: 1258: 1255: 1251: 1250: 1249: 1248: 1245: 1241: 1232: 1228: 1224: 1220: 1218: 1215: 1211: 1207: 1206: 1205: 1202: 1197: 1196: 1195: 1194: 1191: 1186: 1181: 1177: 1173: 1169: 1168: 1167: 1166: 1163: 1159: 1155: 1148:Suggest split 1147: 1145: 1142: 1141: 1138: 1133: 1128: 1118: 1117: 1114: 1109: 1093: 1089: 1079: 1063: 1059: 1036: 1032: 1022: 1018: 1015: 1012: 1006: 1000: 999: 998: 997: 993: 989: 980: 976: 973: 972:Johan Jönsson 969: 965: 964: 963: 962: 959: 958:Johan Jönsson 955: 948: 944: 941: 937: 933: 929: 928: 927: 920: 918: 914: 911: 905: 903: 902: 899: 895: 891: 879: 876: 858: 836: 814: 806: 801: 800: 799: 798: 797: 796: 791: 787: 783: 779: 774: 773: 772: 771: 768: 763: 759: 755: 748: 744: 743: 742: 741: 738: 729: 725: 724: 723: 722: 719: 710: 709: 706:constructive! 704: 700: 699: 695: 694:impredicative 691: 690: 685: 684: 683: 682: 679: 673: 669: 665: 663: 655: 651: 650:limit ordinal 647: 646: 642: 638: 634: 633: 628: 627: 626: 625: 622: 617: 613: 610: 604: 593: 576: 572: 563: 562: 560: 540: 530: 503: 499: 490: 486: 485: 484: 481: 477: 476: 475: 474: 470: 467: 464: 463: 457: 456: 438: 435: 430: 426: 422: 417: 413: 409: 402: 398: 393: 382: 381: 380: 379: 376: 375:157.181.80.93 356: 352: 347: 343: 338: 334: 324: 322: 309: 307: 306: 303: 299: 295: 292: 289: 286: 283: 275: 271: 270: 269: 267: 263: 258: 256: 249: 246: 245: 244: 238: 235: 234: 233: 225: 222: 217: 213: 210: 206: 205: 189: 185: 176: 175: 174: 167: 163: 160: 156: 155: 154: 153: 150: 141: 140: 139: 136: 130: 126: 122: 121: 120: 114: 109: 105: 104: 103: 98: 95: 89: 83: 80: 78: 75: 73: 70: 67: 63: 61: 58: 57: 49: 45: 41: 40: 35: 28: 27: 19: 4220: 4203: 4199: 4131:an open ray 4115: 4077: 4048: 4014: 3991: 3985: 3978: 3974: 3970: 3953: 3938: 3928: 3875: 3864: 3793: 3791: 3785:}}: checksum 3778:0-387-900092 3773:. Springer. 3770: 3767: 3764: 3725:Arthur Rubin 3708: 3681:, I believe) 3649: 3549: 3529: 3497: 3491: 3412: 3409:mathematical 3399: 3378: 3374: 3346: 3338: 3267: 3260: 3162: 3136: 3114: 3110: 3106: 3102: 3098: 3094: 3090: 3086: 3082: 3078: 3074: 3070: 2978: 2974: 2970: 2966: 2962: 2958: 2954: 2950: 2946: 2942: 2938: 2934: 2930: 2894: 2890: 2886: 2874:64.42.233.61 2869: 2834: 2830: 2826: 2822: 2818: 2813: 2811: 2806: 2805:and thus in 2802: 2798: 2793:. So by the 2790: 2786: 2756: 2752: 2748: 2744: 2740: 2698: 2644: 2640: 2636: 2632: 2628: 2624: 2620: 2616: 2612: 2608: 2604: 2600: 2596: 2592: 2588: 2584: 2580: 2576: 2572: 2568: 2413: 2409: 2405: 2401: 2397: 2395: 2390: 2386: 2329: 2325: 2321: 2253: 2249: 2245: 2241: 2218: 2210: 2196:64.42.233.61 2186: 2175: 2171: 2167: 2163: 2101: 2100:applied to { 1867: 1861: 1857: 1853: 1849: 1847: 1568: 1467: 1443: 1386:wellordering 1367: 1365: 1345: 1330: 1237: 1187: 1184: 1151: 1143: 1131: 1124: 1110: 1080: 1023: 1019: 1016: 1013: 1010: 984: 953: 952: 932:real numbers 924: 915: 912: 909: 886: 804: 789: 785: 781: 777: 764: 760: 756: 752: 733: 714: 702: 674: 670: 666: 661: 659: 653: 640: 636: 618: 614: 611: 608: 558: 488: 325: 313: 300: 296: 293: 290: 287: 284: 280: 261: 259: 252: 247: 242: 236: 231: 215: 171: 145: 137: 134: 118: 99: 96: 93: 65: 43: 37: 3411:meaning of 3264:GA criteria 3089:, and thus 917:Mr. Nelson 243:contradict 36:This is an 4098:Salix alba 4025:Salix alba 3874:I guess I 3635:Salix alba 3565:Salix alba 3276:Salix alba 3234:0 (number) 3151:Salix alba 2785:and hence 2240:. Suppose 2136:and hence 1394:cofinality 1372:Hairy Dude 1254:Jon Awbrey 641:consistent 637:foundation 559:Thank you. 274:SirJective 255:SirJective 4241:JRSpriggs 4231:JRSpriggs 4208:JRSpriggs 4070:JRSpriggs 4041:Trovatore 3884:Trovatore 3844:JRSpriggs 3813:Trovatore 3755:Trovatore 3614:JRSpriggs 3610:pollution 3597:JRSpriggs 3584:JRSpriggs 3438:JRSpriggs 3387:JRSpriggs 3351:JRSpriggs 3327:JRSpriggs 3219:Year zero 3184:HyperCard 3165:JRSpriggs 3119:JRSpriggs 2860:JRSpriggs 2180:JRSpriggs 2070:and thus 1986:? Either 1815:JRSpriggs 1683:JRSpriggs 1460:JRSpriggs 1427:JRSpriggs 1410:JRSpriggs 1358:JRSpriggs 1334:JRSpriggs 1300:JRSpriggs 1269:JRSpriggs 1244:Trovatore 1190:JRSpriggs 1127:section 6 1113:JRSpriggs 966:Article: 747:this page 480:Trovatore 266:AxelBoldt 221:Trovatore 129:AxelBoldt 108:AxelBoldt 82:Archive 5 77:Archive 4 72:Archive 3 66:Archive 2 60:Archive 1 3662:Ordinals 3460:Melchoir 3362:Melchoir 3341:Addition 3319:Addition 3298:Melchoir 3294:Addition 3251:Addition 3097:. Since 2396:Suppose 2320:because 1485:Gro-Tsen 1450:Gro-Tsen 1390:club set 1319:Gro-Tsen 1283:Gro-Tsen 1201:Gro-Tsen 1162:Gro-Tsen 1137:Gro-Tsen 728:this one 446:\omega } 262:sequence 232:Doesn't 159:Gro-Tsen 4002:Mooncow 3942:Mooncow 3932:Mooncow 3909:ordinal 3742:ordinal 3715:, then 3546:Ordinal 3400:content 3379:already 3143:Ordinal 3069:, then 2945:. Then 2759:. Thus 2739:, then 2619:. Then 2603:. Thus 2587:not in 2294:, then 1908:, then 1515:Mooncow 1499:Mooncow 1382:WP:LEAD 981:Cleanup 954:Ordinal 898:Apoorv1 767:Apoorv1 737:Apoorv1 718:Apoorv1 678:Apoorv1 621:Apoorv1 321:Fropuff 216:ordinal 106:Fixed. 39:archive 3981:+1)-st 3828:(Talk) 3729:(talk) 3707:. If 3375:almost 3321:, and 3214:Exit 0 2897:. Let 2751:as in 2743:is in 2530:. Let 1396:, and 1368:at all 1348:Elroch 875:Elroch 125:subset 3973:th, ( 3876:would 3723:. — 3480:KSmrq 3425:KSmrq 3339:From 3315:Chomp 3199:Chomp 3189:Array 3081:. So 3031:. If 2829:. So 2701:. If 2207:Proof 1836:Gödel 1779:Gödel 1695:Gödel 1639:Gödel 1471:Luqui 1214:lethe 1002:(UTC) 940:lethe 896:and-- 778:never 489:major 436:: --> 410:: --> 326:It's 319:? -- 302:CSTAR 16:< 4174:< 4102:talk 4029:talk 4019:and 3865:text 3783:ISBN 3775:ISBN 3709:this 3650:This 3639:talk 3629:Yes 3569:talk 3280:talk 3274:. -- 3266:1a) 3155:talk 3101:and 2953:and 1934:and 1227:talk 1212:. - 1176:talk 1132:this 1078:·2. 1051:,0)= 992:talk 892:and 790:only 518:and 253:? -- 149:DRLB 3825:C S 3787:-6. 3744:or 3727:| 3703:or 3505:or 3296:!) 2973:of 2937:of 2801:is 2643:in 2567:If 2166:of 2012:or 1864:.". 805:not 786:any 782:all 4168:∣ 4147:∞ 4144:− 4104:) 4031:) 3996:. 3823:-- 3798:CH 3641:) 3571:) 3423:-- 3415:." 3317:, 3282:) 3157:) 3048:∨ 3042:∈ 3010:∨ 3004:∈ 2998:∨ 2992:∈ 2914:∖ 2846:∈ 2809:. 2797:, 2770:∈ 2724:∈ 2718:∨ 2682:∈ 2676:∨ 2664:∨ 2658:∈ 2647:, 2547:∖ 2515:∈ 2489:≠ 2483:∧ 2477:∉ 2445:∨ 2439:∈ 2433:∨ 2427:∈ 2361:∨ 2355:∈ 2349:∨ 2343:∈ 2305:∈ 2279:∈ 2273:∧ 2267:∈ 2248:, 2244:, 2228:∈ 2147:∈ 2121:⊆ 2115:∈ 2081:∈ 2055:⊆ 2049:∈ 2023:⊆ 1997:⊆ 1971:∈ 1945:∈ 1919:∈ 1893:∈ 1887:∈ 1881:∈ 1801:∈ 1765:⊆ 1745:⊊ 1725:∈ 1705:∈ 1681:. 1669:∈ 1649:⊊ 1625:⊆ 1597:⊂ 1577:∈ 1554:∈ 1534:⊂ 1392:, 1388:, 1281:-- 1267:. 1229:) 1178:) 1135:-- 1108:. 1090:ϵ 1060:ϵ 1033:ϵ 994:) 970:./ 859:ω 837:ω 815:ω 765:-- 703:do 676:-- 619:-- 577:ω 573:ω 537:ℵ 527:ℵ 504:ω 500:ω 439:ω 427:ω 423:≥ 418:ω 414:ω 399:ω 394:ω 353:ω 348:ω 335:ω 190:ω 186:ω 4204:b 4200:b 4180:} 4177:b 4171:x 4165:x 4162:{ 4159:= 4156:) 4153:b 4150:, 4141:( 4100:( 4027:( 3988:) 3979:n 3975:n 3971:n 3694:. 3637:( 3567:( 3278:( 3153:( 3115:S 3111:S 3107:T 3103:u 3099:T 3095:u 3091:T 3087:u 3083:t 3079:u 3075:T 3071:u 3057:u 3054:= 3051:t 3045:t 3039:u 3019:u 3016:= 3013:t 3007:t 3001:u 2995:u 2989:t 2979:S 2975:T 2971:t 2967:S 2963:T 2959:T 2955:T 2951:S 2947:u 2943:U 2939:U 2935:u 2931:U 2917:T 2911:S 2908:= 2905:U 2895:T 2891:S 2887:T 2870:T 2835:S 2831:S 2827:S 2823:S 2819:S 2814:S 2807:T 2803:z 2799:S 2791:z 2787:S 2773:z 2767:w 2757:S 2753:Z 2749:z 2745:S 2741:z 2727:w 2721:z 2715:z 2712:= 2709:w 2699:w 2685:w 2679:z 2673:z 2670:= 2667:w 2661:z 2655:w 2645:S 2641:w 2637:S 2633:Z 2629:T 2625:T 2621:z 2617:Z 2613:Z 2609:z 2605:Z 2601:S 2597:T 2593:T 2589:T 2585:S 2581:S 2577:S 2573:T 2569:Z 2555:. 2550:S 2544:T 2541:= 2538:Z 2518:T 2512:S 2492:T 2486:S 2480:S 2474:T 2454:T 2451:= 2448:S 2442:S 2436:T 2430:T 2424:S 2414:T 2410:S 2406:S 2402:S 2398:S 2391:S 2387:S 2373:. 2370:T 2367:= 2364:S 2358:S 2352:T 2346:T 2340:S 2330:T 2326:S 2322:z 2308:z 2302:x 2282:z 2276:y 2270:y 2264:x 2254:S 2250:z 2246:y 2242:x 2176:S 2172:S 2168:S 2164:z 2150:z 2144:x 2124:z 2118:y 2112:x 2102:y 2084:y 2078:y 2058:y 2052:z 2046:y 2026:y 2020:z 2000:z 1994:y 1974:z 1968:x 1948:S 1942:x 1922:S 1916:y 1896:S 1890:z 1884:y 1878:x 1868:S 1862:S 1858:S 1854:S 1850:S 1400:. 1225:( 1174:( 1094:0 1064:0 1037:0 990:( 873:? 541:0 531:0 431:1 403:1 357:1 344:= 339:1 317:1 50:.

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