Talk:Ordinal number/Archive 3

2008:. Other examples of transfinite sequences are the sequence-union of countable parallel or nested enumerations of potentially infinite mathematical objects — for example, taking into consideration the appropriate respective name-listing rate (say, 1 name per 5 seconds), birth rate (say, 10 persons per second), death rate (say, 2 persons per second), and transient rate (say, 20 persons per second) for each country and its states, provinces, counties, cities, and municipalities, the listing of all living inhabitants of the planet Earth is a theoretivally infinite human task and would require transfinite ordinal numbers to index the list of names (this last example provides a simple natural manifestation of transfinite sequences and ordinal numbers which belie their “abstract-only” Platonist-view existence). 1227:

atomic symbol created by invocation of an axiom put there just for the purpose, and you get ω·2 by invoking another axiom and sup λ (for arbitrary ordinal λ) with yet another axiom. This never made sense to me until I actually tried to piece it together from the axioms. These completed infinities aren't there made from an iterative construction, they're there because they're postulated. It's less like "iterate the successor function infinitely many times" and more like "iterate til you get tired, then invoke a completion axiom to create a limit ordinal". As mentioned I think I just sort of get it now, but IMHO the presentation would benefit from being explicit about the axiomatic justification for saying these limit ordinals actually exist. Btw, another oddity: we have

644:. Natural ways of constructing such a set tend to well-order it without needing to appeal to the well-ordering of every isomorphic copy of it, and moreover a natural way to prove that any finite set can be well-ordered is via its isomorphism with the set of that cardinality whose existence has been previously established. To me an argument that reverses that arrow seems less natural. But that's just me, and I can appreciate that other organizations of what rests on what might justifiably reverse that arrow (not that I have one in mind in this case). -- 798:, "ordinal" wikilinks to this article when it clearly needs to link to the article about linguistics. This wouldn't usually be a problem, but there are literally hundreds of these pages, and all the ones I've looked at need modifying. In this case, not even an ordinal disambig page would be appropriate, as a straight link would obviously be preferred. This confusion is futher augmented by the fact that the linguistic page is at present no more than a stub. Also confusingly linked is the cardinal link on the number pages. Any ideas of how to proceed? 1162:

ordinal (I edited the article to mention this but I hope it's correct) giving ω+1, ω+2, etc.; getting ω·2 and beyond needs the axiom of replacement. After some head scratching I think I understand the idea of this construction, but I haven't worked out the details well enough yet (I'll keep trying since it seems like a good exercise) to put them into the article, so it would be great if someone else could. Right now what's there comes across like "start with 1,2,3,..., keep going til you get to infinity, then stop for a moment and then

676:'Ordinal number' or 'ordinal' is a concept that is understood not only by small children, but even by rhesus monkeys !!! So I understand the frustration of Pcu123456789 and respectfully disagree with user Trovatore who says that "It's a mathematics article" : No, it is totally wrong to assume user typing common english word like 'ordinal' into Knowledge has a PhD in math and wants to read only about set theory. I also disagree that all languages that have ordinal numbers are 'English' so I removed otheruses tag to 31: 1634:. Is that consistent? Who cares, everyone knows that naive set theory of Cantor's era is inconsistent regardless ;-). What I don't see is how to get uncountable ordinals out of this. The whole concept of iteration connotes countability, as does the word "ordinal" ("first, second, third..."). I know the diagonal proof that 406:

self-contained. The hardest concept might be that of order type. For pedagogical completeness in explaining order type one should mention the negative integers and the rationals between 0 and 1 to make the point that there exist linear orders that aren't well orders, but this bit of the explanation should be delegated to the

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I probably have some of the above wrong (along with some known missing), and the above wording is not in presentable form for the article, but I think the questions are pretty basic and that the article ought to treat them one way or the other, so I'll try to write something once I've made a bit more

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The theory of ordinals can be developed without regularity just fine, but one has to define ordinals as transitive sets well-ordered by set membership. The definition given in the article, which refers to just total order, does not work without the axiom of regularity. So, the caveat in the article

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My knowledge of set theory isn't really good, but if I remember correctly, these statements are false: the axiom of regularity is not actually needed at all for the theory of ordinals, and while it may simplify some proofs as shown in that section, those proofs can actually be done from the rest of

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be defined) is that a) negation is not defined for ordinals in general, hence even though ω -1 +1 might be parsed as ω + (-1+1), (which gives a different value than (ω-1)+1, which would have to be ω+1 by your proposed definition), for infinite ordinals negation is not even defined. And b) addition

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I agree with the conclusion there. Sorry for prolonging a discussion that was settling down; I just wanted to make sure the opinion that subtraction isn't defined on the natural numbers, either, was represented here. I think that not mentioning (truncated) subtraction is the most natural choice, as

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I'm not an expert in the area. I do not know whether ordinal subtraction (if definable as claimed) is of sufficient importance to mention it, but if it is, mentioning it does require a citation of a reliable source. Conversely, the statement that subtraction etcetera is not definable (as claimed in

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To JRSpriggs: what exactly do you mean by "not closed"? The operations are (uniquely) defined under the same restrictions as the usual operations on natural numbers. The context of the statement suggests the reading "addition, multiplication, and exponentiation of natural numbers can be extended to

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can be well-ordered. Cantor spent years trying to establish that there was such a wellordering, without success, so in Cantor's system we have to say this is an unanswered question. It took Zermelo's axiomitization of set theory along with introduction of the axiom of choice to create a theory in

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A. I think Cantor had some way to get uncountable ordinals but I don't see exactly how. Trovatore gives a philosophical argument that there should be such an ordinal but I don't see how to get it explicitly from Cantor's rules as described in SEP. Cantor's paper is translated online so maybe I'll

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all countable ordinals (in the informal sense that's hopefully clear) can be gathered together into a completed whole, then that completed whole is itself wellordered, and its length is an ordinal, and it's not a countable ordinal, because otherwise you could stick another point at the end, and get

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and reference criteria. Currently this article does not include in-line citations. It is recommended that the article's editors take a look at the inclusion of in-line citations as well as how the article stacks up against the rest of the Good Article criteria. GA reviewers will give you at least a

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week's time from the date of this notice to work on the in-line citations before doing a full re-review and deciding if the article still merits being considered a Good Article or would need to be de-listed. If you have any questions, please don't hesitate to contact us on the Good Article project

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I'd like to request that the intro section (that describes ω, ω+1, ω·2, etc.) describe more explicitly (based on the axioms of set theory) why these ordinals exist and aren't all the same. It's a bit counterintuitive to have a countable ordinal like ω different from ω, since that means there's a

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which vacuously satisfies the condition that all of its members have the property in question. Consequently, 0 has the property. So all members of {0} have the property. But {0} is the set of ordinals less than 1, thus 1 has the property. So all members of {0, 1} have the property. Thus 2 has the

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that way to a casual reader, because the intro paragraph at least twice mentions the idea that ordinal numbers are an extension of the finite, natural numbers. In such a context, saying that addition etc can be defined, but subtraction can't, sounds like saying that something special about these

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A. In Cantor's system, this relies on principle 2 as mentioned in the SEP article but which is missing from the Knowledge article, so IMO the Knowledge article has a gap. Principle 2 basically says that whenever you see a sequence going "..." you can plop a limit ordinal at the end of the dots,

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is and always has been about mathematics. If you can find references discussing a non-mathematical meaning other than just the linguistic thing about words like "seventeenth" (I'm curious how one might do an experiment showing that Rhesus monkeys distinguish ordinals from cardinals), then by all

3737:: the axiom of foundation by Skolem ("Some remarks on axiomatized set theory" from 1922), and the von Neumann definition of ordinals by von Neumann ("On the introduction of transfinite numbers" from 1923)? I don't have access to the book and can't check myself now if this is indeed the case. -- 1226:

Thanks, that explains the issue pretty well, that the article's presentation of iterating the successor operation out to ω and then continuing the iteration is missing some steps, and evokes a picture of an order-preserving bijection which is of course silly. Really, ω is more like a brand new

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which says that all countable infinities (of course that really means cardinals) are the same. So I think the explanation should begin with finite ordinals 0={}, 1={0}, 2={1,0}, etc.; then get ω by invoking the axiom of infinity explicitly; then use the axiom of union to get the supremum of an

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You 'have been over this issue many times', yet are not willing to fix the problem ? Why ? In Google, just "Ordinal number" "database" alone gives two times more hits than "ordinal number" "set theory", so other meanings clearly deserve short explanation or disambiguation. If you do not like my

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tag. It's a mathematics article, after all, so the complaint about "diving into mathematics" is a bit odd. The tag should be reserved for articles whose logical structure is unclear, not those that need to be made more accessible. Discuss the latter issue on the talk page, not the article page.

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

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Cantor defined them by means of two “generating principles”: the first (1) yields the successor a+1 for any given number a, while the second (2) stipulates that there is a number b which follows immediately after any given sequence of numbers without a last element. Thus, after all the finite

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

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A. My personal answer is no — since, if a limits exists, then it must be the same from all approach-directions . I resolutely believe that there is no ordinal number greater than or equal to ω^ω. Transfinite ordinal numbers are useful for indexing a transfinite sequence . An example of a

3690:, so unlike extensionality, it is not very wise to rely on the foundation axiom without a good reason. IMO it would not hurt to start with the general definition which is adequate without foundation, and then give the other possible definitions as alternative equivalent characterizations. 405:

It's a nice question how to introduce the ordinal concept. Just now I rewrote part of the introduction with the goal of at least making it more precise, if not more intuitive, on the theory that those likely ever to really grasp the concept should find it easier when the definitions are

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infinite thingy numbers precludes them from being subtracted, whereas in fact they behave exactly the same as natural numbers in this respect (the only restriction on the subtraction operation being β ≥ α), just like for addition. So, while the statement may be formally correct, it is

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

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numbers comes, by (2), the first transfinite number, ω (read: omega); and this is followed by ω+1, ω+2, …, ω+ω = ω·2, …, ω·n, ω·n +1, …,ω2, ω2+1, …, ωω, … and so on and on. Whenever a sequence without last element appears, one can go on and, so to say, jump to a higher stage by (2).

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article since otherwise the pedagogically optimal presentation of the ordinals would almost surely be too long to serve as a Knowledge-style introduction. The premise of my attempt at an introduction is that it should at least mention the main facts about the ordinals.

3813:(A notification about this fact was removed with the inappropriate editing summary "rv trolling", apparently in response to an equally inappropriate editing summary by the editor referred to. It was clearly not trolling, and I believe technically it didn't violate 1835:

was an aleph (that required Zermelo's introduction of AC). I'll see if I can contribute a fix this weekend if nobody else does by then, but given my weak understanding of the subject, I'm likely to get something wrong and leave the rest of you to clean it up ;).

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OK, I understood your 'mathematics' meaning 'hard core set theory'. I you are willing to include things like simple counting of numbers to 'mathematics' then I am with you and apologize for misunderstanding. About ordinal numerical abilities of rhesus monkeys,

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I think without stating principle 2 explicitly, there's not really an argument that ω exists. It's like describing a "biggest natural number". I can see with principle 2 that you can get ω+ω and so forth, and upwards in a tree-like fashion to ordinals like

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

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Sorry, I missed "set of ordinals smaller" and thought it said "ordinal smaller." So yeah, obviously if the property's already taken to satisfy all ordinals smaller ordinals, and passes to the ordinal itself, it satisfies them all. I jus misinterpreted

3817:, either. However, the very non-neutral tone of the original notification was likely to cause disruption, so I decided to replace it rather than restore it. I very much hope that's OK. I encourage everybody to stay calm and constructive.) -- 3187:

To EJ: The point is that those operations do not have all the properties normally expected for subtraction, division, logarithm, etc.; in particular they are not closed, i.e. do not give values for all inputs. If you look at the article on

1854:. That article has a detailed explanation of the role of the axiom in constructing large ordinals in ZFC. But, as Trovatore has also pointed out, this is independent of the construction of such ordinals in unformalized mathematics. — Carl 3762:

I have the book here, and in Skolem's paper you refer to it looks like replacement is being proposed, not foundation. Von Neumann's paper does refer to augmenting Zermelo's system with replacement, but does not seem to assume foundation.

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there can be no such completed totality. The claim that such a completed totality exists is a falsifiable conjecture in Popper's sense -- in the strongest possible way, given that it's already been falsified. Whereas for the case of all

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itself, so, being an increasing sequence bounded above they would have a limit, so the subset is not closed-and-discrete as remarked previously). However, any countable ordinal is the order type of a subset of the reals which is closed

1450:"iterate the successor function infinitely many times, and then do it some more". The axiomatics are secondary (remember for example that when Cantor first described the ordinal numbers, he wasn't using any axiomatic system at all). -- 844:

to look for. For example, how could I ever discover the meaning of Phthongometer if I didn't know (or forgot) that it started with a PH and the search options were too narrowly confined. Anyway I stole this modified definition from

2564:) is the order type of a well-ordered subset of the reals (which can be chosen either closed or discrete, but not both). So it is in principle possible to give such a "matchstick" representation of some extremely large ordinals. 123:

or you may contact me personally. On behalf of the Good Articles Project, I want to thank you for all the time and effort that you have put into working on this article and improving the overall quality of the Knowledge project.

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No, I'm not willing to include those. I'm saying "ordinal number" has a specific meaning to mathematicians, and that meaning is what this article is about, and I would like it to stay that way. Whether this article is correctly

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is uncountable, but what reason is there to think there is an ordinal that large, i.e. that they don't run out long before they get to even the first uncountable cardinal (which depending on CH might or might not be as big as

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discrete (proof: the implication ⇒ is obvious, and to prove ⇐, assume there is a limit in the reals, that limit must be in the subset because it is closed, contradicting discreteness). No subset of the reals which is closed

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are not worth the disk space they occupy. In any case, it is the responsibility of the editors who maintain those articles to get the links from them correct, not our responsibility to fulfill their mistaken expectations.

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In any case, I think the somewhat fine point may be lost on the very reader who might need the help – assuming the same somewhat clueless reader feels a compelling urge to understand why the statement is necessarily true.

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I do not think I have anything to argue about and yes, Trovatore, my last comment was not for you. Somebody restored my 'otheruses' to linguistics and I am happy with that. Perhaps there should be also otheruses

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Fixes subsequent to my rewrite greatly appreciated. Does it matter whether the list of transfinites splits? I deleted the spaces just now but by all means put them or equivalent back. My only question is to

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progress in understanding, unless you guys really think it's inappropriate. I appreciate very much your all taking the time to answer this stuff from me and I hope it can lead to making the article better.

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The usual construction of 7 is as 6∪{6}, which totally orders 7 by induction. That every finite set can be totally ordered is true but irrelevant, and moreover either circular (if you use the definition of

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has order type ω. Add in the number 1.5 and the set now has order type ω+1. It's easy to keep going, adding more elements, until you get to ω + ω. Then you add in 2.5 to get ω + ω + 1, and so on. — Carl

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I'm not a math geek. Personally, I looked up Ordinal to make sure I was using it correctly to distinquish from Nominal, Interval and Ratio level data. I did not find this entry helpful in this regard.

3192:, you will see that the decompositions to which you referred are mentioned. The main thing is that those ordinal operations are not used by many people for the simple reason that they are not useful. 2411: 605:(unindent) To Vaughan Pratt: It is difficult to argue about this because the dispute is not over what is true or what is provable, but over what is obvious. Let me just point out that the article on 562:). That any two total (and hence well-) orderings of a finite set are order isomorphic is on the other hand a crucial distinction between finite and infinite sets when distinguishing ω from ω+1. -- 482: 1540: 2302: 2374: 3228:

ordinals, but subtraction and division cannot". If that's not intended, it should be stated explicitly. Also, "not being used by many people" is quite different from "cannot be defined". —

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Also, now that I come to think about it, the article attributes the definition given to von Neumann, but that does not sound right. As far as I now, von Neumann's definition of ordinals

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

2568:, it is useless, because the representations in question get too crowded to make any sense of. Already I had a hard time fiddling with the exponential constants to get a satisfactory 890:

Because both ordinals ω and ω + 1 have the same cardinality but have distinct order types. The finite cardinalities are the only ones that have a unique possible well-ordering. — Carl

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and created a linguistic stub instead. I have seen other examples in Knowledge, where small explanatory sections like this were used instead of full disambig page so I hope that is OK

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

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I modified the "Common use meaning" slightly not because it was wrong, but only because I didn't immediately see the connection between readin', 'riting & 'rithmatic. Numbers are

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are defined independently of and prior to set theory. Thus 7 is not identified with 6∪{6} until after the finite ordinals have been seen to be equivalent to the natural numbers. And

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the axiom of foundation by a couple of years, hence he probably stated the definition in the way suitable for nonwellfounded theories. So that's another reason to use the latter. —

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

1802: 1677: 1654: 1155: 3510:(Undent) I can't help the distinctive feeling of going in circles in this discussion. The point, once again, is that whatever the original editor was thinking, it does not 3476:. At least that's the way a significant number of people would look at it; my guess is that this was what the person who originally edited the article was thinking. — Carl 2593: 1756: 2909: 2834: 2740: 632:

Agreed. However it is not obvious to me why the existence of specific well-ordered finite sets, namely the finite ordinals, should follow from the well-orderability of

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transfinite sequence is the right-sequence-union of a convergent potentially infinite sequence with its limit — say, <3, 3.1, 3.14, 3.141, 3.1415, 3.14159, …, π: -->

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

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and add either 0 if you want a closed subset, or 1 if you want a discrete one — but you can't have both at once. I hope I've made this excruciatingly clear now. --

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finite sets. In general arguing from a universal to an existential usually involves a tacit existential, in this case the existence of a finite set of cardinality

2960: 1679:)? For a reader just trying to grok the basics of this subject (that would be me) these important points seem to be missing so I wish they could be filled in. 761:

I'm not exactly sure whom you're arguing with, but I don't think it's me. I'm quite willing to have articles on other meanings, and disambiguations to them. --

836:! (yes, there are a million things wrong with that thought. lol) Reference materials are not only about containing the "data" but also about being able to 955:). For instance, it states that ordinals were introduced by Cantor in 1897, but does not give a reference. Can someone go through the article and fix it? 1157:

which means (since ω is an element of ω) that f(ω) "should" be some specific natural number, which clearly doesn't fly. Also, schoolchildren are taught

576:(Incidentally, thanks, JRSpriggs, for changing ω to {ω} in the bit I added about topology---I had the sign reversed in my mind, closed instead of open, 3472:

Right; there is no subtraction operation defied on the natural numbers, for the same reason. There is a partial subtraction function, but that isn't a

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In this case, though, there's nothing wrong with what they actually wrote. You need to look at it a little closer. That isn't a fraction sign, and

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This article is very nice, but even though the citation requirements for GA have been relaxed, this article doesn't meet them at the moment (see

2033:? I.e. how do we know that iterating the successor and limit operations indefinitely don't result in more and more countable ordinals forever? 1946: 107: 3631:

is justified, but it really just concerns details of the definition, you do not need the axiom of regularity once you get past such issues. —

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

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No, it is not necessary to have a separate "basis". Consider what happens if we take α to be 0. The set of ordinals smaller than 0 is the

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follows from the idea of concatenation, which is possible to perform given 2 ordinals, but it is meaningless to "unconcatenate" ω by 1.

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Nice!! Incidently, now that this page has a bit of attention on it, is there any chance that someone could add a few cites to meet the

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reach an uncountable instead of staying countable no matter how far you go? Also according to a good historical article I just found

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any two total orderings of a finite set are order isomorphic." with the part I added underlined. Your version explains why there is

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To Vaughan: The sentence in question is "The finite ordinals (and the finite cardinals) are the natural numbers: 0, 1, 2, …, since

734:. (By the way, from a brief glance at the abstract, what the monkeys seem to have understood is what mathematicians would call a 1230: 505:

ordered set, if there did exist such a thing as a finite set that couldn't be well-ordered, how would that change anything?) --

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The article needs a nontechnical section at the top of the introduction. Currently, it immediately dives into the mathematics.

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

2379: 1446:, yes, surprising or not. I don't much agree with the rest of your comments, though. The best way to think about it really 617:

argument from mathematical induction ("which totally orders 7 by induction") cannot be considered obvious to most readers.

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Ok, thanks. I think I get it, but not well enough to incorporate into the article. Do you think you could do that? --

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Yes, there are bijections between the elements of ω and the elements of ω. So they have the same cardinality. However,

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

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ordinals, the hypothesis that there's a completed totality of them is a potentially falsifiable proposition that has

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properties in common with subtraction on the integers to justify calling it subtraction. I imagine the reason why

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Subtraction and division with remainder of natural numbers can also be extended to ordinals, but it is rarely used.

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that it's true; we don't want to limit our view just to countable ordinals, if there are bigger ones out there. --

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which for the first time in 44 years made sense to me. (I'm adding a copy of this to cardinal discussion page too)

110:). A significant change to the GA criteria is the mandatory use of some sort of in-line citation (In accordance to 38: 726:

is a separate issue -- you can see in the archive of this talk page that I suggested it should be at a title like

3768: 3519:. It should not be there unless it includes a proper discussion of why and where the subtraction is not total. — 539:

one finite ordinal per natural number. Neither explanation is strictly necessary, but we are here to be helpful.

2485: 2209: 772: 169: 3612: 2707: 1113: 3649: 1166:" which is bogus. The point to be made is that these infinite sets are logical consequences of the axioms. 649: 589: 567: 510: 416: 341: 3574: 2747: 2534: 2521: 1804:). That is, you can repeat the ordinal iteration construction for as long as you want, but why does that 1582: 1044: 957: 488: 3288:". It was wrong anyway because the remainder (when it exists) is not constrained to be a natural number. 2037:

try to read it. In ZFC, this is again done with the replacement axiom, need to figure out the specifics.

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

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I think 75 might be looking for a more "philosophical" answer. Here's how I look at it: it's clear that

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another countable ordinal larger than it, and by assumption it's larger than all countable ordinals.

1814: 1761: 328:, unlike the situation with the natural numbers. For example ω−1 = ω−2 (= ω). You do however have 259:

What you saw there is NOT an underline indicating division, it is a brace indicating that there are

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as one in bijection with a natural number) or nontrivial (if you use some other definition such as

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Such a set S is automatically well-ordered with respect to set containment. This relies on the

880:"one has to distinguish between the two" but I couldn't figure it out. Can anyone clarify? -- 385:

I agree that it wouldn't hurt to add some intuitive motivation. However I don't agree with the

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Just like subtraction of natural numbers, ordinal subtraction is only defined for β ≥ α. And ω

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means, write it up and put a disambiguation link to it. But please don't muddle the issue in

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What do you mean by discrete - no limit points in the image or no limit points in R? — Carl

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even appeared anywhere in his aleph hierarchy. ZFC can prove that there's an ordinal κ with

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Unless I am mistaken, a graphic similar to the one in the article can also be created for

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though the exact formation rules aren't totally clear to me once you get to ordinals like

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subset of the reals means discrete with the induced topology, so, no accumulation points

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

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

1906: 1718: 1702: 1451: 1213: 1066: 1003: 817: 762: 739: 700: 618: 610: 540: 498: 397: 264: 251: 111: 2945: 3542: 3483: 3372: 2963: 2763: 2685: 1861: 1594: 1577: 1550: 952: 897: 748:

version of the clarification, why not make your own? And what was wrong with link to

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the ZFC axioms. Could someone who is more competent in this area confirm this? –

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In a hypothetical subtraction on ordinals, what ordinal would correspond to ω - ω

2182:

can be well-ordered, and that's how it's done in ZFC today. As for the value of

928:

Thanks for your addition. I'm not sure I'm 100% crystal clear, but it helps. --

3832: 221: 217: 125: 115: 46:

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

3259:

requires a citation. A middle ground is not to mention subtraction at all. --

2414: 606: 555: 407: 1188:

such bijection is order preserving. Which is why they are different ordinals.

1181:

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

3284:

To Lambiam: Good point. Following the middle way, I took out the sentence, "

1713:. In particular, aleph_null has aleph_one as its successor cardinal. Since ω 1580:? This is both for the benefit of readers and for preserving the GA status. 994: 157: 1990:{\displaystyle \varepsilon _{0\varepsilon _{0\varepsilon _{0_{\ldots }}}}} 1888:

But the thing is, for sets, or for ordinals in general, we know a precise

979:

Is it ok to put it like this? Isn't it necessary that there be a 'basis'?

3733:

Aren't both introduced in some preliminary form in two articles found in

3538: 3479: 3368: 2759: 1857: 1546: 1065:

To Eebster: No. Are you paying attention? Try reading the article again.

893: 535:

one finite ordinal per natural number. My addition explains why there is

1039:" After all, you can't prove inductively a property works for ordinals 3708: 3632: 3520: 3427: 3324: 3229: 3159: 2433: 1809: 3648:

However, as far as I am concerned, if one does not have at least the

2595:(I drew the picture in the article), I never managed to build a good 1936:

questions I had, and where I currently am in terms of having answers:

2040:

Q. There's one supremely interesting uncountable set, the real line

3174:

What keeps you from correcting this by removing the false part? --

2531:

Indeed, or even larger. Any countable ordinal (i.e., any ordinal

3849:. I assume that this means the move will not occur. Fortunately. 3615:: every nonempty set B has an element b which is disjoint from B. 2206:, that's the continuum hypothesis. Cantor wanted to prove that 1731:

The point I don't see is why there's an ordinal corresponding to

1439:{\displaystyle \omega ^{\varepsilon _{0}}<\varepsilon _{0}+1.} 1372:{\displaystyle \omega ^{\varepsilon _{0}}<\varepsilon _{0}+1.} 876:

I then read quite a bit more of the article trying to figure out

1476:

Indeed, no formal set theory is needed. The set of real numbers

3660:. (If you do not have a home plate, it is just not baseball.) 25: 3094:

One can also define a sort of "logarithm with remainder": if

2436:

is a good source to start with. I may try to add something.

2742:

into R so that it is discrete? I'm quite certain you cannot.

3806:

there is a proposal to rename the present article, so that

3214:

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

2856:

would be bounded by the point corresponding to the element

1267:{\displaystyle \omega ^{\varepsilon _{0}}=\varepsilon _{0}} 3535:

it is essentially undiscussed in set theory texts. — Carl

2702:

Maybe we could use colour and/or the second dimension? --

2304:, but κ can be any ordinal (Cohen's independence proof). 2023:

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

1850:

I think that I see your question now. The answer is the

1015:

Shouldn't it actually read "... is true of all ordinals

106:. (Discussion of the changes and re-review can be found 102:

listings to ensure compliance with the standards of the

2881:

discrete (again, discrete means no accumulation points

1997:. In ZFC, it's there because of the axiom of infinity. 794:

On a related issue, on individual number pages such as

2406:{\displaystyle {\mathfrak {c}}\neq \aleph _{\lambda }} 2335:

into a given set (such as ω). That is the proof that ω

2137:

itself under the von Neumann construction) exists iff

2948: 2917: 2891: 2862: 2842: 2816: 2722: 2659: 2628: 2601: 2574: 2537: 2488: 2461: 2382: 2345: 2273: 2249: 2212: 2188: 2166: 2143: 2119: 2092: 2068: 2046: 1949: 1817: 1788: 1764: 1737: 1663: 1640: 1613: 1482: 1399: 1332: 1281: 1233: 1141: 1094: 1023: 477:{\displaystyle \epsilon _{0}=\omega ^{\epsilon _{0}}} 443: 173: 2432:I think the article should have a history section. 1535:{\displaystyle \{1-1/(n+2)\mid n\in \mathbf {N} \}} 752:stub other than it was written by "wrong" person ? 98:are in the process of doing a re-review of current 2954: 2934: 2903: 2868: 2848: 2828: 2734: 2672: 2641: 2614: 2587: 2556: 2508: 2474: 2405: 2368: 2296: 2259: 2235: 2198: 2174: 2151: 2129: 2102: 2078: 2054: 1989: 1827: 1796: 1774: 1750: 1717:exists as a set, it is not a proper class like ∞. 1671: 1648: 1626: 1534: 1438: 1371: 1316: 1266: 1149: 1100: 1029: 476: 201: 2297:{\displaystyle {\mathfrak {c}}=\aleph _{\kappa }} 1758:, and even more why there's one corresponding to 114:) to be used in order for an article to pass the 2516:and any integer number of iterations thereof. -- 250:is presumably not an (arbitrary) real number. -- 2369:{\displaystyle {\mathfrak {c}}\neq \aleph _{0}} 870: 263:copies of ω being added together to get ω·r . 2836:(proof: the points corresponding to elements 1901:been falsified. And if it's true, we want to 8: 3055:(for a lack of a better notation) such that 1529: 1483: 1379:Is that correct? It's a little surprising. 2509:{\displaystyle \omega ^{\omega ^{\omega }}} 2236:{\displaystyle {\mathfrak {c}}=\aleph _{1}} 202:{\displaystyle \omega +...+\omega \over r} 3030:Division with remainder: for any ordinals 580:the infinite subsets of ω are the only non 2947: 2924: 2916: 2890: 2861: 2841: 2815: 2721: 2664: 2658: 2633: 2627: 2606: 2600: 2579: 2573: 2548: 2536: 2498: 2493: 2487: 2466: 2460: 2397: 2384: 2383: 2381: 2360: 2347: 2346: 2344: 2323:If and when someone fixes the "proof" in 2288: 2275: 2274: 2272: 2251: 2250: 2248: 2227: 2214: 2213: 2211: 2190: 2189: 2187: 2168: 2167: 2165: 2145: 2144: 2142: 2121: 2120: 2118: 2094: 2093: 2091: 2070: 2069: 2067: 2048: 2047: 2045: 1975: 1970: 1962: 1954: 1948: 1819: 1818: 1816: 1790: 1789: 1787: 1766: 1765: 1763: 1742: 1736: 1665: 1664: 1662: 1642: 1641: 1639: 1618: 1612: 1524: 1495: 1481: 1424: 1409: 1404: 1398: 1357: 1342: 1337: 1331: 1305: 1286: 1280: 1258: 1243: 1238: 1232: 1199:. One bijection maps that element to < 1143: 1142: 1140: 1093: 1022: 466: 461: 448: 442: 171: 1210:Pairing function#Cantor pairing function 529:each finite set can be well-ordered and 3787:Thanks for all the replies so far. – 2557:{\displaystyle \alpha <\omega _{1}} 1811:, Cantor was never able to prove that 1317:{\displaystyle \varepsilon _{0}+1: --> 847:Japanese counter word: Ordinal numbers 44:Do not edit the contents of this page. 3039:, there is a unique pair of ordinals 2327:, it will show that there is a least 156:This seems to be contradicted by the 7: 2029:Q. Is there an uncountable ordinal ω 1159:Hilbert's paradox of the Grand Hotel 2385: 2348: 2276: 2252: 2215: 2191: 2122: 2095: 2086:. Is there an ordinal the size of 2071: 1820: 1767: 1593:Thanks for suggesting Cantor. The 96:Knowledge:WikiProject Good articles 3587:Relying on the axiom of regularity 2394: 2357: 2285: 2224: 1739: 1701:one can show that every set has a 1191:The elements of ω have the form ω· 280:is well defined (delete the first 90:GA Re-Review and In-line citations 24: 3804:Talk:Ordinal number (linguistics) 3607:and very similarly it says that 2673:{\displaystyle \omega ^{\omega }} 2615:{\displaystyle \omega ^{\omega }} 2475:{\displaystyle \omega ^{\omega }} 2451:Scope of the graphical matchstick 3104:0, there are unique ordinals log 2642:{\displaystyle \varepsilon _{0}} 1627:{\displaystyle \varepsilon _{0}} 1525: 351: 29: 2260:{\displaystyle {\mathfrak {c}}} 2199:{\displaystyle {\mathfrak {c}}} 2130:{\displaystyle {\mathfrak {c}}} 2103:{\displaystyle {\mathfrak {c}}} 2079:{\displaystyle {\mathfrak {c}}} 1828:{\displaystyle {\mathfrak {c}}} 1775:{\displaystyle {\mathfrak {c}}} 712:should make things very clear. 350:See also the discussion titled 220:. How is this to be explained? 3003:there exists a unique ordinal 2995:Subtraction: for any ordinals 2684:, but it's just worthless. -- 2110:and if yes, what is its value? 1578:scientific citation guidelines 1512: 1500: 998:property. And so on. See also 962:21:24, 20 September 2007 (UTC) 129:05:48, 26 September 2006 (UTC) 1: 2911:for example, take the set of 2446:09:54, 29 November 2007 (UTC) 2422:20:23, 30 November 2007 (UTC) 2318:10:52, 30 November 2007 (UTC) 2243:but was unable to prove that 2018:07:45, 17 December 2008 (UTC) 1915:02:24, 30 November 2007 (UTC) 1867:13:54, 29 November 2007 (UTC) 1846:09:52, 29 November 2007 (UTC) 1727:02:27, 29 November 2007 (UTC) 1689:10:43, 28 November 2007 (UTC) 1587:19:48, 27 November 2007 (UTC) 1556:13:54, 27 November 2007 (UTC) 1460:03:31, 27 November 2007 (UTC) 1389:09:24, 26 November 2007 (UTC) 1222:09:01, 25 November 2007 (UTC) 1176:05:45, 25 November 2007 (UTC) 811:In my opinion, articles like 750:Ordinal numbers (linguistics) 272:Actually ordinal subtraction 268:09:07, 25 November 2006 (UTC) 255:05:57, 25 November 2006 (UTC) 237:05:28, 25 November 2006 (UTC) 163:, which gives as an example: 3808:Ordinal number (linguistics) 3579:03:07, 29 January 2009 (UTC) 2712:06:09, 4 February 2008 (UTC) 2694:17:09, 1 February 2008 (UTC) 2526:13:25, 31 January 2008 (UTC) 2175:{\displaystyle \mathbb {R} } 2152:{\displaystyle \mathbb {R} } 2062:whose cardinality is called 2055:{\displaystyle \mathbb {R} } 1797:{\displaystyle \mathbb {R} } 1672:{\displaystyle \mathbb {R} } 1649:{\displaystyle \mathbb {R} } 1150:{\displaystyle \mathbb {N} } 1007:03:30, 1 November 2007 (UTC) 988:21:19, 31 October 2007 (UTC) 728:ordinal number (mathematics) 437:= ω doesn't quite look like 401:05:23, 27 January 2007 (UTC) 380:04:39, 27 January 2007 (UTC) 3688:non-well-founded set theory 3656:, one is not talking about 2797:. "No accumulation points 2588:{\displaystyle \omega ^{2}} 1751:{\displaystyle \aleph _{1}} 678:Names of numbers in English 296:). However you don't have 3875: 3845:) closed this proposal as 3548:13:03, 25 April 2008 (UTC) 3529:12:27, 25 April 2008 (UTC) 3489:12:08, 25 April 2008 (UTC) 3436:09:59, 25 April 2008 (UTC) 3378:22:11, 24 April 2008 (UTC) 3333:12:55, 24 April 2008 (UTC) 3298:05:44, 24 April 2008 (UTC) 3264:19:14, 23 April 2008 (UTC) 3238:09:23, 13 April 2008 (UTC) 3202:06:49, 12 April 2008 (UTC) 3179:00:21, 12 April 2008 (UTC) 3168:14:45, 11 April 2008 (UTC) 2972:14:53, 19 March 2009 (UTC) 2801:" is equivalent to closed 2769:12:54, 19 March 2009 (UTC) 2752:12:50, 19 March 2009 (UTC) 1697:< ∞? Because using the 1324:\varepsilon _{0},}" /: --> 859:22:05, 26 March 2009 (UTC) 821:02:18, 14 April 2007 (UTC) 803:10:42, 12 April 2007 (UTC) 780:23:39, 10 April 2007 (UTC) 773:Ordinal number (databases) 766:18:40, 10 April 2007 (UTC) 757:12:59, 10 April 2007 (UTC) 743:04:26, 10 April 2007 (UTC) 717:04:17, 10 April 2007 (UTC) 704:02:02, 10 April 2007 (UTC) 685:01:18, 10 April 2007 (UTC) 139:14:27, 7 August 2007 (UTC) 3827:08:36, 17 June 2008 (UTC) 2904:{\displaystyle \omega +1} 2829:{\displaystyle \omega +1} 2735:{\displaystyle \omega +1} 933:03:38, 25 June 2007 (UTC) 916:12:06, 16 June 2007 (UTC) 903:16:47, 12 June 2007 (UTC) 885:16:23, 12 June 2007 (UTC) 664:23:59, 29 June 2008 (UTC) 654:15:12, 23 June 2008 (UTC) 640:for every natural number 627:11:46, 22 June 2008 (UTC) 594:00:38, 22 June 2008 (UTC) 572:21:43, 19 June 2008 (UTC) 549:21:20, 17 June 2008 (UTC) 515:09:24, 17 June 2008 (UTC) 492:12:40, 15 June 2008 (UTC) 421:09:33, 15 June 2008 (UTC) 359:19:46, 15 June 2008 (UTC) 346:10:18, 15 June 2008 (UTC) 3859:05:30, 3 July 2008 (UTC) 3792:21:22, 19 May 2008 (UTC) 3773:01:32, 22 May 2008 (UTC) 3742:00:36, 22 May 2008 (UTC) 3717:13:16, 19 May 2008 (UTC) 3670:21:18, 16 May 2008 (UTC) 3641:14:10, 16 May 2008 (UTC) 3625:13:17, 16 May 2008 (UTC) 3613:axiom of well foundation 2810:discrete has order type 1278:\varepsilon _{0},}": --> 1118:00:17, 16 May 2008 (UTC) 1075:04:08, 15 May 2008 (UTC) 1053:02:20, 15 May 2008 (UTC) 672:Common use meaning added 3650:axiom of extensionality 2869:{\displaystyle \omega } 2849:{\displaystyle \omega } 2331:which cannot be mapped 1101:{\displaystyle \alpha } 1030:{\displaystyle \alpha } 868:The article says this: 354:lower on this page. -- 3682:Well, a lot of people 2956: 2942:for positive integers 2936: 2905: 2870: 2850: 2830: 2736: 2716:Really, you can embed 2674: 2643: 2616: 2589: 2558: 2510: 2476: 2407: 2370: 2298: 2261: 2237: 2200: 2176: 2153: 2131: 2104: 2080: 2056: 1991: 1829: 1798: 1776: 1752: 1705:, and thus that every 1673: 1650: 1628: 1536: 1440: 1373: 1319: 1268: 1151: 1102: 1031: 874: 478: 210: 203: 154: 2957: 2937: 2906: 2871: 2851: 2831: 2737: 2675: 2644: 2617: 2590: 2559: 2511: 2477: 2408: 2371: 2299: 2262: 2238: 2201: 2177: 2154: 2132: 2105: 2081: 2057: 1992: 1830: 1799: 1777: 1753: 1674: 1651: 1629: 1537: 1441: 1374: 1320: 1269: 1152: 1126:clarification request 1103: 1032: 1000:Transfinite induction 967:Transfinite induction 732:ordinal (mathematics) 560:Dedekind-infinite set 479: 204: 165: 150: 144:Division undefinable? 104:Good Article Criteria 42:of past discussions. 2946: 2935:{\displaystyle -1/n} 2915: 2889: 2860: 2840: 2814: 2720: 2682:an explanation of it 2657: 2626: 2599: 2572: 2535: 2486: 2459: 2380: 2343: 2271: 2247: 2210: 2186: 2164: 2141: 2117: 2090: 2066: 2044: 1947: 1881:So the question is, 1852:axiom of replacement 1815: 1786: 1782:(the cardinality of 1762: 1735: 1661: 1638: 1611: 1480: 1397: 1330: 1279: 1231: 1139: 1092: 1021: 441: 170: 3810:can be moved here. 3735:From Frege to Gödel 3654:axiom of regularity 3601:axiom of regularity 3517:terribly misleading 584:subsets of ω+1.) -- 371:Intro too technical 288:, returning 0 when 160:article on ordinals 18:Talk:Ordinal number 3556:But there must be 3190:Ordinal arithmetic 2952: 2932: 2901: 2866: 2846: 2826: 2732: 2670: 2639: 2612: 2585: 2554: 2506: 2472: 2403: 2366: 2294: 2257: 2233: 2196: 2172: 2149: 2127: 2100: 2076: 2052: 1987: 1825: 1794: 1772: 1748: 1711:successor cardinal 1669: 1646: 1624: 1532: 1436: 1393:It's correct that 1369: 1318:\varepsilon _{0},} 1314: 1264: 1147: 1098: 1027: 971:The article says: 474: 195: 148:From the article: 3562:it is not defined 3546: 3487: 3376: 3121:such that 0 < 2955:{\displaystyle n} 2767: 2113:A. That ordinal ( 1865: 1699:axiom of powerset 1554: 1045:Eebster the Great 1043:than the anchor. 901: 609:assumes that the 199: 87: 86: 54: 53: 48:current talk page 3866: 3536: 3477: 3474:binary operation 3366: 2961: 2959: 2958: 2953: 2941: 2939: 2938: 2933: 2928: 2910: 2908: 2907: 2902: 2875: 2873: 2872: 2867: 2855: 2853: 2852: 2847: 2835: 2833: 2832: 2827: 2757: 2741: 2739: 2738: 2733: 2679: 2677: 2676: 2671: 2669: 2668: 2648: 2646: 2645: 2640: 2638: 2637: 2621: 2619: 2618: 2613: 2611: 2610: 2594: 2592: 2591: 2586: 2584: 2583: 2563: 2561: 2560: 2555: 2553: 2552: 2515: 2513: 2512: 2507: 2505: 2504: 2503: 2502: 2481: 2479: 2478: 2473: 2471: 2470: 2412: 2410: 2409: 2404: 2402: 2401: 2389: 2388: 2375: 2373: 2372: 2367: 2365: 2364: 2352: 2351: 2303: 2301: 2300: 2295: 2293: 2292: 2280: 2279: 2266: 2264: 2263: 2258: 2256: 2255: 2242: 2240: 2239: 2234: 2232: 2231: 2219: 2218: 2205: 2203: 2202: 2197: 2195: 2194: 2181: 2179: 2178: 2173: 2171: 2158: 2156: 2155: 2150: 2148: 2136: 2134: 2133: 2128: 2126: 2125: 2109: 2107: 2106: 2101: 2099: 2098: 2085: 2083: 2082: 2077: 2075: 2074: 2061: 2059: 2058: 2053: 2051: 1996: 1994: 1993: 1988: 1986: 1985: 1984: 1983: 1982: 1981: 1980: 1979: 1855: 1834: 1832: 1831: 1826: 1824: 1823: 1803: 1801: 1800: 1795: 1793: 1781: 1779: 1778: 1773: 1771: 1770: 1757: 1755: 1754: 1749: 1747: 1746: 1678: 1676: 1675: 1670: 1668: 1655: 1653: 1652: 1647: 1645: 1633: 1631: 1630: 1625: 1623: 1622: 1544: 1541: 1539: 1538: 1533: 1528: 1499: 1445: 1443: 1442: 1437: 1429: 1428: 1416: 1415: 1414: 1413: 1378: 1376: 1375: 1370: 1362: 1361: 1349: 1348: 1347: 1346: 1325: 1322: 1321: 1315: 1310: 1309: 1291: 1290: 1273: 1271: 1270: 1265: 1263: 1262: 1250: 1249: 1248: 1247: 1156: 1154: 1153: 1148: 1146: 1107: 1105: 1104: 1099: 1036: 1034: 1033: 1028: 891: 483: 481: 480: 475: 473: 472: 471: 470: 453: 452: 394: 388: 234: 208: 206: 205: 200: 172: 73: 56: 55: 33: 32: 26: 3874: 3873: 3869: 3868: 3867: 3865: 3864: 3863: 3800: 3798:Rename proposal 3765:Unzerlegbarkeit 3589: 3425: 3421: 3417: 3413: 3364: 3138: 3109: 2988: 2944: 2943: 2913: 2912: 2887: 2886: 2885:): if you want 2858: 2857: 2838: 2837: 2812: 2811: 2718: 2717: 2660: 2655: 2654: 2651:an attempt here 2649:. You can see 2629: 2624: 2623: 2602: 2597: 2596: 2575: 2570: 2569: 2544: 2533: 2532: 2494: 2489: 2484: 2483: 2462: 2457: 2456: 2453: 2430: 2428:history section 2393: 2378: 2377: 2356: 2341: 2340: 2338: 2284: 2269: 2268: 2245: 2244: 2223: 2208: 2207: 2184: 2183: 2162: 2161: 2139: 2138: 2115: 2114: 2088: 2087: 2064: 2063: 2042: 2041: 2032: 1971: 1966: 1958: 1950: 1945: 1944: 1933: 1813: 1812: 1784: 1783: 1760: 1759: 1738: 1733: 1732: 1716: 1707:cardinal number 1696: 1659: 1658: 1636: 1635: 1614: 1609: 1608: 1478: 1477: 1420: 1405: 1400: 1395: 1394: 1353: 1338: 1333: 1328: 1327: 1301: 1282: 1276: 1275: 1254: 1239: 1234: 1229: 1228: 1137: 1136: 1128: 1090: 1089: 1019: 1018: 969: 945: 866: 864:Why distinguish 840:the data we're 674: 611:natural numbers 462: 457: 444: 439: 438: 436: 392: 386: 373: 233: 227: 215: 168: 167: 146: 94:Members of the 92: 69: 30: 22: 21: 20: 12: 11: 5: 3872: 3870: 3862: 3861: 3799: 3796: 3795: 3794: 3784: 3783: 3782: 3781: 3780: 3779: 3778: 3777: 3776: 3775: 3751: 3750: 3749: 3748: 3747: 3746: 3745: 3744: 3724: 3723: 3722: 3721: 3720: 3719: 3696: 3695: 3694: 3693: 3692: 3691: 3675: 3674: 3673: 3672: 3646: 3617: 3616: 3605: 3604: 3588: 3585: 3584: 3583: 3582: 3581: 3551: 3550: 3508: 3507: 3506: 3505: 3504: 3503: 3502: 3501: 3500: 3499: 3498: 3497: 3496: 3495: 3494: 3493: 3492: 3491: 3453: 3452: 3451: 3450: 3449: 3448: 3447: 3446: 3445: 3444: 3443: 3442: 3441: 3440: 3439: 3438: 3423: 3419: 3418:(because ω + ω 3415: 3411: 3393: 3392: 3391: 3390: 3389: 3388: 3387: 3386: 3385: 3384: 3383: 3382: 3381: 3380: 3362: 3346: 3345: 3344: 3343: 3342: 3341: 3340: 3339: 3338: 3337: 3336: 3335: 3309: 3308: 3307: 3306: 3305: 3304: 3303: 3302: 3301: 3300: 3273: 3272: 3271: 3270: 3269: 3268: 3267: 3266: 3245: 3244: 3243: 3242: 3241: 3240: 3220: 3219: 3218: 3217: 3216: 3215: 3207: 3206: 3205: 3204: 3182: 3181: 3171: 3170: 3134: 3105: 3092: 3028: 2987: 2984: 2983: 2982: 2981: 2980: 2979: 2978: 2977: 2976: 2975: 2974: 2951: 2931: 2927: 2923: 2920: 2900: 2897: 2894: 2865: 2845: 2825: 2822: 2819: 2789:To clarify: a 2778: 2777: 2776: 2775: 2774: 2773: 2772: 2771: 2731: 2728: 2725: 2697: 2696: 2667: 2663: 2636: 2632: 2609: 2605: 2582: 2578: 2551: 2547: 2543: 2540: 2501: 2497: 2492: 2469: 2465: 2452: 2449: 2429: 2426: 2425: 2424: 2400: 2396: 2392: 2387: 2363: 2359: 2355: 2350: 2336: 2325:Hartogs number 2291: 2287: 2283: 2278: 2254: 2230: 2226: 2222: 2217: 2193: 2170: 2147: 2124: 2097: 2073: 2050: 2030: 2021: 2020: 2004: 1978: 1974: 1969: 1965: 1961: 1957: 1953: 1932: 1929: 1928: 1927: 1926: 1925: 1924: 1923: 1922: 1921: 1920: 1919: 1918: 1917: 1886: 1879: 1871: 1870: 1869: 1822: 1792: 1769: 1745: 1741: 1714: 1703:Hartogs number 1694: 1667: 1644: 1621: 1617: 1604: 1603: 1602: 1591: 1590: 1589: 1565: 1564: 1563: 1562: 1561: 1560: 1559: 1558: 1531: 1527: 1523: 1520: 1517: 1514: 1511: 1508: 1505: 1502: 1498: 1494: 1491: 1488: 1485: 1467: 1466: 1465: 1464: 1463: 1462: 1435: 1432: 1427: 1423: 1419: 1412: 1408: 1403: 1368: 1365: 1360: 1356: 1352: 1345: 1341: 1336: 1326:and therefore 1313: 1308: 1304: 1300: 1297: 1294: 1289: 1285: 1261: 1257: 1253: 1246: 1242: 1237: 1189: 1182: 1145: 1135:between ω and 1127: 1124: 1123: 1122: 1121: 1120: 1097: 1082: 1081: 1080: 1079: 1078: 1077: 1058: 1057: 1056: 1055: 1026: 1010: 1009: 977: 976: 968: 965: 944: 941: 940: 939: 938: 937: 936: 935: 921: 920: 919: 918: 906: 905: 865: 862: 826: 825: 824: 823: 806: 805: 792: 791: 790: 789: 788: 787: 786: 785: 784: 783: 782: 673: 670: 669: 668: 667: 666: 603: 602: 601: 600: 599: 598: 597: 596: 574: 520: 519: 518: 517: 499:User:JRSpriggs 469: 465: 460: 456: 451: 447: 434: 431: 426: 425: 424: 423: 372: 369: 368: 367: 366: 365: 364: 363: 362: 361: 244: 229: 213: 198: 194: 191: 188: 185: 182: 179: 176: 145: 142: 91: 88: 85: 84: 79: 74: 67: 62: 52: 51: 34: 23: 15: 14: 13: 10: 9: 6: 4: 3: 2: 3871: 3860: 3856: 3852: 3848: 3844: 3841: 3838: 3834: 3831: 3830: 3829: 3828: 3824: 3820: 3816: 3811: 3809: 3805: 3797: 3793: 3790: 3786: 3785: 3774: 3770: 3766: 3761: 3760: 3759: 3758: 3757: 3756: 3755: 3754: 3753: 3752: 3743: 3740: 3736: 3732: 3731: 3730: 3729: 3728: 3727: 3726: 3725: 3718: 3714: 3710: 3706: 3702: 3701: 3700: 3699: 3698: 3697: 3689: 3685: 3681: 3680: 3679: 3678: 3677: 3676: 3671: 3667: 3663: 3659: 3655: 3651: 3647: 3644: 3643: 3642: 3638: 3634: 3629: 3628: 3627: 3626: 3623: 3614: 3610: 3609: 3608: 3602: 3598: 3597: 3596: 3594: 3586: 3580: 3576: 3572: 3567: 3564:(not that it 3563: 3559: 3555: 3554: 3553: 3552: 3549: 3544: 3540: 3533: 3532: 3531: 3530: 3526: 3522: 3518: 3513: 3490: 3485: 3481: 3475: 3471: 3470: 3469: 3468: 3467: 3466: 3465: 3464: 3463: 3462: 3461: 3460: 3459: 3458: 3457: 3456: 3455: 3454: 3437: 3433: 3429: 3409: 3408: 3407: 3406: 3405: 3404: 3403: 3402: 3401: 3400: 3399: 3398: 3397: 3396: 3395: 3394: 3379: 3374: 3370: 3360: 3359: 3358: 3357: 3356: 3355: 3354: 3353: 3352: 3351: 3350: 3349: 3348: 3347: 3334: 3330: 3326: 3321: 3320: 3319: 3318: 3317: 3316: 3315: 3314: 3313: 3312: 3311: 3310: 3299: 3295: 3291: 3287: 3283: 3282: 3281: 3280: 3279: 3278: 3277: 3276: 3275: 3274: 3265: 3262: 3258: 3255:the article) 3253: 3252: 3251: 3250: 3249: 3248: 3247: 3246: 3239: 3235: 3231: 3226: 3225: 3224: 3223: 3222: 3221: 3213: 3212: 3211: 3210: 3209: 3208: 3203: 3199: 3195: 3191: 3186: 3185: 3184: 3183: 3180: 3177: 3173: 3172: 3169: 3165: 3161: 3157: 3153: 3149: 3145: 3141: 3137: 3132: 3128: 3124: 3120: 3116: 3112: 3108: 3102: 3097: 3093: 3090: 3086: 3082: 3078: 3074: 3070: 3066: 3062: 3058: 3054: 3050: 3046: 3042: 3038: 3033: 3029: 3026: 3022: 3018: 3014: 3010: 3006: 3002: 2998: 2994: 2993: 2992: 2985: 2973: 2969: 2965: 2949: 2929: 2925: 2921: 2918: 2898: 2895: 2892: 2884: 2883:in the subset 2880: 2863: 2843: 2823: 2820: 2817: 2809: 2804: 2800: 2796: 2795:in the subset 2792: 2788: 2787: 2786: 2785: 2784: 2783: 2782: 2781: 2780: 2779: 2770: 2765: 2761: 2755: 2754: 2753: 2749: 2745: 2729: 2726: 2723: 2715: 2714: 2713: 2709: 2705: 2704:99.234.59.230 2701: 2700: 2699: 2698: 2695: 2691: 2687: 2683: 2665: 2661: 2652: 2634: 2630: 2607: 2603: 2580: 2576: 2567: 2566:Unfortunately 2549: 2545: 2541: 2538: 2530: 2529: 2528: 2527: 2523: 2519: 2499: 2495: 2490: 2467: 2463: 2450: 2448: 2447: 2443: 2439: 2435: 2427: 2423: 2420: 2416: 2398: 2390: 2361: 2353: 2334: 2330: 2326: 2322: 2321: 2320: 2319: 2315: 2311: 2305: 2289: 2281: 2228: 2220: 2111: 2038: 2034: 2027: 2024: 2019: 2015: 2011: 2005: 2002: 2001: 2000: 1998: 1976: 1972: 1967: 1963: 1959: 1955: 1951: 1940: 1937: 1931:section break 1930: 1916: 1912: 1908: 1904: 1900: 1896: 1891: 1887: 1884: 1880: 1876: 1872: 1868: 1863: 1859: 1853: 1849: 1848: 1847: 1843: 1839: 1810: 1807: 1743: 1730: 1729: 1728: 1724: 1720: 1712: 1708: 1704: 1700: 1692: 1691: 1690: 1686: 1682: 1619: 1615: 1605: 1599: 1598: 1596: 1592: 1588: 1585: 1584: 1579: 1575: 1574: 1573: 1572: 1571: 1570: 1569: 1568: 1567: 1566: 1557: 1552: 1548: 1521: 1518: 1515: 1509: 1506: 1503: 1496: 1492: 1489: 1486: 1475: 1474: 1473: 1472: 1471: 1470: 1469: 1468: 1461: 1457: 1453: 1449: 1433: 1430: 1425: 1421: 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705: 702: 698: 693: 689: 688: 687: 686: 683: 679: 671: 665: 662: 657: 656: 655: 651: 647: 646:Vaughan Pratt 643: 639: 635: 631: 630: 629: 628: 624: 620: 616: 612: 608: 595: 591: 587: 586:Vaughan Pratt 583: 579: 575: 573: 569: 565: 564:Vaughan Pratt 561: 557: 552: 551: 550: 546: 542: 538: 534: 530: 526: 525: 524: 523: 522: 521: 516: 512: 508: 507:Vaughan Pratt 504: 500: 495: 494: 493: 490: 487: 467: 463: 458: 454: 449: 445: 432: 428: 427: 422: 418: 414: 413:Vaughan Pratt 409: 404: 403: 402: 399: 391: 384: 383: 382: 381: 378: 370: 360: 357: 353: 349: 348: 347: 343: 339: 338:Vaughan Pratt 335: 331: 327: 323: 319: 315: 311: 307: 303: 299: 295: 291: 287: 283: 279: 275: 271: 270: 269: 266: 262: 258: 257: 256: 253: 249: 245: 241: 240: 239: 238: 232: 225: 224: 219: 209: 196: 192: 189: 186: 183: 180: 177: 174: 164: 162: 159: 153: 149: 143: 141: 140: 137: 131: 130: 127: 122: 117: 113: 109: 105: 101: 97: 89: 83: 80: 78: 75: 72: 68: 66: 63: 61: 58: 57: 49: 45: 41: 40: 35: 28: 27: 19: 3847:no consensus 3846: 3839: 3812: 3801: 3734: 3704: 3683: 3657: 3618: 3606: 3595:states that 3592: 3591:The section 3590: 3565: 3561: 3557: 3516: 3511: 3509: 3285: 3256: 3155: 3151: 3147: 3143: 3139: 3135: 3130: 3126: 3122: 3118: 3114: 3110: 3106: 3100: 3095: 3088: 3084: 3080: 3076: 3072: 3068: 3064: 3060: 3056: 3052: 3048: 3044: 3040: 3036: 3031: 3024: 3020: 3016: 3012: 3008: 3004: 3000: 2996: 2989: 2882: 2878: 2807: 2802: 2799:in the reals 2798: 2794: 2790: 2622:, let alone 2565: 2518:Dan Polansky 2454: 2431: 2332: 2328: 2306: 2112: 2039: 2035: 2028: 2025: 2022: 1999: 1941: 1938: 1934: 1902: 1898: 1894: 1889: 1882: 1874: 1805: 1583:Geometry guy 1581: 1447: 1204: 1200: 1196: 1192: 1185: 1163: 1132: 1129: 1040: 1016: 981: 978: 970: 958:Geometry guy 956: 946: 877: 875: 871: 867: 841: 837: 833: 829: 827: 813:100 (number) 796:100 (number) 736:linear order 723: 696: 692:this article 691: 675: 641: 637: 633: 614: 604: 581: 577: 536: 532: 528: 502: 486:Arthur Rubin 377:Pcu123456789 374: 333: 329: 325: 321: 320:in the case 317: 313: 309: 305: 301: 297: 293: 289: 285: 284:elements of 281: 277: 273: 260: 247: 222: 211: 166: 155: 151: 147: 132: 116:verification 100:Good Article 93: 70: 43: 37: 3593:Definitions 3571:24.58.63.18 2986:Subtraction 2744:24.58.63.18 2438:75.62.4.229 2413:when λ has 2310:75.62.4.229 2010:BenCawaling 1838:75.62.4.229 1681:75.62.4.229 1595:SEP article 1381:75.62.4.229 1168:75.62.4.229 699:article. -- 607:finite sets 352:Subtraction 243:statements. 218:real number 136:216.158.5.4 36:This is an 3819:Hans Adler 3011:such that 2415:cofinality 2333:one-to-one 1164:keep going 1131:bijection 556:finite set 484:to me. — 408:order type 3851:JRSpriggs 3815:WP:CANVAS 3662:JRSpriggs 3365:? — Carl 3290:JRSpriggs 3194:JRSpriggs 2419:JRSpriggs 1907:Trovatore 1895:countable 1719:JRSpriggs 1452:Trovatore 1214:JRSpriggs 1067:JRSpriggs 1004:JRSpriggs 995:empty set 943:GA status 842:intending 818:JRSpriggs 763:Trovatore 740:Trovatore 701:Trovatore 619:JRSpriggs 541:JRSpriggs 430:notation? 398:Trovatore 390:confusing 265:JRSpriggs 252:Trovatore 158:MathWorld 121:talk page 82:Archive 5 77:Archive 4 71:Archive 3 65:Archive 2 60:Archive 1 3843:contribs 3705:predates 3686:work in 3652:and the 3566:couldn't 3133:< log 2964:Gro-Tsen 2791:discrete 2686:Gro-Tsen 2339:exists. 1693:Why is ω 949:WP:WIAGA 834:language 537:at least 336:= 0. -- 312:implies 231:contribs 3789:b_jonas 3739:Lambiam 3622:b_jonas 3414:− ω = ω 3261:Lambiam 3176:Lambiam 3099:1, and 3035:0, and 2329:ordinal 982:Arthena 930:Doradus 913:Doradus 882:Doradus 851:J-puppy 800:Poojean 777:Warbola 754:Warbola 714:Warbola 682:Warbola 661:Lambiam 533:at most 503:founded 433:Also, ε 356:Lambiam 112:WP:CITE 39:archive 3833:JPG-GR 3142:, and 3067:, and 2160:which 1890:reason 1709:has a 1208:using 953:WP:SCG 690:Well, 582:closed 489:(talk) 223:Simões 212:where 3426:). — 3125:< 3103:: --> 3098:: --> 3083:) + ( 3063:< 3034:: --> 1903:know' 1299:: --> 1207:: --> 738:.) -- 724:named 324:< 216:is a 16:< 3855:talk 3837:talk 3823:talk 3769:talk 3713:talk 3666:talk 3658:sets 3637:talk 3599:the 3575:talk 3558:some 3543:talk 3525:talk 3512:look 3484:talk 3432:talk 3373:talk 3329:talk 3294:talk 3257:also 3234:talk 3198:talk 3164:talk 3158:. — 3047:and 3023:) = 2968:talk 2764:talk 2748:talk 2708:talk 2690:talk 2680:and 2653:for 2542:< 2522:talk 2442:talk 2434:This 2376:and 2314:talk 2014:talk 1911:talk 1862:talk 1842:talk 1806:ever 1723:talk 1685:talk 1551:talk 1456:talk 1418:< 1385:talk 1351:< 1274:and 1218:talk 1172:talk 1114:talk 1071:talk 1049:talk 1041:less 951:and 898:talk 855:talk 838:find 832:not 830:math 697:this 650:talk 623:talk 590:talk 578:ω is 568:talk 545:talk 511:talk 417:talk 342:talk 126:Agne 108:here 3802:At 3539:CBM 3480:CBM 3422:= ω 3369:CBM 3015:+ ( 2808:and 2803:and 2760:CBM 2417:ω. 1899:not 1883:why 1858:CBM 1547:CBM 894:CBM 878:why 730:or 659:-- 634:all 615:any 3857:) 3825:) 3771:) 3763:-- 3715:) 3709:EJ 3684:do 3668:) 3639:) 3633:EJ 3577:) 3541:· 3527:) 3521:EJ 3482:· 3434:) 3428:EJ 3371:· 3331:) 3325:EJ 3296:) 3236:) 3230:EJ 3200:) 3166:) 3160:EJ 3154:+ 3146:= 3129:, 3117:, 3113:, 3091:). 3087:% 3079:/ 3075:·( 3071:= 3059:% 3051:% 3043:/ 3019:− 3007:− 2999:≤ 2970:) 2919:− 2893:ω 2879:or 2864:ω 2844:ω 2818:ω 2762:· 2750:) 2724:ω 2710:) 2692:) 2666:ω 2662:ω 2631:ε 2608:ω 2604:ω 2577:ω 2546:ω 2539:α 2524:) 2500:ω 2496:ω 2491:ω 2482:, 2468:ω 2464:ω 2444:) 2399:λ 2395:ℵ 2391:≠ 2358:ℵ 2354:≠ 2316:) 2290:κ 2286:ℵ 2225:ℵ 2016:) 1977:… 1968:ε 1960:ε 1952:ε 1913:) 1875:if 1860:· 1844:) 1740:ℵ 1725:) 1687:) 1616:ε 1549:· 1522:∈ 1516:∣ 1490:− 1458:) 1448:is 1434:1. 1422:ε 1407:ε 1402:ω 1387:) 1367:1. 1355:ε 1340:ε 1335:ω 1303:ε 1284:ε 1256:ε 1241:ε 1236:ω 1220:) 1212:. 1203:, 1195:+ 1186:no 1174:) 1116:) 1108:. 1096:α 1073:) 1051:) 1025:α 1002:. 896:· 857:) 775:. 771:to 652:) 625:) 592:) 570:) 547:) 513:) 464:ϵ 459:ω 446:ϵ 419:) 411:-- 396:-- 393:}} 387:{{ 344:) 316:= 304:= 292:≤ 235:) 193:ω 175:ω 3853:( 3840:· 3835:( 3821:( 3767:( 3711:( 3664:( 3635:( 3573:( 3545:) 3537:( 3523:( 3486:) 3478:( 3430:( 3424:1 3420:1 3416:1 3412:1 3375:) 3367:( 3363:1 3327:( 3292:( 3232:( 3196:( 3162:( 3156:δ 3152:γ 3150:· 3148:α 3144:β 3140:β 3136:α 3131:δ 3127:α 3123:γ 3119:δ 3115:γ 3111:β 3107:α 3101:β 3096:α 3089:α 3085:β 3081:α 3077:β 3073:α 3069:β 3065:α 3061:α 3057:β 3053:α 3049:β 3045:α 3041:β 3037:β 3032:α 3027:. 3025:β 3021:α 3017:β 3013:α 3009:α 3005:β 3001:β 2997:α 2966:( 2950:n 2930:n 2926:/ 2922:1 2899:1 2896:+ 2824:1 2821:+ 2766:) 2758:( 2746:( 2730:1 2727:+ 2706:( 2688:( 2635:0 2581:2 2550:1 2520:( 2440:( 2386:c 2362:0 2349:c 2337:1 2312:( 2282:= 2277:c 2253:c 2229:1 2221:= 2216:c 2192:c 2169:R 2146:R 2123:c 2096:c 2072:c 2049:R 2031:1 2012:( 1973:0 1964:0 1956:0 1909:( 1864:) 1856:( 1840:( 1821:c 1791:R 1768:c 1744:1 1721:( 1715:1 1695:1 1683:( 1666:R 1643:R 1620:0 1553:) 1545:( 1530:} 1526:N 1519:n 1513:) 1510:2 1507:+ 1504:n 1501:( 1497:/ 1493:1 1487:1 1484:{ 1454:( 1431:+ 1426:0 1411:0 1383:( 1364:+ 1359:0 1344:0 1312:, 1307:0 1296:1 1293:+ 1288:0 1260:0 1252:= 1245:0 1216:( 1205:k 1201:n 1197:k 1193:n 1170:( 1144:N 1133:f 1112:( 1069:( 1047:( 1037:? 900:) 892:( 853:( 648:( 642:n 638:n 621:( 588:( 566:( 543:( 509:( 468:0 455:= 450:0 435:0 415:( 340:( 334:x 332:− 330:x 326:x 322:y 318:z 314:y 310:z 308:− 306:x 302:y 300:− 298:x 294:y 290:x 286:x 282:y 278:y 276:− 274:x 261:r 248:r 228:/ 226:( 214:r 197:r 190:+ 187:. 184:. 181:. 178:+ 50:.

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