Talk:Cardinal number

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[edit] Older discussion

Summarising:

The original assertion about the inaccessibles was inaccurate and has been amended by AxelBoldt. The page has moved from Cardinal to Cardinal number.

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Someone should explain what the following types of cardinals are: inaccessible, Mahlo, indescribable, ineffable, partition, Ramsey, measurable, strongly compact, supercompact, extendible, huge.

Yeah, I second that. I want a description of indescribable numbers, and quick! The huge ones are easy though. They are just HUGE, man.

The definition of n-hyperinaccessible doesn't seem to be complete; so far, it only works for natural numbers n but I think we want it for all cardinal numbers n. --AxelBoldt 19:15, August 29, 2001 (UTC)

Does anyone have a more formal definition of Mahlo cardinals? 20:10, (unsigned by 62.8.212.xxx December 31, 2001 (UTC))

I think we should start with the everyday definition of cardinal (as opposed to ordinal) numbers. Only then should we progress to the more general (and abtruse) case described here. This is a deeply scary article :-)

A cardinal is 0-hyperinaccessible if it is a weakly inaccesible cardinal... It reminds me of undergraduate days. Pauk 20:37, September 27, 2002 (UTC)

Can I just say this article is beautifully written. Impeccably clear: it brings tears to my eyes to see this elegance and simplicity. From my student days I vaguely recall other kinds: hyper-hyper-Mahlo? Robinson? Gritchka 12:49, June 19, 2003 (UTC)

[edit] Algebra with cardinals?

Hi... I've been using the Wiki as a method of independent study on some areas of interest, such as this. I'm just a first-year calculus student with some curiosity and ambition, so some of the technical points are slightly beyond me. I'm posting this comment because I seem to remember seeing somewhere a list something like:
Aleph-null + Alpeh-null = Alpeh-null
2 * Aleph-null = Aleph-null
2^Aleph-null = Aleph-one (my understanding is that this last one is somewhat murky -- cannot be [dis]proven)
Anyway, I can do longer find this table, and I'm interesting in pursuing what kind of operations are legal on cardinals and what the results are. Can anyone help me out some? My inferences would expand this list to be something like:
Alpeh-a + Aleph-b = Aleph-b, b >= a
Also, I keep seeing things like: "Cantor's diagonal argument shows that 2^| X | > | X | for any set X." I could make a proof that shows that a set's cardinality is greater when I add one element to it; but adding one element wouldn't change the size of an infinite set. At what fundamental level is addition with transfinite cardinals and the power operation different?
Also, if anyone is kind enough to field these questions, I'm curious would what happen if to transfinite cardinals were multiplied.
Thanks, -David 02:29, September 1, 2004 (UTC)

This article does actually cover quite a bit of cardinal arithmetic already. You may also want to look at the PlanetMath article on cardinal arithmetic, which goes into a bit more detail. You are right that 2^{\aleph_0} = \aleph_1 can neither be proved nor disproved in ZFC (see continuum hypothesis). The other equations you list are correct. I'm not sure I really understand your question about Cantor's Theorem. Addition or multiplication of two infinite cardinals is trivial in ZFC, as the result is merely the larger of the two cardinals. Cardinal exponentiation is more complicated. --Zundark 08:14, 1 Sep 2004 (UTC)

[edit] cut paragraph re axiom of choice

Removed this from the formal section as the previous definition does include the case "no axiom of choice". Some of this could still be useful, though:

"Note that without the axiom of choice there are sets which can not be well-ordered, and the informal definition of cardinal number given above does not work. It is still possible to define cardinal numbers (a mapping from sets to sets such that sets with the same cardinality have the same image), but it is slightly more complicated. One can also easily study cardinality without referring to cardinal numbers."

What's wrong with that paragraph? -- Schneelocke (cheeks clone) [[]] 17:42, 27 Nov 2004 (UTC)

[edit] Redundant article

With the information provided in this article, cardinality is redundant. It would make sense to merge one into the other, but I'm not sure which way... Fredrik | talk 23:30, 2 Dec 2004 (UTC)


[edit] "Makes no sense to me"

(every member of the set is a set of numbers of its own (a_0, a_1, ..., a_n),\;\; a_i \in \mathbb{N}, like an extended ordered pair)

This wording, from the history section, makes no sense to me. I would simply delete it, but destroying something just because I don't understand it seems dangerous :)
67.164.12.169 00:16, 14 October 2005 (UTC)

The wording here is saying that each member of the set (eg "apple banana cabbage") is a set of its own ("a" "p" "p" "l" "e"), ("b" "a" "n" "a" "n" "a"), ("c" "a" "b" "b" "a" "g" "e"), etc. The members of the "superset" each contain members of their own "subsets". In a sense, this is words versus letters. Words are units, letters are a smaller unit that are contained within words, and so on. --anon 70.92.174.251 04:21, 17 December 2005 (UTC)

[edit] One more question of cardinals in ZF

If the axiom of choice is not assumed and X does not have a well-ordering, the cardinality of X is defined to be the set of all sets which are equinumerous with X and have the least rank that a set equinumerous with X can have.

Is this indeed a set, not a class? (I'm not familiar with the definition of cardinal numbers without AC - I tried to find it here.) --Kompik 18:40, 29 November 2005 (UTC)

Yes it's a set, because for any ordinal r there is a set of all sets of rank r. This is the point of restricting to sets of least rank ("Scott's trick") - you can reduce any non-empty class to a representative set (but you need the Axiom of Regularity). --Zundark 20:15, 29 November 2005 (UTC)

[edit] Counting and cardinality

At a very basic level, there is an inconsistency in terminology which needs to be addressed for the benefit of the non-mathematical reader of this article.

In an eponymous wikipedia article, counting is defined as a matching process (tallying), while in this article counting numbers are equated to a set's cardinality. This latter includes the empty set (zero) that cannot be tallied. Material objects are required for the process of one-to-one matching.

It seems to me that it would be better to use the Peano definition of natural number, with counting numbers then being the set of all natural numbers that are successors. John Morgan 28 Jan 06

[edit] Jaina mathematics

The following material was added April 8, 2006 by user:Jagged 85, and removed the same day by User:JRSpriggs with the brief edit summary "revert useless "history" about the Jains". Now, if the material is correct, it is hugely relevant to the article — but I find it hard to believe that it is correct, despite the numerous references added by Jagged 85. Someone should check these! Anyway, here is the material & references:

An early concept of infinite cardinal numbers is found in India in the works of Jaina mathematicians from the 4th century BC to the 2nd century CE. They classified all numbers into three groups: enumerable, innumerable and infinite. The highest enumerable number N of the Jains corresponds to the concept of aleph-null \aleph_0, the smallest infinite cardinal number. The Jains defined a system of infinite numbers, of which \aleph_0 is the smallest. In the Jaina work on the theory of sets, two basic types of infinite cardinal numbers are distinguished. On both physical and ontological grounds, a distinction was made between asmkhyata and ananata, between rigidly bounded and loosely bounded infinities.
==References==

To Noe: This stuff is purely an attempt to gain credit for priority for the Jaina. Even if it is true historically (I do not know whether it is), it contains no useful mathematical information. It just clutters up the article with irrelavant stuff. If the authors of this material must put it in Wikipedia, let them put it into a purely historical article, NOT IN AN ARTICLE ON MATHEMATICS. JRSpriggs 11:55, 9 April 2006 (UTC)

To JJSpriggs: Like you, I think this stuff probably doesn't belong in the article, because it most likely is untrue. But you seem to think that even if it's true, it doesn't belong - and I must object to that. An article on a mathematical subject cannot be limited to "useful mathematical information" - e.g., the following isn't useful mathematical information either: "The cardinal numbers were invented by Georg Cantor [...] in 1874–1884".--Niels Ø 17:58, 9 April 2006 (UTC)
If it is true that the Jains had notions of transfinite numbers, then it deserves some mention in the history section, but it depends on what exactly their notion was. I recently reverted the Jainist addition at aleph number, and it's harder to justify its presense there: aleph number is about a specific concept that Cantor invented. Even if the Jains had something, was it really equivalent to Cantor's definition? Anyway, I followed some of those links, and read about infinity being the number of grains of sand in a cylinder the size of the Earth, and about distinguishing between infinity in 1, 2, 3, or more dimensions. Neither of these concepts has anything to do with transfinite numbers, so I remain wholly unconvinced that this stuff belongs anywhere other than an article about ancient indian mathematics. -lethe talk + 18:16, 9 April 2006 (UTC)
To Noe & Lethe: Well, I am not too keen on the historical stuff about Cantor either, but I realize that we must give credit to those who created the mathematics which we are using for three reasons: (1) because it is an ethical standard of our society (otherwise we are plagarists); (2) to provide a way for people to find additional reference materials on the subject which were not explicitly mentioned in the article; and (3) to disambiguate notions with similar names by qualifying them with the author's name. I do not think that any of these reasons apply to the Jaina material because it was not a source used by Western mathematicians in developing the mathematics which we are writing about. If the user who put in this stuff had stated a valid and original theorem or other mathematical fact which had been developed by the Jaina, then that would change this judgement. JRSpriggs 04:25, 10 April 2006 (UTC)
I agree with your suggestion that the Jaina material be put into "an article about ancient indian mathematics". Let them create ONE article called Jaina mathematics. Then they can put a SINGLE pointer to that article in each mathematics article where they think that it would be appropriate. It should be *[[Jaina mathematics]] in the "See also" section. JRSpriggs 06:00, 10 April 2006 (UTC)
I agree with your points. -lethe talk + 11:31, 10 April 2006 (UTC)

I think the important point here is that it is inappropriate to insert mention of vaguely-related Jaina math concepts into very specific Cantor-formulated articles. For example, with transfinite number, clearly the article is about the concept originated by Cantor, and indeed that's what the term is used in modern mathematics to refer to. From the included material on Jaina mathematics, it's clear that their concepts are in fact distinct from Cantor's. Just because one author (in the given sources) has seen fit to call the Jaina number concept "transfinite" does not mean it is the same as Cantor's. It certainly does not justify including material that is only vaguely related into the article.

I agree with the above proposal: in articles where Jaina mathematics are directly relevant, some mention (via a link) would be appropriate, but excessive mentions should be avoided. After all, other cultures have come up with their own mathematics also that has direct bearing on some of these concepts. Why ignore them and give Jaina mathematics all the attention? --Chan-Ho (Talk) 06:37, 11 April 2006 (UTC)

I wouldn't object to these mentions in the article infinity (in addition to obviously any article on ancient indian mathematics). But any idea that they arrived at Cantor's definitions I will view with extreme skepticism. I saw nothing of the sort in the few links I followed. -lethe talk + 06:53, 11 April 2006 (UTC)

[edit] Why not to split article in two (linguistics and mathematics)?

Hi! Excuse my English! I'm a russian speaker. We have completely different terms for these notions in linguistics and mathematics (also as for term "ordinal number"). So, trying to decide what to do with interwiki, I realized, that it is also wrong in English to put both notions in one article. Wikipedia has disambig feature, so why not to use it? It seems to me, that it should be one common disambig article and two separate articles for two spheres, may be with linguistics one redirected to "Names of numbers in English". But I can't edit article myself, because I'm not sure I'm right. I don't know English well, so may be there are some weighty reasons for merging two notions? Dims 23:22, 23 September 2006 (UTC)

Zdorovo, tovarish! :)
I would very much disagree with moving the math part of this article to cardinal number (mathematics), as the main usage of cardinal numbers is in math. Perhaps the linguistics term needs to go to cardinal number (linguistics) while the math article staying where it is, at cardinal number. Anybody willing to create such a a linguistics article? Oleg Alexandrov (talk) 03:09, 24 September 2006 (UTC)
Oleg, this is similar to the dispute at Talk:Ordinal number#Should the article really be called transfinite ordinal numbers over renaming that article. Notice that the ugly paragraph which you moved to the bottom of this article contains the disambiguation information which is usually at the top to allow people who got here by mistake to find their way without reading this whole article or becoming discouraged and giving up. JRSpriggs 03:33, 24 September 2006 (UTC)
Well, the linguistics concept is not much more different from the math one (at least that's what I can see with my untrained eyes). So the question remains the same, anybody willing to write a cardinal number (linguistics) article? (By the way, I doubt that many linguists actually look at the linguistics concept of a cardinal number). Oleg Alexandrov (talk) 03:39, 24 September 2006 (UTC)
I think there's a good deal less distinction between the linguistic and mathematical notions for the "cardinal" concepts than there is for the "ordinal" ones. The notion of an "ordinal number" considerably predates Cantor, and to the best of my knowledge always meant words like "first" and "seventeenth". I don't see that there was any explicable idea there that got generalized into Cantorian ordinals; AFAIK it was pretty much exclusively a linguistic concept, not one with an underlying noumenal referent. Note that the concept captured by Cantorian ordinals is not so much "order in which something occurs" as "length of a sequence"; that isn't something you'd express by st/rd/th-type words, and I don't know that it had a name before Cantor.
On the other hand the earlier notion of "cardinal number" -- that is, just, you know, ordinary numbers, counting how many of something there are, does generalize directly to Cantorian cardinals. --Trovatore 08:07, 24 September 2006 (UTC)
Perhaps adding a little note like "For the linguistic concept, see names of numbers in English." at the top (and deleting the linguistics section of the article altogether, or placing the material in that little section in the linked-to article) would work. -Grick(talk to me!) 09:34, 6 December 2006 (UTC)

[edit] An idea about cardinality of an infinite set

I've read over this site about cardinal number. And I don't know whether the sequence (aleph 0, aleph 1, aleph 2,...) exhausts all the cardinality of any infinite set X. How about your opinions? --Frejer

Yes, every infinite cardinal is of the form \aleph_\alpha for some ordinal α (assuming the Axiom of Choice, of course). --Zundark 14:30, 3 November 2006 (UTC)

[edit] Linguistic usage link

I have replaced the linguistics section with a link to Names of numbers in English at the top of the article. The content of this article is about infinite cardinals in mathematics. However, to those unfamiliar with mathematics, "cardinal number" refers to the linguistic meaning, so the link must be at the top of the article, not hidden at the bottom. Many other Wikipedia articles use the same "this article ..." link.

If you disagree with what I did, or think it violated Wikipedia policy, please tell me on my talk page. Pcu123456789 04:14, 27 January 2007 (UTC)

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