# Talk:Cardinal number

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## 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.

---

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)

## 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 ${\displaystyle 2^{\aleph _{0}}=\aleph _{1}}$ 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)
> I could make a proof that shows that a set's cardinality is greater when I add one element to it
What you can't do is make a valid proof of that.
Also, note that most of this stuff assumes at least some choice (Countable Choice, Axiom of Choice).
For example, "For all infinite X, |X cross X| = |X|." is equivalent to the Axiom of Choice.
See http://www.math.niu.edu/~rusin/known-math/99/infin_arith for a proof that it implies AC.
Even saying "If X is infinite then Aleph-null ≤ |X|." uses Countable Choice.

## 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)

## 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)

## "Makes no sense to me"

(every member of the set is a set of numbers of its own ${\displaystyle (a_{0},a_{1},...,a_{n}),\;\;a_{i}\in \mathbb {N} }$, 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)

## 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)

## 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

I'm not as troubled by this as John, but I have a suggestion if others are. During the discussion of the definition of cardinal numbers in terms of bijective functions, you can make the link clear with a small aside like: "If there is a bijective function between two sets, then the elements of those sets can be 'matched' 1-1." - Dennis K. 11 Aug 11 — Preceding unsigned comment added by 71.167.112.122 (talk) 12:14, 11 August 2011 (UTC)

## 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 ${\displaystyle \aleph _{0}}$, the smallest infinite cardinal number. The Jains defined a system of infinite numbers, of which ${\displaystyle \aleph _{0}}$ 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==
• Jain, L. C. (1982). Exact Sciences from Jaina Sources.
• Joseph, George Gheverghese (2000). The crest of the peacock : the non-European roots of mathematics. Princeton University Press. ISBN 0691006598.
• Singh, N. (1987). Jain Theory of Actual Infinity and Transfinite Numbers.
• Jain, L. C. (1973). "Set theory in the Jaina school of mathematics", Indian Journal of History of Science.
• Agrawal, D. P. (2000). Ancient Jaina Mathematics: an Introduction, Infinity Foundation.
• O'Connor, J. J. and E. F. Robertson (1998). "Georg Ferdinand Ludwig Philipp Cantor", MacTutor History of Mathematics archive.

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? --C S (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)

## 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)

## 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 ${\displaystyle \aleph _{\alpha }}$ for some ordinal ${\displaystyle \alpha }$ (assuming the Axiom of Choice, of course). --Zundark 14:30, 3 November 2006 (UTC)
Of course, even assuming the Axiom of Choice (which most everybody is inclined to do), it can be impossible to prove which ordinal number a is. Without the continuum hypothesis, examples like ${\displaystyle 2^{\aleph _{0}}}$ could be equal to ${\displaystyle \aleph _{1}}$, ${\displaystyle \aleph _{42}}$, or even ${\displaystyle \aleph _{\omega }}$. However, the axiom of choice requires it be SOME ${\displaystyle \aleph _{\alpha }}$, where ${\displaystyle \alpha }$ is an ordinal number. Eebster the Great (talk) 04:19, 8 May 2008 (UTC)
There are still limits on what ${\displaystyle 2^{\omega }}$ can be; per König's theorem (set theory), ${\displaystyle 2^{\omega }}$ has confinality greater than ω and so cannot equal ${\displaystyle \aleph _{\omega }}$. — Carl (CBM · talk) 11:46, 8 May 2008 (UTC)
But ${\displaystyle 2^{\omega _{0}}}$${\displaystyle 2^{\aleph _{0}}}$; one is ordinal and the other cardinal. But I might be wrong; is there an upper limit to ${\displaystyle \alpha }$ in ${\displaystyle \aleph _{\alpha }=2^{\aleph _{0}}}$? Eebster the Great (talk) 21:11, 16 May 2008 (UTC)
I tend to follow the usual convention of using ordinals and cardinal interchangeably in exponents. What I was saying is that the cofinality of ${\displaystyle 2^{\aleph _{0}}}$ can't be ω. There is no upper limit on what ${\displaystyle 2^{\aleph _{0}}}$ can be; it can be any cardinal with an uncountable cofinality. — Carl (CBM · talk) 21:19, 16 May 2008 (UTC)
So for example it could be ${\displaystyle {2^{\aleph _{0}}}^{+}}$ ? :-) --Trovatore (talk) 09:00, 17 May 2008 (UTC)

I'm under the impression due to a statement by Cohen that without the AC the continuum can be larger than any aleph. Does anyone think there are that many real numbers? Breath of the Dying (talk) 13:26, 24 September 2009 (UTC)

Even without AC, there is still an aleph that is not less than the continuum (see Hartogs number). What exactly did Cohen say? --Zundark (talk) 14:05, 24 September 2009 (UTC)
Well look Cohen just keeps on saying he disbelieved the continuum hypothesis because the powerset operation is very "rich", so I had that thought. I personally think it's pointless attempting to imagine how "big" these cardinals are, since nowhere in the real world do we see them. They're just a product of axiomatic set theory, i.e. logic.Breath of the Dying (talk) 16:42, 24 September 2009 (UTC)
Easton's theorem indicates that the continuum can be made larger than any given aleph by appropriate use of forcing. However, as Zundark said, the continuum cannot be made larger than all alephs simultaneously because it cannot be greater than its own Hartogs number. JRSpriggs (talk) 10:04, 25 September 2009 (UTC)

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)

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 math not language! (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 find the data we're intending 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 Japanese counter word: Ordinal numbers which for the first time in 44 years made sense to me. (I'm adding this copy from ordinal discussion page) J-puppy (talk) 22:10, 26 March 2009 (UTC)
I don't like the current note. Cardinal numbers always indicate quantity, whether in specialized mathematical usage or ordinary usage. I can't presently think of a better way of putting it, though. Algebraist 22:14, 26 March 2009 (UTC)
Cardinal numbers always indicate quantity? wow and all this time I thought that such a fancy name must have a special meaning. I can't presently think of a better way of putting it either. Hopefully someone cleverer than me will. Thank you for teaching me. =) J-puppy (talk) 22:27, 26 March 2009 (UTC)

## Doubts over the recently added 'Simplified Explanation' section.

This section was added about an hour ago[1]. It does not cover infinite cardinals and is therefore incorrect or incomplete. The introductory paragraph already mentions the natural numbers, hence this new section is redundant. I would suggest removing this new section. Aaron McDaid (talk - contribs) 10:44, 27 May 2007 (UTC)

Maybe you are right, perhaps the simplified section does contain inaccuracies, I don't know. I came to this Wikipedia page to find out what a cardinal number was, andn after reading the entire page became totally confused. I may be thick, but the article is far to complex for me to understand. Whiulst it now makes more sense (after I found out what a cardinal number is elsewhere on the net), it is not useful to a beginner. We need a simple explanation. By all means correct my text if it is wrong, but do not over-complicate it to the point of being un-readable to the mathematically ignorant. Thanks for commenting rather than the more rash instant removal.
Although 11:17, 27 May 2007 (UTC)

Simplified Mathematics
To: JRSpriggs
You have reversed the "simplified Explanation in the article Cardinal Numbers.
I came to this Wikipedia page to find out what a cardinal number was, andn after reading the entire page became totally confused. I may be thick, but the article is far to complex for me to understand. Whilst it now makes more sense (after I found out what a cardinal number is elsewhere on the net), it is not useful to a beginner. We need a simple explanation. By all means correct my text if it is wrong, but do not over-complicate it to the point of being un-readable to the mathematically ignorant. Commenting rather than the more rash instant removal would have been more polite.
I am not a mathematician, but I do like to understand maths. Unfortunately, all too often the mathematical topics in Wikipedia go into so much detail that the novice is left completely confused by massive in-depth examination of every minute detail. For all maths topics we need a simplified explanation, as if you were teaching a child of 7. Then we can have the more complex explanations that only a trained mathematician will understand.
By all means remiove a simple explanation if you are willing to clean up the original over-complicated article to make it understandable. Otherwise just pressing the delete button is lazy and arrogant. How would you like it if somone did that to you? - I guess you know already because it seems to happen all the time to everyone.
I hope you will take the time to make the article more understandable.
Thanks in advance for any help you can give in this area. Although 13:58, 28 May 2007 (UTC)
I think this is the crucial sentence which could be improved:
While for finite sets the size is given by a natural number - the number of elements - cardinal numbers (cardinality) can also classify degrees of infinity.
We need to mention, as this does, that cardinal numbers describe the size of sets - natural numbers (1,2,3) for finite sets and others for infinite sets. And also that not all infinite sets are the same size. Perhaps give an example about the relative sizes of naturals, rationals and reals? I think the fact that infinite sets can be compared for size is the interesting concept and which could be explained better. Any ideas? Aaron McDaid (talk - contribs) 22:37, 28 May 2007 (UTC)
The information you are asking for is already in the lead, if you read it with an open mind. However, the examples and explanations must and do come later.
The added section which I removed was not helpful because it ignores what cardinal numbers are, and instead talks about what they are not. This will only confuse most people. I have been working (as needed) on this article for over a year, so do not treat my attempts to protect it as interference with your supposedly needed, but actually haphazard changes. JRSpriggs 07:46, 29 May 2007 (UTC)
Hi, it wasn't me that made the original big change. I agree that the new paragraph made the article worse not better, but that doesn't mean the current article is perfect. The current article is correct, and perfect for people who already know the subject like myself but just maybe it could be a little clearer in the introduction. For example the phrase 'degrees of infinity' isn't as clear as it could be - it's just talking about the sizes of infinite sets. It could simply be used to say that cardinal numbers measure the size of sets, both finite sets and infinite sets? The word 'cardinality' is introduced without explaining that it's just the technical term for 'size'. Aaron McDaid (talk - contribs) 18:43, 29 May 2007 (UTC)

The cardinal numbers are: ${\displaystyle 0,1,2,3,\cdots n,\cdots ;\aleph _{0},\aleph _{1},\aleph _{2},\cdots \aleph _{\alpha },\cdots }$. That is, they are the natural numbers (finite cardinals) followed by the aleph numbers (infinite cardinals). The aleph numbers are indexed by ordinal numbers. The natural numbers and aleph numbers are subclasses of the ordinal numbers. If the axiom of choice fails, the situation becomes more complicated — there are additional infinite cardinals which are not alephs.

I hope that this satisfies you. JRSpriggs 05:00, 30 May 2007 (UTC)

Thanks for the improvement. Although 21:33, 31 May 2007 (UTC)

## Linguistics, again

I tend to think that some of the linguistic material should be moved back to this page; at minimum, there should be two articles, one for cardinal number (mathematics) and cardinal number (linguistics). The article on Names of numbers in English doesn't cover the field adequately; cardinal numbers appear in languages other than English. Linguists broadly distinguish four different kinds of numbers:

- Smerdis of Tlön 17:29, 5 November 2007 (UTC)

## Move proposal

I just noticed a move proposal at Talk:Cardinal (Catholicism) that may concern editors of this page. People who watch this page might want to comment there. Sam Staton 14:16, 4 December 2007 (UTC)

To clarify, that proposal is about the disambiguation page cardinal, not this page. — Carl (CBM · talk) 14:20, 4 December 2007 (UTC)

## Original definition

I remember that I read somewhere that Cantor's original definition of a "cardinal number" was the set of all sets equipotent with a given set. Something like:

${\displaystyle Card(X)=\{x|\exists f:x\to X,f{\mbox{ is bijective}}\}\,}$

of course now we know that this monster (except when X is the empty set) is not a set, but a proper class, but with this definition it makes sense to call Cantor's paradox a paradox... Does anyone know where is the reference for this? It would be appropriate to include (if sourced) in the article. Albmont (talk) 14:22, 14 May 2009 (UTC)

In article New Foundations, this definition is attributed (without sources!) to Gottlob Frege. Albmont (talk) 17:41, 14 May 2009 (UTC)
This seems to confirm my belief that Cantor did not define ordinals or cardinals to be sets at all: for him they were entirely new objects. The definition you give seems to be (a rather anachronistic version of) something Frege says in the Grundlagen. Algebraist 18:26, 14 May 2009 (UTC)
See Scott's trick. Each proper equivalence class is replaced by its subset consisting of the elements having minimal rank. Thus cardinal numbers can be defined as sets without assuming the axiom of choice. JRSpriggs (talk) 00:01, 15 May 2009 (UTC)
Seen Scott's trick. It sounds like a circular definition to me. Yickes, these math articles in Wikipedia are circular! Maybe they should all be moved to Wikibooks and organized in a hierarchical (sp?) way, with article index α rigorously based on all articles index x for every ordinal number x < α... (Wikipedia is not well-founded). Albmont (talk) 14:20, 15 May 2009 (UTC)
The circularity you see is from confusing an internal definition of ordinals or cardinals with an external one. On the one hand, if one already knows what the ordinals are, one can define the cumulative hierarchy by transfinite iteration. On the other hand, if one starts with a model of ZFC, one can define within it (representatives for) ordinals. But the former construction is an external, meta-level construction, while the latter is an internal, object-level definition. — Carl (CBM · talk) 16:19, 15 May 2009 (UTC)

## Different definitions

Hmm, should this page discuss different definitions of cardinals (Von Neumann, Frege-Russell, Scott-Potter)? -- Schneelocke (talk) 23:35, 27 November 2009 (UTC)

They're all mentioned briefly. A proper discussion seems a good idea, though. Algebraist 23:38, 27 November 2009 (UTC)
There are other articles such as https://wiki.alquds.edu/?query=Measurable_cardinal that speak of the "subsets" of a cardinal number in such a way that "the subsets of the cardinal K" are treated as if they are the same things as the subsets of a set S that "has cardinality K". It would be useful to reconcile this terminology with the definition of cardinality given in the current article. The definition in the current article treats a cardinal as an equivalence relation defined on a collection of sets. From this point of view, a cardinal K is a collection of ordered pairs of sets. So if the set S "has cardinality K", this indicates that S is an member of at least one ordered pair in the collection K. So "the subsets of K" are not the same set as "the subsets of S". Tashiro~enwiki (talk) 17:32, 28 March 2017 (UTC)
A cardinal is not an equivalence relationship; it is an equivalence class — a class of sets which are equinumerous to each other.
Those other articles are using "cardinal number Κ" as a shorthand (like using "crown" for King Henry VIII) for the "initial ordinal of Κ", one member of the equivalence class. In this case, talking about its subsets in this way is appropriate. JRSpriggs (talk) 20:18, 28 March 2017 (UTC)

## Why the class of sets equinumerous to a given nonempty set is not a set

In an edit summary, PAStheLoD (talk · contribs) said "Formal definition: "too large to be a set" is not defined anywhere on Wikipedia, so this should be clarified. Also, there's no argument for why that class is too large.". See Limitation of size. Suppose X is any nonempty set (the empty set does not have this problem since { {} } is a set). If

${\displaystyle Z=\{Y\,\vert \,Y\approx X\}\,}$

were a set, then for some fixed aX let

${\displaystyle W=\{b\,\vert \,b=Z\lor (b\in X\land b\neq a)\}\,.}$

We get

${\displaystyle W\approx X\,}$
${\displaystyle W\in Z\,}$
${\displaystyle Z\in W\,.}$

Consequently

${\displaystyle A=\{W,Z\}\,}$

is a nonempty set containing no element disjoint from A itself, which is a violation of the axiom of foundation. JRSpriggs (talk) 08:04, 23 October 2010 (UTC)

By the way, if ZX already (so that W defined above does not work), then just set W=X. JRSpriggs (talk) 08:10, 23 October 2010 (UTC)

Looking back over this I see that I have not explained why the class Z is too large to be a set, merely that it cannot be well-founded. To see that it is too large, let us define a class function

${\displaystyle F(c)=X\times \{c\}=\{\langle b,c\rangle \,\vert \,b\in X\}\approx X\,}$

whose domain is the universe V and each value of which is a distinct member of the class Z. Thus Z is at least as large as the universe and so cannot be a set. The reason this argument does not apply to the empty set is that {}×{c}={} so the values in that case are all equal instead of being distinct. JRSpriggs (talk) 11:15, 23 October 2010 (UTC)

## Incorrect statement in article

The article says:

If the axiom of choice holds and ${\displaystyle \aleph _{0}\leq \kappa }$ and ${\displaystyle \mu <\kappa \,}$ and ${\displaystyle {\mu }^{\lambda }=\kappa \,,}$ then ${\displaystyle \aleph _{0}\leq \lambda <\kappa \,}$ and for any cardinal ${\displaystyle \nu \,,}$ ${\displaystyle 1<\nu \leq \kappa \,}$ implies ${\displaystyle {\nu }^{\lambda }=\kappa \,.}$

But this is wrong. For example, if ${\displaystyle \mu =\beth _{\omega }}$ and ${\displaystyle \lambda =\aleph _{0}}$ and ${\displaystyle \kappa =(\beth _{\omega })^{\aleph _{0}}}$ and ${\displaystyle \nu =2}$, then ${\displaystyle \nu ^{\lambda }<\kappa }$. --Zundark (talk) 16:12, 25 January 2011 (UTC)

To Zundark: Thank you for pointing that out. It was my mistake. I have now reverted myself. JRSpriggs (talk) 20:03, 25 January 2011 (UTC)

## least rank-definition

However, if we restrict from this class to those equinumerous with X that have the least rank,
then it will work (this is a trick due to Dana Scott: it works because the collection of objects
with any given rank is a set).


I think that this trick is much older. It is mentioned in Suppes, "Axiomatic Set Theory", (1960), but I'm triyng to find a better reference. kismalac 22:02, 14 April 2011 (UTC)

I don't know what that proves. Scott was active in 1960. --Trovatore (talk) 22:20, 14 April 2011 (UTC)
It doesn't prove anything. By "better" I meant "earlier". kismalac 06:27, 15 April 2011 (UTC)

## Remaking of the History section

The section history is very awful and has incorrect affirmations.

I will read the only historical source. The most important historical sources, the work of Cantor, Dauben, etc. is not cited and I have serious about the interpretation of the very only source.

I will work, when I have some time, on the Portuguese wikipedia and re-work the historical section.

The article is very weak and would be remaked in most part.

--Gonzalcg (talk) 23:40, 22 June 2012 (UTC)

For example, the statement:

"Cantor first established cardinality as an instrument to compare finite sets; e.g. the sets {1,2,3} and {2,3,4} are not equal, but have the same cardinality: three."

is false.

--Gonzalcg (talk) 00:20, 23 June 2012 (UTC)

Would prefer a citation for this. — Arthur Rubin (talk) 08:27, 23 June 2012 (UTC)

### Citation

Of course: G. Cantor, 1932. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. E. Zermelo, Ed. Berlin: Springer; reprinted Hildesheim: Olms, 1962; Berlin/Heidelberg/New York: Springer, 1980. http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN237853094&IDDOC=49439

But who actually must charge with the onus of the proof is the people which have written the above affirmation.

Does not matter. In page 125 of the cited paper by Deiser, the analysis of the research of Cantor 1873-74 this work is considered the first. In the first paper by Cantor, he has considered only infinite sets. See p. 115 of the work above, in which Cantor has begun with algebraic reals, i.e. an infinite set.

I don't understand, please explain to me: why you and others do not want that the mistakes were corrected? Why systematically my editions are reverted?

I insist: the history section is awful and it must be totally rewritten. The reader must be warned that the contents is not reliable.

--Gonzalcg (talk) 21:28, 23 June 2012 (UTC)

### Some problems of the section

The section contributes to the following misunderstandings:

1) Cantor was the first to consider the one-to-one correspondence as a way of quantity (because previous authors are not cited).
2) Cantor first has formulated the one-to-one correspondence for finite sets and then has extended it for infinite ones.
3) Cantor was the first to think about how to extend the notion of quantity based on one-to-one correspondences to infinite sets.
4) Cantor has given a suitable notion of cardinal number
5) The current notion of cardinal number is essentially the Cantor concept.

If you think that some of the above (1-5) statements would be true, please talk me.

--Gonzalcg (talk) 17:46, 24 June 2012 (UTC)

I think it's correct to say that (one of the) current concepts of cardinal number is a formalization of the Cantor concept, although that may also be true of some of the other contemporaneous concepts of cardinal number. The history section needs sources, but I do not believe Deiser represents current thinking among mathematical historians. — Arthur Rubin (talk) 18:18, 24 June 2012 (UTC)
Perhaps Deiser does not represents. The problem of the section is bigger that this: what is the Cantorian concept of cardinal number? In both senses: the Cantorian double line definition, ${\displaystyle {\overline {\overline {A}}}}$, and the mathematical use of the notion of cardinal. In other words: what was the main contribution of Cantor for the concept (and not the theory) of cardinal numbers? Dauben, Hallett, Hao Wang, Fraenkel, Bar Hillel, etc. can help us in this task, but the work of Cantor is the primary source. Moreover, at least Galileo's paradox and the work of Bolzano should be cited. — Preceding unsigned comment added by Gonzalcg (talkcontribs) 19:11, 24 June 2012 (UTC)
I certainly think you're correct that the history section is garbled, and in particular that 1, 2, and 3 above are wrong (I'd have to check to see if the text actually says those things, exactly, but it definitely says some wrong things). Numbers 4 and 5 are a little more iffy. On 4, suitable for what, exactly? And on 5, it depends on what you mean by "essentially". I think it is essentially the same concept; the use of initial ordinals, which Cantor did not use, is a convenient way of coding that concept. --Trovatore (talk) 16:19, 27 June 2012 (UTC)
I corrected History section to avoid making the inaccurate statement 2) above. Weux (talk) 19:47, 29 April 2013 (UTC)

### The cause

The first edition of the section has more mistakes and there is not citation nor sources:

Note that the "1)" above is explicitly asserted:

"Cantor invented the one-to-one mapping, which easily showed that two sets had the same cardinality if there was a one-to-one correspondence between the members of the set."

Further corrections were not enough to avoid misunderstandings. In similar cases it is advised to rewrite all the section.

Notice also that the above (false) statement: "Cantor first established cardinality as an instrument to compare finite sets; e.g. the sets {1,2,3} and {2,3,4} are not equal, but have the same cardinality: three." already was in http://en.wikipedia.org/w/index.php?title=Cardinal_number&oldid=4362151

It was not deleted. But my contributions were deleted almost immediately, although they are serious and well founded.

--Gonzalcg (talk) 18:43, 24 June 2012 (UTC)

## SIMPLIFY INTRO!

I had to look stuff up to follow your intro. Go to any length you want in the body of the article, but keep it simple in the intro. This was written by a mathematician, to a mathematician...hardly appropriate for an encyclopedia. Pb8bije6a7b6a3w (talk) 15:22, 10 June 2014 (UTC)

What stuff did you look up? -Jochen Burghardt (talk) 17:32, 10 June 2014 (UTC)
I guessed that bijection was a problematic term and put the idea in simpler terms. --Mark viking (talk) 20:48, 5 September 2014 (UTC)

last chances with my life........i will never came back here in my one whole human one ............but i always be with my world in tack with me to breath with my favorite beautiful views......this are only way to feel my refreshing into my love life in lives....... — Preceding unsigned comment added by 203.198.248.59 (talk) 04:34, 25 October 2014 (UTC)

I wholeheartedly concur. Please DO keep in mind Wikipedia is used by elementary school children as well as those in higher learning. As such, you must NOT begin with collegiate-level mathematical language, as 90% of readers are NOT familiar with it. Instead, begin with a simple definition and a simple example suitable for elementary school children. Then, move the higher-level description to the Formal Definition section where it belongs. DO NOT repeate the Formal Definition section in the intro. Clepsydrae (talk) 16:21, 22 January 2022 (UTC)

## Truncated subtraction

It is possible to define a total subtraction operation on cardinal numbers that extends the truncated subtraction operation on the finite cardinal numbers. Using the same variable names as in the section Cardinal number § Subtraction, it is defined by

σ ∸ μ = inf {κ | μ + κ ≥ σ}.

If μ < σ, this is the unique cardinal number κ such that μ + κ = σ. If μ ≥ σ, σ ∸ μ = 0. This can then be used to define (truncated) multiset subtraction, extending set subtraction. Since OR is a no-no, this cannot be added without a reference to published work. The idea is so obvious that it is unlikely no one has written it down, but I could not find it described.  --Lambiam 13:03, 14 December 2020 (UTC)

## Sourcing for our definition of cardinal arithmetic

In the discussion on Talk:Infinity plus one#Broad-concept article?, I'm worried about the lack of sourcing for our definitions in the Cardinal arithmetic section. I'd be grateful if someone could confirm they are established in the literature. — Charles Stewart (talk) 14:41, 21 January 2022 (UTC)

Added a citation there from the book that happens to be on my shelf here, at least for addition, multiplication and exponentiation.
Since addition and multiplication are trivial, "Cardinal arithmetic" in works by Shelah and other set theorists usually refers to cardinal exponentiation; but introductory sections like the one in Schindler (2014) :I cited there use it to refer to all the operations. When I am home tonight I can check my copy of Kunen's Set Theory, which is more widely used than Schindler's I think. Felix QW (talk) 14:57, 21 January 2022 (UTC)
Thanks very much! I note also that Thomas Forster's list of research avenues has one on cardinal arithmetic and permutation models: "Cardinal arithmetic is that part of set theory for which equipollence is a congruence relation" [2]. That's a nice way of putting it: it's an unconventional source, but maybe it's usable attributed. — Charles Stewart (talk) 15:24, 21 January 2022 (UTC)