Talk:Cardinal number

message: 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)
thread_title:  cut paragraph re axiom of choice 

message: 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)
thread_title: Redundant article

message: :(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)
thread_title:  "Makes no sense to me" 

message: :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)
replies: text: 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)
thread_title: One more question of cardinals in ZF

message: 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)
thread_title:  Counting and cardinality 

message: 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==
:*Jain, L. C. (1982). Exact Sciences from Jaina Sources.
:*
:*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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
replies: text: I agree with your points. -lethe talk [ +] 11:31, 10 April 2006 (UTC)

text: 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)
thread_title:  Jaina mathematics 

message: 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 \alpha (assuming the Axiom of Choice, of course). --Zundark 14:30, 3 November 2006 (UTC)
replies: text: 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 2^{\aleph_0} could be equal to \aleph_1, \aleph_{42}, or even \aleph_\omega.  However, the axiom of choice requires it be SOME \aleph_\alpha, where \alpha is an ordinal number.  Eebster the Great (talk) 04:19, 8 May 2008 (UTC)
replies: text: There are still limits on what 2^\omega can be; per König's theorem (set theory), 2^\omega has confinality greater than ω and so cannot equal \aleph_\omega. — Carl (CBM · talk) 11:46, 8 May 2008 (UTC)
replies: text: But 2^{\omega_0} ≠ 2^{\aleph_0}; one is ordinal and the other cardinal.  But I might be wrong; is there an upper limit to \alpha in \aleph_\alpha = 2^{\aleph_0}? Eebster the Great (talk) 21:11, 16 May 2008 (UTC)
replies: text: I tend to follow the usual convention of using ordinals and cardinal interchangeably in exponents. What I was saying is that the cofinality of 2^{\aleph_0} can't be ω. There is no upper limit on what 2^{\aleph_0} can be; it can be any cardinal with an uncountable cofinality. — Carl (CBM · talk) 21:19, 16 May 2008 (UTC)
replies: text: So for example it could be {2^{\aleph_0}}^+ ? :-) --Trovatore (talk) 09:00, 17 May 2008 (UTC)

text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  An idea about cardinality of an infinite set 

message: 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:
*cardinal numbers
*ordinal numbers
*fractional numbers, and
*serial numbers, used in counting.
- Smerdis of Tlön 17:29, 5 November 2007 (UTC)
thread_title:  Linguistics, again 

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

message: 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:
: 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)
replies: text: In article New Foundations, this definition is attributed (without sources!) to Gottlob Frege. Albmont (talk) 17:41, 14 May 2009 (UTC)
replies: text: 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)
replies: text: 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)
replies: text:  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)
replies: text: 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)
thread_title:  Original definition 

message: In an edit summary,  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 
:Z = \{ Y \, \vert \, Y \approx X \} \,
were a set, then for some fixed a∈X let
:W = \{ b \, \vert \, b = Z \lor ( b \in X \land b \neq a ) \} \,.
We get
:W \approx X \,
:W \in Z \,
:Z \in W \,.
Consequently
: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)
replies: text: By the way, if Z∈X already (so that W defined above does not work), then just set W=X. JRSpriggs (talk) 08:10, 23 October 2010 (UTC)

text: 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)
thread_title:  Why the class of sets equinumerous to a given nonempty set is not a set 

message: The article says:
:If the axiom of choice holds and \aleph_0 \leq \kappa and \mu < \kappa \, and {\mu}^{\lambda} = \kappa \,, then \aleph_0 \leq \lambda < \kappa \, and for any cardinal \nu \,, 1 < \nu \leq \kappa \, implies {\nu}^{\lambda} = \kappa \,.
But this is wrong. For example, if \mu=\beth_\omega and \lambda=\aleph_0 and \kappa=(\beth_\omega)^{\aleph_0} and \nu=2, then \nu^\lambda<\kappa. --Zundark (talk) 16:12, 25 January 2011 (UTC)
replies: text: 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)
thread_title:  Incorrect statement in article 

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

message: 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.
Apologize the bad English
--Gonzalcg (talk) 21:28, 23 June 2012 (UTC)
thread_title:  Citation 

message: 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)
replies: text: 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)
replies: text: 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,  \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 (talk • contribs) 19:11, 24 June 2012 (UTC)
replies: text: 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)

text: I corrected History section to avoid making the inaccurate statement 2) above. Weux (talk) 19:47, 29 April 2013 (UTC)
thread_title:  Some problems of the section 

message: The first edition of the section has more mistakes and there is not citation nor sources:
http://en.wikipedia.org/w/index.php?title=Cardinal_number&diff=4362151&oldid=4352334
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. 
Apologize the bad English
--Gonzalcg (talk) 18:43, 24 June 2012 (UTC)
thread_title:  The cause 

message: 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)
replies: text: What stuff did you look up? -Jochen Burghardt (talk) 17:32, 10 June 2014 (UTC)

text: I guessed that bijection was a problematic term and put the idea in simpler terms. --Mark viking (talk) 20:48, 5 September 2014 (UTC)

text: 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)
thread_title:  SIMPLIFY INTRO! 

message: 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 , it is defined by
:σ ∸ μ = inf {κ | μ + κ ≥ σ}.
If μ < σ, this is the unique cardinal number κ such that . If , . 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)
thread_title:  Truncated subtraction