Talk:Uncountable set

message: The term "non-denumerable" is less common than uncountable, but it is used.  I added it because I found a wiki link to the non-existant page "non-denumerable."  (It now redirects here.)  Here are a few pages on the Internet that use the term: 
http://www.google.com/search?hl=en&q=non-denumerable+mathematics
Crunchy Frog 16:58, 24 August 2005 (UTC)
thread_title:  Denumerable 

message: This may sound very obvious, but with regard to the 'lack of referencing' flag at the top of the page, I notice that the 'countable set' page has a good number of references - presumably if someone enthusiastic can look up those books, they might provide suitable citations for this page also.  (Especially as, as I understand it, 'uncountable' strictly just means 'not countable'.)
OK, this has probably already been thought of though!!  Ab0u5061 18:59, 10 January 2007 (UTC)
thread_title:  lack of references 

message: In his 1963/64 paper On the axiom of determinateness (Fund.Math), Jan Mycielski considers implications between various statements associated with AD, among them a weak version of countable choice, and also the statement 
:Every non-denumerable set of reals contains a perfect subset. 
On the other hand, Solovay shows (in his 1970 Annals paper) that the sentence 
:every uncountable set of reals contains a perfect set
is consistent with ZF+DC.   (Emphasis added by me, in both statements).  
So Mycielski does not assume a version of AC, and uses the word "non-denumerable"; Solovay constructs a model of DC, in which it is reasonable
to talk about "uncountable" sets.  (Solovay mentions Mycielski several times in the introduction.) 
I think that these two quotes give weak support to Arthur Rubin's suggestion that the word "uncountable" should be replaced by "non-countable"
if we do not assume AC (or a sufficiently strong fragment of AC).  (I say "weak" because the number two is very small, and both papers
are rather old.)
Aleph4 10:16, 22 January 2007 (UTC)
thread_title: Two quotes from the literature

message: I think that the definition 
:\Omega_1 \! is the least regular ordinal greater than \omega.\!
is not standard notation.   Note that \Omega_1 could be undefined:  There is a 1985 paper  of Moti Gitik (Regular cardinals in models of ZF, Transactions AMS) which proves
that the sentence
: "No  \aleph_\alpha>\aleph_0 is regular"
is consistent. --Aleph4 17:45, 22 January 2007 (UTC)
replies: text: I think the whole section should be deleted. It's mostly just confusing. If someone can show that there's a standard agreement as to whether infinite-but-Dedekind-finite sets are "uncountable", that would be different, but I don't believe there is any such standard agreement, and in its absence there's no point in giving a discussion of what the possible agreements might be. In fact it's worse than pointless; it's borderline OR. --Trovatore 23:20, 26 January 2007 (UTC)
thread_title:  Omega_1? 

message: In mathematics, an uncountable set is an infinite set which is too big to be countable.
To:
In mathematics, an uncountable set is an infinite set which contains too many elements to be countable.
I believe this is slightly more precise, and slightly more understandable. The word "big" may seem intuitive, but it may also lead to misunderstanding and complication. 128.163.129.168 (talk) 22:09, 24 March 2009 (UTC)
replies: text: Seems reasonable.  I changed which to that. --Trovatore (talk) 23:57, 24 March 2009 (UTC)

text: The Dedekind-finite infinite sets of cardinality \kappa \, satisfy \kappa \nleq \aleph_0 \, and \aleph_0 \nleq \kappa \,. I think that this is what you mean by "disorganized" (i.e. lacking structure in its powerset). While such sets might be called "uncountable" because they are not countable sets, many people do not use the word "uncountable" that way. Rather "uncountable" is often understood to mean that it can be counted as far as counting can go but it still has more elements, that is, \kappa \nleq \aleph_0 \, and \aleph_0 \leq \kappa \,, also written as \aleph_0 < \kappa \,. JRSpriggs (talk) 03:22, 4 October 2010 (UTC)

text: I don't think that we should focus too much on the non-choice case, and particularly not in the first sentence of the article. There is a section lower down for that. The first paragraph should develop the intuitions that will be more useful, in general. Those will make use of the axiom of choice. — Carl (CBM · talk) 12:25, 4 October 2010 (UTC)
thread_title:  Change to first line? 

message: Does it make sense to add this article to :Category:Unknown content? Possibly, but I don't know enough about the subject matter. I thought about adding Pi and then Irrational number and that led me here. Any thoughts? --Northernhenge (talk) 20:01, 5 March 2013 (UTC)
replies: text: Definitely do not add this article, or either of the other two you mentioned.  That makes no sense at all as far as I can tell. --Trovatore (talk) 20:15, 5 March 2013 (UTC)
replies: text: Thanks Trovatore. I'll take your advice. --Northernhenge (talk) 18:47, 6 March 2013 (UTC)
thread_title:  Query about possible category 

message: The comments here, and the article seem, to concur that uncountable means "too-big" to count.  An uncountable set is defined as having "too many" elements to count.  This seems incorrect as the notion of "bigness" (or "too many-ness") by definition applies as an absolute to the concept of infinity/infinite sets.  In terms of "bigness"/"too many-ness", every infinite set is too big to count/has too many elements to count.  My understanding of the concept of an uncoutable set is as a set that it is not possible to EVEN TO BEGIN to count.  For instance, there are an infinite number of integers, but the set is countable: 1, 2, 3.... etc ad infinitum.  There are also an infinite number of numbers between zero and 1 but these are uncountable.  We can start with the number zero ..... but then that's it, we cannot get to the next number in order to begin our count.   Would the next number be 0.1, or 0.01 or 0.001 or etc.?   Surely this is the "definition" of uncountable?  The article as is seems indecipherable, continuing this further by introducing the notion of a "degree of uncountability" without explaining this other than by a statement that it is the case.  If the article is not actually incorrect then it is too technical and with an inadequate introduction for a non-technical publication.  LookingGlass (talk) 08:10, 4 January 2014 (UTC)
replies: text: You need to give up your preconceived notions which are getting in the way of understand this. Just accept that, in this context, "countable" means that there is an injection from the set into the natural numbers, i.e. there is a way of listing the elements such that you could count up to any one of them by going thru that same list in the same order. JRSpriggs (talk) 09:25, 4 January 2014 (UTC)

text: Your argument about there being no "next" number after zero applies equally well to the rationals, yet the rationals are countable.  It's true that a countable set can be placed in an order where each element has a "next" element, but this ordering need not have any relation to any "normal" ordering that may be defined on the set.  Mnudelman (talk) 21:08, 5 September 2015 (UTC)
thread_title:  Uncountable and "bigness"