Talk:Uncountable set

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[edit] Denumerable

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)

[edit] lack of references

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)

[edit] Uncountable in non-AC settings

The article claimed

The foregoing assumes the axiom of choice. Without the axiom of choice, there might exist cardinalities incomparable to \aleph_0 (such as the cardinality of a Dedekind-finite infinite set). These are not considered to be uncountable.

I have never seen this claim - I have always thought uncountable was simply the negation of countable. But I am used to working in settings where AC is assumed. Is the quote above correct? CMummert · talk 20:52, 19 January 2007 (UTC)

The point is that uncountable sets are TOO BIG to be countable. Without the axiom of choice there may be sets which are not countable for other reasons than because they are too big. They cannot be counted simply because you are unable to systematically choose the next element to count, rather than because you run out of natural numbers with which to count them. JRSpriggs 08:24, 20 January 2007 (UTC)
I understand about non-well-orderable sets and the failure of AC. But every time I can remember seeing "uncountable" defined, it was defined as the simple negation of countable. This is likely because every time I have seen it defined it was defined in the context of ZFC. If there is a convention somewhere that a cardinal must be comparable with aleph_0 to be uncountable, then we should certainly include it here, but I have never seen that convention in print.
Basically, I'm asking if a reference can be found for the "without AC" section. It's especially important to get articles like this right because they are probably used as "the" definition by people doing searches on Google. CMummert · talk 14:03, 20 January 2007 (UTC)
You have a point, I could not find a reference which defines it. However, is it more natural to say that a cardinal number \kappa\! satisfies \kappa > \aleph_0 or \kappa \nleq \aleph_0? Perhaps we should ask Arthur Rubin (talk · contribs) how it is defined when ~AC. JRSpriggs 10:04, 21 January 2007 (UTC)
Yes, I would be glad to hear Rubin's opinion on the matter. I was going to leave a comment on his talk page but see you already left one.
What you are saying is a perfectly reasonable interpretation of uncountability, and it shows that there is not a unique way to generalize the definition when switching from AC to not-AC. So I am in no hurry to delete the non-AC paragraph, and I wouldn't say we disagree over any of the facts, only (possibly) over their presentation. CMummert · talk 14:07, 21 January 2007 (UTC)
I would probably use it as \kappa > \aleph_0, or possibly even \kappa \ge \aleph_1, but I really don't recall seeing "uncountable" used in contexts where it wasn't applied to a well-orderable set, even without the axiom of choice. In other words, I really can't help you. I think the term that I would use for \kappa \nleq \aleph_0 is non-countable, but that may just be me. (I'm reminded of a model in which \beth_1 covers \aleph_0 but is not equal to \aleph_1). — Arthur Rubin | (talk) 16:45, 21 January 2007 (UTC)
I'd say just cut the section out, unless someone can be found who actually has some experience with models that have these pathologies and knows the literature on them well. We a similar problem at transfinite number with someone who wanted to claim that "transfinite" specifically meant "Dedekind infinite", apparently an attested usage, but not standard in my experience. I don't really think there is any widely agreed convention on these usages. From my point of view that's not surprising; AC is true in the real world, and even most widely-studied models of ¬AC, such as L(R), all satisfy at least countable choice. --Trovatore 23:01, 21 January 2007 (UTC)
I think we should remark early on that the notion of "uncountability" is, in practice, only used when we have (enough of) AC available. (Unlike the notion of "countability", which makes sense already in a (fragment of) ZF.) I have read a few (very few) papers involving Dedekind finite sets, but have never seen "uncountable" mentioned there.
But note that under the definition
κ uncountable ⇔ \aleph_1\le \kappa,
you would need the axiom of choice to show that the reals are uncountable, which I find weird. (And I think Cantor would agree.)
--Aleph4 01:01, 22 January 2007 (UTC)

I added a list of the four possible generalizations of "uncountable" to not-AC and suggested that it would be best to be specific. JRSpriggs 05:18, 22 January 2007 (UTC)

Good advice, I think, but is this really the place to give it? I'm still not convinced the section should be there, given that none of us has really seen it used in this way or can give a reference showing it to be a standard usage. If we just delete the section, problem solved. --Trovatore 05:58, 22 January 2007 (UTC)

[edit] Two quotes from the literature

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)

[edit] Omega_1?

I think that the definition

\Omega_1 \! is the least regular ordinal greater than \omega.\!

is not standard notation. Note that Ω1 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)

To put it more clearly: I think that the fourth clause (about Omega_1) should be deleted. While it may not be obvious which of the first three clauses are the appropriate generalisation of "uncountable, it seems obvious to me that the fourth is not appropriate. It should be included only if it somebody finds a reference. --Aleph4 22:43, 26 January 2007 (UTC)

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)