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There is this statement: "Moreover, any cofinal subset of B whose cardinality is equal to the cofinality of B is well-ordered and order isomorphic to its own cardinality." Is this correct? I don't think so... The last part "order isomorphic to its own cardinality." would imply that B is a cardinal?! Anyone? 82.157.131.133 21:21, 30 Mar 2005 (UTC)
Yup, that sentence didn't make sense. I think I have it stated correctly now.-- Luke Gustafson 11:05, 22 December 2005 (UTC)
Countable union of countable sets is countable - this requires at least the Countable Axiom of Choice, as far as I know. Correct?
Correct, I added a mention of that.-- Luke Gustafson 10:36, 22 December 2005 (UTC)
How does the countable union of countable sets help in establishing that cf(card(R)) is uncountable? Without CH we don't know that sets smaller than R are countable - so maybe we can get R with a countable union of uncountable sets which still are smaller than R. -- SirJective ( 84.151.224.202) 23:03, 2 January 2006 (UTC)
Isn't every cardinal well-ordered? In particular, isn't 0 a least element of every cardinal? - lethe talk + 12:49, 31 January 2006 (UTC)
The article says: "one can prove that for a limit ordinal δ ". But I have a problem: ℵδ is a limit cardinal iff δ is a limit ordinal. And I think ℵδ is always greater than δ, while cf(δ) ≤ δ. Therefore, this statement implies that no inaccessible cardinal exists. Somethings not right. Or am I mucking it up? - lethe talk + 08:51, 1 February 2006 (UTC)
You-all are using a character (in "think ℵδ") which just looks like a square-box to me. And I do not think that my browser is defective. Why? If it is supposed to be an , then you should use <math>\aleph</math> to create it. JRSpriggs 12:25, 19 March 2006 (UTC)
Like the user at 151.200.whatever (who is probably my nephew Edward), I could not read the small alephs in Internet Explorer, but now that I am using Firefox (without any change in fonts), they are clear. JRSpriggs 09:02, 25 March 2006 (UTC)
I've reorganized the article a bit. I moved the definition of regular and singular into the lead. I added some content to the "Examples" section (please double check my additions), and moved that section to follow immediately after the lead (I think it helps to have examples as soon as possible). And I titled the remaining content "Properties". Comments? Paul August ☎ 16:51, 21 February 2006 (UTC)
This section was copied with minor changes from the Wikipedia article on ordinal numbers. JRSpriggs 07:48, 18 March 2006 (UTC)
The second sentence, "Moreover, any cofinal subset of B whose cardinality is equal to the cofinality of B is well-ordered, and these sets are all order isomorphic.", in the section "Properties" seems (to me) to be incorrect. Let us take B to be the Integers with the usual order. Their cardinality is aleph-null. And so is the cardinality of the obvious co-final subset which is the natural numbers (which are well-ordered). But one could also choose the cofinal subset to be the integers themselves (after all, B is cofinal in itself, right?), since they also have cardinality aleph-null which is the cofinality. And the integers are not well-ordered, contrary to the sentence. Also the integers are not order-isomorhpic to the natural numbers, again contrary to the sentence. JRSpriggs 12:16, 19 March 2006 (UTC)
about merging cofinality with cofinal (mathematics), I say absolutely. support. - lethe talk + 20:30, 6 July 2006 (UTC)
I have removed the overlap between cofinality and cofinal (mathematics). Now I am not sure about the merge anymore. I think that enough could be said about the properties of cofinal subsets to justify separate articles. -- Tobias Bergemann 20:48, 6 July 2006 (UTC)
Right now there is a certain asymmetry between the content on regular cardinals and regular ordinals: regular ordinal redirects to cofinality while regular cardinal has its own (very short) article. Maybe regular cardinal should be merged into cofinality. — Tobias Bergemann 11:49, 7 July 2006 (UTC)
"This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member." - This is a way to avoid the axiom of choice.
From Axiom of choice:
—Preceding unsigned comment added by Drysh ( talk • contribs)
As a matter of fact, quoting axiom of choice again, "Trichotomy: If two sets are given, then they either have the same cardinality, or one has a smaller cardinality than the other." is one of the theorems which are equivalent to the axiom. In other words, without the axiom there are pairs of cardinal numbers which are incomparable -- neither is smaller than the other nor are they the same. JRSpriggs 03:48, 18 August 2006 (UTC)
This depends on your definition of a cardinal. You can define cardinals such that cardinals are well-ordered independent of choice, but the drawback is that it is then possible to have sets with undefined cardinality. —Preceding unsigned comment added by 24.196.91.135 ( talk) 00:11, 13 July 2010 (UTC)
Are there any rules to compute the cofinality of the sum, product or power of ordinals? For example, what is the cofinality of ? Albmont ( talk) 20:25, 3 April 2009 (UTC)
It is written : "This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member" Isn't it always true by the well ordering of the ordinals that every set of ordinals has a minimum ? Why do one need the axiom of choice for that ? — Preceding unsigned comment added by 79.182.240.117 ( talk) 16:30, 30 June 2012 (UTC)
Does this have excess words>
cf(κ) = the cardinality of the smallest collection of sets of strictly smaller cardinals such that their sum is κ
Why do we need 'sets of' in there?
173.25.54.191 ( talk) 22:03, 13 March 2014 (UTC)
In the section "Cofinality of cardinals", the text says "cf(κ) = the cardinality of the smallest collection of sets of strictly smaller cardinals whose sum is κ". But the equation directly following it defines it as the infimum, i.e. a greatest lower bound rather than the least element from the set. Which is true? If they are always the same due to properties of cardinalities, it is better to use "min" than "inf", since "inf" implies that a limit must be taken whereas "min" just picks the smallest element, which is then assumed to exist. If they are not the same, then I believe the textual definition should be corrected to use the infimum. 212.242.115.68 ( talk) 15:02, 12 November 2014 (UTC)
This article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
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There is this statement: "Moreover, any cofinal subset of B whose cardinality is equal to the cofinality of B is well-ordered and order isomorphic to its own cardinality." Is this correct? I don't think so... The last part "order isomorphic to its own cardinality." would imply that B is a cardinal?! Anyone? 82.157.131.133 21:21, 30 Mar 2005 (UTC)
Yup, that sentence didn't make sense. I think I have it stated correctly now.-- Luke Gustafson 11:05, 22 December 2005 (UTC)
Countable union of countable sets is countable - this requires at least the Countable Axiom of Choice, as far as I know. Correct?
Correct, I added a mention of that.-- Luke Gustafson 10:36, 22 December 2005 (UTC)
How does the countable union of countable sets help in establishing that cf(card(R)) is uncountable? Without CH we don't know that sets smaller than R are countable - so maybe we can get R with a countable union of uncountable sets which still are smaller than R. -- SirJective ( 84.151.224.202) 23:03, 2 January 2006 (UTC)
Isn't every cardinal well-ordered? In particular, isn't 0 a least element of every cardinal? - lethe talk + 12:49, 31 January 2006 (UTC)
The article says: "one can prove that for a limit ordinal δ ". But I have a problem: ℵδ is a limit cardinal iff δ is a limit ordinal. And I think ℵδ is always greater than δ, while cf(δ) ≤ δ. Therefore, this statement implies that no inaccessible cardinal exists. Somethings not right. Or am I mucking it up? - lethe talk + 08:51, 1 February 2006 (UTC)
You-all are using a character (in "think ℵδ") which just looks like a square-box to me. And I do not think that my browser is defective. Why? If it is supposed to be an , then you should use <math>\aleph</math> to create it. JRSpriggs 12:25, 19 March 2006 (UTC)
Like the user at 151.200.whatever (who is probably my nephew Edward), I could not read the small alephs in Internet Explorer, but now that I am using Firefox (without any change in fonts), they are clear. JRSpriggs 09:02, 25 March 2006 (UTC)
I've reorganized the article a bit. I moved the definition of regular and singular into the lead. I added some content to the "Examples" section (please double check my additions), and moved that section to follow immediately after the lead (I think it helps to have examples as soon as possible). And I titled the remaining content "Properties". Comments? Paul August ☎ 16:51, 21 February 2006 (UTC)
This section was copied with minor changes from the Wikipedia article on ordinal numbers. JRSpriggs 07:48, 18 March 2006 (UTC)
The second sentence, "Moreover, any cofinal subset of B whose cardinality is equal to the cofinality of B is well-ordered, and these sets are all order isomorphic.", in the section "Properties" seems (to me) to be incorrect. Let us take B to be the Integers with the usual order. Their cardinality is aleph-null. And so is the cardinality of the obvious co-final subset which is the natural numbers (which are well-ordered). But one could also choose the cofinal subset to be the integers themselves (after all, B is cofinal in itself, right?), since they also have cardinality aleph-null which is the cofinality. And the integers are not well-ordered, contrary to the sentence. Also the integers are not order-isomorhpic to the natural numbers, again contrary to the sentence. JRSpriggs 12:16, 19 March 2006 (UTC)
about merging cofinality with cofinal (mathematics), I say absolutely. support. - lethe talk + 20:30, 6 July 2006 (UTC)
I have removed the overlap between cofinality and cofinal (mathematics). Now I am not sure about the merge anymore. I think that enough could be said about the properties of cofinal subsets to justify separate articles. -- Tobias Bergemann 20:48, 6 July 2006 (UTC)
Right now there is a certain asymmetry between the content on regular cardinals and regular ordinals: regular ordinal redirects to cofinality while regular cardinal has its own (very short) article. Maybe regular cardinal should be merged into cofinality. — Tobias Bergemann 11:49, 7 July 2006 (UTC)
"This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member." - This is a way to avoid the axiom of choice.
From Axiom of choice:
—Preceding unsigned comment added by Drysh ( talk • contribs)
As a matter of fact, quoting axiom of choice again, "Trichotomy: If two sets are given, then they either have the same cardinality, or one has a smaller cardinality than the other." is one of the theorems which are equivalent to the axiom. In other words, without the axiom there are pairs of cardinal numbers which are incomparable -- neither is smaller than the other nor are they the same. JRSpriggs 03:48, 18 August 2006 (UTC)
This depends on your definition of a cardinal. You can define cardinals such that cardinals are well-ordered independent of choice, but the drawback is that it is then possible to have sets with undefined cardinality. —Preceding unsigned comment added by 24.196.91.135 ( talk) 00:11, 13 July 2010 (UTC)
Are there any rules to compute the cofinality of the sum, product or power of ordinals? For example, what is the cofinality of ? Albmont ( talk) 20:25, 3 April 2009 (UTC)
It is written : "This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member" Isn't it always true by the well ordering of the ordinals that every set of ordinals has a minimum ? Why do one need the axiom of choice for that ? — Preceding unsigned comment added by 79.182.240.117 ( talk) 16:30, 30 June 2012 (UTC)
Does this have excess words>
cf(κ) = the cardinality of the smallest collection of sets of strictly smaller cardinals such that their sum is κ
Why do we need 'sets of' in there?
173.25.54.191 ( talk) 22:03, 13 March 2014 (UTC)
In the section "Cofinality of cardinals", the text says "cf(κ) = the cardinality of the smallest collection of sets of strictly smaller cardinals whose sum is κ". But the equation directly following it defines it as the infimum, i.e. a greatest lower bound rather than the least element from the set. Which is true? If they are always the same due to properties of cardinalities, it is better to use "min" than "inf", since "inf" implies that a limit must be taken whereas "min" just picks the smallest element, which is then assumed to exist. If they are not the same, then I believe the textual definition should be corrected to use the infimum. 212.242.115.68 ( talk) 15:02, 12 November 2014 (UTC)