Talk:Regular cardinal

message: Responding to the remarks above: In L(R) (or any other model of the axiom of determinacy), not all successor cardinals are regular.  For example, ℵ3 is singular, with cofinality ℵ2. In even more pathological models where countable choice fails, I think ℵ1 can be singular, though I'm not as sure about that. This could be an interesting addition, but the article is already written a bit confusingly, so it should be done carefully (I don't really like the way the point about singular limit cardinals is phrased; it strikes me as an excessive attempt at terseness). --Trovatore 15:08, 7 July 2006 (UTC)
replies: text: I see (sort of). Hmm. What if the continuum is the union of a countable collection of countable sets? (And that's possible. Yes?) That makes ℵ1 singular right? And in any case, the article's implication that we can talk about the regularity of an aleph even without AC (cofinality of its initial ordinal equal to itself), and that successor alephs can be singluar, is correct. 192.75.48.150 20:00, 7 July 2006 (UTC)

text: Is this idea of dividing cardinals without AC into aleph numbers and non-aleph numbers used?  I have never heard of it. -lethe talk [ +] 23:19, 10 July 2006 (UTC)
thread_title: Without AC

message: The new additions by anon (Dan?) look good to me.  But he writes "Thus the regularity or singularity of every number is determined up to the first fixed point of the aleph function. (And further, I think! But where?)"
The first fixed point \kappa=\aleph_\kappa of the aleph function is singular; it is the limit of the ω-sequence of cardinals \aleph_0, \aleph_{\aleph_0}, \aleph_{\aleph_{\aleph_0}},... and so has cofinality \aleph_0.  Beyond that, we still have lots of singular and regular cardinals.  For example, the successor cardinal \aleph_{\kappa+1} is regular for the same reason that all successor cardinals are, and the limit cardinal \aleph_{\kappa+\omega} is singular as it has cofinality \aleph_0.  Actually I don't see anything special about this fixed point in terms of knowledge of regularity of cardinals. -lethe talk [ +] 22:33, 10 July 2006 (UTC)
replies: text: Makes sense. Is there a first cardinal which ZFC proves to exist, but does not prove that it is regular or singular? --anon (who's Dan?) 192.75.48.150 14:25, 11 July 2006 (UTC)
replies: text: Aren't you Dan? -lethe talk [ +] 05:27, 12 July 2006 (UTC)
replies: text: Saw right through my disguise, eh. Damn. -Dan 192.75.48.150 17:31, 12 July 2006 (UTC)

text: If I had to guess, I would say that any cardinal that can be proved to exist in ZFC can also be proved to be regular or singular.  Most cardinals in ZFC are constructed with the axiom of replacement which gives you direct information about the cofinality of that cardinal.  But it's just a guess. -lethe talk [ +] 05:34, 12 July 2006 (UTC)

text: Sure, with ZFC, we know that every cardinal is regular or singular, but that doesn't mean that we know which one a particular cardinal is.  I argued above that we can always determine the cofinality, but actually I think we can answer Dan's question in the negative: the cardinality of the continuum exists in ZFC, but ZFC doesn't prove whether it's regular or singular (it could be either).  I don't know whether it is the first such cardinal, but it is certainly one such.  Similar comments would apply to every beth number. -lethe talk [ +] 07:08, 12 July 2006 (UTC)
replies: text: In light of this conversation, I think the section about the axiom of choice in this article needs to be rewritten to get rid of the stuff about fixed points of the aleph function, which seems irrelevant.  In particular, the statement that every cardinal up to the first fixed point has known regularity is not true.  The cardinality of the continuum might be less than this fixed point, but can still be regular or singular. -lethe talk [ +] 07:54, 12 July 2006 (UTC)
replies: text: Well, by transfinite induction, it is true. Sort of. Grrr. Okay, is this better? 192.75.48.150 17:31, 12 July 2006 (UTC)
replies: text:  A little better, but not satisfactory.  I have revised the section.  How is it now?-lethe talk [ +] 17:42, 12 July 2006 (UTC)
thread_title:  first fixed point of aleph 

message: Is it really consistent with ZF that ω1 be the countable union of countable ordinals?  That seems to contradict its very definition. -lethe talk [ +] 22:53, 10 July 2006 (UTC)
replies: text: It could be the limit of a strictly increasing ω-sequence of countable ordinals, IF one cannot choose a bijection with ω for every one of those ordinals simultaniously. JRSpriggs 06:40, 11 July 2006 (UTC)
replies: text: Oh yes, of course.  Assuming AC, the countable union of countables is countable.  Without AC, there may be a countable union of countables which is not countable (and therefore can be ω1).  Thank you. -lethe talk [ +] 06:47, 11 July 2006 (UTC)
thread_title:  getting uncountables from countables 

message: Since a regular ordinal is always an initial ordinal, it seems to me that the concept of regular ordinal and regular cardinal are inseparable. Perhaps regular ordinal should redirect here, or this should be moved over the redirect currently occupying regular ordinal? --Dan 192.75.48.150 14:25, 11 July 2006 (UTC)
replies: text: Or maybe you could just leave it alone? Yes, they are closely related concepts. But we have many different articles on closely related concepts. It is not necessary to merge everything that is related together. Indeed, it would be impossible. JRSpriggs 03:30, 12 July 2006 (UTC)
replies: text: But it's not a question of merging; it's a question of where the redirect should point. Regular ordinals and regular cardinals are precisely the same thing; there's not even an intensional difference, as far as I can see. Besides which, in my experience no one talks about "regular ordinals" (I mean, why would you?). So I think regular ordinal should redirect here, rather than to cofinality. --Trovatore 03:41, 12 July 2006 (UTC)
thread_title:  Regular ordinal 

message: The article never defines what a singular ordinal is.  It defines a regular ordinal as a limit ordinal that satisfies ....  In particular, a successor ordinal is not a regular ordinal because it's not even a limit ordinal.  But it makes no sense to say that such ordinals are singular.  If there is such a thing, it would probably be a limit ordinal that does not satisfy the property ...  So verbiage in the Examples section saying things like \omega+1 etc. is singular is misleading and should be replaced just saying it's a successor ordinal for example.  Hope someone well versed in set theory can comment. PatrickR2 (talk) 06:20, 17 October 2023 (UTC)
thread_title:  Singular ordinals?