Talk:Regular cardinal

regularity of 2

rv: 2 has cofinality 1 (as does any nonzero finite ordinal) so it is singular

It seems to me that 2 "cannot be expressed as the union (supremum) of a collection of fewer than [2] smaller cardinals." Now, I don't doubt you, but perhaps the article's definition is mistaken? Or am I missing something basic? -Dan 192.75.48.150 20:51, 5 July 2006 (UTC)

I think I see your point: we can write 2 = 1 ∪ {1}. 1 is a smaller ordinal, but {1} is not an ordinal. As for the definition in the article, yes. I think there's something wrong with it. Every finite ordinal n has cofinality 1 since it has a greatest element n – 1, but the supremum of any cofinal set in a finite ordinal is the greatest element, which is strictly smaller. This definition probably works for limit ordinals, and therefore also for infinite cardinals (which are limit ordinals). Maybe someone can confirm this, and we can fix the article. -lethe ^{talk +} 21:17, 5 July 2006 (UTC)

I don't believe there is complete uniformity on this point. You can define the cofinality of α either to be the length of the shortest sequence of ordinals less than α that is unbounded below α (which makes the cofinality of infinite successor ordinals 1), or you can define it to be the cardinality of the smallest partition of α into pieces of cardinality less than that of α (in which case the cofinality of infinite successor ordinals is 2). It almost never makes a difference, so there hasn't been much need to standardize. However this may be related to MJH's point in the old debate over three forms of mathematical induction. --Trovatore 23:12, 5 July 2006 (UTC)

The correct formal definition is in terms of cofinality. The sentence about a supremum is a condition which is equivalent for limit ordinals (i.e. ordinals whose cofinality is greater than 1). JRSpriggs 06:22, 6 July 2006 (UTC)

Yeah, so the definition in the article doesn't work for finite cardinals. What's to be done about this? One solution might be to just exclude finite cardinals from the discussion. -lethe ^{talk +} 06:53, 6 July 2006 (UTC)

I think JRSpriggs' comment misses the point a little bit; I said the ambiguity is in the definition of cofinality itself, not just the definition of "regular". If everyone agreed what the cofinality of a successor ordinal is, there'd be no problem, but there is no such agreement.

I think Lethe's suggestion is the correct one. As a general rule, cofinality is of interest only for limit ordinals (therefore necessarily infinite, unless you want to quibble about zero). Let's just remove all references to the finite case; by doing so we'll be reflecting the literature. --Trovatore 14:55, 6 July 2006 (UTC)

If I understand your second definition of cofinality (using a partition) right, and it looks like I don't, then cf(0) = 0, cf(1) = 1, cf(2) = 2, cf(3) = 2, ... cf(ω) = ω, cf(ω+1) = ω, cf(ω+2) = ω, right? Why would infinite successor ordinals have cf = 2? How can ω+1 be partitioned into 2 pieces of cardinality less than that of ω+1? 192.75.48.150 16:03, 6 July 2006 (UTC)

Hm, you're right, I don't have that quite right. I had a memory there was this alternative definition; I'll have to think about what it might have been. Maybe it was just for cardinals. --Trovatore 16:14, 6 July 2006 (UTC)

ω + 1 = ω ∪ {ω}, so using this partition notion, you have cf(ω) = 2, as with any successor ordinal. -lethe ^{talk +} 16:33, 6 July 2006 (UTC)

It looks like the first piece of your partition is of the same cardinality as that of the whole. But I don't want to prolong something which sort of sounds like it might not matter. Is regularity ever used in a context where this makes a difference? (Is regularity ever used other than to define inaccessibility? If not, we might even want to merge.) 192.75.48.150 17:23, 6 July 2006 (UTC)

There are all kinds of contexts where you worry about whether cardinals are regular, and about the cofinality of ordinals that aren't cardinals. But I think we can all get along very nicely without agreeing on whether 2 is a regular cardinal, or what the cofinality of successor ordinals is. --Trovatore 17:51, 6 July 2006 (UTC)

Okay. Perhaps this article could be motivated better? 192.75.48.150 14:29, 7 July 2006 (UTC)

According to Cofinal (mathematics), "In mathematics, a subset B of a partially ordered set A is cofinal if for every a in A there is b in B such that a ≤ b.". And as it says in Cofinality, "The cofinality of an ordinal α is the smallest ordinal δ which is the order type of a cofinal subset of α." and "An ordinal which is equal to its cofinality is called regular and it is always the initial ordinal of a cardinal.". I consider these to be the correct definitions. As far as 2 = {0,1} is concerned, it has two cofinal subsets, namely {0,1} and {1}. The order type of {0,1} is 2; and the order type of {1} is 1 ({1} is order isomorphic to {0} = 1) which is smaller than 2. So the cofinality of 2 is 1 which is not 2. Thus 2 is singular. JRSpriggs 06:40, 7 July 2006 (UTC)

Cardinals as partially ordered sets

I am struggling to find a succinct, correct wording for the definition of the cofinality of a cardinal, as the cardinal has to be identified with a partially ordered set before on can speak of its cofinal subsets. Reverse von Neumann cardinal assignment, so to speak. How is this usually done without referring to the Axiom of choice? Are there any unobvious technicalities? — Tobias Bergemann 07:16, 7 July 2006 (UTC)

You don't need ordinals to assert that a cardinal is a union of smaller cardinals. This is the definition that's currently in the article, so what's the problem? -lethe ^{talk +} 07:39, 7 July 2006 (UTC)

The definition that is currently in the article is strictly in terms of cofinality:

In mathematics, an infinite cardinal number κ is called regular if it is equal to its own cofinality.

I think I would prefer to give the union property as the definition and to use cofinality only for the definition of regular ordinals.

In mathematics, an infinite cardinal number κ is called regular if it cannot be expressed as the union (supremum) of a collection of fewer than κ smaller cardinals.

Or maybe even

In mathematics, an infinite cardinal number κ is called regular if it cannot be expressed as the cardinality of a union (supremum) of a collection of fewer than κ sets with cardinality smaller than κ.

Generally I think the article should be clearer about the differences between regular cardinals and regular ordinals. — Tobias Bergemann 08:07, 7 July 2006 (UTC)

So you want to promote the third sentence of this article to the first sentence. I've no objection. The further modification in your third proposal looks good as well. Actually, using cardinal arithmetic, we can be more economical. The cardinality of a union is, by definition, the sum of the cardinals. So we could say:

In mathematics, an infinite cardinal number κ is called regular if it cannot be expressed as the sum (supremum) of a collection of fewer than κ sets with cardinality smaller than κ.

Since we're getting a bit long-winded, which probably means harder to understand. -lethe ^{talk +} 08:18, 7 July 2006 (UTC)

The sum of an arbitrary collection of cardinals doesn't work without some choice principle (possibly full AC, my understanding is that this is an open problem). Not to be too much of a constructivist nuisance, but we do have a non-AC result in this article. On the other hand, I'm not too sure the result is correct. I might suggest dropping the non-AC qualifier, and carrying on without worrying about well-ordering cardinals. 192.75.48.150 14:29, 7 July 2006 (UTC)

Tobias wants a concept of cofinality for cardinals which have no given ordering and cannot be well-ordered due to a lack of the axiom of choice. I do not see how this can be done. As User:Arthur Rubin pointed out in connection with König's theorem (set theory), we cannot choose a set to represent each of an infinite collection of cardinals without using the axiom of choice. JRSpriggs 08:44, 7 July 2006 (UTC)

Thank you for clarifying this point. When I first encountered this page I did not understand how the concept of cofinality applied to cardinals at all. It still is not obvious to me that the cofinality of a cardinal is well-defined independently of the chosen ordering for the cardinal. — Tobias Bergemann 08:57, 7 July 2006 (UTC)

Would it make sense to have regular ordinal defined in terms of cofinality (which should be defined in terms of ordinal), regular cardinal defined in terms of cardinal sum, and then a statement that a ordinal is regular iff it is the initial ordinal of a cardinal? (Regular ordinal would redirect here.) 192.75.48.150 14:29, 7 July 2006 (UTC)

To Tobias: I think that "regular" is essentially a property of ordinals and it applies to cardinals only to the extent that they are identified with their initial ordinals. So we should simply not apply the concepts of "regular" and "singular" to non-well-orderable cardinals. To 192.75.48.150: A statement that an "ordinal is regular iff it is the initial ordinal of a cardinal" would be false since many initial ordinals are not regular. For example, 2, 3, 4, ω_ω, and ω_ω1. JRSpriggs 01:42, 8 July 2006 (UTC)

Of course, my bad. Make that: ...initial ordinal of a regular cardinal. -Dan 70.81.251.195 22:06, 8 July 2006 (UTC)

As pointed out in one of the examples at Cofinality, the real numbers have cofinality aleph-null when considered with the usual ordering; and they have an uncountable cofinality when one uses a well-ordering of them. So a set cannot have a cofinality independent of the chosen ordering. So "regular" is meaningless for non-well-ordered sets or cardinals. JRSpriggs 01:55, 10 July 2006 (UTC)

Could you add an appropriate explanation to the article? I really think this point should be made clearer. For now I am going to remove the reference to cofinality from the article. (Feel free to revert if you disagreee.) — Tobias Bergemann 08:16, 10 July 2006 (UTC)

Without AC

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)

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)

I know the less about cardinal numbers than any of you. So naturally, it was me who re-edited the article... I might have all sorts of stuff wrong. If it's totally out to lunch, revert, that's fine. It still suffers from being seriously undermotivated. Other than $\aleph _{\omega }$ motivating Fraenkel's axiom of replacement (and I don't recall that he referred to regularity), and defining inaccessible cardinals, who cares? I get the feeling that some people here could say a lot more, and it sort of sounds like the interesting stuff might happen without AC, where the regularity or singularity of some alephs are "free". But I don't know, that is out of my league. Maybe someone who knows could just briefly sketch something out? 192.75.48.150 20:46, 10 July 2006 (UTC)

Oh, I also redirected regular/singular ordinal here. -Dan 192.75.48.150 20:56, 10 July 2006 (UTC)

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)

first fixed point of aleph

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)

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)

Aren't you Dan? -lethe ^{talk +} 05:27, 12 July 2006 (UTC)

Saw right through my disguise, eh. Damn. -Dan 192.75.48.150 17:31, 12 July 2006 (UTC)

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)

Every infinite cardinal which is neither aleph-null nor a successor cardinal nor a weak inaccessible is singular, if the axiom of choice holds. But perhaps that begs the question, since that relies on the definition of a weak inaccessible as an uncountable regular limit cardinal. JRSpriggs 06:52, 12 July 2006 (UTC)

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)

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)

Well, by transfinite induction, it is true. Sort of. Grrr. Okay, is this better? 192.75.48.150 17:31, 12 July 2006 (UTC)

A little better, but not satisfactory. I have revised the section. How is it now?-lethe ^{talk +} 17:42, 12 July 2006 (UTC)

getting uncountables from countables

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)

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)

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)

Regular ordinal

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)

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)

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)

I just feel that cofinality does a better job of explaining this than regular cardinal does. And I feel that the notion involving a sum of cardinals is derivative, since it is connected to regularity by using the equipollence of A and A×A (= axiom of choice). JRSpriggs 06:57, 12 July 2006 (UTC)

The comment about whether you can prove an aleph regular

The last sentence of the current article states

Indeed, the existence of any regular aleph other than $\aleph _{0}$ is not provable without the axiom of choice (a result proved by Moti Gitik).

This is surely not correct as stated, is it? It would imply something similar to "The existence of an aleph other than $\aleph _{0}$, implies the axiom of choice".... Hurkyl (talk) 05:41, 1 September 2011 (UTC)

I don't see how it would imply that, no. The piece you may be missing (which also seems to be missing from the article) is that the proof that successor cardinals are regular uses AC. In fact, it follows from the axiom of determinacy (inconsistent with AC) that $\aleph _{3}$ through $\aleph _{\omega }$ are all singular, with the $\aleph _{n+2}$, for n a natural number, always having cofinality $\aleph _{2}$.

Models where $\aleph _{1}$ is singular are less natural than models of AD, but they do exist, I think, though I don't know how to construct one. If $\aleph _{1}$ is singular then even countable choice fails. --Trovatore (talk) 05:58, 1 September 2011 (UTC)

The article does say "If the axiom of choice holds, then every successor cardinal is regular.". One of the equivalents of the axiom of choice is "Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A×A.". If $\aleph _{\alpha +1}$ were singular, then it would be the sum of $\aleph _{\alpha }$ or fewer sets with $\aleph _{\alpha }$ or fewer elements, and thus $\aleph _{\alpha +1}\leq \aleph _{\alpha }$ which is impossible. JRSpriggs (talk) 07:58, 1 September 2011 (UTC)

It's slightly subtler than the Tarski thing. You don't need AC to show that there's a bijection between $\aleph _{\alpha }$ and $\aleph _{\alpha }\times \aleph _{\alpha }$. You do need it to infer from that that the union of $\aleph _{\alpha }$ sets of cardinality $\aleph _{\alpha }$ has cardinality $\aleph _{\alpha }$. --Trovatore (talk) 08:49, 1 September 2011 (UTC)

As stated, I read it as analogous to a claim like "You can't prove well-orderings exist without choice" (even if I drop choice and posit well-orderings as an axiom, choice is still there!), rather than merely stating the axioms of ZF are not enough to prove it. There exist models where you have regular alephs and you don't have choice, right? I'm going to rephrase if there aren't any objections -- "Furthermore, it is consistent with ZF that every aleph bigger than $\aleph _{0}$ is singular". Hurkyl (talk) 06:24, 3 September 2011 (UTC)

Oh, I see what you're saying, I think. Of course you never need a given axiom to prove something, if you just assume something else instead. You could even assume 0=1 and then prove everything.

I don't think it's extremely unclear as stated to readers familiar with the field, but your version is more precise. Or you could drop the reference to models and just add "in ZF" to the current formulation, before "without the axiom of choice". --Trovatore (talk) 21:15, 3 September 2011 (UTC)

You're probably right. I'm often in the situation (which I find quite frustrating) where I know enough of a field to know some statement smells funny when taken at face value, but not enough to be sure to know if there's some subtlety or convention I'm unaware of. (or in some cases if my intuition is flat-out wrong) Since I'm sure I'm not the only one in that position, I wanted to clear up the article (since I figured it was an easy fix). Hurkyl (talk) 01:00, 20 September 2011 (UTC)

Omega-one countable?

Isn't $\omega _{1}$ defined as the first ordinal that is NOT a successor ordinal or a countable limit of countable ordinals? OneWeirdDude (talk) 01:06, 5 February 2014 (UTC)

Please specify which sentence in the article is the subject of your complaint. JRSpriggs (talk) 08:31, 5 February 2014 (UTC)

"For instance, the union of a countable set of countable sets need not be countable. It is consistent with ZF that $\omega _{1}$ be the limit of a countable sequence of countable ordinals [...]." This is false. If the countable sets are countable in number, then at worst this is like matching up ℕ×ℕ with ℕ, skipping over any duplicates. OneWeirdDude (talk) 19:12, 9 February 2014 (UTC)

It says "consistent with ZF", not "consistent with ZFC". You're right, ω₁ cannot in fact be a limit of a countable sequence of countable ordinals. But you need some fragment of AC to prove it.

Or at least it rings true that you need some fragment of AC. Now that you bring it up, I don't know the proof or where to find a reference. Presumably it would be something similar to the argument that it's consistent with ZF that the reals are a countable union of countable sets, maybe the same model in fact. But I don't know where to find that one either. --Trovatore (talk) 20:18, 9 February 2014 (UTC)

We could ask Talk:Arthur Rubin, who is kind of our local authority on the axiom of choice, for a reference to a model of the proposition that all limit ordinals have cofinality ω (a strong negation of the axiom of choice).

Choosing a particular bijection with ω for each of an ω-sequence of sets requires the axiom of countable choice (or something close to it). Such a choice must be made before the bijections can be combined together into a single bijection of the union of those sets with ω (using the pairing function). JRSpriggs (talk) 07:25, 10 February 2014 (UTC)

Singular ordinals?

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)

Questionable claim about Σ₁-elementary embeddings

In the Wikipedia article, it is mentioned that a limit ordinal κ is a regular cardinal iff the set of α < κ with a Σ₁-elementary embedding j and j(α) = κ is club in κ. However, this condition seems to be much weaker than being regular as I've managed to prove that an uncountable regular cardinal must be a limit of ordinals with this condition (I'll share this proof if someone asks for it). Unfortunately, I cannot find the first source of this claim (as in, I cannot find the article "Bounds on provability in set theories") so I cannot check if this claim might be wrongly copied over.

If anyone knows where I can find "Bounds on provability in set theories", can you please tell me? Thank you in advance! 213.93.13.10 (talk) 14:36, 5 September 2024 (UTC)