Talk:Measurable cardinal

comments

I love your comment on the mistake in previus version. Another way is to use a model in which κ is not measurable but it becomes measurable after a forcing. Kunen has such a model. 157.181.80.93 11:15, 11 June 2006 (UTC)[reply]

I removed the following passage: "A previous version of this entry contained the following (false) claim: "A regular cardinal is measurable if there is a cardinal preserving generic extension of V in which the cardinal is singular." This is an interesting error, so it deserves a couple of comments:

1. There is a forcing notion, due to Prikry and usually called Prikry forcing, that shows that if a regular cardinal is measurable, then there is an extension of V as claimed. In fact, Prikry showed that if a regular cardinal κ admits a Rowbottom filter, then the version of Prikry forcing for this filter preserves all cardinals while making κ singular. However, not only measurable cardinals, but also, for example, real-valued measurable cardinals, admit such a filter. Real-valued measurable cardinals are not measurable.

2. However, if a cardinal is regular in V and singular in a cardinal preserving extension of the universe, then there is an inner model of V with a measurable cardinal." b/c I don't believe it's Kosher for an article to refer to itself as an article -- if this discussion can be framed in a different way, then it could be reinserted. Zero sharp 20:03, 6 August 2006 (UTC)[reply]

Yeah, I think 0# is right here. I also think the point made by the removed text, while interesting, is not one of the first 2,543 things you'd want to tell someone about measurable cardinals. Let's try to keep a little encyclopedic focus. --Trovatore 21:01, 6 August 2006 (UTC)[reply]

Just happened to be looking here, and it occurred to me I should probably clarify -- I was saying Zero sharp was right that it's better for an article not to talk about itself, when it can be avoided. I don't actually know the result mentioned in point 2, and can't comment on the correctness of its statement (though I have no reason to doubt it). --Trovatore (talk) 03:56, 20 December 2007 (UTC)[reply]

It's in the Dodd-Jensen covering lemma.Kope (talk) 07:50, 22 December 2007 (UTC)[reply]

Never the less its something I would find useful for my research. Where can I find a statement of the ammended version of this result? Barnaby dawson 16:58, 12 November 2007 (UTC)[reply]

κ-additivity or <κ-additivity?

This is really not clear. At least, it needs a detailed comment, and a clear definition of what is κ-additivity in measure theory, as it is clear that the measure of any singleton is 0, and true κ-additivity would imply the whole set is of measure 0... --Dfeldmann (talk) 16:03, 27 December 2009 (UTC)[reply]

I think it's standard to say κ-additivity for what you're calling "<κ-additivity". See e.g. Jech. Maybe it shouldn't be standard, but that's not up to us as encyclopedists.

I do agree that the point should be made more explicitly. --Trovatore (talk) 22:33, 27 December 2009 (UTC)[reply]

Is AxCh needed for strong inaccessibility?

The text says that the axiom of choice is used in the proof that a measurable cardinal κ is strongly inaccessible. However, I do not see where it is used. Can anyone show me? JRSpriggs (talk) 08:09, 12 February 2010 (UTC)[reply]

I'd have to think about how it's used, but it's clearly needed. For example AD implies that ω₁ is a measurable cardinal. So you need something that contradicts AD even to establish that a measurable cardinal must be weakly inaccessible. --Trovatore (talk) 06:42, 15 February 2010 (UTC)[reply]

I must have been tired when I wrote that. The assumption AxCh is used implicitly in the form — either 2^λ < κ or κ ≤ 2^λ. JRSpriggs (talk) 08:27, 15 February 2010 (UTC)[reply]

Definition

In the first sentence, the article talks about a measure on the power set of κ. To me, a measure on a set X is a function on the power set of X. A measure on the power set of κ is thus a function on the power set of the power set of κ. Too much power? YohanN7 (talk) 17:35, 10 December 2013 (UTC)[reply]

I'm not sure this is completely standardized in mathematical usage. It's occasionally convenient, though, to be able to talk about a measure on a (countably complete) Boolean algebra, and in that sense, the measure in question is on the Boolean algebra P(κ). I agree that "measure on κ" would also be understood. --Trovatore (talk) 20:24, 10 December 2013 (UTC)[reply]

It makes sense as it is, but there are pitfalls. I scanned around and my interpretation of "measure" is apparently as in outer measure, where the σ-algebra of measure by definition is all of P(κ). Either way, the point to be made is that all subsets are supposed to be measurable. This is not automatic with the outer measure concept, but it seems to be so with the measure concept with Σ=P(κ).

I thought about another formulation, but it seems to either make matters worse or to be too long winded. YohanN7 (talk) 01:13, 11 December 2013 (UTC)[reply]

Philosophy

So, I introduced a philosopher of science into the reference list. Why not? When it comes to large cardinals, mathematicians are little more than technically skilled philosophers themselves, and the referenced paper is good and mathematically "not wrong at all" as I understand it. Besides, I needed a reference for the layman description of a two-valued measure in the lead. I also added Jech, and hope to expand the article with some technical details in a (for a layman) readable way.

Next to come, definition (mathematical) of two-valued measure and probably the Ulam matrix (will be detailed in that article) to support actual proof outlines of statements made here.

Then probably an illustration of the interconnection between two-valued measures and filters (and also ideals).

Violent protests are welcome! YohanN7 (talk) 17:45, 19 August 2015 (UTC)[reply]

I don't see any problem with adding the link to Maddy's work. (It doesn't reflect her current view, but that's OK; it's still a notable and influential one.) I don't really know why you would expect "violent protests"; it seems quite a natural addition to me. --Trovatore (talk) 19:13, 19 August 2015 (UTC)[reply]

"Violent protests are welcome!" should be read as "opinions on my plans are welcome". But I was tbh a little concerned about the Maddy reference, since it comes first of all, and might have been controversial.. YohanN7 (talk) 19:27, 19 August 2015 (UTC)[reply]

Witnessing measure in M

The article states: "Notice that the ultrafilter or measure which witnesses that κ is measurable cannot be in M since the smallest such measurable cardinal would have to have another such below it which is impossible."

It's possible that I am mistaken, but doesn't this statement imply the inconsistency of at least rank-into-rank I2? This statement seems to assume that if the measure still exists in M then M still sees that it will be in the transitive class it makes. 107.77.212.218 (talk) 00:49, 11 January 2017 (UTC)[reply]

Rank-into-rank can't be witnessed by a single measure. You need a collection of extenders. For future reference, please ask such questions at Wikipedia:Reference desk/Mathematics rather than the talk page of the article — those are supposed to be for discussing what should appear in the article. --Trovatore (talk) 01:52, 11 January 2017 (UTC)[reply]

I asked it here because I thought it was possible that the statement was either mistaken or poorly worded. If the statement means to say that there is no elementary embedding of V into a transitive class M such that the measure induced on the critical point by that embedding exists in M, then it seems to follow that the power set of the power set of the critical point cannot be in M for any elementary embedding of V into a transitive class M. Whatever the statement means to say, it could be clearer, since it speaks of "the" measure witnessing that the cardinal is measurable and expects the reader to figure out whether it means any measure or just the one associated with that particular embedding. 2601:645:8301:A565:114A:D7B9:D47E:CFD2 (talk) 02:37, 11 January 2017 (UTC)[reply]

I think you're correct that it's poorly worded. As stated the argument doesn't directly imply anything about any κ except the least measurable.

However it is the case that when j : V-> M is an elementary embedding with critical point κ given by an ultrapower by the measure μ on κ, M cannot contain μ as an element. That's probably what the text is really trying to get at. With some elaboration it may even be correct: If M contains μ, then κ must be measurable in M as witnessed by μ, so you can take the ultrapower again, and so on, and then the pullbacks of all these critical points form an infinite descending chain in V. Or something like that. It's been a while. I agree that the current text as worded does not really convey this argument, even assuming it's correct. --Trovatore (talk) 03:03, 11 January 2017 (UTC)[reply]

I'm not knowledgeable enough in the area to be comfortable editing the article, but the way the lead speaks of any embedding M and then the paragraph containing the statement in question only mentions that M is the ultrapower parenthetically doesn't make it clear that we are placing restrictions on M that aren't present in the equivalence statement. 2601:645:8301:A565:114A:D7B9:D47E:CFD2 (talk) 03:42, 11 January 2017 (UTC)[reply]

"Subsets" of a cardinal number?

The current Wikipedia article on cardinal number, https://en.wikipedia.org/wiki/Cardinal_number, defines cardinality in terms of an equivalence relation that is defined on a collection of sets. By that definition, if the set S "has cardinality K", this indicates that S is a member of some ordered pair P of sets and that P is an member of the set K (since a "relation" like K consists of a set of ordered pairs). So "subsets of S" are not the same collection of sets as "the subsets of K".

However, the current article on "Measurable cardinal" seems to treat "the subsets of K" as if they were the same sets as "the subsets of a set S that has cardinality K". This terminology may be a cultural tradition in discussing measurable sets. If so, it should be mentioned. Tashiro~enwiki (talk) 17:43, 28 March 2017 (UTC)[reply]

Please see my response to a related question here. JRSpriggs (talk) 20:24, 28 March 2017 (UTC)[reply]

Proof of inaccessibility of real-valued measurable cardinals

I added a proof for two reasons:

The proof is in itself notable in my opinion.
The proof demonstrates that the mathematics of large cardinals need not always be totally inaccessible.

YohanN7 (talk) 11:11, 10 July 2017 (UTC)[reply]

Gobbledygoop

The Wikipedia article on Measurable cardinals will be all but gobbledygoop to anyone who knows only basic Set Theory and Real Analysis or General Topology. No beginner can possibly grasp what the Hell a measurable cardinal is either from the standard definition using a two valued K-additive on an ultrafilter etc., or from the still more obscure definition in terms of embeddings into the Von Neuman universe. Since such folks, -- and I would to some extent include myself here -- have no initial way of grasping what such a cardinal might be composed of, or better, what in the world it is supposed to enumerate, and also why it is so very large compared to lesser cardinals, the rest of the article will be worthless gibberish to them. That doesn't mean that it won't remain a useful reference for more advanced Set Theorists, but I have always assumed it to be the purpose of the Wikipedia to explain things clearly and simply enough, and with sufficient illustrations where needed, that anyone with a basic grasp of the larger subject, in this case Set Theory, should be able to understand the more advanced topics discussed in the Wikipedia fairly quickly. This article, however, does not do that. Indeed, here the reader is given no good intuitive way to grasp what in the world a measurable cardinal is, and where it comes from etc., before being spirited away into proofs about its many known qualities. — Preceding unsigned comment added by Boscolovich (talk • contribs) 22:42, 20 September 2019 (UTC)[reply]

(Assuming ZFC is consistent.) One cannot prove the existence of a Measurable cardinal nor provide any examples of one. So talking about it is kind of like saying "suppose there is a magical fire-breathing flying dragon, what can we infer from this?". In other words, the point of defining it is not to point out actual examples, but to give one a way of deducing things about it, if it exists. JRSpriggs (talk) 02:09, 21 September 2019 (UTC)[reply]

I think workers from a fair range of foundational views will at least take the position that, given the current state of knowledge, it is more natural to assume that measurables do exist than that they don't. An "example" of a measurable cardinal could then be "the least measurable cardinal", which is of course not very helpful vis a vis Boscolovich's concerns.

The general complaint that the article is not as useful as it could be, even to readers who have sufficient background to understand the topic, is always worth considering and seeing if we can do better. At a quick look, though, no clear plan for improvement presents itself to my mind. Possibly we could add motivation by adding some history, fleshing out the real-valued measurable section and starting with Lebesgue measure, move to 2-valued measures on cardinals, talk about what the additivity of such a measure can be (if it's uncountable, it must be measurable). Would that help? If done right, maybe it could help a little, but I'm kind of unsure. In any case it's unlikely to appear in the lead. --Trovatore (talk) 00:31, 22 September 2019 (UTC)[reply]

See List of large cardinal properties where it says "Existence of a cardinal number κ of a given type implies the existence of cardinals of most of the types listed above that type, and for most listed cardinal descriptions φ of lesser consistency strength, Vκ satisfies 'there is an unbounded class of cardinals satisfying φ'.". So one is dancing on the edge of contradiction, coming as close as possible without falling in.

You should try to see how each of these definitions of large cardinals seeks to establish the existence of similar large cardinals smaller than it. JRSpriggs (talk) 08:05, 22 September 2019 (UTC)[reply]

JR, I have a decent background in the subject, and I would argue with the claim that the definitions "seek" to show the similar large cardinals; it seems to me more to flow from a natural big picture. But that's neither here nor there; if you want we can talk about it on our personal talk pages. The question at hand is how to make the article more useful to more people (it being understood, of course, that there is only so much we can realistically do). --Trovatore (talk) 21:33, 22 September 2019 (UTC)[reply]

Gobbledygoop. The emperor has no clothes. These guys are smoking weed in the math department of some university or another, there are certain basic facts and logical consequences of accepted axioms they are not willing to accept, and they keep contradicting themselves. Thw fact of the matter is, that Lebesgue measure is not defined for all subsets of the reals, and no amount of expostulating and smoking ultra slim filtered cigarettes can change that. 24.237.158.249 (talk) 13:28, 17 January 2022 (UTC)[reply]

Regarding the "Gobbledygoop" comment above

The above comments section is damning, so lets work this out, and see where things first go wrong. The informal definition in the lede is just fine, and entirely understandable (to naive readers). The formal definition currently reads like so:

Here, κ-additive means: For every λ < κ and every λ-sized set {A_β}_β<λ of pairwise disjoint subsets A_β ⊆ κ, we have

μ(⋃_β<λ A_β) = Σ_β<λ μ(A_β).

The left-hand side looks fine: any naive understanding of set union yields acceptable intuitive results. The right-hand side is, however, confusing. What kind of addition is this? This is a confusion for the naive reader: if there are two sets A_β with μ(A_β)=1 then 1+1=2 so this contradicts the idea that the measure is 0-1-valued. So maybe Σ_β<λ μ(A_β) actually means MAX_β<λ μ(A_β)? Wildly guessing that some form of cardinal arithmetic is intended for the summation, and then clicking into that article, does not resolve the confusion. Perhaps the intent is that one and only one of the A_β has measure 1, and all the others have measure zero? Perhaps the sum is meant to be "logical-OR" (So that 0=F and 1=T is an indicator function indicating when A_β is small or not?) The mental juggling needed to explore this interpretation of this sentence seems pointless. So this is the first "gobbledygoop" issue to be fixed.

Subsidiary to this, it might be useful to state that ZFC is 'equivalent'(?) to first-order logic, and so indicator functions can be cleanly defined in ZFC for all cardinalities. i.e. there is no danger or risk of working with indicator functions (they are just predicates in FOL). This, at least, avoids potential confusion for a different class of naive readers, who might be trying to map concepts of classifying functors (see Yoneda lemma) onto this article. 84.15.187.169 (talk) 14:25, 10 July 2024 (UTC)[reply]

More "Gobbledygoop"

The next point of confusion appears in the "properties" section, which currently reads as follows:

It is trivial to note that if κ admits a non-trivial κ-additive measure, then κ must be regular. (By non-triviality and κ-additivity, any subset of cardinality less than κ must have measure 0, and then by κ-additivity again, this means that the entire set must not be a union of fewer than κ sets of cardinality less than κ.) Finally, if λ < κ, then it can't be the case that κ ≤ 2^λ. If this were the case, we could identify κ with some collection of 0-1 sequences of length λ. For each position in the sequence, either the subset of sequences with 1 in that position or the subset with 0 in that position would have to have measure 1. The intersection of these λ-many measure 1 subsets would thus also have to have measure 1, but it would contain exactly one sequence, which would contradict the non-triviality of the measure. Thus, assuming the Axiom of Choice, we can infer that κ is a strong limit cardinal, which completes the proof of its inaccessibility.

Lets work this out. The first sentence is fine, although perhaps the word "trivial" should be struck. Next, open paren seems to delimit the start of a proof of the first sentence, and the close paren means the poof is concluded. OK, fine. Next it says "Finally". Finally what? Is this part two of the proof encased by the parenthesis? If so, the parenthesis should be moved. Or maybe this is a new paragraph discussing a new topic? Who can say?

Whatever, lets keep going. We read: identify κ with some collection of 0-1 sequences of length λ. Uhhh, is the "collection" of length λ, or are the sequences of length λ? Maybe both? Can I have a collection of five sequences? Why is it possible to "identify" the collection with κ? What kind of "identification" is this?

Without any clear idea of what these sequences are supposed to be, the trouble snowballs. Here: The intersection of these λ-many measure 1 subsets ... would contain exactly one sequence, which would contradict the non-triviality of the measure. Huh? What's the contradiction? So the intersection is one sequence. What's the big deal? Why would this imply that the measure is trivial?

The obvious (counter-)example that a newbie might want to think about is the case where λ=ω countable infinity and κ is the cardinality of the continuum, for which one has κ = 2^λ if CH holds. If CH does not hold, then I guess that the cardinality of the continuum κ is still $\kappa =2^{\omega }=\aleph _{2}$ and, at any rate the "collection" of "sequences" would refer to the Cantor set or possibly some subset of the Cantor set. Each countable-length sequence is a single point in Cantor space (i.e. the fine topology) and there's an uncountable number of points, and so if one tries to plug through this case into the gobbledeygoop paragraph, one gets strung up with no clear idea of how this "proof" results in the desired conclusion that κ must be regular (i.e. that the cardinality of the continuum is not a regular cardinal.) Clean this up, and maybe the rest of the article becomes "accessible". 84.15.187.169 (talk) 15:15, 10 July 2024 (UTC)[reply]