Talk:Ultrafilter on a set

Principal at a point

In the "Free or principal" subsection (Subsection 3.2), should "principal at X" be "principal at x" (lowercase)? I'm not sure.... Thatsme314 (talk) 19:55, 27 June 2021 (UTC)

Yes, should be lower-case. Looks like it's been fixed already. 67.198.37.16 (talk) 17:32, 1 May 2023 (UTC)

Is the measure-theory interpretation needed?

This article contains a section called "Interpretation in the measure theory" - is this really needed? It seems to be rather trite, and occupies a fair chunk of space in an already much-too-long article. Can it be removed? In it's entirety? This same topic is also covered in Ultrafilter#Ultrafilter on the power set of a set where it similarly seems to add no value. My (negative) assessment is that these paragraphs are the result of some ah-ha moment some student had, and that's very nice and everything, but it really doesn't belong in an encyclopedia. 67.198.37.16 (talk) 05:52, 1 May 2023 (UTC)

I'm blanking the entire section now. I mean, you could do this "measure-theoretic interpretation" on anything that has an indicator function, because indicator functions are one except where they're zero, and calling the indicated area "large" and the rest "small" strikes me as silly. 67.198.37.16 (talk) 06:01, 1 May 2023 (UTC)

Well, first of all, I'd like to clarify that we are not calling the indicated area "large", we are calling each set in the indicated set (which is itself a set of sets) "large". And for filters, this makes sense (at the very least) because any superset of a "large" set is still "large", which isn't true for an arbitrary indicator function. Bbbbbbbbba (talk) 02:54, 2 May 2023 (UTC)

In my opinion, the value of that section was that it tells the reader (kind of) intuitively what a set $A\in U$ is. The definitions give some properties sets in $U$ must satisfy, but don't give any intuitive picture. About everything about filters make more sense when one understands filters as "collections of large sets". Bbbbbbbbba (talk) 03:11, 2 May 2023 (UTC)

The part that stuck in my craw was that any principal filter at a point x assigns a measure of 1 to that point (and thus, all other points a measure of zero.) But that seemed goofy, as one could have picked one of these other points, and generated a different ultrafilter from that. This vaguely resembles the case of the Dirac delta function, with all of the measure concentrated on a single point. But at least the Dirac delta allows you to do calculus; ultrafilters do not have enough to accomplish that. So I was trying to imagine where such an interpretation could be useful, and could not think of any. 67.198.37.16 (talk) 04:25, 2 May 2023 (UTC)

Of course, principal filters are the "boring" ones. Personally my earliest understanding of ultrafilters came from the proof of the compactness theorem through ultraproducts (and specifically the construction of a non-standard model of arithmetic as a ultrapower of the standard $\mathbb {N}$). Here, loosely speaking, a statement holds in the ultraproduct if and only if it holds in a "large" set of factors. Obviously a principal ultrafilter won't work because then the ultraproduct would just be isometric to one of the factors. Bbbbbbbbba (talk) 07:18, 2 May 2023 (UTC)

The other thing that bugged me is that the conventional definition of measures and sigma-additivity requires the use of Borel sets, which requires both countable unions and countable intersections. This requirement is stronger than what ultrafilters need, which seems to be finite intersection, only. So this kind of mixing up of axioms from different systems was jarring. 67.198.37.16 (talk) 05:16, 2 May 2023 (UTC)

I can agree on this point (although that's why the original paragraph pointed out that a content is more accurate). I think the good part of the original paragraph was the intuition that sets in the filter are "large" sets, not necessarily the detailed math stuff. Bbbbbbbbba (talk) 06:57, 2 May 2023 (UTC)

I agree on this point that the broad intuition of "large" set is what matters here. That seems to be mentioned adequately in the section "Characteristics", item 3: "So an ultrafilter decides for every ... whether ... is "large" (i.e. ) or "small" (i.e. )", which also has a reference where the interested reader can get more information. Removing the previous section on "Interpretation in the measure theory" was the right thing to do in my opinion (and as a reminder, the encyclopedia in not meant to "teach": WP:NOTTEXTBOOK).

The removal also makes the article more easily manageable, as it was overly long. PatrickR2 (talk) 17:26, 27 May 2023 (UTC)

My worry is that the section "Characteristics" is already going into some technical details, so if a reader could not intuitively grasp the concept of a filter or an ultrafilter, they may not be able to get to that section at all. Bbbbbbbbba (talk) 02:42, 28 May 2023 (UTC)

I just realized that Filter (set theory) does mention the characterization "collection of large subsets" upfront, and we are now linking to that page, and interested readers would likely follow the link. So I now agree there should be no problem with leaving the previous section deleted. Bbbbbbbbba (talk) 06:21, 28 May 2023 (UTC)

Canonical example of a free ultrafilter

I thought that a canonical example of a free ultrafilter was the set of all cofinite subsets of the natural numbers. I found it curious that it's not included in this article. It's clearly 'ultra', according to the definitions, and its obviously atomless, but I don't know how to prove that its free, e.g. by proving that the intersection of *all* cofinite sets is the empty set. This because I don't know how to prove that the infinite union of all finite subsets of the natural numbers is the entirety of all natural numbers. Because I guess this proof requires the axiom of countable choice. For every finite set, I can choose the least element in that set, and so for every n there is a finite subset of N where n is the least element (for example, the singleton {n}), and thus, the countable union of all finite subsets is N and thus the intersection of all cofinite sets is the empty set. But I don't understand set theory well enough to claim this is error-free; perhaps the definition of a cofinte set requires something more than ZF? I dunno. At any rate, I figure this "basic example" should appear as an example in this article, yes? (That is, it shows that free ultrafilters can be constructed with ZF+ACC but not ZF+ultrafilter lemma.) 67.198.37.16 (talk) 04:53, 2 May 2023 (UTC)

No, it's not ultra. It contains neither the set of all odd numbers nor the set of all even numbers. Bbbbbbbbba (talk) 06:50, 2 May 2023 (UTC)

Oh, ahh, in this example, the universe of discourse is F union N\F where F is the set of all finite subsets of N. The evens and odds are neither finite nor are they cofinite; they are not a part of the discourse. I guess I should have mentioned that. I assumed this example was "widely known" (I didn't make it up; it dates back to the dawn of time.) 67.198.37.16 (talk) 14:30, 2 May 2023 (UTC)

Oh, so the thing is that as far as I know, when people talk about ultrafilters on sets (instead of ultrafilters on general posets), the universe of discourse is always the power set of some $X$. The definitions in this article are based on that assumption and imply it too, but looking back I guess the introduction is talking about this in a confusing way. Bbbbbbbbba (talk) 18:26, 2 May 2023 (UTC)

Ah, of course. "Set theory" is always about sets, and, necessarily about power sets (due to the axiom of whatever its called). Now, things get weird with Paul Cohen's forcing, and I know that boolean-valued models of ZFC are used to do that. But I have no clue if the above example could be "forced" into ZFC. Anyway, even if it was, that's an advanced topic. OK. This article should probably say that its mostly about ZFC unless it is pretending to cover NF or other systems. But that is way out of my league. 67.198.37.16 (talk) 18:32, 14 May 2023 (UTC)

Well you are probably reading into it too much. Filters on sets are a special case of filters on posets (where the poset is the powerset of a given set ordered by inclusion, i.e. a powerset lattice), and they happen to be useful enough that people sometimes specifically study them under the same name, which can be confusing. Bbbbbbbbba (talk) 04:22, 17 May 2023 (UTC)

OK, whatever. I should erase my last comment. Upshot is I think this is a "standard textbook example" of a free ultrafilter even though I couldn't tell you what textbook it might be in. It should be reviewed in one of the WP articles on ultrafilters; it counter-acts the implication made in this article that "free ultrafilters exist but no one can construct an example of one". If you insist that the universe consists only sets and powersets, then the example won't work, but standard textbooks in general topology never-ever make such assumptions. 67.198.37.16 (talk) 09:23, 18 May 2023 (UTC)

Actually Ultrafilter does mention:

If $X$ is infinite then the Fréchet filter is not an ultrafilter on the power set of $X$ but it is an ultrafilter on the finite–cofinite algebra of $X.$

Still, in this case I'd like you to state explicitly why the finite-cofinite algebra in useful in some practical use case, i.e. why you don't need to consider those sets that are neither finite nor cofinite. You say "[I] insist that the universe consists [of] only sets and powersets", but in reality since we are using set inclusion as the order relation we are already considering sets, and I am only trying to account for every subset of the domain $X$. It seems to me more like that you are insisting that your universe consists of only finite sets and cofinite sets, without adequate justification. Bbbbbbbbba (talk) 10:51, 22 May 2023 (UTC)

Let me provide an example I've already mentioned in another topic: ultraproducts. The intuition is that, in general, a sentence $\varphi$ may be true in some of the factors of the ultraproduct, and false in others. We say $\varphi$ is in the ultraproduct if the set of (indices of) factors where $\varphi$ is true is "large", i.e. if that set is in a given ultrafilter $U$. (The definition given in the ultraproduct page only states this explicitly for sentences of the form $a=b$, but the point is that it can be shown that this works for any sentence.) In particular, the condition that $U$ is ultra guarantees that if the set of factors satisfying $\varphi$ is not "large", then the set of factors satisfying $\neg \varphi$ will be "large". This requires $U$ to be over the power set of the index set $I$, because the set of factors that satisfy $\varphi$ may be any subset of $I$; I cannot just assume that it would be either finite or cofinite. Bbbbbbbbba (talk) 11:13, 22 May 2023 (UTC)

I think we're in general agreement, except for some minor miscommunication. One miscommunication is about power sets and set theory. One of the axioms of set theory (I forget what it's name is) tells you that power sets are also sets. So if the conversation is about set theory, then the universe is necessarily all possible sets, including power sets. Therefore, any claims, statements or theorems must necessarily hold for the entire universe, and not some subclass. The example I gave (of a free filter) only holds for the finite/cofinite subset; in other words, it cannot be a free filter on all of set theory. So, as you correctly pointed out, it doesn't hold when including the set of odd integers.

If this article is an article about theorems, claims and statements that that only hold in "all of set theory", and not on subsets, then you are correct; my free filter example does not belong in this article. In this case, I have only two (minor) complaints: first, it wasn't obvious that this is what this article was about (silly me), and now that it is clear, I'm wondering: what fractions of this article hold true for all set theories, and what fraction of this article holds true only for certain set theories, but not others?

Next let me respond to this:

> I'd like you to state explicitly why the finite-cofinite algebra in useful in some practical use case

Well, I can think of several.

• When studying Boolean algebras, there are questions like "how many different kinds of boolean algebras are there?" and "what are their models?" and the finite-cofinite algebra is an example of a boolean algebra that is unlike others.
• Finite and cofinite sets are referred to as "cylinder sets" in the product topology, so if you are studying the Bernoulli process or other kinds of ergodic processes, you'll promptly encounter this.
• There's an analogous situation with periodic and non-periodic (aka "chaotic") geodesics on compact Riemann surfaces (and/or other Riemannian manifolds). In general, the periodic ones are countable, the chaotic ones are uncountable. In terms of measure theory, the uncountable ones form a measure of one, the periodic ones are of measure zero. There's more: this measure is the "invariant measure" on the manifold (e.g. the Haar measure, if its a group, the "ground state" if its a quantum-mechanical system) and it is always the Frobenius-Perron eigenvalue of the corresponding transfer operator. Now, you can dismiss what I just wrote by saying "that's far-fetched", but if you actually go out and try to think hard about ergodic systems, you will have to, at some point, confront the finite-cofinite algebra.

I'll leave it at that. I won't tell you why I'm thinking about ergodicity, you will find it far-fetched if I told you.

Oh, and a stupid post-script, for entertainment purposes only: If you study spin-glasses, you routinely work with ultrametrics. If you then ask "what the heck is an ultrametric"? you promptly get the answer that its an ultraproduct. In physics, it goes under a different name, it is called "the replica trick". (Phil Anderson covers this in a sequence of articles in "Physics Today", circa 1980's) Why are spin-glasses interesting? Well, the underlying theory is almost isomorphic to the theory of deep-learning neural nets. (Talagrand. Roughly speaking, geometry of extremely high-dimensional spaces is effing weird, and in a certain sense, "explains" the weirdness of ChatGPT. So there.) So, in a certain sense, the diameter of the known universe is very small. 67.198.37.16 (talk) 23:43, 23 May 2023 (UTC)

There is probably still some misunderstanding, or at least you are phrasing your understanding in a very weird way. No one is trying to talk about filters "on all of set theory". The very first sentence of this article mentions the set $X$, which would be a very limited subset of the universe (e.g. the set of all natural numbers). Then in the "Definition" section, the 4th condition (ultraness) talks about $A\subseteq X$, and I'm saying there is no obvious reason why $A$ should not range over all subsets of $X$. It's from this desire to capture all the subsets that the power set $\wp (X)$ naturally appears. So the reason I talk about power sets is not because of the axiom of power set, or even the axiom schema of separation (which asserts that any definable subset of $X$ is indeed a set); instead, the role played by those axioms is that they enable me to say what I want to say within the language of set theory, which is that, for example, that the property of being odd is a thing I want to care about in many contexts. Bbbbbbbbba (talk) 01:30, 25 May 2023 (UTC)

A way of thinking that may help is that in mathematics, generally all theorems, claims and statements should hold in "all of the theory". The way mathematicians talk about a limited subclass of the theory is by applying restrictions to variables, such as $A\subseteq X$ (or equivalently, $A\in \wp (X)$). You are free to replace this restriction with $A\in S$ where $S\in \wp (X)$ is a subalgebra of the power set lattice of $X$; it is simply not what this specific article is talking about.

I guess you could instead consider a limited theory which only include finite and co-finite subsets of $X$, and sets of those sets (which the ultrafilter $U$ would be an instance of). Something like this probably could be best built up as a many-sorted theory like the typed set theory. But that strikes me as a very weird way of doing things. Bbbbbbbbba (talk) 01:58, 25 May 2023 (UTC)

Umm, I don't know how to phrase this politely, but ... are you arguing for the sake of arguing? The finite/cofinite ultrafilter example I gave is a standard textbook example from standard textbooks that you can find on the shelves of standard university math libraries. But, for some reason, you don't like it. After discussion, you don't like it because "power sets". Given the way you talk about things, it seems be be because power sets are needed to model set theory. Have you ever studied model theory? (I have.) So when you repeat back to me, what I just told you:

> A way of thinking that may help is that in mathematics, generally all theorems, claims and statements should hold in "all of the theory".

Well, that seems weird. Why are you repeating back to me, what I just told you, and why are you phrasing it as advice? In fact, there is a very formal way of making that statement; it is given on page 10 or 20 of standard textbooks on model theory. Very roughly, "theorems of a language must hold in all models of that language". The way that you talk about things, it sounds like you are trying to model the language of set theory, where the $\subseteq$ symbol has a specific meaning that follows from the power set axiom. As you yourself point out, there's also a concept of posets and lattices, in which the symbol $\subseteq$ has an entirely different meaning: it only includes those things that are reachable by the partial order. This is as it should be: this is because lattices do not have a powerset axiom as a part of their language, and the theorems that hold on the language of lattices are going to be different from the theorems that hold on the language of set theory. This is all well, and normal and good.

You continue with this:

> I guess you could instead consider a limited theory which only include finite and co-finite subsets of $X$, and sets of those sets (which the ultrafilter $U$ would be an instance of).

To be precise, it is not the theory that is limited, it is the language that is limited. That language is the language of lattices and order theory, which does not include the power set axiom, and therefore, the symbol $\subseteq$ has a different meaning in that language. However, the theorems of that language still hold; and, in particular, the definition of the free ultrafilter still holds.

You conclude by saying:

> Something like this probably could be best built up as a many-sorted theory like the typed set theory. But that strikes me as a very weird way of doing things.

Huh? I'm quite sure that lattices and posets are not built up from "many-sorted (set) theories". Lattices and posets are built up from axioms, which are combined to form a language. The theories are built from the language, and then there are specific models in which the language holds. The standard textbook example of this is group theory, which is defined by three axioms: identity, inverse and product. Specific groups, say permutation groups, are models of the language of group theory, and all theorems of group theory hold for all permutation groups (and for all other models of group theory, i.e. all other groups, too). There isn't any textbook on the planet that suggests that "Something like group theory probably could be best built up as a many-sorted theory like the typed set theory." That's not how its done. Now, maybe you could do it that way, I don't know. But it would be very weird to define group theory in that way. It would be very weird to define order theory in that way. It would be very weird to define the axioms of any branch of mathematics in that way. So why are you suggesting this?

Frankly, I'm sort of embarrassed to have to write all of this. I've concluded that I know a heck of a lot more mathematics than you do. You've placed me in the uncomfortable position of having to listen to you try to explain something to me, that I understand better than you do. Now, if you wish, we can continue to have this conversation, but if so, don't accuse me of "weirdness" when I say perfectly ordinary things. When you don't understand something, phrase it as a question, and not as an incorrect statement which I then have to knock down. 67.198.37.16 (talk) 16:12, 26 May 2023 (UTC)

I concede that you have a point. I was arguing with this paragraph you had written:

> If this article is an article about theorems, claims and statements that that only hold in "all of set theory", and not on subsets, then you are correct; my free filter example does not belong in this article. In this case, I have only two (minor) complaints: first, it wasn't obvious that this is what this article was about (silly me), and now that it is clear, I'm wondering: what fractions of this article hold true for all set theories, and what fraction of this article holds true only for certain set theories, but not others?

That was a very confusing way to put that for me, because to me it felt very clear that, in the same sense that your free filter example holds in the finite-cofinite algebra, everything in this article holds in the power set lattice of $X$, not "all of set theory". The use of "subset" in this context was also a little too informal for me to process properly. The word made me think of subsets of $X$ or some other specific set, even though you were probably talking about subclasses of the set theoretic universe.

However, then I realized that you are not even really talking about subclasses of the set theoretic universe; you are talking about working in a theory (the theory of posets/algebras) that is independent of set theory, even though the symbol $\subseteq$ happens to mean something similar (which could have been avoided by using $\leq$ for the poset in the first place). I admit that, when you put it that way, it was starting to make sense. It was just taking some time for me to adjust my thinking.

I admit that you understand model theory better than I do, but it does not seem that you understand set theory better than I do, which probably was what had caused us to take wildly different viewpoints in the first place.

Regardless, I want to continue this conversation because I think it has been valuable. It has unveiled some confusion you and I have, and the whole point of Wikipedia talk pages is to improve the article to be less confusing (at least that's a big part of it). You said you were wondering "what fractions of this article hold true for all set theories, and what fraction of this article holds true only for certain set theories, but not others", and I think with discussion we can clarify this too, but only if you first formalize your question in a language we could agree on. (In your viewpoint, you are studying $A\leq X$ in a theory of posets, but you would still need to study the set $U\subseteq \wp (X)$ in some sort of set theory, even if informally. Even when we switch to talking about only set theories, we could certainly study the two sorts of objects in two different set theories, such as NF and ZFC. So the phrase "holds true only for certain set theories" needs some clarification.) Bbbbbbbbba (talk) 18:13, 26 May 2023 (UTC)

I guess you want a concrete question instead of statements, so here you go. How would you formulate the sentence "If $A\subseteq X$ then either $A$ or its relative complement $X\setminus A$ is an element of $U$" in the language of lattices? How do you talk about $U$ at all in the language of lattices? Bbbbbbbbba (talk) 20:07, 26 May 2023 (UTC)

Sorry I did not want to read this hugely long conversation between you guys. But there are two article: Ultrafilter (set theory) which covers "ultrafilters on a set $X$", that is, ultrafilters in the poset $\wp (X),$ = power set of the set $X$ ordered by inclusion; and then the article Ultrafilter, which covers ultrafilters in posets in general. The Frechet filter (on the integers for example) that you mention is not an ultrafilter in the power set of the integers. It is an ultrafilter in the poset of finite and cofinite subsets of the integers. Hence it belongs in the second article, not the first one. What am I missing? PatrickR2 (talk) 17:39, 27 May 2023 (UTC)

Well one of the problem is that why ultrafilters on a set are so important that they deserves a separate article covering them, and that's apparently where our viewpoints differ.

• Their viewpoint seems to be that "if the conversation is about set theory, then the universe is necessarily all possible sets, including power sets", and when studying ultrafilters in posets in general one should be using a different theory than a set theory (the theory of lattices for example).
• My viewpoint is that everything fits in the framework of set theory, and there is no need to introduce another theory. It's only that the power set lattice is implied by the specification $A\subseteq X$, whereas a general poset deserves more explicit mention; and the former also happens to be an particularly important special case.

Bbbbbbbbba (talk) 18:19, 27 May 2023 (UTC)

I see. Ultrafilter in a poset is the more general notion that subsumes ultrafilters on a set, that is, ultrafilters in a power set lattice, covered in Ultrafilter (set theory). The latter is a bit of a misnomer then. What do you guys think of renaming it Ultrafilter (on a set) to be more precise and bypass some of this debate?

As for the other option of combining both articles into one, it makes some sense. On the other hand the article about ultrafilters on a set is already very large (thankfully less so than before because of your recent trimming, but still) and has a lot (too much?) of information specific to that case. Combining both articles into one would make it more unmanageable, too daunting to read, etc. I think it probably makes sense to keep them separate due to the amount of information. But the general ultrafilter article could maybe be trimmed a little, as most of the applications seem to deal with the power set lattice case. All this can be debated. PatrickR2 (talk) 22:41, 27 May 2023 (UTC)

This is part of the reason I want to understand more about the applications of ultrafilters in general posets. One feeling I was having is that ultraness in general posets is kinda "cheap", since it seems to me that any filter $F$ on $X$ could be "made ultra" by taking $F\cup \{X\setminus A\mid A\in F\}$ as the poset. Of course it should be more subtle than that since in a "real" application one would fix a poset before studying its ultrafilters, also also filters in sub-posets (sub-lattices?) of $\wp (X)$ may not even be filters in $\wp (X)$. But I just don't have a clear enough intuition on those ideas. Bbbbbbbbba (talk) 00:43, 28 May 2023 (UTC)

Oh, about renaming it to "Ultrafilter (on a set)": From my understanding it's the "rarely" option in WP:NCDAB, but personally I feel that it might be appropriate (if somewhat awkward), since using "(set theory)" as the disambiguating phrase (and stating in the hat note that "This article is about the mathematical concept in set theory.") feels misleading to me too. However, that's also a point of contention between me and 67.198.37.16, since they seem to take the position that "being about set theory" is an appropriate characterization of the difference between this article and Ultrafilter. Bbbbbbbbba (talk) 01:39, 28 May 2023 (UTC)

If we go with the "rarely" guideline of WP:NCDAB, the better name would then be "Ultrafilter on a set" without disambiguating parentheses. PatrickR2 (talk) 04:39, 28 May 2023 (UTC)

Hmm... That's certainly also an option, although I'm not sure whether that counts as a less commonly used term, or a descriptive title. I cannot pinpoint how "Ultrafilter on a set" is different from "English language" but it surely feels different. Bbbbbbbbba (talk) 05:48, 28 May 2023 (UTC)

"Ultrafilter on a set" versus "Ultrafilter (set theory)". PatrickR2 (talk) 05:57, 28 May 2023 (UTC)

I have streamlined the lead of Ultrafilter (set theory) a little bit, removing much of the redundant verbiage to get a more focused introduction. Also mentioned in the lead the alternate definition in terms of complementary subsets. And updated the hatnote to say This article is about specific collections of subsets of a given set. instead of This article is about the mathematical concept in set theory. That should make it clear to the readers that we are talking here about ultrafilters on a set and not a more general concept. And maybe make it more palatable to change the title to "Ultrafilter on a set" later on.

(But we can still debate whether we should combine both articles or not.) PatrickR2 (talk) 05:53, 28 May 2023 (UTC)

Certainly a step in a good direction. I am now contemplating mentioning the fact "it is not possible to construct an explicit example of a free ultrafilter" in the lead, both because this fact is specific to ultrafilters on sets, and also because this fact feels relatively profound within this topic. Bbbbbbbbba (talk) 06:28, 28 May 2023 (UTC)

Yeah, good idea. PatrickR2 (talk) 16:09, 28 May 2023 (UTC)

One thing I've been thinking about is that we need to:

• either establish at some point in this article that we are only talking about (ultra)filters on sets here,
• or, for each occurrence of "(ultra)filter", we need to either explicitly say "on a set" or make sure that the sentence is not wrong even if interpreted as a general statement (at a quick glance, it seems that the article is actually not doing so bad in this regard).

The former solution would lean towards renaming to "Ultrafilter (on a set)" and the latter would lean towards "Ultrafilter on a set". Bbbbbbbbba (talk) 01:51, 31 May 2023 (UTC)

It is already kind of indicated in the hatnote at the top. But I guess we could make that even more explicit in the lead and elsewhere. In any case, I think we should (1) change the title of the article to "Ultrafilter on a set" to make the scope clear from the very top; (2) introduce a redirect "Ultrafilter (on a set)" to be used in other articles if desired; (3) change "Ultrafilter (set theory)" to a redirect to this same article. Also, in the Definitions section the word "ultrafilter" (maybe even "ultrafilter on a set") should be in bold at the place it is defined. That should help draw the reader's attention to the fact we are talking about ultrafilter on a set here.

Does that sound reasonable? And if you agree, would you like to do it, or should I give it a try?

I am also interested in your opinion about the following. In my opinion this article contains a lot of useful information, but also a lot of tangential minutiae that are maybe too technical for this encyclopedia. A lot of these details were added by editor Mgkrupa a while ago, who used to go on editing sprees that were sometimes controversial. I have not found the energy to read all the details in this article. But just glancing at it, have you ever heard of a "grill" or a "filter-grill" for example? I have never seen this in any application of ultrafilters. I know he cites Dolecki and Mynard, which is one of the books he read, but does this need to be here in full detail? PatrickR2 (talk) 05:00, 31 May 2023 (UTC)

Let's try it and see how it turns out. Maybe it would be better for you to do it since you are the one who came up with the plan.

About grill/filter-grill: I've never heard of them either. Maybe I'll try to take a look at Dolecki and Mynard if I could find some time and energy, but for now I think it's fair to remove the section altogether until someone could provide some motivation/applications for the concepts. In general, too much technical details can be troublesome to handle since we may want to preserve some high-level ideas demonstrated, but in the case of this specific section I don't see any high-level ideas shown here. Bbbbbbbbba (talk) 17:36, 31 May 2023 (UTC)

@Mgkrupa: Hi. Can you comment on the section about grills/filter-grills? I have never seen that concept except here. Apart from the book by Dolecki and Mynard, is this something that has wider use somehow? Could you say a few words about the importance of that concept? PatrickR2 (talk) 20:18, 31 May 2023 (UTC)

@Bbbbbbbbba Would you mind adding your opinion to the Requested move section below? I.e. Support, Oppose , etc. as done in many of the other move discussion? (see https://en.wikipedia.org/wiki/Wikipedia:Requested_moves/Current_discussions) This is needed to "build consensus" before the move. PatrickR2 (talk) 22:46, 3 June 2023 (UTC)

Regarding the extra paragraph in the lead, it's ok. The information starting at the sentence "The existence of free ultrafilters ..." is a good summary. For the first two sentences though, no need to go into full technical details in the lead. I would maybe not have mentioned the Frechet filter for example, that's for the rest of the article. But we need to explain the principal vs. free ultrafilter. That should probably be spelled out in more detail for a novice reader, without forcing them to follow a link to the rest of the article. Something like "There are two types of ultrafilter on a set. Principal ultrafilters are those ... Ultrafilters that are not principal are called free. These are the ultrafilters for which there is not element x ... Those are the ultrafilters of most interest in applications to ... ... (and then your stuff about existence) " Or similar stuff. Just a suggestion. PatrickR2 (talk) 05:08, 31 May 2023 (UTC)

The thing is that the concept of "free" is defined for filters (and even general families of sets), where a principal filter is necessarily not free, but the converse is not true: There are filters that are neither principal nor free. The detailed logic goes like this:

• "Free" is defined as "the intersection of all sets in this family is the empty set", or equivalently "for any $x\in X$ there is a set in this family that does not contain $x$".
• For filters, since they are upward closed, this is equivalent to "$X\setminus \{x\}$ is in the filter for all $x$", i.e. the filter includes the Fréchet filter, which is a definition I find both concise and close to the original definition.
• A fixed (non-free) filter is necessarily contained in some upper set $\uparrow \{x\}$. Since an ultrafilter is maximal, it must be $\uparrow \{x\}$, and thus principal at a point. (On the other hand, a fixed filter can have a more complex structure. It may also be principal, but not at a single point.)

So in short, the terms are a little involved. Bbbbbbbbba (talk) 18:13, 31 May 2023 (UTC)

I understand what you are saying. But in the context of ultrafilters, which is what this article is about, it's quite a lot simpler. Either the ultrafilter consists of all subsets of $X$ containing a specific point (the principal ultrafilter case), or not. In the second case, it's called a free ultrafilter. That's all we need for the lead. Additional nuances are covered in the rest of the article, specifically in the Proposition at the end of the Characterizations section. The lead does not need to get too technical. PatrickR2 (talk) 19:37, 31 May 2023 (UTC)

Oh, I guess I just wanted to rephrase the sentence "ultrafilters that are not principal are called free". Maybe along the lines of: "Free ultrafilters are exactly those ultrafilters that are not principal." Bbbbbbbbba (talk) 01:18, 1 June 2023 (UTC)

Or even simpler, without "called" and without "exactly", something like "The ultrafilters that are not principal are the free ultrafilters." (leaving it up to the rest of the article to define/characterize them) PatrickR2 (talk) 22:07, 1 June 2023 (UTC)

Requested move 1 June 2023

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: moved. (closed by non-admin page mover) CLYDE ^{TALK TO ME}/_{STUFF DONE} (please mention me on reply) 00:14, 9 June 2023 (UTC)

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Ultrafilter (set theory) → Ultrafilter on a set – There are two similarly named articles: Ultrafilter (set theory) and Ultrafilter From the titles alone it is not entirely clear what the respective scope of the articles are. The article "Ultrafilter (set theory)" is specifically about ultrafilters on a set, that is, ultrafilters on the power set of an underlying set, ordered by subset inclusion; "Ultrafilter" covers ultrafilters on a general poset. But in a recent discussion in https://en.wikipedia.org/wiki/Talk:Ultrafilter_(set_theory), one of the editors was arguing that "Ultrafilter (set theory)" could refer to many more situations having to do with/(expressible in?) set theory. Another editor and I want to remove any such ambiguity by changing the title to "Ultrafilter on a set". Please see the full discussion on the Talk page. One possibility we also considered was "Ultrafilter (on a set)", but that did not seem better according to the the "rarely" guideline of WP:NCDAB. Instead, we could add that as a redirect if desired. PatrickR2 (talk) 23:00, 1 June 2023 (UTC)

• Support. The disambiguation phrase "(set theory)" does not distinguish clearly between ultrafilters on sets and ultrafilters on posets, so some change is necessary. Bbbbbbbbba (talk) 00:31, 4 June 2023 (UTC)
• Support. Per nom. PatrickR2 (talk) 21:46, 6 June 2023 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Which set theory?

There continues to be confusion about the idea of a "set theory". There are more than one: There's

• Zermelo–Fraenkel set theory with and without choice,
• New Foundations
• Von Neumann–Bernays–Gödel set theory
• Morse–Kelley set theory
• Tarski–Grothendieck set theory
• constructive set theory

and even more ... these differ in the kinds of axioms they are built out of, and differ in what they can say, and in how ultrafilters are defined. The much-too-long conversation above continues to ignore this. 67.198.37.16 (talk) 06:09, 6 June 2023 (UTC)

How does "how ultrafilters are defined" differ between different set theories? It's not like the definition of an ultrafilter contains any non-stratified application of comprehension that would make NF complain, or it talks about proper classes in a way that would necessitate NBG or MK. Bbbbbbbbba (talk) 21:26, 6 June 2023 (UTC)

I'm not expert on set theory, and I might be misremembering something I misunderstood a decade ago, but I can't begin to imagine how an ultrafilter in constructive set theory is the same as in ZFC. The existing article already says this, more or less, something about free ultrafilters not being constructable. This whole conversation started because I happened to know of such a construction *on posets*. Realistically, I'm not going to run to the library to find some chapter in some book from long ago, so this conversation is moot. But overall, none of this is confidence-installing as to it's correctness. I'd slap an "expert attention required" template on this article, except that I know the experts never arrive, even decades later. 67.198.37.16 (talk) 03:48, 7 June 2023 (UTC)

Well, with constructive set theory it's not just a different set theory, but a different logic, which I admit might change things a little more. (I'm not familiar at all with intuitionistic logic; I won't say I'm any "expert" in set theory, but at least I know ZF(C) and NF(U) fairly well.) Then again, for all I know, it could be working largely like in ZF without Choice: It has most of the same definitions and properties, except a free one may not exist. Bbbbbbbbba (talk) 04:09, 7 June 2023 (UTC)

Clarifying the impossibility to "construct an explicit example" of a free ultrafilter

@67.198.37.16: You have a point here, although I'd like to first clearly say what I don't think is correct. With ultrafilters on a set $X$ (which I'll assume $X=\mathbb {N}$ for concreteness), one cannot just reject something like "the set of all odd numbers" as a valid candidate for $A\subseteq X$; no reasonable set theory would reject the existence of the set of all odd numbers, at least not if it accepts that $\mathbb {N}$ is a set in the first place. (I believe this has less to do with the axiom of power set than the axiom schema of separation.)

However, what one could reasonably reject is the existence of non-constructible sets, i.e. one could accept V=L. Paradoxically (at least to me), this proves the axiom of choice, a very non-constructive principle. Specifically, the constructible universe can be well-ordered in a way that can be, in some sense, "explicitly defined" (to compare two constructible sets, one first compares the earliest stage at which they appear in the constructible hierarchy, then compares the smallest Gödel numbers of formulas that can be used to define them (at that stage), and finally recursively compares the parameters used in said formula one by one).

From such a well-ordering, one could extract a free ultrafilter from the proof of the ultrafilter lemma: Basically, starting from the Fréchet filter, one inspects each set $A\subseteq X$ in the well-order, and if $X\setminus A$ is not already in the current filter $F$, one extends the $F$ with $A$ (which means adding $A\cap B$ for each $B\in F$, as well as all supersets of those sets). Once one goes through all sets $A\subseteq X$, the resulting filter will be an ultrafilter.

So in a sense, one can "construct an explicit example" of a free ultrafilter in ZF + V=L. The word "explicit" is debatable: Even if given an explicit Gödel numbering scheme, it would be very difficult to find out whether, say, the set of all odd numbers or the set of all even numbers is in this ultrafilter. But the point is exactly that the phrase "explicit example" is too subtle to use lightly.

For what it's worth, this answer on Mathematics Stack Exchange states that when people argue that a free ultrafilter is "hard to define", the relevant field is descriptive set theory. I don't understand enough of descriptive set theory to make sense of that, though. Bbbbbbbbba (talk) 09:40, 7 June 2023 (UTC)

One should be careful to distinguish "in set theory" from "in topology". In topology, one is free to define all sorts of crazy spaces, having all sorts of pieces-parts added, removed, doubled, built out of cartesian products, tensor products, coproducts, whatnot. In topology, where one is free to define what is in and not in one's universe, the original example holds. (That is, one can declare that "the universe of everything consists only of finite subsets of N" and it is obvious that "the set of odd numbers" is not a part of that universe, by definition.) Books on topology provide definitions for ultrafilters that are 98% identical to what is in this article. The confusion is about the remaining 2%: what is generic and holds for topology, but fails for set theory, and vice-versa? Perhaps WP should have an article about ultrafilters "in general" (i.e. "in topology"), together with another article that explains what changes when restricting to set theory.

Yes, you are right, "axiom schema of specification" is the primary stumbling-block. In the example of the set of all finite and cofinite subsets of N, the axiom of power set does not allow you to build the set of odd numbers. The axiom schema of specification does. That is because it allows you to create a predicate defining the odd numbers. Keyword "predicate". This is perhaps the biggest difference between set theory and topology. Set theory is all of the set-theory axioms, plus all of logic (the sigma-pi hierarchy.) By contrast, in topology, one does not "extend" the definition of some topological space by including all definable subsets of it. For example, if you took the weak topology, and then added all definable subsets, you'd get the strong topology, which is utterly different. Almost any and every theorem you could state for a weak topology is utterly false in the strong topology. You cannot, in general, adjoin some arbitrary predicates to a topology, without wreaking that topology. (In the model-theoretic sense, set theory is "smaller" than topology, and, from what I can tell, all of the verbiage about topoi is about how to say this correctly, without violating the "common sense" notion that set theory is somehow "everything".)

I don't want to overwhelm you with a wall of text. However, there's one or two more things to say. Next post. 67.198.37.16 (talk) 19:51, 10 June 2023 (UTC)

OK, so two more minor remarks. First, about intuitionist logic. This is popular in artificial intelligence and machine learning, because it allows for a three-state logic: true, false, and dont-know. An excellent example of this is answer-set programming (ASP). A classic beginner mistake is to add "A=not not A" as a rule -- it will crater performance and solvability, changing run-time from milliseconds to hours. The solution to this mistake is to defines two sets, A and NA using only (mostly) positive and negative predicates, and then later on, declare that NA is "not A". I can't quite explain it clearly, here, but if you play with ASP for a while, it will bonk you in the head, and you'll have an "A ha" moment. Similar scenarios show up in SQL and GraphQL and in constraint-solving, constraint-satisfaction and motion-planning systems. Intuitionist logic is pervasive in AI/ML, although it bubbles so far under the surface, its mostly invisible. If you're a software person who is theory minded, or a theory person interested in software, then studying intuitionistic logic is worthwhile and enlightening.

As to free ultrafilters... well, the reason I decided to read this particular article is because I was reading a PDF "A beginner's guide to forcing" by Timothy Chow. I got to section 6, "Boolean-valued models" and felt an uncontrollable urge to read about ultrafilters, instead. So, I dunnno, perhaps there's some crazy way to "construct" a free ultrafilter (for set theory), using forcing, and specifically, using the boolean-valued models approach to it. But I dunno. That's over my head. If its possible, it's not something I could do in a few hours or a few weeks. 67.198.37.16 (talk) 20:17, 10 June 2023 (UTC)

Regarding "A beginner's guide to forcing", two things: First, in the Boolean-valued models approach, the "ultrafilter" one talks about is an ultrafilter on $\mathbb {B}$ (a Boolean algebra, i.e. a poset) instead of an ultrafilter on any set, so strictly speaking we are already outside the scope of this article (in this case, this is even indicated by the use of $\leq$ instead of $\subseteq$ in the definition). Secondly, why do you think that article cares about an "explicit example" of a free filter? The word "choose" in "choose a subset $U$ of $\mathbb {B}$..." does not signify explicitness. If anything, it signifies a kind of implicitness as in "axiom of choice". Bbbbbbbbba (talk) 00:49, 11 June 2023 (UTC)

I strongly suspect that an "ultrafilter in topology" can easily be described in set theory by taking (maybe part of) what you regard as the "universe" as a poset, and defining it as an ultrafilter on that poset/that kind of posets. From topological space:

> More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness.

So everything can be defined as nested sets, and those axioms can be instead thought of as premises regarding the "universe" (the topology). Bbbbbbbbba (talk) 00:29, 11 June 2023 (UTC)

Dear ip.address user: you seem to have a lot of fuzziness about the notion of "universe in set theory" versus "universe in a topology" (this last thing does not make sense) versus a bunch of other related concepts (logic, ultrafilters, etc.) I suggest you will find it helpful to go to https://math.stackexchange.com/ and browse about these topics and ask a few targeted questions of your own. There are some very qualified people (including professional mathematicians) who answer questions specifically about logic and set theory, ultrafilters, etc. They always have very informative answers. You will get more well founded and comprehensive answers than this discussion here. Regards. PatrickR2 (talk) 04:16, 11 June 2023 (UTC)

Thanks for the advice. I've being fuzzy because I wanted to communicate the concepts quickly, instead of writing impenetrably long walls of text. I'm flabbergasted that this discussion started about what is literally a (well-known) textbook example (from textbooks in topology and/or boolean algebra) but somehow requires me to go to stack exchange to talk to the people there. I honestly don't understand why so many words have to be spilled on these talk pages on such a relatively simple example. 67.198.37.16 (talk) 07:58, 11 June 2023 (UTC)

Well, we already discussed this. The current article is specifically about "Ultrafilter on a set" (with the definition as given in the article, i.e., it's strictly about ultrafilters in the powerset of a set). Your simple example is not an "ultrafilter on a set" and thus does not belong in this article. Instead, it belongs in the other Ultrafilter article, as it's an example of an ultrafilter in a poset that is not the powerset of a set. I don't know if your example is already in Ultrafilter or not, but if not, feel free to add it there. PatrickR2 (talk) 17:42, 11 June 2023 (UTC)

Actually, I cannot find a source for this example even when I Google "topology ultrafilter" and click into multiple results (the page Ultrafilter and Boolean algebra (structure) are the only other places where I've seen it), so a citation might be in order. Again, the example is obviously correct if it is talking about free ultrafilters on the finite-cofinite algebra (i.e. not the kind talked about on this page), but I'm just wondering under what context one could unironically discuss "free ultrafilters on the finite-cofinite algebra". Bbbbbbbbba (talk) 08:05, 12 June 2023 (UTC)

\wp as power set

Carrying over a proposal from elsewhere about changing $\wp$ to ${\mathcal {P}}$ for the power set. Tule-hog (talk) 01:27, 2 December 2024 (UTC)