Wikipedia:Articles for deletion/Conglomerate (set theory)

The following discussion is an archived debate of the proposed deletion of the article below. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a deletion review). No further edits should be made to this page.

The result was keep. Have Mercy! I have a math degree, but by no means do I consider myself a mathematician. How much worse for the average editor! Much of this seems more like a content dispute than an AfD review. There seems to be some consensus, hardly overwhelming and after significant article revision, that the topic should be kept. 78.26 _{(spin me / revolutions)} 14:23, 3 May 2019 (UTC)

Conglomerate (set theory)

Note: This discussion has been included in the list of Mathematics-related deletion discussions. Eozhik (talk) 04:18, 15 April 2019 (UTC)

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Conglomerate (set theory) (edit | talk | history | protect | delete | links | watch | logs | views) – (View log · Stats)

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The term conglomerate does not have rigorous definition in mathematics. The authors, Adamek, Herrlich, Strecker, don't define it as an object in a known axiomatic first order theory, and don't construct a special axiomatic theory for it. Their book (the only source for this article) is a reliable source in the field of category theory, but not in set theory, and not in foundations of mathematics, as it should be in this article. 7 years ago we discussed this at Mathoverflow, where specialists in logic explained the details. As far as I can see nothing changed since that time. I believe, mentioning this term in an encyclopedia doesn't help people to understand mathematics, but creates extra problems. In particular, in Russian Wikipedia this article serves as a justification for absurd declarations, like this

Можно пойти дальше и рассматривать совокупности классов — конгломераты, совокупности конгломератов и так далее. (One can go further and consider sets of classes — conglomerates, sets of conglomerates, and so on.)

-- and it's impossible to persuade the authors there to edit this. I think, this article should be deleted, and I invite mathematicians to take part in this discussion. Eozhik (talk) 04:13, 15 April 2019 (UTC)

• Keep Hilbert's program to place all of mathematics on a rigorous, axiomatic basis famously failed and so arguments of that sort are not reasons to delete. The concept is notable – see The Conception of Conglomerate, for example. As for Mathoverflow and the Russian wikipedia, they not reliable sources and seem to be a source of canvassing. Andrew D. (talk) 09:38, 15 April 2019 (UTC)

@Andrew Davidson: 1. Your link doesn't work. What is written in the "Advances in Chinese Computer Science" that can be useful here? 2. This:

Hilbert's program to place all of mathematics on a rigorous, axiomatic basis famously failed and so arguments of that sort are not reasons to delete.

-- sounds loud and needs an explanation. I am a mathematician, and I don't know mathematical terms that mathematicians use, and that are not rigorously defined. Which ones do you know? 3. Mathoverflow is a place where mathematicians discuss problems like the one we discuss here. And it presents opinions of professionals. When a professional mathematician, a specialist in logic and in foundations of mathematics, writes this

The main reason why you're not getting an answer is not necessarily the meaning of "justification" but the meaning of "conglomerate." None of the standard set theories (ZF, NBG, MK) admit those.

-- for me this means something. 4. Russian Wikipedia is indeed a strange place, but this does not mean that people must agree with the nonsense about mathematics that is written there. Eozhik (talk) 19:19, 15 April 2019 (UTC)

The link works fine for me – such links may give different results for people in different locations. The source has now been incorporated into the article, thanks to good work by Alexei Kopylov. My !vote stands. Andrew D. (talk) 09:14, 26 April 2019 (UTC)

• Primarily Weak Keep, but wouldn't object to a Merge to Class (set theory) either. Some more authors that talk about them a little: Osborne's widely used textbook on homological algebra, [1], and [2]. So this is a term that's in use (albeit not widely). What I've found isn't as in-depth as I think it should ideally be, though. I didn't really look too hard, though, and there might be more out there that would justify an outright keep. But if not, maybe merging this as a natural extension of a class would make sense. –Deacon Vorbis (carbon • videos) 14:38, 15 April 2019 (UTC)

@Deacon Vorbis: Do you understand Osborne's reasoning about conglomerates? How does he define them? Eozhik (talk) 19:19, 15 April 2019 (UTC)

Is this a test? What do I win if I answer correctly? –Deacon Vorbis (carbon • videos) 19:52, 15 April 2019 (UTC)

This is a question. I believe when people defend mentioning something in encyclopedia they should understand the meaning of this word. Osborne seems to try to construct the notion of conglomerate inside the Von Neumann–Bernays–Gödel set theory (at least he mentiones this theory at page 1). But (according to Mendelson) all objects in NBG are classes, and proper classes can't belong to anything, including what Osborne constructs as special objects (=special classes if everything happens in NBG) which he calls conglomerates. This contradicts to Osborne's claim (at page 152)

If A is a proper class, then A belongs to the power conglomerate of A

The contradiction could be resolved by introducing some special new membership relation, but I don't see a definition of this relation. This looks frivolous. And the same with the other authors. Eozhik (talk) 20:43, 15 April 2019 (UTC)

I don't see how any of that is relevant. We're here to decide if the topic is notable enough to warrant keeping an article about it, not to debate its mathematical validity. –Deacon Vorbis (carbon • videos) 21:01, 15 April 2019 (UTC)

Hm, this is impressive. So for you notability does not depend on meaningfulness. If several people utter words, the meaning of which they cannot explain, then, even if experts call it nonsense, the article about this will be notable, and its authors and those who decide whether this article is useful or not in the encyclopedia, they even are not obliged to understand what it is about. Eozhik (talk) 21:39, 15 April 2019 (UTC)

Of course notability does not depend on meaningfulness — plenty of wrong or nonsensical things are the subject of sufficient discussion in reliable sources that they can support an encyclopedia article. A few examples are creationism, Time Cube, angle trisection, cold fusion, etc. If there are reliable sources that are critical of the conglomerate idea (i.e., if this is not your own original research) then they can be included as well, and the discussion of the idea (whether in its own article or some other article) should give appropriate weight to the various perspectives as they appear in reliable sources. —JBL (talk) 11:56, 16 April 2019 (UTC)

@Joel B. Lewis: I must say, I don't find your list of examples convincing. The qualitative difference is that in each of those cases the reader understands what the question is about. There are no problems in understanding and explaining to other people what are the views of those who believe in divine creation, or what is the problem of angle trisection, or the idea of cold fusion, or even in retelling the story of Time Cube. (The example with the trisection in this list looks really strange, because it is absolutely clear, there is nothing "nonsensical" in it. The absence of solution doesn't make the problem senseless.) Independently on whether the ideas in these examples are "wrong" or useful, everybody understands them. On the contrary in the case of conglomerates even specialists do not understand what this is. For example, you, being a mathematician, can you explain what is meant in the declaration that "a proper class must belong to its power conglomerate"? Despite the fact that "conglomerates" are constructed in NBG, and therefore must be classes, and at the same time proper classes are defined exactly as those which do not belong to any other class (see Elliott Mendelson, 1997, page 226). The situation is scandalous as if somebody said "Consider a number $x$ which is positive and negative at the same time. Then following my tricky and ingenious reasoning we obtain... — and then something spectacular, like — inconsistency of mathematics!" And Wikipedia willingly publishes an article about this revolutionary scientific event. My reproach to the authors of this article is that they do not check the declarations and do not even understand what the article is about. Eozhik (talk) 18:38, 16 April 2019 (UTC)

For example, you, being a mathematician, can you explain ... I don't have any interest in this question, sorry. I also don't think it's relevant to whether there should be an article on this topic in Wikipedia. To put it in very silly language, AfD is a kind of logical system with its own formal grammar, and your arguments do not obey that formal grammar. (About trisection, I didn't recheck the article before posting it, I had remembered (apparently wrongly) that it had some in-depth discussion of crankery. Maybe it's somewhere else, maybe I am just imagining things.) --JBL (talk) 19:26, 16 April 2019 (UTC)

I did not understand your explanation about AfD. Eozhik (talk) 07:03, 17 April 2019 (UTC)

See WP:DISCUSSAFD, and note the repeated emphasis on whether the article meets or violates Wikipedia policy, something unaddressed in your nomination and comments. --JBL (talk) 11:43, 17 April 2019 (UTC)

Oh! It turns out that there is a formal procedure. It is not enough to say what you need in human language, you must express this with the club's euphemisms system. Joel, may I ask you to translate my speeches into the local dialect? Eozhik (talk) 17:01, 17 April 2019 (UTC)

I think it will be useful if I make a little summary with the attempt to translate what I say into the local Wikipedia dialect following Joel's recommendations. As far as I understand, in Wikipedia this is called a fringe theory. The definition of fringe theory in mathematics I think must be simple: a theory is said to be fringe if it uses terms or notions that are not accurately formalized. As I told before, the idea of conglomerate is exactlty this case: conglomerates are constructed as objects in Von Neumann–Bernays–Gödel set theory (or in Morse-Kelley set theory) with the aim to endow proper classes (i.e. classes which are not sets) with the same properties as sets have, for making it possible to consider "families of proper classes", but this directly contradicts to the definition of proper class (as a class that is not an element of any other class, see Elliott Mendelson, 1997, page 226 or definition of proper class in MK). This is an inaccurate, and moreover, a contradictory definition, and that is why people at Mathoverflow treated this term with scepticism. The objects defined like this can't exist in mathematics. Andrew Davidson writes that this is normal, but this is not normal. I don't know other mathematical terms that are defined with contradictions and at the same time are used in mathematics. I would not protest if in the article the notion of conglomerate was described as "an idea which is not yet properly formalized". But it is described differently, as if it was a common mathematical concept with a normal definition. This is a cheating, a reader hoax. Eozhik (talk) 15:37, 21 April 2019 (UTC)

I would like to make an additional comment to this

Hilbert's program to place all of mathematics on a rigorous, axiomatic basis famously failed and so arguments of that sort...

This is a big exaggeration. Hilbert and his followers did not manage to prove that in the reconstructed axiomatized mathematics new paradoxes will never appear. That is all their "failure". The other aims that they declared were achieved. In particular, they removed all the existing paradoxes. So now (in contrast to the beginning of 20th century) there are no contradictions in mathematics, and this is the merit of Hilbert and his followers. It is true that they found unexpectedly that in their approach there is no (and there can't be) a guarantee that some new paradoxes will not appear in future. But up to now nobody found these new paradoxes (despite numerous attempts). And modern mathematicians follow the standards of rigor exposition established by Hilbert. In particular, contradictory definitions are not allowed. Eozhik (talk) 16:20, 21 April 2019 (UTC)

"Failure of Hilbert's program" is undoubtedly a fringe theory, and does not deserve to be discussed.

Please, focus on the subject of this discussion, that is arguments for or against deleting this article.

• Comment Searching on Google book with "conglomerate set theory class" one gets about ten books that use these terms. Most are about category theory or related subjects, such as homological algebra and topology. So the term is notable and this cannot be considered as a fringe theory. However, I am not sure that all these books use the same definition. Moreover it seems that, at least for above link [2], a conglomerate is another name of a Grothendieck universe. In any case, it seems that none of the references provides a formal theory of conglomerates, and that the definition of conglomerates inside ZFC needs either the existence of an inaccessible cardinal or, equivalently of a Grothendieck universe.

Also, conglomerate appears twice as a red link in Class (set theory), in unsourced sentences that seem weakly related to the content of their sections. D.Lazard (talk) 16:48, 21 April 2019 (UTC)

• Weak delete or merge somewhere. By the preceding comment, it is clear that "conglomerate" must appear in WP, and that a merge is a good solution. However I am unable to decide in which article it should be merged, and I also unable to merge properly (that is to clarify the relationship between the "merged from" and the "merged to" contents. As the article under AfD has almost no content except for a single reference, the best is probably to delete, until someone will be able to fix the redlinks in Class (set theory), and to clarify my above guesses. 16:48, 21 April 2019 (UTC) — Preceding unsigned comment added by D.Lazard (talk • contribs)
• Weak delete or merge somewhere. Obsolete; see below. I agree with User:D.Lazard (whose signature is missing in the line above). Boris Tsirelson (talk) 17:40, 21 April 2019 (UTC)
• Delete The phrasing in the current stub isn't worth saving; if the topic does merit discussion, it should be within an article on a better-established subject. (I don't think that using the reference given here in another article would count as "merging" by any meaningful definition.) XOR'easter (talk) 21:53, 21 April 2019 (UTC)
• Keep/merge: the term does get used in mathematics; the article can be expanded or merged to somewhere and I don’t think the deletion encourages such an action. I think D.Lazard’s comment can be the starting point; we can say there is no fixed definition but the term is often used for example in category theory (to mean x, y, z, etc.) —- Taku (talk) 12:51, 22 April 2019 (UTC)

@TakuyaMurata: which definition would you use if the article were saved? Eozhik (talk) 16:55, 22 April 2019 (UTC)

Relisted to generate a more thorough discussion and clearer consensus.
Please add new comments below this notice. Thanks, Randykitty (talk) 17:08, 22 April 2019 (UTC)

I don’t think we need to choose a definition; in fact, the article can simply be a collection of all definitions that appear in literature. That’s informative. —- Taku (talk)

Taku, so, as far as I understand, you don't see a problem in the fact that these "definitions" contradict the existence of the objects that they define? Eozhik (talk) 17:29, 22 April 2019 (UTC)

No, that’s a math problem not a Wikipedia problem; in fact, the article can state such a math issue; an issue is interesting and not a ground for the deletion. In fact, your nomination reason itself is a good material for the article, as I see. —- Taku (talk) 17:38, 22 April 2019 (UTC)

Taku, that is strange for me. From what you say it follows that the authors of Wikipedia are not obliged to understand the essence of the matter and to check if the information has any sence, is it? If there is no sense at all, that is normal? Eozhik (talk) 17:49, 22 April 2019 (UTC)

That’s not what I’m saying. I’m saying that if a definition is problematic but is used in practice, it deserves to be mentioned (and so the editors need to understand and be aware of the problem). If there is some inconsistency, we mention the inconsistency instead of hiding it from the readers; inconsistency itself is useful information to be mentioned. —— Taku (talk) 18:29, 22 April 2019 (UTC)

I do agree with the others that there does not seem to be a standard definition. But mentioning that “there is no standard def” is encyclopedic if not math information. —- Taku (talk) 18:31, 22 April 2019 (UTC)

Taku, the problem is not that there is no standard definition. The problem is that there are no definitions that meet the standards of mathematical rigor. All definitions (that I know) are absurd. Eozhik (talk) 18:39, 22 April 2019 (UTC)

But that’s ok; Wikipedia can still mention math that is not up to today’s standard with clear documentation of looseness/issue. A naive set theory is problematic but is still a perfectly valid topic. We don’t need to mention defs that no one use but the term is used at least in some category-theory textbooks. If the problem is saying this is a topic in set theory (cf. below), that can be changed (no need to claim this is a part of set theory). —- Taku (talk) 18:50, 22 April 2019 (UTC)

Taku, in my understanding, for stating looseness we must have reliable sources. In contrast to naive set theory, the "theory of conglomerates" is very young, nobody (as far as I know) wrote any analysis of whether it is consistent or not. As I told here I would not be against this article if it contained accurate explanations of looseness. But where will we find references for this explanation? It is clear for me (and I believe for every mathematician who studied first order logic) that the existing "definitions of conglomerates" are absurd. But does this mean that for protecting people from reading this absurdity in Wikipedia I must write a special article in "Mathematical Reviews" or in other similar journal with criticism on conglomerates? Wouldn't it be easier (and more honest with respect to the readers) just to delete this article until the authers of the idea will create a reliable text with accurate definitions (according to the usual mathematical standards)? Eozhik (talk) 19:26, 22 April 2019 (UTC)

If there is no reliable source on the looseness, the looseness cannot be mentioned here in Wikipedia (otherwise we violate WP:OR). You’re asking what Wikipedia should do when no good quality reliable sources are available: my view is that we still need to present some unsatisfactory definitions or theories with some explanations. It sometimes happens that math research undergoes without proper rigor. For example, it is common to ignore set-theoretic issues in the theory of stack. It is not Wikipedia’s role to protect the readers from possible set-theoretic or logic issues. It should be up to the readers to decide whether they are ok with some defs or theories without proper rigor not to the editors; Wikipedia is ultimately a summary of literature not the place to exercise quality control on the literature.

But the style of the article must be like: Author X defines “conglomerate” to be Y. It should be very clear that Wikipedia editors are not endorsing Author X‘s def. —- Taku (talk) 19:47, 22 April 2019 (UTC)

Hm. @TakuyaMurata: Taku, I did not understand, what are the set-theoretic problems in the theory of stack? Eozhik (talk) 06:12, 23 April 2019 (UTC)

Ah, yes, I see. They seem to be overcame, however. Eozhik (talk) 08:30, 23 April 2019 (UTC)

• Delete. Rarely used (under this definition) in set theory, and not used by experts. Possibly used in category of sets, which isn't my field. I find other definitions of "Conglomerate" in set theory much more credible. Alternatively, move without redirect to Conglomerate (category of sets). If it were to be kept as part of a quasi-disambiguation page, there is little accurate in the article, even if potentially sourced. For example, "The subclasses of any class, and in particular, the collection of all classes (every class is a subclass of the class of all sets), form a conglomerate." is false in set theories not proven to be inconsistent. — Arthur Rubin (talk) 18:11, 22 April 2019 (UTC)

Arthur, excuse me, I did not find any credible definitions of conglomerates. Which ones are you speaking about? Eozhik (talk) 18:28, 22 April 2019 (UTC)

@Eozhik:. Perhaps you're correct.

Still delete. It seems well-sourced, but it's still a fringe theory. I'm not sure what conglomerates being "closed" under pairing, etc., might mean. The obvious definitions would make non-class conglomerates members of other conglomerates. — Arthur Rubin (talk) 03:19, 27 April 2019 (UTC)

Fringe? Well, maybe. Then, according to WP:FRINGE#Spectrum of fringe theories, we should classify it as either pseudoscience, or questionable science, or alternative theoretical formulation. Which one? Boris Tsirelson (talk) 06:35, 27 April 2019 (UTC)

After a search on MathSciNet I feel astonished. Conglomerates are used in more than 10 publications reviewed (and reviews). No one of these defines this notion, nor cites a source where it is defined (or did I miss something?); all just accept it to be well-known. I guess, J.Adamek, H.Herrlich, G.Strecker (The Joy of Cats...) is the implicit source. The review of this source is linked 528 times from refs and 10 times from reviews. In the end of the review the source is criticized for esoteric terminology (but not conglomerates). What does it all mean?? (For detail see Talk:Conglomerate (set theory)#Puzzling.) Boris Tsirelson (talk) 07:56, 27 April 2019 (UTC)

I shared there my opinion about what this can mean. This list lacks the case when somebody reports that he proved something important in mathematics while what he presents turns out be trivially erroneous. I think this can be classified as "fermatism" (I don't see mentioning of this term in English, in Russian it means attempts to prove the Fermat's Last Theorem, or something similarly loud, without the necessary education and skill even for seeing mistakes in your own text). Eozhik (talk) 09:46, 27 April 2019 (UTC)

Mostly "questionable science"; those publications which attempt to define the term, and produce formulations without clear contradictions, qualify as alternative theoretical formulations. However, pseudomathematics is different from pseudoscience, is that it can describe the consequences of something that does not exist, as Eozhik points out above. (That can end up being interesting mathematics, if the thing being nonexistent is interesting.) Set theory is unique (even in mathematics) in having alternative definitions, all reasonable, which produce completely different results. At this point, though, naive set theory is known to be contradictory, so any extension of set theory without a proposed axiom system (or axiom schema system, is pretty much worthless, whether or not it falls into our WP:FRINGE classification. And I speak as an expert in set theory. — Arthur Rubin (talk) 22:44, 27 April 2019 (UTC)

Sure, but there are axiomatic systems for conglomerates. E.g. [3]. They are not proven inconsistent, and I have not seen it is described as questionable science. So, I don't think it is fringe. Alexei Kopylov (talk) 00:46, 28 April 2019 (UTC)

• This is not an axiomatic system. I explained this at the talk page. Eozhik (talk) 09:13, 28 April 2019 (UTC)

• Comment I am not an expert in logic/set theory, but I guess that it is not so difficult to repeat the transition from ZFC to NBG, getting "the axiomatic conglomerates theory". In terms of the cumulative hierarchy such transition means just one more level added. Is it a problem to add two more levels, likewise? (Of course, some definitions mentioned by Eozhik will be modified in this theory.) Boris Tsirelson (talk) 18:37, 22 April 2019 (UTC)

Boris, but they did not do this! I also can believe that it is possible (although not easy) to construct a first order theory, like NBG or MK, where "collections of proper classes" would have sense. But where is this theory? Eozhik (talk) 18:45, 22 April 2019 (UTC)

OK, but are you sure that wikipedia should describe only successfully finished projects? A counterexample: Gerard 't_ Hooft#Fundamental aspects of quantum mechanics. Boris Tsirelson (talk) 18:49, 22 April 2019 (UTC)

Boris, I would not protest if the article contained an explanation like "conglomerates are just an idea which did not find a rigor definition yet..." and something like this. But the article does not contain such explanations. And this is a cheating, as I wrote above. Creating a parallel reality like what people do in Russian Wikipedia, see, for example, this thread. Eozhik (talk) 19:01, 22 April 2019 (UTC)

OK again. The article needs to be edited, which does not imply that it has to be deleted. And by the way, looking at the stackexchange discussion I do not see such diatribe toward the Russian Wikipedia. Boris Tsirelson (talk) 19:04, 22 April 2019 (UTC)

Boris, I gave some bright citations here, here and my question itself contains an absurd quotation from Russian Wikipedia:

David Hilbert... accepted the intuitionistic views.

As to editing the article do you think it will be OK to write that there is no a definition of conglomerate that satisfies the usual mathematical standards of rigor? I suspect for this we should ask a specialist in logic to write a review in Mathematical Reviews or in a similar journal. Eozhik (talk) 19:40, 22 April 2019 (UTC)

Yes, I see the problem. But we could write only uncontroversial facts like this: "...but this source does not propose an axiomatic theory of conglomerates".

And, by the way, about the trouble with "conglomerate of pairs of classes": this could be dealt similarly to "Analysis in J_2" by Nik Weaver. Boris Tsirelson (talk) 19:52, 22 April 2019 (UTC)

And about Hilbert: "From our finitary viewpoint, therefore, we cannot argue that an equation like the one just given, where an arbitrary numerical symbol occurs, either holds for every symbol or is disproved by a counter example." (From the highest-score answer there.) Boris Tsirelson (talk) 19:57, 22 April 2019 (UTC)

Boris, I don't understand your hints about Hilbert. My point is that phrases like this

Чтобы сделать свою идеологию общеприемлемой, Гильберт исключил из числа допустимых логических действий многие из самых спорных моментов — доказательство от противного, актуально бесконечные множества... (In order to make his ideology generally acceptable, Hilbert eliminated many of the most controversial points from the list of admissible logical actions - proof by contradiction, actually infinite sets ...)

— are lies. It is not nice to write them in an encyclopedia. Eozhik (talk) 05:36, 23 April 2019 (UTC)

Rather off-topic here; for now I reply, but if you want to continue in this direction, it is better to do it on my (or your) talk page. Well, the above phrase is written there not in context of the "working mathematics" but in the context of metamathematics of that (overly optimistic) time; not for examining triangles, polynomials etc., but for examining formal texts (proofs). Maybe, there one should emphasize more clearly this context difference. Boris Tsirelson (talk) 05:53, 23 April 2019 (UTC)

I think we can stop discussing Hilbert here, but I would like this talk to be preserved here as an illustration. Eozhik (talk) 06:46, 23 April 2019 (UTC)

The axiomatization of conglomerates are given in the link given by Andrew Davidson in the first comment: Zhang, Jinwen (1991). "THE AXIOM SYSTEM ACG AND THE PROOF OF CONSISTENCY OF THE SYSTEM QM AND ZF#". Advances in Chinese Computer Science. Vol. 3. pp. 153–171. doi:10.1142/9789812812407_0009.. This is rigorous and well defined system. Alexei Kopylov (talk) 21:28, 22 April 2019 (UTC)

I see, thank you. So, the book cited in our article is not the only source. Boris Tsirelson (talk) 04:16, 23 April 2019 (UTC)

I guess, a model (and moreover, "the intended model") of the conglomerates theory exists in ZFC plus large cardinal (any large cardinal; a worldly cardinal is enough), which gives a kind of justification to existence of conglomerates. Of course, just existence; their notability is another story. Boris Tsirelson (talk) 04:51, 23 April 2019 (UTC)

As far as I can see, in this article the author, Zhang Jinwen, tries to present the theory of conglomerates as an extension by definition (see K.Kunen, II.15) of the Von Neumann–Bernays–Gödel set theory. This means in particularly, that all the objects in his theory are still classes in the sense of NBG. If so, then by definition (see Elliott Mendelson, 1997, page 226) the proper classes can't belong to anything, including any special objects constructed by the author. The problem is still unsolved. (And the constructed theory is inconsistent from the very beginning.) Eozhik (talk) 05:57, 23 April 2019 (UTC)

Wow... Maybe. I am not enthusiastic to read that paper, the more so that it is not available to me (and google books gives not all relevant pages). And still, my note about the (intended) model is believable. I return to my vote "weak delete or merge", plus "we could write only uncontroversial facts like this..." We are not the Higher Attestation Commission anyway. Boris Tsirelson (talk) 06:10, 23 April 2019 (UTC)

Boris, how are large cardinals connected to all this? Eozhik (talk) 06:01, 23 April 2019 (UTC)

A strange question. They are evidently related to proposed extensions of the set theory. Boris Tsirelson (talk) 06:13, 23 April 2019 (UTC)

Boris, I think our education in logic was focused on very different things. I must say, I don't understand what you mean. Moreover in my understanding if the model in ZFC+large cardinals that you are talking about existed, that would mean that ZFC+large cardinals was inconsistent as well (since this "paradox with proper classes that belong to something" would be transferred into ZFC+large cardinals). Eozhik (talk) 06:40, 23 April 2019 (UTC)

In regard large cardinals, and models, see Von Neumann–Bernays–Gödel set theory#Models. But I don't see V_κ+2 as likely having a sensible theory. — Arthur Rubin (talk) 09:29, 23 April 2019 (UTC)

It depends on your idea of "sensible theory". It may satisfy a theory specially designed for such case, like that of Jinwen Zhang. And, I guess, it satisfies some more axioms. Though, probably, V_κ+2 should be replaced with something in the spirit of "Analysis in J_2" by Nik Weaver, in order to adapt "conglomerate of pairs of classes" etc. (For links to Zhang and Weaver see above.) Boris Tsirelson (talk) 07:03, 24 April 2019 (UTC)

• Delete: I change my !vote because of the lengthy above discussion: Presently the article contains only a non-workable definition and a reference containing one of several published definitions. Thus is does not satisfies the requirements of WP:NPOV. The length of the discussion shows that expanding into a decent article would be WP:Original synthesis, which is not accepted in WP. Also, merging would be original synthesis, as this would need choosing between several definitions. On the other hand, deleting the article would not provide any trouble, as it seems that all references provide their definition of a conglomerate. So a normal reader would never be tempted to come here for having more details. So, deleting seems the only way for respecting WP policies without trouble for readers. D.Lazard (talk) 08:49, 23 April 2019 (UTC) (Obsolete, because of the recent expansion of the article, see below. D.Lazard (talk) 10:02, 25 April 2019 (UTC) )
• Delete. I am convinced by D.Lazard's arguments that this does not neutrally represent the subject and cannot reasonably be turned into something that does. —David Eppstein (talk) 00:15, 25 April 2019 (UTC)
  • Ok, fine, for consistency I'll strike this too. —David Eppstein (talk) 06:08, 30 April 2019 (UTC)

New version of the article

• Keep. Thanks to the discussion above, I have dig into the sources and significantly expanded the article. Please take a look. Even if there are different approaches to introduce conglomerates, their essential properties are independent of the definition. I also believe that the current name is adequate: it is a set-theoretic foundation for categories of categories, like Class (set theory). Alexei Kopylov (talk) 03:03, 25 April 2019 (UTC)

@Alexei Kopylov: your edits don't solve the problem. Proper classes can't be elements of other classes as this is stated in the extensions by definition of NBG or MK that you refer to. And all such extensions are inconsistent. And their models in ZFC, if they exist, don't prove the consistency of these theories. They could prove inconsistency of ZFC with the buildups that are used for these constructions, if these models indeed existed, but there are no evidences of this existence. As in the case of Hilbert, about whom you wrote in Russian Wikipedia that he "accepted intuitionistic views", this is a fringe theory. Or in the case of Grothendieck whom you now present in English Wikipedia as a person having some relation to conglomerates. Or in the case of mathematics, about which you defended the thesis from a marginal source that

Математика была вынуждена бесповоротно отказаться от претензий на абсолютную достоверность или значимость своих результатов. (Mathematics had to irrecoverably abandon claims for absolute certainty or significance of its results.)

All this is called "creating a paralllel reality". Eozhik (talk) 04:25, 25 April 2019 (UTC)

Eozhik, stop personal attacks against Alexei Kopylov (they are punishable). Voice your opinion on the discussed article, not on behavior of others in general. "весь в иголках ежик"   :-) Boris Tsirelson (talk) 04:30, 25 April 2019 (UTC)

Борис, ну Вы, видимо, не в России живете поэтому от Вас это так далеко. Но проблема здесь есть. Eozhik (talk) 05:25, 25 April 2019 (UTC)

• Keep. Thanks to Alexei Kopylov it is now good enough. Boris Tsirelson (talk) 04:21, 25 April 2019 (UTC)

@Tsirel: Boris, as far as I understand, this atinomy with the proper classes that belong to other classes does not confuse you. Eozhik (talk) 04:29, 25 April 2019 (UTC)

Think again. These are not extensions by definitions. And conglomerates are not classes. Boris Tsirelson (talk) 04:32, 25 April 2019 (UTC)

Not extensions by definition? Then what is it? These objects are constructed inside NBG, and are not classes? Eozhik (talk) 04:47, 25 April 2019 (UTC)

I never said that proper class belongs to other classes. They belong to conglomerates, but not all conglomerates are classes. And they are certainly not constructed inside NBG. Alexei Kopylov (talk) 04:53, 25 April 2019 (UTC)

It is not enough to say that something belongs (or does not belong) to something. There must be a theory that satisfies the usual mathematical standards of rigor. Eozhik (talk) 05:22, 25 April 2019 (UTC)

• Delete. OK, as far as I understand I can vote as well. Here is my vote. As I wrote above, this theory is not presented as a first order theory, independent from the existing ones, ZFC, NBG, or MK. The attempts to present it as extension of NBG or MK lead either to the theories, where conglomerates evidently don't exist (if these are extensions by definition as in the given references), since the idea of "proper class that belongs to other classes" contradicts the definition of proper class. Or to inconsistent theories (if the axiom of existence of conglomerates is added to NBG or MK). If, on the other hand, this theory is constructed inside ZFC then the usage of the word "class" is confusing, since this refers the reader to NBG and MK. I believe, it is not honest to ignore this. Eozhik (talk) 04:59, 25 April 2019 (UTC)

Eozhik, for some general ideas about extensions of the set universe, in relation to large cardinals etc., you may look here (near the end). Boris Tsirelson (talk) 05:04, 25 April 2019 (UTC)

Boris, this is not an answer. This is again a hint. They are speaking about classes in NBG and MK, pretending to extend these theories with the aim that "proper classes would be elements of new objects, conglomerates". But what I see by now is either extentions by definitions of NBG (with the assumed supplementary axiom of existence of conglomerates, that evidently makes these theories inconsistent), or constructions inside ZFC where classes are not defined at all. With such loud declarations they should have constructed a serious first order theory, independent from NBG, MK and ZFC. (And to explain in which sense it extends NBG and/or MK.) Where is this theory? Eozhik (talk) 05:17, 25 April 2019 (UTC)

Well, maybe some writings on this matter are incorrect, I did not real them all carefully. But the project is evidently doable, and very probably, some writings on it are correct. It is a pity that our experts in logic/set theory are rather silent. Boris Tsirelson (talk) 05:26, 25 April 2019 (UTC)

Boris, this is what separates us. You think that for writing something in an encyclopedia it is enough that someone got a feeling that something stands behing the loud declarations. But I need a possibility to check the details. (Which often turn out to be false, by the way.) Eozhik (talk) 05:39, 25 April 2019 (UTC)

As for me, it is not a loud declaration. Two more levels over the ZFC set hierarchy is not at all a big deal for set theory; large cardinals give much, much more. But these two more levels appear to be somewhat convenient in category theory and computer science; well, why not? A kind of applied set theory. It happens often that some math is actively used in applications while is of little interest for pure math. Boris Tsirelson (talk) 05:46, 25 April 2019 (UTC)

And our article, as of now, is definitely useful for an interested reader; it gives him, at least, an annotated collection of sources that are not so easy to find. Let him judge. Boris Tsirelson (talk) 05:57, 25 April 2019 (UTC)

The article about Hilbert who accepted intuitionistic views also contained a lot of unexpected sources that are not so easy to find. I would say it is honest in such cases to warn the reader about the traps, so that it would not look like a cheating of non-specialists. And to remove evident absurdities, like inconsistent theories which are presented as normal. Eozhik (talk) 06:16, 25 April 2019 (UTC)

OK. If you see an evident error in one (or more) of these sources, for instance, if it is written there that the extension is by definitions (thus, conservative), or that each conglomerate is a class, then please let us know, and add this fact to the article. Boris Tsirelson (talk) 06:23, 25 April 2019 (UTC)

Boris, I already told this: in this article Zhang Jinwen constructs his theory as an extension by definition of GB:

In this article we develope the axiom system of conglomerates denoted as ACG. The system ACG falls into two parts, the first part is just GB, the second part is composed of five groups A,B,C,D,E.

(page 157). He just adds to GB new predicate symbols $scl$, $cog$, with a series of formulas "that define them" (and a new term, "conglomerate"). This is an extension by definition. The new formulas don't cancel the old theorems of GB, everything what was true in GB, is true in the new theory as well. In particular, proper classes still can't belong to anything. If he would define a new relation "X belongs to Y", we could understand this phrase — "proper class belongs to this object" — differently. But he does not do this (and he uses the same symbol for this relation $\in$). This means that he understands this relation in the same sense. Eozhik (talk) 07:04, 25 April 2019 (UTC)

See my reply there. Boris Tsirelson (talk) 20:23, 25 April 2019 (UTC)

Is there a theorem of GB that says that any proper class do not belong to anything? I don't think so. Also, they use multi sorted logic, so you can't get contradiction to a statement that a class is a member of conglomerate in GB, because there is no sorts for conglomerate in GB. Anyway, we are not here to judge other work. Alexei Kopylov (talk) 07:27, 25 April 2019 (UTC)

For proper classes see Elliott Mendelson, 1997, page 226, I cited him many times. Of course, there is no necessity to judge, if there is nothing to say. Eozhik (talk) 07:42, 25 April 2019 (UTC)

In that theory you can't define "X is a set" as "X belongs to something" as in NBG. Alexei Kopylov (talk) 08:02, 25 April 2019 (UTC)

Extensions by definition don't cancel the previous definitions. Eozhik (talk) 08:20, 25 April 2019 (UTC)

This can be considerd as a conservative extension (in the sense that it says nothing new about GB) if we notice that the new predicate symbols define empty classes (because of the aprrearing contradiction with "proper class that belongs to..."). Eozhik (talk) 07:13, 25 April 2019 (UTC)

They are explicitly say that the extension is non-conservative. Alexei Kopylov (talk) 07:27, 25 April 2019 (UTC)

It is not enough to say this. There must be a theory that shows this. Eozhik (talk) 07:42, 25 April 2019 (UTC)

Formally, this depends on whether he writes anywhere that conglomerates exist, or not. I don't see this in the text. If he does not write this, then his theory becomes a trivial extension of GB, since the symbols that he adds define empty classes, and therefore, his "conglomerates" don't exist. But if he writes that they exist (what I assume, must be somewhere) then this means that he adds this formula as a supplemenary axiom, and this automatically makes his theory inconsistent. Eozhik (talk) 07:42, 25 April 2019 (UTC)

And the same with the other sources. It is evident for me that the authors don't understand the difference between naive set theory and axiomatic set theories. And all these "misunderstandings" grow from this. Eozhik (talk) 07:52, 25 April 2019 (UTC)

• Keep The article has been largely expanded, and is correctly sourced, which makes my preceding !vote obsolete. This does not means that all issues raised in this discussion have been solved, but that they can be solved by seeking a consensus on the talk page of the article. D.Lazard (talk) 10:02, 25 April 2019 (UTC)
• Comment There are several kind of issues raised in the above discussion. Some of then are minor, such as that about the section heading "Conglomerates in Grothendieck’s approach"; it can be raised by simply renaming it "Conglomerates and Grothendieck universes". Some are about a possible misinterpretation of sources; these must be discussed and raised by a discussion on the talk page of the article, not here. I understand from this discussion that some editors have a doubt about the logical correction of some sources. If this doubt can be sourced, this must appear in the article. If this doubt is original research, it must not appear as a part of the article. In this case, I suggest to use the tag {{unreliable source?}}, and to explain the doubt in details in the talk page. D.Lazard (talk) 10:02, 25 April 2019 (UTC)

OK, I'll put this template. Eozhik (talk) 05:09, 26 April 2019 (UTC)

• Comment Yes, I just started a continuation of the discussion of the doubt there. Boris Tsirelson (talk) 20:23, 25 April 2019 (UTC)
• Keep With the expansion of the article by Alexei Kopylov (nice work!), the topic has been shown to have been discussed in multiple independent sources, reliable as far as I can tell. I respect Eozhik's concerns about foundations supporting this concept--developing this criticism into a paper published in a math journal could be a real contribution to the field. It would then be entirely reasonable to add a summary of that paper to the article. But at present, the article should reflect the current state of the literature. It can be debated whether there is enough secondary sourcing (I'd count the Large category entry in nLab among those sources, others might not), but I think it is enough to support an article. Hence keep, or merge if that would help achieve consensus. --{{u|Mark viking}} {Talk} 21:41, 25 April 2019 (UTC)
• Comment What impresses me in this picture is where are specialists in logic? Eozhik (talk) 05:22, 26 April 2019 (UTC)
• Comment On this point we agree, completely. Moreover, I wrote an "invitation" here, but unsuccessfully. Boris Tsirelson (talk) 06:56, 26 April 2019 (UTC)

Fringe theory of non-fringe objects?

Arthur Rubin, speaking as an expert in set theory, at 18:11, 22 April 2019 (UTC) (and later, again) votes "delete" and argues that it is a fringe theory. I am inclined to agree that the relevant "theory of conglomerates" is fringe, but I think that this notion "conglomerate" is not fringe. And if so then, given that the article is entitled "Conglomerate (set theory)" rather than "Theory of conglomerates", the article should not be deleted. (But what about moving it to "Conglomerate (category theory)"?

Indeed, on one hand, we observe that a number of sources that propose a first order theory of conglomerates are unreliable. On the other hand, such theory is not needed! Instead one may take the usual theory "ZFC plus one of well-known large cardinals", and interpret (in this theory) "conglomerates" as just sets, "classes" as sets satisfying a special condition, while redefining "sets" as sets satisfying another special condition. This way follows a well-known trick, first used in geometry in 1868, see Beltrami–Klein model. For detail, see Talk:Conglomerate (set theory)#Puzzling. Boris Tsirelson (talk) 19:11, 28 April 2019 (UTC)

The comparison with Beltrami–Klein model is a big exaggeration. See the same discussion. Eozhik (talk) 20:20, 28 April 2019 (UTC)

Set theory is not about the concept of set, but of the theory of sets. Hence there needs to be a theory of conglomerates for conglomerate (set theory) to be appropriate. I don't understand category theory well enough to assert that the concept of conglomerates, even in the absence of a credible theory, is not useful in category theory. My alternative to deletion is to move, without redirect, to conglomerate (category theory), and to excise (the remaining) link(s) from set theory articles, without prejudice to deletion if category theory experts think it inappropriate. — Arthur Rubin (talk) 10:02, 2 May 2019 (UTC)

I agree: move to "conglomerate (category theory)". Boris Tsirelson (talk) 10:59, 2 May 2019 (UTC)

(edit conflict)I agree with Arthur that conglomerates are uniquely motivated by categories, and more precisely by the need (for homological algebra) of algebraic computation with categories (derived categories, for example). As the objects of most usual categories are not sets (category of sets, for example), some extension of ZFC is needed. One such extension has been done by adding an axiom of existence of very large sets (inaccessible cardinals), allowing Grothendieck universes. I interpret the introduction of conglomerates as a kind of intuitionistic approach by people thinking that conglomerates are easier to accept than inaccessible cardinals. This interpretation is supported by the fact that most books that use the term are elementary textbooks on applications of homological algebra. So, in my opinion

1. Conglomerate theory is a fringe theory, since it is not used by the main stream of mathematical research, which uses Grothendieck universes.
2. The application field of conglomerates is category theory, but a theory of conglomerate (if any) is an extension of set theory, and thus the title conglomerate (set theory) if convenient.
3. There are no secondary source validating the existence of a non-contradictory conglomerate theory
4. There are a sufficient coverage of the term "conglomerate" by reliable sources (published books) for allowing an article despite it is a fringe theory that is not reliably validated.

So, I maintain my !vote of keep, although, IMO, the article must be edited for clarifying the above points. D.Lazard (talk) 11:17, 2 May 2019 (UTC)

After that discussion at the talk page I understand at last what people can have in mind when pronoucing this word, "conglomerate". (@Tsirel: Boris, I am impressed by your patience. Thank you and forgive me for my unbridledness.) I don't insist on deleting now. But, independently on whether this text will be moved or not, I think it must be rewritten so that people like me, i.e. not specialists in logic, but with a common mathematical education, that includes logic, would understand it. There must not be traps that provoke misunderstanding and controversy. In particular, the meaning of the words that each class can be considered as an element of more general objects ("conglomerates") must be clarified. I don't know how this can be done without breaking the interdiction of the original research, but if there is a possibility, I would welcome it. Eozhik (talk) 16:45, 2 May 2019 (UTC)

Nice. Now, about "how this can be done without breaking the interdiction of the original research". Here is a quote from the annotation to the book by Horst Herrlich and George E. Strecker:

"The attempt is made to present category theory mainly as a convenient language..."

In this spirit, it appears to be convenient (in category theory) to reserve the word "set" for sets that belong to a given universe (for instance, sets of rank less than a chosen worldly cardinal), the word "class" for subsets of this universe (for instance, sets of rank at most this worldly cardinal), and the word "conglomerate" for all sets (of any rank whatsoever).

"Universes are of critical importance to formalizing concepts in category theory inside set-theoretical foundations." (Quoted from "Universe (mathematics)".) Boris Tsirelson (talk) 17:45, 2 May 2019 (UTC)

Boris, I did not understand, are you saying that it is possible to write this?

...it appears to be convenient (in category theory) to reserve the word "set" for sets that belong to a given universe (for instance, sets of rank less than a chosen worldly cardinal), the word "class" for subsets of this universe (for instance, sets of rank at most this worldly cardinal), and the word "conglomerate" for all sets (of any rank whatsoever).

Eozhik (talk) 05:13, 3 May 2019 (UTC)

If yes, that would be an acceptable solution. Eozhik (talk) 05:18, 3 May 2019 (UTC)

As for me, yes it is, why not. If others do not object... Boris Tsirelson (talk) 05:39, 3 May 2019 (UTC)

The above discussion is preserved as an archive of the debate. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a deletion review). No further edits should be made to this page.