Talk:Skolem's paradox/GA1

GA Review

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Nominator: Pagliaccious (talk · contribs) 04:15, 22 August 2024 (UTC)

Reviewer: David Eppstein (talk · contribs) 23:23, 26 August 2024 (UTC)

First read-through: meaning and completeness

First sentence: should "first-order model of set theory" maybe be "model of first-order set theory"? And how can a model prove anything? It is mathematicians who prove things, not models. Or by extension you could maybe say that an axiomatization proves something (its theorems) but it is still not the model that is the subject of "prove".

Somewhere in the article, and maybe also in the lead, it should be clarified what a model is. I am familiar with them but I think we should not expect all readers to be. I think the punch line of the paradox should also be made more explicit: because an uncountable set exists, such a set is an element of the model, despite the model being (externally) countable.

In the lead "Formally, Skolem's paradox is that every countable axiomatization of set theory in first-order logic, if it is consistent, has a model that is countable": isn't this really just the Löwenheim–Skolem theorem, not the paradox itself? What meaning does "Formally" add? And in the next sentence, about proving existence of uncountable sets, maybe Cantor should be briefly credited?

I suspect that the last line of the lead 'More recently, the paper "Models and Reality"...' is intended to be a summary of the last paragraph of the "Later opinions" section, on Putnam and reactions to Putnam, but this is unclear because that paragraph never mentions "Models and Reality" by name.

You have the dates of Cantor's uncountability theory and the Löwenheim–Skolem theorem, but not of Skolem's observation that they are (philosophically at least) contradictory. Can it be dated?

Doesn't the second paragraph of "The result and its implications" lead to a different paradox? The paradox as stated earlier is "this model is countable but yet it (its domain) contains an element that (in the theory of the model) is uncountable". But the second paragraph doesn't talk about an element that is uncountable in the theory of the model; it talks about a set that is not in the model (the set of all subsets of the model). And I don't see how the part about "meaning we cannot put each set of the model B in relation with some natural number" follows. It somehow seems to be assuming that each subset of the model is a set of the model; why? This paragraph seems to be a mixed-up combination of the Skolem paradox and a different paradox: if B is a countable model, then there exists a set of all subsets of B (with the cardinality of the continuum) but we know that "the set of all sets" cannot actually exist. (The resolution to this paradox being that this set of all subsets of B is not an element of B so it does not belong to the model.)

In the third paragraph of the same section "Skolem resolved the paradox by concluding that the existence of such a set cannot be proven in a countable model". What do you mean by proving something in a model? I think the resolution is that, if U is an element of B that models an uncountable set, and C is an element of B that models a countable set, then there does not exist an element phi of B that models a bijection from U to C. It's not merely that there is no proof element in B that phi models such a bijection, but that this bijection is not part of the model at all.

How does the "formally" of the fourth paragraph introduce a different level of formality than the "formally" of the second paragraph? Especially as both paragraphs appear to consist of informal prose rather than formalized logical deductions? In any case I think this fourth paragraph much more closely describes the paradox and its resolution than the second. One quibble: "There are two special elements of M; they are": maybe instead of "they are", more accurate would be "they model"?

In the fifth paragraph, we again have this confusing issue of whether a model contains an element that models a bijection, or whether the bijection can be proven to exist. Why not "relative to one model, no enumerating function puts some set into correspondence with the natural numbers, but relative to another model, this correspondence may exist"? What does provability have to do with it, except for the side point that if an object is actually proven to exist (in the theory) then it must exist (in the model that models the theory)?

In the "Reception" section, how exactly does "Skolem's result" (by which I imagine is intended the paradox, not the Lowenheim–Skolem theorem) prove that first-order set theory cannot be categorical? I mean, it cannot be categorical, but how does that follow from the paradox?

The van Dalen & Ebbinghaus reference at the end of the Reception paper discusses a 1937 paper of Zermelo which, van Dalen & Ebbinghaus state, was intended to refute the Skolem paradox. Why is this work not discussed in more detail in this section? The sentence "It is now known that Skolem's paradox is unique to first-order logic; if set theory is studied using higher-order logic with full semantics, then it does not have any countable models, due to the semantics being used." appears to be referenced to van Dalen & Ebbinghaus p145 but that page says nothing about higher-order logic not having countable models; can this be sourced properly?

A minor formatting note: the |30em in the reflist of the References section is no longer needed (reflists are put into columns by default), but Template:Reflist suggests using 20em instead for articles with shortened footnotes, as used here. When I view this article on my laptop I get only one tall column of references, so a smaller number like 20em would make it more likely that a compact two-column format could be used.

I considered the possibility of discussing the proofs of both Lowenheim-Skolem and Cantor's uncountability theorem, but ultimately decided against requesting that. I think it would be too duplicative of material that belongs better on the articles on those theorems. The only important part here is merely that both are proofs in the first-order theory and therefore must be true of any model of the theory.

The illustrations are more decorative than informative, but they are properly licensed and have informative and relevant captions; I think they're ok, and it's hard to imagine what else might be used as an illustration for this article.

I think that covers WP:GACR criteria 1b, 3, and 6. Criteria 4 and 5 are unlikely to be problematic, but I still need to do another read-through (another day soon, most likely) for low-level copyediting (1a) and for cross-checking the references against the material they reference (2; touched on briefly above but not thoroughly checked).

—David Eppstein (talk) 06:59, 27 August 2024 (UTC)

Hello David Eppstein. Thank you for taking the time to write such a thorough review. I think that I've addressed all of your comments:

• Throughout the article, I've done my best to remove "prove" or "proven" where possible, replacing it with "satisfied" whenever necessary, as you described at several points.
• I've clarified what a model is in the lead. If you think that the article would be improved by a longer explanation, I would gladly move it to the "Background" section and expand it.
• I've removed all the "Formally" introductory phrases, mentioned Cantor in the lead, and fixed the mention of Putnam
• Added date of Skolem's paper
• Removed the second paragraph of the Result section
• Concerning the "first-order set theory cannot be categorical" claim: this article is a bit of a forgotten battleground, as you might tell from the talk page. This claim is an artifact from this era of the page, which I missed when sourcing and cleaning up the existing prose. It seems to only be mentioned on this course webpage, but nothing published. This source was a topic of much discussion on the talk page. I'm happy to remove it from the article.
• I've sourced the "Skolem's paradox is unique to first-order logic" statement properly. The van Dalen & Ebbinghaus only sourced the following sentence, on Zermelo's 1937 paper.
• Concerning this 1937 paper of Zermelo: in fact, it is not so much a paper as a handwritten rough draft. It appears in English on pages 155-156 of the van Dalen and Ebbinghaus paper. From what I understand, this is Zermelo's last attempt at an attack on "finitism" by means of attacking the paradox, and it was left unfinished, unpublished, and uncirculated until the van Dalen and Ebbinghaus paper. It's interesting within the context of the history of the paradox, but I don't know if it's very important to a broader understanding of the paradox's history, as it was left unpublished. I've added a short sentence clarifying that the refutation was unfinished.
• Fixed the ref formatting
• For the images, although you said that they seemed to be fine, I've decided to remove all but Skolem's portrait. I agree that they're only decorative, and Zermelo's and Putnam's pictures were a bit irrelevant.

Kind regards, Pagliaccious (talk) 14:45, 28 August 2024 (UTC)

Second reading: Copyediting and some potential additional sources

The previous edits look good; here are some new, mostly less significant, concerns.

The article isn't very consistent about whether the first time a mathematician is named (rather than merely mentioning something else named after them), or later, it should be just the last name or the full name: "Thoralf Skolem", "Cantor proved" (but then later "Georg Cantor"), "Ernst Zermelo", "Hilary Putnam", "Löwenheim", "Abraham Fraenkel", "John von Neumann", "Stephen Cole Kleene", "Geoffrey Hunter", "Reuben Goodstein", "Hao Wang", "Brouwer", "Carl Posy", "Cohen's method", "Kanamori", "Hilary Putnam", "Timothy Bays", "Tim Button".

Background: "When Zermelo proposed his axioms for set theory": link Zermelo set theory (these are the 1908 axioms). But later in the section you have "Zermelo's axioms of set theory" linked to Zermelo–Fraenkel set theory; this is confusing, because it's not clear whether the 1908 axioms or ZF is the intended meaning. If you called it "the axioms of Zermelo–Fraenkel set theory" here, it would be clearer, and would not make it look like the paradox applies only to an obsolete variant of the theory. The same confusion between two different axiom systems persists into the first paragraph of the next section, which in any case does not need to link them again.

"There is only a countably infinite amount of ordered pairs" reads awkwardly, maybe better "There are only countably many ordered pairs" or maybe something like "The set of ordered pairs is countable"

"Zermelo at first considered": do we need to link Zermelo again here?

"Hilary Putnam considered it": here "it" can only mean the Skolem paradox, but as there is no preceding noun in the same paragraph and the last noun from the previous paragraph is "formal systems", the referent is grammatically unclear.

"the most recent": Is this time-specific wording necessary? They are both over a decade old, so not particularly recent, and I don't think it would be helpful to demand sourcing for the claim that these really are the most recent significant discussions.

Speaking of which, is there anything worthy of mention in the following recent publications?

• Penchev, "Skolem's Paradox and Quantum Information. Relativity of Completeness according to Gödel", Philosophical Alternatives Journal
• Hosseini and Kimiagari, "Higher-Order Skolem's Paradoxes and the Practice of Mathematics: a Note", Disputatio
• Hanna, "A Neo-Organicist Approach to the Löwenheim-Skolem Theorem and “Skolem’s Paradox”", Science for Humans
• Shapiro, The limits of logic: higher-order logic and the Löwenheim-Skolem theorem (a book-length work which calls the paradox "the other main theme of this volume")

I think the following ones really may be worthy of mention:

• Roman Suszko, "Canonic axiomatic systems", Studia philosophica (Poznán), 1951
• Pogonowski, "On the axiom of canonicity", Log. Log. Philos., 2023
• Riccardo Bruni, Review of Pogonowski, MR4562923
• Jan Kalicki, Review of Suszko, J. Symbolic Logic, doi:10.2307/2267712

I have only seen the Kalicki and Bruni reviews, not the original papers. The reviewers both write that Suszko developed an axiomatic theory of sets in which the paradox is obtained without going through the Lowenheim-Skolem theorem. From Bruni's review, the Pogonowski paper clarifies Suszko's and puts it into historical context.

—David Eppstein (talk) 01:42, 31 August 2024 (UTC)

Hello David Eppstein. I believe that I've made all of your suggested changes:

• The full name of mathematicians is given at their first mention and in all wikilinks, and their last name is given at all subsequent mentions.
• I've done my best to clear up the difference in context between Zermelo's axioms, ZFC, and standard models of first-order set theory in general. The paradox holds for all "standard" (to use a nebulous term from Bays and Eklund) models of first-order set theory, probably most notably ZFC; however, Skolem's 1922 paper is written with specifically Zermelo's 1908 axioms in mind, so I chose to leave "Zermelo's axioms" in the paragraph beginning with "In 1922, Skolem pointed out..." The second part of Skolem's paper is his proposal of the axiom of replacement, independent of Fraenkel's own 1922 proposal. After this historical description, I've added a statement clarifying that the result holds for ZFC and other "standard" models.
• I've implemented the rest of your copyediting suggestions.

Thank you for listing these much newer publications. I'll take a look at them and see if I can't find more to include in the article. Kind regards, Pagliaccious (talk) 15:04, 31 August 2024 (UTC)

I've been looking at the first set of publications this afternoon. The Hanna and Penchev papers seem to be short appeals to the paradox in fields outside of logic/mathematics, so I'll include a short sentence on these uses at the end of the "Later opinions" section. The short paper by Hosseini and Kimiagari is about a higher-order class of "paradox" which they associate with Skolem's paradox by calling it an "extension" of his paradox, but I don't know if it's notable enough to mention. Shapiro's book is a collection of papers, many of which are already included as references in the article, but many of which I think ought to be referenced.

As for the "Axiom of Canonicity," Pogonowski's paper seems very interesting, and I'd certainly like to include it in the article, but I cannot find Suszko's work online or in any library catalog. I'll have everything from these publications finished by tomorrow. Pagliaccious (talk) 20:54, 31 August 2024 (UTC)

Third reading: source check (mostly)

I finally took the time to look through the talk page and...wow. Not the sort of thing I would have expected controversy over. Fortunately long done.

Last line "the Skolem's idea": either "Skolem's idea" or "the Skolem idea"

Minor reference formatting details (beyond the scope of the GA requirements): consider adding |language=de to the German-language references. I would not generally include publishers on journal publications but it's mostly harmless. On the other hand, the book source From Frege to Gödel: A Source Book in Mathematical Logic is missing its publisher (Harvard University Press). And we're not very consistent about which journals get their names linked, nor on whether the titles of chapters and journal articles are sentence case or title case. In Hunter 1971, "Macmillan" is misspelled. Kleene 1967 is again missing a publisher. Moore 1980 is unlinked (doi:10.1080/01445348008837006) and its journal is uncapitalized. Klenk 1976 and Resnik 1966 could use the |jstor= parameter rather than merely a link (it would have made it more obvious to me that I had access to the link). Kanemori 2012 is missing page numbers.

I'm not entirely convinced that Penchev 2020 counts as a peer-reviewed and reliable source. The "Epistemology eJournal" that it's in [1] is basically just a topic subclassification of the SSRN preprint server without peer-review [2]. On the other hand, Penchev appears to be an associate professor in the Bulgarian Academy of Science [3] so maybe he counts as a "established subject-matter expert" under WP:EXPERTSPS? In any case if this is kept then the name of the publisher should be split from the name of the sort-of-journal.

Footnote [1], the only footnote in the lead (a direct quote) is appropriate and valid for both its uses.

The first paragraph in "Background" implies that Cantor's theorem is his 1874 proof of the existence of an uncountable set. But checking footnote [2] I learn that the 1874 paper proved only that the reals are uncountable, by a different method than diagonalization, much weaker from how our article on Cantor's theorem states the theorem (that every set has a larger power set).

The claim that Cantor's is "One of the earliest results in set theory" appears unsourced.

Bayes 2007 is linked with |url=, putting the link on the book title. It should be linked with |article-url=, putting the link on the chapter title. Long ago |url= used to be smart and guessed what to link, usually correctly, but the citation template maintainers in their infinite wisdom decided to require different url parameter names for different link locations. This reference might have been formatted before that change, but this issue still bites me all the time. The same thing is also happening for Hanna 2024. There's another issue on footnote [3], that the link pagination does not match the book pagination, I assume because the pagination changed across different editions of the book. So in reading the source from its link I had to do some arithmetic to find the correct page for this note. This footnote doesn't go into much detail about what it is used for (the definition of countability) but this is such a basic and easy to source notion that I think it's ok.

Footnote [4]: I don't see Cantor's theorem on page 203 of Zermelo 1967. That page has a related theorem that a set cannot be its own powerset, and a comment that this avoids Russell's paradox. I ran out of limited-preview pages to check whether maybe Cantor's theorem itself is on a subsequent page.

The claim that Lowenheim's was "the first proof of what Skolem would prove more generally" appears unsourced (sources [5] and [6] are only to their two papers, not to any subsequent source that clarifies the priority). Reference [7] should surely be Bays 2007, not Bays 2000 (which has no pages with this number), and re-used as a named reference from reference [3] rather than given as a separate footnote. The only part of the rest of the paragraph (sourced to [7]) that I didn't see in the source is that this is "the downward form" of LS.

Reference [8]: now I see where the odd "proved by a model" formulation was coming from. Anyway, ok for its use here.

"holds for any standard model of first-order set theory, such as ZFC": the part before the comma is in footnote [9]. The specific application to ZFC is not (the source uses ZF as an example) but can be covered by repeating footnote [3].

Reference [10] led me to read footnote 3 of Bays 2007, which I think is not a correct formulation of being a singleton. Not a big deal, just an illustration of how hard it is to get these formulations correct. Page 4 of the preprint version does raise an issue we don't mention: when we say that we every element \hat m of the model is countable (even the one that models an uncountable set) do we mean that the actual elements of \hat m are countable or that the elements of the model that model elements of \hat m are countable? In a countable model both are true but they could be different in general. The (now third) formalization of our "The result and its implications" picks the second kind of countability, of the elements of the model that model elements of \hat m, and that makes sense because who cares what the elements of a model actually are (they could be natural numbers or ur-elements for all we care), but maybe we should mention this distinction? Anyway, this footnote is good for this content.

Reference [11]: Ok for the description of countability relative to a model, but not for the claim "Skolem used the term "relative"", which could I guess be sourced directly to Skolem. "He described this as the "most important" result" could also use a footnote to Skolem.

[12] Kunen: [page number needed]. [13] Enderton p152: states and discusses the Skolem paradox but does not appear to source its formulation in contemporary set theory as "countability is not an absolute property". So this claim still needs a better source.

Quote to Fraenkel, footnote [14]: our article says Fraenkel's Introduction to set theory (1928), with no page number. The source says Einleitung in die Mengenlehre (3rd ed., 1928, p. 333). Maybe we could at least clarify that this wording is a translation of Fraenkel, and give the page number for Fraenkel?

Von Neumann, [15][16]: Ok

"Zermelo ... spoke against it starting in 1929": contradicted by the source [15], which on that page only says that he gave talks about his foundational views beginning in 1929 (without mentioning any specific position on Skolem's paradox) and then on p.151 states "It is not quite clear whether Zermelo really knew Skolem's precise treat- ment in [10] or whether he was really aware of its scope when he gave his second-order definition, because he does not mention Skolem, whereas later, Skolem is the crystallization point of his criticism." and goes on to quote Skolem as suggesting that Zermelo may have been unaware of this work. It is only on [15]/p153 that one finds a direct response by Zermelo to Skolem, cited to a reference uncertainly dated to 1931.

"argued against the finitary metamathematics" [17]: cited to p.519 of Kanemori but I think this extends across pp. 519–520.

Half-paragraph beginning "Zermelo argued that his axioms should instead be studied in second-order logic", cited to van Dalen p.151": I think that this statement about 2nd-order logic can be sourced to that citation, but I do not see the remaining claims (Skolem does not apply to 2nd-order logic; Zermelo published in 1930; proved categoricity results; led to discovery of cumulative hierarchy; formalized infinitary logic) on that page of that source.

I'm not seeing "now a standard technique for constructing countable models" in footnote [21]; where is it?

"if set theory is studied using higher-order logic with full semantics, then it does not have any countable models": source [9] says only that Lowenheim-Skolem does not hold. Which I guess implies that there are no countable models but it would be better to have an explicit statement for that.

[26]: the relevant quote extends over pp. 304–305.

[28]: I did not have Google Books access to check this page of this source. But I am unsure about the phrasing "absolute countability was first championed by L. E. J. Brouwer from the vantage of mathematical intuitionism": does this mean that Brouwer was the first to connect absolute countability to intuitionism? Or does it mean that Brouwer was the first champion of absolute countability overall, despite [27]'s claims that this position was taken by Skolem himself?

"Both the Skolemites and Brouwer oppose mathematical Platonism": source [29] appears to only discuss the Skolemites and Platonists.

"Posy denies the idea that Brouwer's position was a reaction to any set-theoretic paradox": Posy speaks more specifically of "the set theoretic paradoxes discovered at the turn of the twentieth century", which suggests he was thinking of other paradoxes than this one, formulated somewhat later.

I don't see any discussion of forcing, let alone it being an extension of Skolem's paradox, on [31] Kanemori 2012 pp. 47-48 [4]. His discussion of forcing begins on p.51 and the analogy to Skolem is on p.53.

I think that's the last big read-through; once these are addressed it should pass GA. —David Eppstein (talk) 01:53, 2 September 2024 (UTC)

Hello David Eppstein. I've made all of the changes you suggested, except for looking at the Bays 2007 countability distinction, fixing my "Zermelo ... spoke against it starting in 1929" confusion, and finding a reference for the "if set theory is studied using higher-order logic with full semantics, then it does not have any countable models" claim. I'll hopefully have that sorted in a few days. As for the rest:

• I've included JSTOR and doi links where able
• I've made all titles title case. I had previously been following whatever appeared at the top of the work.
• I used the Penchev paper to back up the claim that Skolem's paradox is used outside of math/logic because I was unable to source an English translation of the other Penchev paper you suggested I take a look at, "Skolem's Paradox and Quantum Information. Relativity of Completeness according to Gödel". I could only find a Bulgarian version. If I'm able to translate it, I can replace the old Penchev reference with this one, but in the mean time I've separated the journal and publisher names like you suggested.
• For the Cantor result, I added a ref from Kanamori ("Set theory was born on that December 1873 day when Cantor established that the collection of real numbers is uncountable", 1873 being the year Cantor wrote to Dedekind describing his yet-unpublished result). To connect this result to the more general "Cantor's theorem" article I wrote a short clarifying sentence.
• You're correct that Cantor's theorem is missing from page 203 of Zermelo 1967; it is on page 200: van Heijenoort writes in the preface that "[Zermelo] then proves theorems about sets. The development goes as far as Cantor's theorem".
• For the Löwenheim–Skolem theorem, I've changed a footnote instead to reference van Heijenoort's prefaces to the articles by the two authors, which explains that Skolem "simplified" and "generalized" the Löwenheim–Skolem theorem which Löwenheim first showed.
• I've added a ref describing the "downward" theorem (Nourani, page 160: "The part of the theorem asserting that a structure has elementary substructures of all smaller infinite cardinalities is known as the downward Löwenheim–Skolem Theorem.")
• I've sourced the Skolem quotes (most important result, the use of "relative")
• I've added the Kunen page notes. Kunen describes the notion of non-absoluteness with respect to countability; you're quite right that Enderton does not describe this. I've added a second reference to replace Enderton which clearly describes it. This claim and the Enderton is another bit which I missed from an older version of the article. In fact, the claim of non-absoluteness predates the Nourani 2014 reference, so I'm a little wary of circular referencing, but the Kunen reference backs this up nonetheless.
• The van Dalen p 151 source should actually be 152. I was referencing footnote 12. This only backs up the "discovery of cumulative hierarchy" and "formalized infinitary logic" claims. For the two preceding sentences and their claims (Zermelo argued that his axioms require 2nd-order logic; Zermelo published in 1930; proved categoricity results) I've added further sources and removed the "categoricity" statement.
• Fixed the "Henkin's proof" reference. The claim "now a standard technique" is due to Baldwin, page 5, and the "for constructing countable models of a consistent first-order theory" claim is more or less explicit in a new Hodges reference.
• I slightly rephrased the statement on Brouwer and paradoxes. For the claim that Brouwer was opposed to Platonism, this is included in that Posy 1974 reference on the same page. I then added a new statement about Skolem's intuitionism. I don't believe that the source is online, so here is the quote: "I mention, only to put aside, the interesting issue of whether Skolem's intuitionism was a product of Brouwer’s influence or had some other source, perhaps the ideas of the school within which he was trained. Since, however, Skolem remarked in (1929a, 217) that the ideas of his 1923 paper were developed ‘independently of Brouwer and without knowing his writings’ I am inclined to discount the first source."

Please let me know if there are any issues with these changes.

I have a few questions about your suggestions. For the quote to Fraenkel, you suggested that I clarify that this is a translation of Fraenkel. I assume that I ought to do this for all of the block quotes, since the Skolem and von Neumann quotes are also originally German. However, since I'm not entirely sure whether this is a translation by van Dalen and Ebbinghaus, I can't write "(translated from the original German by van Dalen and Ebbinghaus)". Do you have any ideas for how to present this more elegantly? For now I've used footnotes, and I'm working on sourcing page numbers in the German originals. On a more minor note, what do you mean by "linking journal names"? And a question about footnotes, since I've never written an article of this sort before: for references to "primary" literature, specifically quoting short phrases of authors ("most important" result, etc), should I include a full quote as a footnote?

Kind regards, Pagliaccious (talk) 03:12, 4 September 2024 (UTC)

Ok, I think that's enough that I can pass this for GA now and trust you to carry on making the remaining changes as you see fit.

Re your questions: If they are translations copied from somewhere then we definitely need to say who translated them somewhere, at least in the footnotes. If they were just translated by some past Wikipedia editor then I think it's adequate to cite only the original German, as long as the citation makes clear that it is not English (for instance by using the language parameter). I don't think we need to provide extended context for the quotes in the footnotes.

As for linking journal names, I meant using something like | journal = [[The Journal of Symbolic Logic]] in the citation template (if you use the source editor), so that readers would be able to click on the link and find out some context about the journal. It's far from necessary, and should only be done when we actually have an article about that journal. But we currently only have a single link like that (to Crelle's journal on Cantor 1874) and it looks lonely.

One more really minor thing, while you're still cleaning up other stuff: Hodges 1985 is listed as being published by "CUP Archive". I don't know why web sources use that name, but this is really Cambridge University Press. —David Eppstein (talk) 06:45, 4 September 2024 (UTC)