View source for Talk:Cardinality

{{talkheader}}
{{WikiProject banner shell|class=C|vital=yes|1=
{{WikiProject Mathematics|importance=high}}
}}
{{Archives}}
<div style="clear:both"></div>

== Merger proposal Infinite sets ==

I propose that [[Infinite set]] be merged into [[Cardinality#Infinite sets]]. I think that the content in the Infinite set article can easily be added to this page, as most of it is already on this page. 

[[User:DrWikiWikiShuttle|DrWikiWikiShuttle]] ([[User talk:DrWikiWikiShuttle|talk]]) 06:35, 26 June 2016 (UTC)

:''Oppose'': No. I do not think that its content is contained here, and it is important enough to deserve its own article. [[User:JRSpriggs|JRSpriggs]] ([[User talk:JRSpriggs|talk]]) 15:01, 26 June 2016 (UTC)
:''Oppose'' - This is a notable article, with a notable context. It should have an article of its own. [[User:ShriSanamKumar|<b><span style="color:red">♥</span>&nbsp;<span style="color:#FF7E00">Shri</span> <span style="color:#00A300">Sanam</span> <span style="color:#5cc5ef">Kumar</span> <span style="color:red;">♥</span></b>]] 17:55, 18 October 2016 (UTC)

== (G)CH and research ==

Lurking through {{ut|Jochen Burghardt}}'s talk page, I happened upon a discussion that took place last November or so, where previous text had mentioned ongoing research into CH, which had been labeled with {{tl|cn}}, then someone made the statement less specific and removed the template, then the whole thing got removed.  Jochen wrote and edit summary that said ''restore ((cn)) after sentence that hasn't changed since it was added: I doubt that Cohen.1963 (independence of GCH) leaves much room for research, but I'm not an expert''

So here's the thing:  independence from ZFC does not close an issue in set theory.  All sorts of set-theoretic research is into questions known to be independent of ZFC, and indeed specifically into the continuum hypothesis.  It would be easy to supply citations for ''examples'' of such research.  Finding a reliable secondary source for the specific statement that research continues into the continuum hypothesis would be harder, as this is at a more "meta" level, but I expect it can be done.

If we're going to continue to talk so much about CH in this article, I do think it's reasonable to mention continuing research efforts.  It seems to me that it ought to make sense to cite specific research directions and cite those, rather than needing to find a secondary source for the "meta" statement; would others find this acceptable?

On the other hand, I'm not sure that the general article on cardinality needs to deep-dive into this at all. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 20:07, 20 September 2023 (UTC)

{{ping|Willondon}}, the other participant in the previous discussion. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 20:22, 20 September 2023 (UTC)

:Seems you're referring to my post at [[User talk:Jochen Burghardt#Continuing research on cardinality?]]. My closing opinion, "I thought (well, assumed, to be honest) that there were surely open questions still. Given that nobody (including 250 page watchers) has addressed the template with an actual citation, I thought it best to delete my essentially original research." That's still my opinion: that my opinion should be dismissed, because it relied on original research by assumption. I have no valid opinion on whether or not research is settled or continues.   <small><sub>''signed'', </sub></small>[[User:Willondon|Willondon]] ([[User Talk:Willondon|talk]])  21:03, 20 September 2023 (UTC)
:: Research does continue, and citations can be found.  The real question is whether it makes sense to talk about it here (or indeed whether this article should talk so much about the cardinality of the continuum, when the article is about cardinality in general).
:: But then again, CH is sort of the first question that came up that wasn't immediately resolvable, so it ''might'' well make sense. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 21:06, 20 September 2023 (UTC)

== Cardinality is size? ==

I am the only decent antisettheorest you have.
Size is precisely defined in original for example ZFC set theory, as well as cardinality and ordinality.  The highest level as well as the original set theorists do not consider cardinality or ordinality the same as size.  In all finite cases they are the same and it is severely the case that they are mistakenly considered the same.  But in infinite cases the same sets can have different size, cardinality, ordinality.
So you Wikipedia editors are going further in their error than the original and highest level set theorists.
They define 'size' as cardinality (the definition of cardinality) to add to 'size' their lies;
 that which applies to the finite by proof does/doesn't apply to the equivalent infinite, the does and separately the doesn't cases defined. 

 And not being able to prove something true, by a particular sort of proof, proves it false.
Consider, you in Wikipedia and in the earliest original literature claim your technique in general for comparing infinite thus all set sizes applies to ≤ and ≥ and combining them by a proof =, but not < or > even though the < and > proofs look just as legitimate.  This should be articled, we (antisettheorests) should get that.
And the diagonal argument is not claimed to nor does prove ≠, but uncountability, thus none of us can prove using your sort of proof = infinite size (though I in particular can using a binary tree proof, don't confuse it or parallel it to your diagonal sort of proofs or your matching proofs), thus the set theorists are claiming disproving proof of = somehow indicates ≠.
Consider also this is how they conclude cardinal a + b is not the same as cardinal b + a.  By lack of proof of countable.  This likewise should be articled.
Thus contrary to your edit cardinality is not at all size, but rather historically is quite often considered size contrary to claims of the set theorists, that is how I want it to read.
I also want cardinal, ordinal, cardinality, ordinality, way better defined. [[User:Victor Kosko|Victor Kosko]] ([[User talk:Victor Kosko|talk]]) 23:23, 23 March 2025 (UTC)

:To Victor Kosko: Your statement is nonsense. [[User:JRSpriggs|JRSpriggs]] ([[User talk:JRSpriggs|talk]]) 04:16, 24 March 2025 (UTC)
::Per the Wikipedia rule you editors use against me your edit should be removed not being editorial comment
::I've seen set theorists comment that cardinality is not size, it's in Wikipedia cantor refused to claim it's size, and I've seen it in the original literature.
::My argument is solid, using the logic of set theorists the proof the set of even numbers is larger (> not ≥) than the set of primes is as valid as the proof their the same size, thus all the (infinite)sets of a given  cardinality are for any 2 less than, greater than, and equal in size
::Likewise exponential of infinity (aleph0) equals infinity (aleph0) in the same sense twice infinity equals infinity is easily proven.  The changes to make these not so are manipulative to make infinitism what cantor and the cantorites wanted infinitism to be, but not proven truth.
::That's not 'cardinality is only technically not the same as size', it's far from the same!
::Can't you see that. [[User:Victor Kosko|Victor Kosko]] ([[User talk:Victor Kosko|talk]]) 22:18, 26 March 2025 (UTC)
:The second sentence says cardinality may be called size "when no confusion with other notions of size is possible", and has an explanatory footnote mentioning other notions.  With that disclaimer, it's an accurate statement.  Whether we should really have it as the second sentence I'm not sure; I'm not convinced it adds much. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 22:31, 26 March 2025 (UTC)

== Infinite hotel ==

the infinite hotel analogy won't work.  Your confused by the Cantorites claiming the set of reals has a size.
IF there could be a seat on the buss for each real then likewise a room also, and the reals would be somehow countable.  Don't think of Aleph1 as if it's a natural number.  The reals are not atomizable so such a buss cannot exist, which is why the hotel cannot accommodate them not because of size.
And the infinite hotel annalogies are not about size, counting one in hotel against another is not part of the arguments.
To do that the Cantorites have to say 2 sets can be =, ≤, or ≥ size if infinite but not < of >, and claim for example Aleph0 × 2 = Aleph0.  Non of these are part of the infinite hotel arguments you just imagine they are.  Rather the infinities are just simply never-ending [[User:Victor Kosko|Victor Kosko]] ([[User talk:Victor Kosko|talk]]) 03:10, 14 April 2025 (UTC)

== Comeasurability and irrational numbers ==

I'm not convinced that comeasurability and the discovery of irrational numbers deserves to be mentioned in the history section. Just because you've found an number outside of your set doesn't mean the set has larger cardinality. For example, the [[dyadic rational]]s <math>(\mathbb{D})</math> are countable and dense in the rationals, but 1/3 is not dyadic, but that doesn't prove <math>|\mathbb{D}|  < |\Q|.</math> 

On the other hand, the fact that that the rationals are strictly smaller than the reals is reasonably important, and mentioned in nearly every source. As written, I don't think it justifies its own inclusion. Could it be rewritten in a way that wouldn't confuse the reader? If not, I'm opting to remove it. <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 23:02, 6 May 2025 (UTC)

:I think I added that (first version, at least); not because it indicated anything about the cardinality of infinite sets, but showed an early ''hint'' at the concept. The ancient Greeks knew the <s>set</s> totality of rational numbers was infinite, and yet it didn't include a number for the length of every possible line segment. The resulting kerfuffles indicate that they knew ''something'' was not quite right. I see it as an important development in the thinking on infinite sets. I'd prefer a rewrite instead of removal.  <small><sub>''signed'', </sub></small>[[User:Willondon|Willondon]] ([[User Talk:Willondon|talk]])  23:18, 6 May 2025 (UTC)

:I considered how to rewrite this paragraph. I think confusion may result from use of the word "set" here: {{tq|even the infinite set of all rational numbers was not enough|q=y}}. The Prehistory section used the layman's terms "group" (not in its mathematical sense) and "instance". The sentence in Ancient History uses "set" prematurely, since at that time, there was no concept of 'set' in its modern sense. The Pre-Cantorian Set Theory section is a better time to introduce the word. So I came up with [https://en.wikipedia.org/w/index.php?title=Cardinality&diff=prev&oldid=1289769809] "it was seen that the combined collection of all rational numbers was infinite, yet it did not have enough numbers to describe the length of every possible line segment". <!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Willondon|Willondon]] ([[User talk:Willondon#top|talk]] • [[Special:Contributions/Willondon|contribs]]) 18:52, 10 May 2025 (UTC)</small>
::That phrasing definitely helps, my only issue now is the historicity of {{tq|"it was seen that"}}. As far as I know, the Greeks didn't think about the rationals as some infinite collection, so the phrasing here seems a bit misleading. Skimming the souce cited there, it doesn't seem to say that either, unless I missed it. <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 19:39, 10 May 2025 (UTC)
::This formulation is totally misleading. It suggest that Greeks did know rational numbers, although they did not. Secondly, all that they know was that there are incommensuralble lengths, and this has nothing to do with cardinality. 
::I suggest to remove every mention of comeasurability, since this has nothing to do with cardinality: the existence of irrational numbers gives not information on their cardinality. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 20:18, 10 May 2025 (UTC)
:::I agree it was misleading, but I think the point is the fact that <math>|\Q| < |\R|</math> was a monumental result, and the Greeks showed the first hint of that fact. 
:::I also think pedagogically keeping it is more helpful. Per [[WP:ONEDOWN]], a reasonable portion of the readers of this article may not know about irrational numbers. This gives a good starting point before jumping into the less intuitive parts and helps fill that section a bit. What's currently there, using terms like "infinity" and "cardinalities" when talkng about how the Greeks thought seems anachronistic and at least as misleading as before.
:::I'll try to rewrite what was originally there to be less misleading, let me know what you think. <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 15:31, 24 May 2025 (UTC)
::::Again, what Greeks discovered is that (stated in modern terminology) there are [[algebraic number]]s that are not rationals, that is, {{tmath|\Q\subsetneq \overline \Q}}. As both sets are countable (<math>|\Q| = |\overline \Q| = |\N|</math>), this has nothing to do with the fact that the cardinality of the continuum is greater than that of the natural numbers. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:15, 24 May 2025 (UTC)
:::::Well, to be pedantic, what they discovered was the [[constructible number]]s, but that's not important. Your response doesn't address the second half of my reply. It is pedagogically useful to keep that in the section. The fact that <math>\Q</math> is dense on the number line and yet there are numbers "missing" is surprising and not common knowledge at the reading level of this article, and seems a necessary prerequisite before the next few sections. 
:::::Talking about algebraic closure and even cardinality seems anachronistic. The Greeks discovered neither of these. The key word here is "hint" at the result. I agree with you, and I am going to make it clear that they did not, in fact, prove anything about cardinality beyond vague foreshadowing. <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 16:40, 24 May 2025 (UTC)

== Diagonal argument ==

@[[User:JRSpriggs|JRSpriggs]], I'm a bit concerned about the changes in [https://en.wikipedia.org/w/index.php?title=Cardinality&diff=prev&oldid=1289765899 this edit]. Formally, it's fine, and more concise than what was there before, but pedogogically, it comes off as "we wave our hands, move some symbols around, and by magic, we've proven what we wanted" for those not accustomed to proof-based math. Worse still, it doesn't really explain why the argument is called the ''diagonal'' argument. 

The average reader of this article is likely taking their first courses with proofs, or is a math-interested high school student. I'd really prefer the old version, which explains that {{tq|"<math>T</math> cannot be mapped to since it differs from each <math>f(n)</math> at least at index <math>n</math>"}}, and takes time to emphasize the "diagonal" of the proof.

Is there something specific about the other proof that you dislike, apart from conciseness? <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 20:16, 10 May 2025 (UTC)

:"we wave our hands, move some symbols around, and by magic, we've proven what we wanted". That was the feeling that I got from the older version. If the diagonal does not work, then try a different diagonal. Sure that will fail by the same argument. But then why not just focus on the particular element T of P (A) which causes the problem? [[User:JRSpriggs|JRSpriggs]] ([[User talk:JRSpriggs|talk]]) 13:32, 11 May 2025 (UTC)
::@[[User:JRSpriggs|JRSpriggs]] {{tq|"If the diagonal does not work, then try a different diagonal."}} I'm confused by this. I suppose I could mention more explicitly "Since <math>f</math> was chosen arbitrarily, no such function <math>f</math> can exist." if that's the issue. 
::I'm also confused by {{tq|"But then why not just focus on the particular element T"}}, is that not how the older version came off? Formally, these seem like the same proof, but the newer version leaves out a lot of the intuition for the proof. My goal when I was writing it was to emphasize the intuition of the proof and downplay the formalities since those can be found in the main article.
::To be clear, I think the newer version is better for anyone with a reasonable amount of undergraduate-level math experience, but per [[WP:ONEDOWN]], the article should be written for a math-intrested highschooler. And I think the lines:
::* Then if the subset <math>T = f(t)</math> was in the image of <math>f</math>, then <math>t \in f(t) \iff t \notin f(t)</math>
::* let the subset <math>T \subseteq A</math> be given by <math>T = \{a \in A | a \notin f(a)\}</math>. If <math>T = f(t)</math>, then <math>t \in f(t) \iff t \notin f(t)</math>.
::are somewhat hard to parse for a reader at that level, and don't give much insight beyond the symbolic logic. Pedagogically speaking, a student would  question the soundess of an argument or the soundess of logic before they question their own intuition; hence my "feels handwavey" bit before.
::Is there a middle ground somewhere here? I think if you could clarify the two bits above where I'm confused, I could probably rewrite it. <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 20:05, 23 May 2025 (UTC)

== Scope of the article ==

I'm more or less finished with the elementary bits of the subject, and I'm hesitant to add much more. My goal when writing this was to make it readable for an advanced high school student. I'm aware some areas are missing, like cofinality and large cardinals, but I'm worried that (1) it would make the article less readable for those at the highschool level, and (2) it seems out of the scope of normal mathematical usage in that I have never encountered these outside of set theory itself. 
I'd suggest keeping those areas limited to the [[Cardinal number]] article, and keeping this article to the normal usage. Beyond some cleaning and rephrasing, does anyone disagree that this article covers all the main aspects of the subject, as in item 3 of [[WP:GACR6]]? <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 19:55, 31 May 2025 (UTC)

:Pinging @[[User:D.Lazard|D.Lazard]] and @[[User:Trovatore|Trovatore]]. Elsewhere, you both suggested a section on non-standard set theories. Given the above, do you still believe a section needs to be added to the article? <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 20:00, 31 May 2025 (UTC)
::Cardinality in models of AD is a very interesting topic, but I don't think there's any hurry to jam in a section on it.  It would be a useful addition at some point. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 02:26, 1 June 2025 (UTC)
:::I think this tips it for me. We could have a section about "Non-standard set theories" with subsections:
:::- Without AC
:::- With AD
:::- Constructive set theories
:::- Class theories
:::- Category theory
:::Each of these can probabaly fill 1-2 paragraphs without going off topic or out of scope, which seems like enough for a section.
:::Not sure if category theory is really appropriate, but it would be a good opportunity to talk about how equinumerosity is "set isomorphism" and therefore defines the strictest definition of "size of a set" independet of the nature of its elements. And how cardinal numbers are the [[skeleton (category theory)|skeleton]] of [[Category of sets|SET]]. 
:::Thoughts? <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 22:34, 3 June 2025 (UTC)
:::: There's not much special to say about cardinality in "class theories", except that I suppose you could treat ON as a cardinal.  Otherwise it's going to be essentially the same as the normal treatment.  Similarly, I think, for category theory, though I'm not a category theorist so I could be misinformed.  Constructivists do have some special considerations (see [[subcountability]]) but I'm not sure whether there's a concept of cardinality that neatly carries over to an article like this; it would be good to find someone expert in the subject.
:::: Pending expert advice on constructivism and categories, I think the most important topic is &not;AC and more specifically AD.  But I think there is no hurry there.  If you're trying to get the article to GA I would focus first on polishing the existing content. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 23:25, 3 June 2025 (UTC)
:::::Gotcha. I'm working on filling the History section first, I think the lead needs to mirror the body, and I need to add in-line citations, but apart from that, are there any specific things you think need polishing? <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 23:51, 3 June 2025 (UTC)
::::::{{ping|Farkle Griffen}} I think the bit that most stands out to me is the "Paradoxes" section.  To be honest I'm not sure we need that section at all in this article.  Most of it is not really about cardinality.  If it is to be kept it should be trimmed rather brutally to keep only the parts most closely related to cardinality.
::::::But that's not even the thing I'm most concerned about.  The biggest problem is that the current text reinforces the idea that the paradoxes come from the informality of naive set theory, which is not true.  The paradoxes do not attach to informal set theory based on the [[cumulative hierarchy]], and formalization is not the cure for them.  This is a problem I have tried to mitigate for years across multiple articles, with perhaps some modest success, but it is still a problem, and I would hate to see it propagate here as well.
::::::(As an aside, you mentioned that you wanted to include a subsection on "axiomatic set theory" to the "history" section.  For reasons similar to the above, I don't think "axiomatic" is a good name for that, and I'm even skeptical that we need a subsection on modern set theory, because not much that is strictly related to cardinality is different.) --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 20:24, 5 June 2025 (UTC)
:::::::What do you mean by {{tq|"Most of it is not really about cardinality"}}? I could see a case for ''Hilbert's paradox'', but the rest seem pretty related IMO? The only reason I started that section was because paradoxes are mentioned in nearly every introductory book on set theory. It's hard to make a case that they're not [[WP:DUE]] in that respect.
:::::::I see what you mean about {{tq|"the current text reinforces the idea that the paradoxes come from the informality of naive set theory"}}. Would it help to replace the lines resembling "formalization is the cure" with something more specific, like "Disallowing unrestricted comprehension or introducing proper classes are two common cures"? I'd like the article to get your point across, but I'd be more willing to add clarification than to remove large portions of that section.
:::::::To the aside, I don't plan on adding a lot to that section, only really mentioning the well-ordering theorem (briefly) and the continuum hypothesis. Both of which are tied to "axiomatic set theory". The section seems incomplete without mentioning those. I was worried "Modern set theory" would attract additions that are too modern and out of scope. <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 21:28, 5 June 2025 (UTC)
:::::::: I took another look and actually there is a case for all of them here.  But it's not about "due weight"; it's about whether it's about cardinality.  This isn't the article on set theory in general.  I would say the Hilbert Hotel is actually the one that's ''most'' about cardinality, followed by Skolem.  
:::::::: I don't agree that the well-ordering theorem and CH are specific to axiomatic set theory.
:::::::: The point I was making is that the resolution of the paradoxes isn't even really about axioms.  The paradoxes come from a conceptualization of sets as extensions of properties, rather than as collections of objects.  If you take essentially Cantor's original view (collections of objects, which themselves become objects to be collected "later") and apply it iteratively, you get the von Neumann hierarchy, without ever running into the paradoxes.  The continuum hypothesis can be formulated in this context, and it's a well-specified question without ever axiomatizing the subject in first-order logic. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 22:09, 5 June 2025 (UTC)
:::::::::{{tq|"[...] you get the von Neumann hierarchy, without ever running into the paradoxes."}} I'm confused how that changes anything? That's just another way of disallowing unrestricted comprehension: you're just limiting set comprehension up to some <math>V_\alpha</math>. I agree that viewing sets as extensions of formulas is problematic, but from personal experience, it's a fairly common view (though maybe not the majority), and iirc was one of the biggest motivations for developing a theory of proper classes. Seems like it would be an [[WP:NPOV]] issue to pretend this view doesn't exist.
:::::::::I agree that the statement of CH doesn't have much to do with axioms, but its "resolution" (if you can call it that?) certainly did. Similarly with well-ordering. The connection between well-ordering and cardinality, at least historically, has been an axiomatic issue. Are you suggesting that the history section shouldn't mention the resolution to CH at all? <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 23:02, 5 June 2025 (UTC)
:::::::::: CH is not resolved.  It remains an open question, and work on it continues.  What is known is that it can't be proved or refuted from certain particular formal theories.
:::::::::: If we treat CH in this article (I am unsure whether we should; it's a little removed from the core topic of cardinality in general) then yes, we will need to mention the independence results, but that does not mean that the question itself is particular to "axiomatic" set theory. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 23:12, 5 June 2025 (UTC)
:::::::::::Okay, now I'm confused. How can one study CH non-axiomatically if its truth is entirely dependent on the axioms you use, and is independent of the standard set of axioms?
:::::::::::I agree, I don't think the article should treat CH on its own, only a quick section on it's history: it seems like the last big event in the history of the "concept of cardinality". I think the coverage in {{alink|Cardinality of the continuum}} is enough for the article aside from the history. <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 23:27, 5 June 2025 (UTC)
:::::::::::: Its truth does not depend on the axioms.  Its provability (or refutability) depends on the axioms.  --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 23:32, 5 June 2025 (UTC)
:::::::::::::What's the difference? If I have one one set of axioms that refutes it, and another that proves it, isn't that the same as saying "It's truth depends on the axioms"? <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 23:35, 5 June 2025 (UTC)
:::::::::::::: No, it isn't.
:::::::::::::: The von Neumann universe is uniquely determined up to a unique isomorphism.  Any statement of set theory is therefore either really true or really false.  In some cases we can't prove which it is, from the axioms we have.  But you need to stop thinking of the axioms as primary.  The ''structure'', the universe of sets, is primary. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 23:39, 5 June 2025 (UTC)
:::::::::::::::Okay, I'm gonna say some things that I'm not expecting you to respond to, since I'm clearly out of my depth, but it's where my head is at. 
:::::::::::::::How? Up to what isomorphism? By Löwenheim–Skolem, I'm not sure internal claims about infinite cardinality translate between isomorphic models faithfully. Since ZFC can construct the von Neumann universe, and CH is independent, wouldn't that mean both ZFC+CH and ZFC-CH can construct it? And therefore the structure doesn't encode CH within the isomorphism?
:::::::::::::::With that out of the way, (a) do you have any resources about that? I.e., do you have a source that talks about CH in the structural way you are? This is for me personally, not really the article. (b) I'm still slightly hesitant to rename it since the whole history being mentioned is about the axioms. Even if CH can be studied independently of axioms, the results mentioned (I.e, Godel and Cohen) are about the axioms. But I'm not too tied to the name anyway. Would you prefer "Modern set theory"? I'm not a huge fan of that as I mentioned above, but I could live with it. <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 00:04, 6 June 2025 (UTC)
:::::::::::::::: I'll respond to the stuff you're asking about personally on your talk page. For the article &mdash; hmm. CH is one of the oldest questions in set theory so "modern" seems odd (by the same token, it was formulated before any axiomatization, which is another strike against "axiomatic"). Does it have to live in the history section? At a glance it's already covered in the "cardinality of the continuum" section (which itself may be too specific for this article), and for that matter there's a nod to it in the "history" section as well. I don't see any need for a new section to host it. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 00:13, 6 June 2025 (UTC)
:::::::::::::::::I'd argue that it should really have more than two or three sentences of overview. If you were trying to give a complete overview of "History of cardinality" to someone just learning the subject, I feel like there's two major items you'd mention every time: Cantor and CH. I don't think this article would make it passed a GA review by anyone in [[WP:WPM]] if we left it out just because of the prevalence of it. <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 00:29, 6 June 2025 (UTC)
:::::::::::::::::Also, to respond to {{tq|"it was formulated before any axiomatization, which is another strike against "axiomatic""}}. With the current formulation, it would be clear that CH itself is not axiomatic but Godel and Cohen's results were. I wouldn't be taking out that Cantor originally phrased CH in the "Early set theory" part. I think I would also mention [[Hilbert's problems|Hilbert's 1900 list of problems]] which he put CH first (which is another reason to have CH) and also mention Zermelo introduced the first axiomatic set theory in 1908. <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 00:44, 6 June 2025 (UTC)

::I did not suggested a section on non-standard set theories. On the opposite, I suggested to assume ZFC once for all, and to add a section on cardinality in other set theoies, only if there is a notable content for such a section. My guess is ''not''. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 08:04, 1 June 2025 (UTC)
::Nevertheless, I have many concerns with the current state of the article: 
::* It is written in a textbook tone (see [[WP:NOTTEXTBOOK]] which makes the most important poperties difficult to find; in paticular a large space is devoted to recalling basic concepts of mathematics; this is distracting, since most readers interested in infinite cardinalities know them already.
::* It does not respect [[WP:TECHNICAL]], as many elementary facts, such as {{alink|Finite sets}}, occur after very technical considerations, such as {{alink|Uncountable sets}}.
::* It is misleading as using non-mathematical terms (such as "pairing") for mathematical definitions, and using mathematical terms with a definition that is not the common one. In particular, it is said that equinumerosity is an equivalence relation on sets, when equivalence relation are commonly (and in particular in Wikipedia) defined on sets, and there is not set of all sets. This is misleading because this is the precise reason for not defining cardinal numbers as equivalence classes under equinumerosity.
::* It is also misleading by hidding most difficulties of the theory, generally related with the fact that there is no set of all sets.
::* <s>[[Aristotle]]'s (I guess) principle</s> [[Euclid]]'s axiom 5, ''"the whole is greater than the part"'', is never mentioned in the history section, although all apparent paradoxes are based on the fact that infinite sets break this <s>principle</s> axiom. This is a great idea of Cantor to accept that this <s>principle</s> axiom does not apply to infinite sets.
::I have the project to restructure the whole aticle in the spirit of the lead, but I have not the time for the moment. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 12:13, 1 June 2025 (UTC)
:::Most of these I was already aware of, and I was planning to work on them soon, don't worry. I was more concerned about the scope. Small details like these are pretty easy to fix.  <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 14:29, 1 June 2025 (UTC)
::::I disagree that these are details and that they are pretty easy to fix. In paticular, IMO, items 1, 2 and 4 require a complete restructuring and rewriting of the article. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 15:02, 1 June 2025 (UTC)
:::::What's the point of this reply, exactly? To tell me I'm wrong about my own opinion on the difficulty to make these edits? <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 15:14, 1 June 2025 (UTC)
:::Going through these after having some coffee:
:::(1) I could be convinced to remove that section, but at this point, I still think it's necessary. In my experience, online at least, most people's experience with the subject is not academic, and comes from popular online videos (especially [https://www.youtube.com/watch?v=SrU9YDoXE88 this one]) and a popular online mantra, often ignorantly repeated, that "Some infinities are bigger than other infinities" then a vague wave to "cardinality". Moreover, basic set notation and terms like injective and bijective aren't part of the usual American high school curriculum. It's two pretty short paragraphs that anyone familiar with the terms can skip, but are absolutely necessary for who I expect are the majority of readers. If you could convince me that they are not the majority, then I would agree to delete them.
:::(2) I agree that {{alink|Uncountable sets}} is technical, but I don't think there's any way around that. "Countable" and "Uncountable" are necessary introductory terms that go together and are far more important than cardinality of finite sets, which (unfortunately) is often treated as trivial, and only mentioned in passing by many books. 
:::Also, I don't think {{alink|Finite sets}} is elementary. Sure, the statement "The cardinal number of a finite set is just its number of elements" is elementary, but it's only one sentence. The rest of the details, showing that cardinality does indeed coincide with the natural numbers uniquely, and finite cardinal arithmetic (i.e., basic combinatorial principles) behaves as intended, are fairly technical. Could you provide an alternative outline for the article to discuss?
:::(3/4) This detail is already mentioned at the bottom of {{alink|Equivalence}}. Nevertheless, I have loosened the language a bit in that section.
:::(5) The History section is not complete. It is still missing and "Axiomatic set theory" section covering (at least) the well-ordering theorem and CH. And the "Ancient history" section is missing a lot of detail. I was planning on getting this done either today or within a couple days. <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 17:12, 1 June 2025 (UTC)
::::The fact that equinumerosity is not an equivalence relation is far to be a detail: this is the fundamental reason for which one cannot define cardinal numbers (and even natural numbers) as equivalence classes. 
::::The content of [[Cardinality#Finite sets|§ Finite sets]] is  elementary, since kids know it before ten, even if they do not how to prove it. It is the presentation that is unnecessarily pedantic and prevents people to recognize what they know.
::::About basic mathematical facts, such as the definition of injective, etc.: I do not oppose to recalling the definition with a "that is," after the wikilink. But more details and examples unrelated with cardinality are out of scope. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 21:26, 1 June 2025 (UTC)
:::::(a) What exactly are you advocating for here? If you would like to add emphasis, go ahead. All I'm saying is that it's already there. 
:::::(b) Sure, again, the single fact that "If a set has n elements then it's cardinality is n" is trivial. And again, that is only one sentence. The presentation says this explicitly and clearly, then in a subsection offers a proof. How is that pedantic? And again, can you offer an alternative outline for the article to discuss?
:::::(c) Great, I don't intend on adding any more details than what is necessary for the article. We seem to be in agreement here <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 22:17, 1 June 2025 (UTC)
::::::''"What exactly are you advocating for here?"'': Simply that this aticle is a confusing, not written in an encyclopedic tone, does not follow Wikipedia guidelines and requies a complete rewriting. This was true before your edits and remains true after them. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 00:44, 2 June 2025 (UTC)
:::::::I'd like to improve the article. Can you be more specific in what exactly you're looking for? For example, can you offer an alternative outline for the article to dicsuss? <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 01:29, 2 June 2025 (UTC)
::::: D., by "equinumerosity is not an equivalence relation", do you just mean that it's not a set of ordered pairs, because that would be a proper class??  I think that really is a detail.  If you ask any set theorist if equinumerosity is an equivalence relation, they'll almost certainly say yes, unless they suspect you're asking a trick question.  You can use equivalence classes as a complete invariant for equinumerosity just fine, via the [[Scott trick]].  They're just not as convenient as initial ordinals in the default context. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 21:52, 3 June 2025 (UTC)
::::::A "detail" that requires to know the theory of ordinals and Scott trick to be resolved is far to be a detail. Moreover, Wikipedia must be coherent across articles, and this is only at the end of the section {{alink|Equivalence}} that it is said that the given definition differs from that of the linked articles. Therefore, I suggests to replace the whole section {{alink|Equivalence}} by:
::::::{{blockquote|Having the same cardinality is an [[equivalence relation]] between sets, with the provisio that equivalences relations are commonly defined on sets, and the [[Class (set theory)|class]] of all sets is not a set. This means that, if one denotes with {{tmath|1=\operatorname{card}(A) = \operatorname{card}(B)}} the equinumerosity of two sets {{tmath|A}} and {{tmath|B}}, one has (these properties result immediately from basic properties of [[bijective function]]s):
::::::* {{tmath|1=\operatorname{card}(A) = \operatorname{card}(A)\qquad}} (reflexitivity)
::::::* {{tmath|1=\operatorname{card}(A) = \operatorname{card}(B) \implies \operatorname{card}(B) = \operatorname{card}(A)\qquad}} (symmetry)
::::::* {{tmath|1=\operatorname{card}(A) = \operatorname{card}(B)}} and {{tmath|1=\operatorname{card}(B) = \operatorname{card}(C) \implies \operatorname{card}(A) = \operatorname{card}(C) \qquad}} (transitivity)}}
::::::This formulation should satisfy people that accept equivalence relations on sets only as well as those that accept equivalence relations on classes [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 10:44, 4 June 2025 (UTC)
:::::::I don't think this is necessary. For example, [[Equivalence relation]], [[Relation]], [[Binary relation]], [[Reflexive relation]], etc. all say [[Equality (mathematics)|Equality]] is an equivalence relation, even though it is never defined like that formally. Further, [[Equivalence relation]] only breifly mentions sets of ordered pairs a few sections down. I don't think any reader will have difficulty with this. <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 17:10, 4 June 2025 (UTC)
::::::::What is not necessay? to warn readers that there are two common definitions of equivalence relations? or to recall the definition of equivalence relations? [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 15:29, 6 June 2025 (UTC)
:::::::::The preemptive warning. I've restructured the section to be closer to what you suggested above. Do you like the current version more? <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 20:34, 6 June 2025 (UTC)
::::::::::I maintain that the section {{alink|Equivalence}} is out of the scope of the article, and a single paragraph (such as that I have suggested above) is sufficient in section {{alink|Equinumerosity}}. Indeed, people reading this section are supposed to know a little of basic mathematics.
::::::::::Moreover this section is confusing by providing links where  the definitions are not used in the section: in the links, definitions are given only for sets, while here they are applied to [[proper class]]es.
::::::::::Also, "one may assign a representative ..." is essentially wrong, since one cannot define cardinal numbers without AC, and AC is not used for defining equinumerosity. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 11:22, 7 June 2025 (UTC)

==Section "Axiomatic set theory"==
@[[User:Trovatore|Trovatore]], I've added the section with the tentative title {{alink|Axiomatic set theory}} and tried to keep it relatively short. LMK if you have a better title or if you think changes need to be made <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 01:32, 11 June 2025 (UTC)
: At a glance (and really just a glance; this is an initial reaction) the material itself looks fine.  I still don't understand why it's in ''this'' article.  I don't believe any of it is closely enough connected with cardinality to belong here.  There could be stuff worth scattering around in other articles, but none of it seems to be about cardinality, and I think it should be removed entirely from this article. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 01:37, 11 June 2025 (UTC)
::The well-ordering theorem and CH seem pretty closely tied to Cardinality IMO, and honestly, the section just feels incomplete without it. How about we open up a discussion at [[WT:WPM]] to get some other opinions? <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 01:50, 11 June 2025 (UTC)
:::<small>I moved this new discussion into a new section, in order to allow interested people to find it easily. Please avoid adding new posts in the middle of a long discussion.</small>
:::I agree with Trovatore that this section is out of the scope of this article, and that the section must be entirely removed, except for the sentences on the continuum hypothesis, whose place is not in the history section. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 09:44, 11 June 2025 (UTC)
::::Why wouldn't it belong in the history section? Everything mentioned in that section happened ~70-100 years ago in a field that is only ~150 years old. Moreover, aside from Cantor, the independence of CH from ZFC it's arguably the biggest development in the history of set theory, and certainly the most well-known. Wording and emphasis can be changed, but to not mention it ''at all'' would leave the section missing a major detail. <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 16:59, 11 June 2025 (UTC)
::::: Of course it's hugely important to set theory.  It's not really about ''cardinality''.  It's certainly of interest to mention CH in this article because CH is stated in terms of cardinality, and at that point, yes, you mention that it's independent of ZFC.  That was already in at least one other section, two if I remember correctly.
::::: But once you start giving the history of ZFC, it seems to me that you've left off talking about cardinality.
::::: It almost seems like you're trying to turn "history of cardinality" into "history of early modern set theory", and respectfully I don't think this is the right article for that. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 18:09, 11 June 2025 (UTC)
::::::We can discuss the first paragraph on its own if that's the issue, but how exactly is the second paragraph not about cardinality? I'm defending that the whole section should not just be deleted. If the short mention in {{alink|Cardinality of the continuum}} is the motivation for deleting the section, then I can just remove that short mention.  <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 18:16, 11 June 2025 (UTC)
:::::::No, I'm saying that the short mention is in the right place and about the right amount of coverage for this article. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 19:02, 11 June 2025 (UTC)
:::::::I don't know.  The new section maybe isn't terrible in the context of the other recent expansion.  I just feel like this article has gotten a bit away from a direct focus on the topic given by the title.  It really is almost a separate "set theory" article, which is just not what I think should really appear under this title. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 19:10, 11 June 2025 (UTC)
::::::::I agree. The History section is written as if set theory would be a subtopic of the theory of cardinalities, while in reality, this is the theory of cardinalities that is a subarea of set theory. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 20:42, 11 June 2025 (UTC)

== Restructuring the article ==

As said several times in the preceding threads, this article has many issues. I have prepared a new version in view of solving some of them, which I'll implement soon. The main changes are the following.
* Moving {{alink|History}} toward the end of the article, since one must know cardinalities for understand their history
* Moving {{alink|Etymology}} as a subsection of {{alink|History}}, since the etymology is the history of the word
* Moving {{alink|Paradoxes}} as a subsection of {{alink|History}}, since the interest of these paradoxes is essentially historical
* Otherwise, I do not change {{alink|History}}, although my opinion is that many changes are needed
* Restructuring the remainder of the article into sections and subsections corresponding to the subtopics of the subject, and sorting these sections by increasing technicalities.
* Adding {{tl|main article}} templates (this was not possible with the preceding version, because of the poor structure)
* Removing trivia and other details that do not belong to this article. For example, the definition of algebraic numbers is not useful for people that know them, nor for those that do not know them, because this is not an example for them.
* Adding definitions and making them visible (no definition of a cardinality was given, or one had to scan the whole article to find it).
[[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 14:04, 17 June 2025 (UTC)

:My boldly change has been reverted with only a procedural argument (the editor did not got the time for discussing). So, I have restored my edit for giving to this editor and to others the time for reading the new version and discussing it. Please, read [[WP:BRD]], which says in particular {{tqq|If you revert, be specific about your reasons in the [[Help:Edit summary|edit summary]] or on the talk page. BRD does ''not'' encourage reverting, but recognizes that reversions happen. Try to [[Wikipedia:Revert only when necessary|revert only when necessary]] and always follow the [[Wikipedia:FIXTHEPROBLEM|editing policy]]}}. Here, the reverting editor did not provide any concern about the new version and did not discuss the above motivations. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:19, 17 June 2025 (UTC)
::Please look at [[WP:EP|the official editing policy]], specifically, [[WP:CAUTIOUS]]. Being BOLD is fine, but for large edits likely to be controversial (such as rewriting the entire article), you should give time to discuss changes first, especially if you edit is reverted. You rewrote the entire article less than 30 minutes after making this post. I am willing to discuss changes, but you did not offer ''any'' time for other editors to offer their opinion. Though, yes, I do oppose your edit and that is why it was reverted. See below. <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 19:00, 17 June 2025 (UTC)
:(1) I'm pretty hesitant to do this. For more technical topics, it's fairly common to put the history section at or near the top since they're less technical, and can help guide the reader toward undarstanding the concepts before the more detailed sections. (E.g. [[General relativity]], [[Electron]], [[Gliding]], [[Oganesson]], etc.) which seems to be a good idea for this article. Though, an "overview" section might be helpful if the History section is not adequate for this purpose.
:(2/3) The history section is bloated as is, and doesn't need more content. Including them ruins the clear chronological structure of the section. Moreover, it is fairly common to give Etymology a seperate section from History, and the items mentioned in the Paradoxes section are also of modern philosophical interest, not just history.
:(4/5) Please be more specific, {{tq|"changes are needed"}} gives me no clue on how to help you. Similarly with "Restructure by technicality"... the article currently has this structure. If you are talking about {{alink|Finite sets}} specifically as you mentioned before that can be discussed, but the rest of the article seems roughly in accordence. 
:(6) No issue with this. Though I disagree that the structure is {{tq|"poor"}}. I do prefer the article with this structure. 
:(7) I'm confused, how is a definition of "algebraic number" not helpful to {{tq|those that do not know them}}? It seems like it would be exceedingly helpful to define what they are for readers who haven't heard the term. I suppose I'm not that attached to that definition, so I could probably be convinced to remove it, but at this point it seems necessary.
:(8) A definition of "Cardinality" is somewhat hard to state uncontroversially. ''Technically'' cardinality means "size", but it is always defined relationally first (i.e. same/less/greater cardinality) then other objects, such as initial ordinals or other sets satisfying the peano axioms (or [[Scott's trick]]), are used to give it a non-relational, concrete meaning. Worse still, the exact "concrete meaning" can change depending on the author or context, which is not the case for the relational definition. This could be mentioned in the "Overview" section as above. But in any case, I think this could be resolved by a rewrite of the lead. Currently the lead does not reflect the article, and a revised version could explain this. 
:The issues mentioned here do not justfy [https://en.wikipedia.org/w/index.php?title=Cardinality&oldid=1296056537 your edit] completely rewriting of the article, removing nearly 17 thousand bytes. Please open seperate discussions for the large changes you wish to implement, rather than forcing discussions of several independent changes into one post and rewriting the whole article in one edit. <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 18:31, 17 June 2025 (UTC)