Talk:Set (mathematics)

@Jochen Burghardt you undid my restructuring which is fair, I was on the bolder side. The current state at 15 main sections (definition until history) is way too much. I am basing this claim in comparison to peer reviewed articles such as field, 1, derivative etc. And at 3500 words short the many sections give this article a non-encyclopedia ring to it. Im positive you too see this issue.

As for HOW we might restructure it I thought of bundling them together under one section called properties. You mentioned that bijection should be defined before cardinality (which is fair), so (because the section is so short) I moved it under definition. Here's what I have:

1 Definition and notation

1.1 Roster notation

1.1.1 Infinite sets in roster notation

1.2 Semantic definition

1.3 Set-builder notation

1.4 Classifying methods of definition

1.5 Functions

2 Properties

2.1 Membership and cardinality

2.1.1 The empty set

2.1.2 Singleton sets

2.1.3 Power sets

2.1.4 Infinite sets and infinite cardinality

2.1.5 The continuum hypothesis

2.2 Basic operations

2.2.1 Principle of inclusion and exclusion

2.3 Subsets

2.4 Special sets of numbers in mathematics

2.5 Partition

3 Euler and Venn diagrams

4 Applications

5 History

Honestly we should remove the very short subsections and integrate into its supersection[?] such as empty set, singleton etc.

What I am dissatisfied with is, even with the current state, is the early placement of the continuum hypothesis. This is way beyond the purpose of properties and should be mentioned after euler diagram. Toukouyori Mimoto (talk) 15:43, 22 January 2025 (UTC)

At a glance this all seems reasonable to me. Paul August ☎ 18:16, 22 January 2025 (UTC)

I have formatted your table of content for easier reading.

This table of content is not exactly the current one, but do not correspond exactly to your above suggestion (supression of short sections). So, I suggest the following structure

1 Definitions

1.1 Specification by enumeration or by a property

Empty set and singletons belong to this section

Roster notation must described, but details must be left to the "main article"

1.2 Operations inside sets

Membership, subsets, inclusion, intersection, etc.

2 Operation on sets

Disjoint union, Cartesian product, set exponentiation, powerset

3 Cardinality

3.1 Comparizon of cardinalities

It is here that the definitions of injections, surgections, and bijections must be linked and shortly recalled

3.2 Finite sets

Inclusion-exclusion principle and Pigeonhole principle can be stated here

3.3 Infinite sets

Existence, countability, continuum, etc

4 Graphical representation

Euler and Venn diagrams, number line

5 History

D.Lazard (talk) 18:19, 22 January 2025 (UTC)

While I agree the number of sections could be reduced, I find it diffucult to come up with a structure immediately. However, each notion should be defined before its first use.

Moreover, since set theory is the most common foundation of mathematics, I'm in favor of demonstrating its formal rigour throughout the article. For example, I'd indicate that the notions of function, injectivity, surjectivity, and bijectivity are usually defined based on sets; this would mean mentioning the (Kuratowski) definition of ordered pair, cartesian product, and relation before them. (In the current article, and seemingly in both Toukouyori Mimoto's and D.Lazard's structure suggestion, the function notion is presented just informally as something external. This might be ok for other articles, to reduce their complexity, but in the set article, it misleads the reader to think "Well, math isn't just based solely on sets, it uses functions as another notion for its foundation".)

Why not start from the structure of a good set theory textbook? E.g. Halmos uses this structure: axiom of extensionality, axiom scheme of specification, pair sets, unions and intersections, complements and power sets, ordered pairs, relations, functions, families of sets, inverse mappings and compositions, numbers, Peano axioms, arithmetic, orderings, axiom of choice, Zorn's lemma, wellorderings, transfinite recursion, ordinal numbers, sets of ordinal numbers, ordinal arithmetic, Schröder-Bernstein theorem, countable sets, cardinal arithmetic, cardinal numbers [my ad-hoc translation of sections title from the German edition]. We could omit some of the later sections, join or split some of the sections, or reorder some of them (provided definition-before-use is guaranteed), but I think it is a good structure to start with. I also looked at Kamke, but he devotes only a few pages to introductory stuff and mainly focuses on cardinal and ordinal arithmetic. - Jochen Burghardt (talk) 11:00, 24 January 2025 (UTC)

Wikipedia is not a textbook, and using a a textbook structure is generally not a good idea. Here, we have three strongly related articles: Set (mathematics), Set theory and Naive set theory. Using Halmos's structure would require to merge these three article, or, at least to have too much WP:content fork between them. I am not in favor of a merge.

My understanding is that this article must focus on what is needed to understand the common usage of sets in mathematical articles. So, axioms must be mentioned only when necessary; for example, it must be said the the definition of powersets, infinite products and set exponentiation require the axiom of choice. Similarly, functions and relations are detailed in separate articles, and are needed here for comparing cardinalities. So, the section on cardinality comparison, may begin with "For comparing cardinalities, we need the concept of injective function that is defined as follows".

Also, pedantry must be avoided. This is why I uses "Specification by enumeration or by a property" is a section title, instead of "axiom of extensionality, axiom scheme of specification". The fact that these ways of specifications are formalized with these axioms does not belong to this article, although if may be said that their formalization are called axiom of extensionality and axiom of specification.

In summary, I disagree with your suggestions. D.Lazard (talk) 12:46, 24 January 2025 (UTC)

I agree. Paul August ☎ 15:43, 24 January 2025 (UTC)

I agree with your structure of placeing operations inside sets in the definitions before operations on sets. I accept this as a blueprint for improvment. But I noticed youd left out the special sets of numbers. I assume this will be defined in 1.1 and then used in 3.3 did I get that right? Im not sure if applications was forgotten but we should keep it and list a handfull of cases when sets are directly involved in other disciplines. The immediate example is the relationship between logic and sets. There may still be more to add once the primary sections are complete, I suggest a mention of ordinals after cardinalitty. Toukouyori Mimoto (talk) 15:31, 25 January 2025 (UTC)

Section § Special sets of numbers in mathematics has many issues. My opinion is that it does not belong to this article. However, it is not harmful to keep it until the other sections will be correct. So I suggest to delay the discussion of this section.

For the moment, my feeling that there is a rough consensus on the structure. So, the first task it to write the sections that have been listed. It is by doing this that one can see how to improve the structure and the content of the article.

The best way for rewrite the article is to proceed incrementally: adding one of the suggested sections without taking care whether its content duplicates parts of the existing content; discuss here the new section; then remove the old content that has been duplicated; then passes to another section. Such an edition scheme allows collaborative editing without too much harm for readers. D.Lazard (talk) 16:29, 25 January 2025 (UTC)

Yes, agreed. Toukouyori Mimoto (talk) 16:46, 25 January 2025 (UTC)

Seems reasonable Paul August ☎ 16:46, 25 January 2025 (UTC)

Agreed.   [ D.Lazard convinced me not to use a textbook structure. I assume, we all will pay attention to define (or link) concepts before use. When a concept isn't defined in detail (like e.g. ordered pair, I guess), I'd still suggest to clearly indicate that it can be defined rigorously from the previously introduced comcepts — maybe it is sufficient to state this once and for all at the end of the lead.]   I don't really understand the distinction between 1.2 and 2; e.g. would union be in 1.2 (like intersection), or in 2 (line disjoint union), and why? - Jochen Burghardt (talk) 17:37, 25 January 2025 (UTC)

"Union" is an operation inside a given set. Disjoint union creates a new set. This is an opeation on sets that is the dual of the Cartesian product, and is the analogue for sets as the direct sum for vector spaces. D.Lazard (talk) 20:47, 25 January 2025 (UTC)

Like D Lazard said, this article is about ser the object, not set theory. It is important to discuss sets in and of themselves first. Going more broadly into set theory is secondary. I think mathematics articles shouldnr, unlike history or social sciences, have background information to build up to somethingg. Thus we should mention the axioms somewhere later after all basic definitions are through. A more appropriate structure is L&L's Gale encyclopedia of science in article set theory: definitions, properties, operations, application of set theoru. This resembles D Lazard's structure. Toukouyori Mimoto (talk) 15:39, 25 January 2025 (UTC)

I just finished to restructure and rewrite the article, following the above lines. As usual, the article can certainly be further improved, and constructive improvements are welcome. D.Lazard (talk) 18:03, 1 April 2025 (UTC)

Recently, Farkle Griffen changed dramatically the beginning of the article without any explanation of the reasons of these changes. IMO, these changes are a strong disimprovement, and I'll revert them. Here are some reasons.

• Moving § Subsets before § Roster notation, without taking care that the former uses the latter. I could agree with this change, if the section would be edited accordingly.
• Removing § Definitions: This section is intended for allowing a quick access to the basic notions that are needed to understand any text on sets. With Farkle Griffen version, a reader who want a quick access to this content is confused by not knowing which section it must read for that. In fact, the content § Definitions is split, in Farkle Griffen version, into several sections, interlaced with the historical context and the definition of the non basic definition of cardinality.
• Very short sections, that break the reading flow.
• Pedantic section headings ("Membership", "Extensionality") that many reader do not understand before reading the section.
• Removal of the fundamental fact that ⁠${\displaystyle \mathbb {N} }$⁠ is an infinite set (this is the axiom of infinity).
• The sentence added at the end of the lead has nothing to do here: It is a philosophical interpretation of the basis of set theory

D.Lazard (talk) 17:13, 12 April 2025 (UTC)

These seem relatively minor and easily fixable. I don't think a revert was necessary. But I'll try to address the issues you've raised here. Though, in the future, please avoid language like "these changes are a strong disimprovment". Obviously I don't think they are, and constantly saying this only adds disdain and unnecessary heat to the conversation.

• I think the section on notation could probably be moved into the introduction, or just below it. Seems reasonable since, checking a few introductory books, they all seem to introduce this notation almost immediately after an intuitive explanation of sets.
• About 80% of the article is necessary for any text on sets. Besides, there's already a common convention on Wikipedia for this kind of thing, an "Introduction" section.
• The heading "Definitions" doesn't really help a reader understand what's in the section. Certainly clear section headings and subsections would be better so the reader can navigate to the section they need. The "Introduction" section is capable of doing exactly what you seem to want the "definition" section to do.
• The subsections are small because this article is severely lacking in content. Trying to find the best way to make the article "Flow" at this point seems worse for growth.
• I would disagree that "membership" is pedantic. It's an extremely common word. But I see your point for "Extensionality". You're right that something like "Equality" would work better. And on that, it might be worth adding the symbol to "Membership (∈)" so readers can navigate to that section if they're trying to identify that symbol.
• The removal of that was accidental, and can easily be put back. Though it might be better in a "notation" section. What's your opinion on renaming "Specifying a set" to "Notation" and including a "Common sets" subsection for these?
• The fact that sets are primitive is included in nearly every introduction to set theory or foundations that I've read. Surely that's enough reason to mention it. – Farkle Griffen (talk) 18:16, 12 April 2025 (UTC)

D.Lazard, As a side note, the fact that sets are primitive has nothing to do with philosophy. For example, imagine you're teaching an introductory proofs course and a student asks "But how is membership (${\displaystyle \in }$) defined?" The only answer is "It's not" or "Ostensively" (without also giving them a crash course in predicate logic). The term "Primitive notion" (or a synonym) is used in evey texbook on mathematical logic for cases like this. It's not philosophical, it's a completely formal and necessary term. – Farkle Griffen (talk) 18:44, 12 April 2025 (UTC)

Firstly, this page is mainly for discussing improvements of the aticle, and this section is specifically for discussing whether your edits improve or disimprove the article. So, when I consider that some edits are disimprovements, I have to say it. This does not means anything against you personally. Here are some answers to your points.

• Sections "Introduction", "Context" and "Definitions": There is no Wikipedia convention for the name of such sections. MOS:LEAD says The lead [...] should [...] establish context, explain why the topic is notable [...]. For this article, there is too much to say on the context for saying it in the lead. This is the reason for a section § Context.
  I agree that the heading "Definitions" may be confusing. So, I'll use above suggestion by Trovatore and rename it § Basic notions
  Apparently, a part of your edits was intended to merge these two sections into a section § Introduction. I disagree with this. In any case, such a restruring requires a discussion and a consensus here.
• The subsections are small because this article is severely lacking in content. Trying to find the best way to make the article "Flow" at this point seems worse for growth.: I do not think that the article is severly lacking of content; please say what is lacking in the beginning of the article. In any case, this is a bad idea to disimprove the article for making easier to expand it.
• The fact that sets are primitive is included in nearly every introduction to set theory or foundations that I've read: This article is not on set theory nor on foundation of mathematics. So mentioning this fact here is not appropriate, when it is not mentioned in Naive set theory nor in Set theory.
• The removal of that was accidental, and can easily be put back. Though it might be better in a "notation" section. What's your opinion on renaming "Specifying a set" to "Notation" and including a "Common sets" subsection for these?" The fact that natural numbers form an infinite set is not a question of notation. "Notation" is too vague for this section, since notation for inclusion and membership is defined elsewhere. "Notation for specifying a set" is unnecessarily long. Putting too many things in a section "Notation", would be confusing by separating concepts from their notation.

If some points need further discussion, please split them in different subsection, per WP:TALK#SEPARATE. D.Lazard (talk) 10:54, 13 April 2025 (UTC)

You didn't address the point that it adds unnecessary heat and disdain. "This does not means anything against you personally." I've seen you around enough to know this, but I've also seen my own and other's reactions to this... add in multiple other instances like "this is a bad idea to disimprove the article" in subsequent replies or elsewhere interactions, and other editors, newcomers especially, tend to feel WP:BITTEN even if you don't intend it. At the very least, you could dampen the blow by taking care to mention "I consider...", "I believe...", or "This seems..."

"If some points need further discussion, please split them in different subsection, per WP:TALK#SEPARATE." I'm not exactly sure what this means, or how to do it. Simply adding 3 ='s to each side of a heading doesn't seem to be working. I'll let you give an example, and I'll follow suit.

• There's not consensus or policy, but, for titles of a first section like that, "Introduction" or "Background" are the usual convention. I've never seen a section titled "Context" for this purpose.
  "Apparently, a part of your edits was intended to merge these two sections into a section § Introduction." Not quite... mostly, my issue was that you start explaining the history of set theory in a section before the section where you explain what sets are. This seems backwards. Further, the previous "Definitions" section moved extremely fast for a beginner. After a quick explanation of what sets are, it immediately moves into the principle of extension and infinite sets. I tried to split the basic "Introductory" bits, like whats sets are and why we care into one section, and other, more formal propoerties into another (hence the section "Basic properties"), trying to make the article more accessible/approachable.
• please say what is lacking in the beginning of the article. Exactly the content of the sections which you stated as being "too small".
• "This article is not on set theory nor on foundation of mathematics." This seems strange. If every book about set theory or foundations mentions this isn't enough to include that fact about sets, then what is? I don't suppose you own a book solely titled "Sets".
  "when it is not mentioned in Naive set theory nor in Set theory." Neither of those is rated GA or FA, so I don't know why we're using them as an example. This talk page is about this article, not those. Though, yes, I would agree it should be mentioned there too.

– Farkle Griffen (talk) 18:39, 13 April 2025 (UTC)

I do not want to answer to the first paragraph, per WP:TALK#contentnotcontrib, which says "Comment on content, not on the contributor or It's the edits that matter, not the editor". If you have concerns with my conduct, there are other pages for that.

By refering to WP:TALK#SEPARATE, I mean:

• If you are not satisfied by the current structure of the article, open a discussion on that (note that the above § Restructuring is about that), in which you will detail and motivate the structure that you suggest. Note also that section § Context has already be discussed in the above section § Context.
• If you are not satisfied with some section name, open a specific discussion. Note that the names § Definition, § Definitions and § Basic notions have been discussed in § Note on formal definition.
• If you think that it is important to mention in the lead that a set is a primitive notion in § Axiomatic set theory, open a discussion about that. Note that this is already in section, § Axiomatic set theory.
• If you have other concerns, open a discussion on each of them.

D.Lazard (talk) 10:24, 14 April 2025 (UTC)

The article currently has a large amount of material that I'm not sure fits completely comfortably at an article whose topic is just supposed to be sets (not even set theory). In particular the stuff on the cardinality of the continuum seems excessively detailed for this article.

But putting that aside for now, it also contains an assertion that I would like to inquire into. It's the claim by D.Lazard that the uncountability of the reals was rejected Prof Lazard originally wrote "refused" by mathematicians for many years after Cantor originally proved it.

I am not sure that is true. It's certainly true that there were vigorous objections, by Kronecker and Brouwer and so on, but I would like to see evidence regarding a general lack of acceptance, or even a lack of general acceptance, which I suppose is a different thing.

It is possible that the whole section should be trimmed back to the point that we don't need to address this here, in which case I would still be interested in hearing more about the question, perhaps in some other venue. --Trovatore (talk) 07:46, 28 December 2025 (UTC)

Yes, Trovatore, that statement is questionable. At the very least it needs clarification: does "rejected by mathematicians" mean "rejected by some mathematicians", "rejected by many mathematicians", "rejected by most mathematicians", "rejected by all mathematicians", or what? I suspect the truth is in the first half of that list, but it is very likely that many readers will take it as meaning something in the second half. Also I may be wrong: the truth may really be in the latter half; either way it is unsatisfactory to have the statement in a form which is so easily liable to misinterpretation. Perhaps either D.Lazard or someone else can clarify the statement, and also provide a citation for it; if not, as unsourced and unclear content which has been challenged, it needs to be removed from the article. JBW (talk) 00:56, 4 January 2026 (UTC)

I suggest to rewrite the end of the sentence as ... these results were so astonishing that they were rejected by many mathematicians, and several decades were needed before their common acceptance., and to move the paragraph after the next one. Indeed, Indeed, the equinumerosity of the line and the plane was certainly much more difficult to accept D.Lazard (talk) 11:58, 4 January 2026 (UTC)

A (couple of) citation(s) about the discussion would be great. I'd suggest to collect them here, for now; lateron we may move them to an own section at Cardinality of the continuum, at Cantor's diagonal argument, or at Cantor's first uncountability proof. Moreover, I'd suggest to keep a brief hint (like D.Lazard's sentence) here in Set (mathematics): To my experience, many non-mathematicians make fun of set theory and doubt it can have any nontrivial results - uncountability of the reals is one counterexample that may still be unterstandable to them. - Jochen Burghardt (talk) 12:49, 4 January 2026 (UTC)

I feel that it is appropriate to continue here a point recently discussed on the talk page of the Sequence article, which applies to the opening sentence of this article. @Jacobolus wrote that the word collection is not precisely defined. I responded by pointing out that outside of any specific academic community the word has very clearly understood meaning. The opening sentence of this article states that a set is a collection. How can this sentence be improved to account for the generally understood meaning of the word collection? Kevincook13 (talk) 17:08, 10 January 2026 (UTC)

I added the word "called" for clarity, but other than that, I think the first sentence is good as it stands. In many formal treatments, "set" is a primitive notion, not defined in terms of other things, so it doesn't make sense to try to define it precisely here. Calling a set a "collection of different things" conveys well what a set is supposed to be.—Ebony Jackson (talk) 18:25, 10 January 2026 (UTC)

I think you dramatically underestimate ordinary people's ability to reason by analogy without applying extreme levels of misplaced nitpicking pedantry. Most people don't come to math articles brandishing their etymological dictionaries as weapons, and are willing to accept that some abstract formal topics cannot be exactly defined in a single plain-language sentence. –jacobolus (t) 19:32, 10 January 2026 (UTC)

Geach referred to the idea that one can't have any knowledge of something unless one can precisely define it as "the Socratic fallacy". Some philosophers disagree that this is a fallacy, but I am with Geach on this. Often, as in this case, giving a general and imprecise description of what one is referring to is much more helpful than trying to make a rigorous definition. JBW (talk) 22:57, 10 January 2026 (UTC)

I agree that the sentence does not need to convey meaning in great detail, but it needs to be accurate. I also agree that it conveys well what a set is supposed to be. Let's plainly write that a set is supposed to be a collection, and that a finite set is a collection. Kevincook13 (talk) 00:58, 11 January 2026 (UTC)

Let's move the existing first sentence of this article to become the new first sentence of Finite set, changing set to finite set. Kevincook13 (talk) 13:20, 12 January 2026 (UTC)

Oppose: The 1st sentence here is fine. @Kevincook13: I'm afraid I don't get you point - do you mean that a collection is always finite, or that a set is always finite, or that collection should have a formal mathematical definition, or what else is your problem with this article? - Jochen Burghardt (talk) 13:52, 12 January 2026 (UTC)

The word collection is one of the words we use to specify finiteness: antique cars are cool, but this collection is the best. Kevincook13 (talk) 03:56, 13 January 2026 (UTC)

I don't see why if you had infinitely many antique cars, assuming you could, it wouldn't be a "collection". To be honest I'm losing patience with this discussion. If you really want to move forward with this idea that seems to be intuitive to you and no one else, please provide sources that would be usable in the article. --Trovatore (talk) 06:39, 13 January 2026 (UTC)

Sources abound regarding the meaning of the word collection. It is a tool which belongs to everyone, and in this article is being used incorrectly. Kevincook13 (talk) 15:24, 13 January 2026 (UTC)

Yes, Sources abound regarding the meaning of the word collection, e.g. "a group of things". What is the source for your assumption that a collection must be finite? Paul August ☎ 17:18, 13 January 2026 (UTC)

The word finite is not so well understood: a finite music collection might be misunderstood to refer to a new "finite" genre of music. You are right that sources do not abound which publish the words "a collection must be finite". The word finite is primarily used in mathematics, which uses the word collection differently. Kevincook13 (talk) 18:07, 13 January 2026 (UTC)

Mathematics does not in fact use the word "collection" differently, and in ordinary usage finiteness is not in fact part of the meaning of the word "collection". It is true that the collections most people are likely to think of are finite, but that is a different thing.

Looking back at another discussion you contributed to, I suspect that the disconnect here is that you don't believe in completed infinity. I'm afraid that's a you problem. Mathematics does believe in it, to the extent that mathematics can be said to believe in anything. That is, completed infinities are used routinely in mathematics and considered unremarkable. --Trovatore (talk) 20:10, 13 January 2026 (UTC)

Those first two points are essentially the same point, because if finiteness is part of the meaning of the word collection, then mathematics does use the word differently. This can be seen in the second sentence which states that a set may be finite or infinite.

I believe that defining a set as a collection which may be finite or infinite is the reason that the word finite is ubiquitous in mathematics, and evidence that mathematics is using the word collection differently.

I believe that the word completed is also used to specify finiteness. If so, the phrase completed infinity is finite infinity. It shouldn't be surprising that some people might not believe it. The challenge for mathematics is to more effectively communicate the details of what is so unremarkable to you.

I believe that it is uncontroversial that a finite set is a collection. Let's plainly say so! Kevincook13 (talk) 13:40, 14 January 2026 (UTC)

Yeah, so you're basically just wrong. --Trovatore (talk) 19:04, 14 January 2026 (UTC)

I think it is much more common to use "collection" as a synonym for "set" than as a synonym for "finite set". So I think it is fine to describe a set as a collection of different things, and to leave the first sentences of Set_(mathematics) and Finite set as they are. —Ebony Jackson (talk) 16:09, 12 January 2026 (UTC)

"Collection of objects" is fine. It's used in countless introductory textbooks with no asterisk about finiteness. That said, I can sympathize with @Kevincook13 in being frustrated with only a vague, informal definition.

It could be worth adding a section "Definition" since a clear, formal definition isn't trivial. For example, "The collection of all sets that don't contain themselves", certainly seems to fit the definition here. Similarly, "The collection of all ordinal numbers". There could also be more effort given to convince the average reader about the non-contradictory nature of infinite sets, since the average reader likely has no experience with actually infinite collections, and is probably skeptical of them.

One difficulty though is that in nearly all foundations, "set" is never explicitly defined, but taken as a primitive notion. – Farkle Griffen (talk) 20:41, 14 January 2026 (UTC)

A vague, informal definition is exactly all we should have here. --Trovatore (talk) 04:20, 15 January 2026 (UTC)

@Trovatore, I'm not against having an informal definition as an introduction to the subject. To be clear, you're saying you oppose including any definition that's not a "vague informal" one, anywhere in the article? – Farkle Griffen (talk) 04:41, 15 January 2026 (UTC)

Yes. --Trovatore (talk) 06:11, 15 January 2026 (UTC)

(I should say, though, we could get somewhat less vague. For example we could start with the Cantorian description from the Beiträge and/or the Grundlagen, and segue into the von Neumann hierarchy. But "set" as you note is generally taken as a primitive notion, so these are always going to be informal in some sense.) --Trovatore (talk) 06:14, 15 January 2026 (UTC)

The axiomatic method allows us to be perfectly precise without giving a definition. In geometry, Euclid's original definitions of point, line, plane have turned out unnecessary in the presence of Hilberts axiomatization. - Jochen Burghardt (talk) 10:49, 15 January 2026 (UTC)

First-order axioms, like ZFC, can't characterize sets. This article is supposed to be about sets, not about set theory. Of course we will mention ZFC, but I don't think we should hit it heavily. (A second-order axiomatization can characterize sets up to, say, the rank of an inaccessible cardinal, but I think that goes even further beyond what we probably want the scope of this article to be.) --Trovatore (talk) 20:29, 15 January 2026 (UTC)

@Kevincook13: I just wanted to point out that the article already contains several citations for saying that a set is a collection of different things. This is the common understanding of what a set is outside of Wikipedia, so I don't think this should be changed, unless there is a preponderance of notable sources suggesting otherwise. —Ebony Jackson (talk) 17:58, 15 January 2026 (UTC)

My edit is supported by all of those sources, and aligns with the consensus expressed here that we do attempt to clearly define the term. Kevincook13 (talk) 22:19, 15 January 2026 (UTC)

do not attempt Kevincook13 (talk) 22:19, 15 January 2026 (UTC)

What you are objecting to is being more forthright with our readers. Kevincook13 (talk) 03:51, 16 January 2026 (UTC)

We have an important decision to make. Either we change our consensus and define the term in the opening sentence, however broadly, or in that sentence we inform our readers that we are not defining the term. Kevincook13 (talk) 15:26, 16 January 2026 (UTC)

We are not defining the term. ZFC does not define it either, just as Euclid does not give a meaningful definition of point. I think the appropriate way to address your concern would be not to change the lead sentence, but to mention when Zermelo-Fraenkel set theory is introduced that it does not define what a set is, but that it is a system of axioms intended to express how they behave, intended to capture properties that agree with our intuitive understanding of collections. —Ebony Jackson (talk) 03:26, 17 January 2026 (UTC)

As I say above, I don't think we should hit the formal-theory aspect hard in this article. This article is about sets, not about axiomatic set theory. And you really can't substitute a first-order theory for a characterization of sets. For the more advanced parts of the article, the von Neumann hierarchy gives a better template for what I think should appear here than ZFC does (though certainly we will at least mention ZFC). --Trovatore (talk) 04:55, 17 January 2026 (UTC)

What I mean by defining the term, is writing the first sentence of this article so that people will have some idea of what a set is. We don't have to go into great detail.

By stating that a set is a collection, the first sentence currently conveys the idea that a set is essentially the same as a finite set, which would be fine if it were true. A reader who understands the meaning of the term finite will be confused by the second sentence, which states that a set may be finite or infinite, because a finite set cannot be finite or infinite. Most readers will have no idea what the second sentence means.

My edit informs readers using plain language. Kevincook13 (talk) 15:38, 17 January 2026 (UTC)

I made some edits to address your concern about being clear that we are not defining "set". For most people, "set" and "collection" are synonyms. In any case, the first paragraph makes it clear that sets are allowed to be infinite, so there should be no confusion. —Ebony Jackson (talk) 19:28, 17 January 2026 (UTC)

You are right that for most people, "set" and "collection" are synonyms, and I now agree with your undoing of my edit. I think that I understand better now. A set is essentially the same as a finite set. Let me explain. Mathematics uses the term formal to signify class or description, rather than collection or object. Often, the term set is used to to refer to something other than a collection. The word formal signals this other use of the term. This other meaning is class or description.

Axioms are descriptions, rather than objects. It is true that axioms can uniquely describe objects, but axioms are neither objects nor some sort of substitute for objects, as if objects would not be necessary if axioms were understood. The concept of Object is what gives meaning to the concept of Axiom. Kevincook13 (talk) 15:42, 21 January 2026 (UTC)

I have reverted your changes to both this article and Finite set, which were based entirely on your personal ideas, rather than on reliable sources, and which have repeatedly failed to acquire any support from other editors. --JBL (talk) 18:49, 21 January 2026 (UTC)

Notice that with the help of my edits it is plainly evident that both this article and the Finite Set article describe exactly the same concept, and should be merged. Also notice that what I have written is abundantly supported by reliable sources! Kevincook13 (talk) 20:28, 21 January 2026 (UTC)

With the help of your edits it is clear that you don't understand mathematics very well, and are just a WP:FRINGE WP:POV-pusher who is unwilling to abide by consensus; if you attempt to reinstate any of your edits, you can expect to be taken to the administrator's noticeboard. For complete avoidance of doubt, what is your opinion about the fact that every other editor who has commented has opposed your edits (including recently but also to your past edits going back many years involving the article 0.999...), and whether that will prevent you from continuing? --JBL (talk) 20:33, 21 January 2026 (UTC)

Well, you just reverted my edit to the short description,in which I changed the word mathematical to distinct. Did you do that only because of your assessment of my personal understanding of mathematics? Kevincook13 (talk) 21:25, 21 January 2026 (UTC)

JBL reverted to the last version prior to your two most recent edits. That incidentally undid the change to the short description. I personally think "collection of distinct objects" is a reasonable short description for this article, but the version with "mathematical" does have the advantage of making it clear that this is a math article. Since short descriptions are mostly for giving broad context to mobile users rather than precise definitions, that is an important advantage. --Trovatore (talk) 21:43, 21 January 2026 (UTC)

The objects are not all mathematical. Kevincook13 (talk) 21:49, 21 January 2026 (UTC)

But the article is. We could say "mathematical collection of distinct objects" but that's 44 characters, slightly over the "soft limit". Maybe "mathematical collection of objects" would be slightly better. I think any of these would be acceptable and it isn't really worth worrying about too much, but I do see at least a small advantage in including the word "mathematical" in the SD. --Trovatore (talk) 22:10, 21 January 2026 (UTC)

The word mathematical is already part of the name of the article. The word distinct greatly clarifies meaning, providing value to our readers. I will put it back now. Kevincook13 (talk) 22:28, 21 January 2026 (UTC)

I personally am OK with this change. --Trovatore (talk) 22:37, 21 January 2026 (UTC)

Me, too. --JBL (talk) 23:35, 21 January 2026 (UTC)

I think that I can see how it may seem that my edit was based on my own personal ideas. I am trying to write to a general audience. I write about your topics, but not in your language, and not in language found in your reliable sources. But they aren't just your topics. These same topics have been written about extensively in very different kinds of reliable sources. Wikipedia challenges people like us to bridge gaps of understanding between various communities, which keep largely to themselves, and yet who routinely deal with the exact concept this article attempts to describe. Kevincook13 (talk) 15:33, 23 January 2026 (UTC)

All well and good. But whatever content you wish to add is required to be supported by reliable sources (as defined by WP:RS). Paul August ☎ 19:06, 23 January 2026 (UTC)

I am going to go ahead and make some changes. Before I do, I want to give some explanation here.

This article needs focus. Articles should primarily be encyclopedic, rather than lexical. In other words, it should be focused on a concept, rather than on a term. Although the concept described to a general audience in the first sentence of this article is not the only concept referred to in mathematics by the term "set", it is still appropriate to have an article dedicated to this specific concept.

Other concepts also referred to as sets should, or already do, have articles dedicated to them, such as Hereditary set. Kevincook13 (talk) 17:53, 28 January 2026 (UTC)

@Kevincook13: Are your changes going to be supported by citations to specific sources that other editors will agree qualify under WP:RS? Because if not then I can tell you in advance what the response is going to be. --JBL (talk) 18:17, 28 January 2026 (UTC)

I suggest that we remove this article from the Set disambiguation page, because this article fails to help disambiguate the term. This article does not attempt to describe any one definite and separate object of our intuition or thought, and the collection of concepts it does attempt to describe most likely already have a Wikipedia article of their own. I am thinking specifically of: Finite set, Infinite set, and Hereditary set, which I suggest we add to the disambiguation page. Kevincook13 (talk) 13:48, 29 January 2026 (UTC)

I don't think you understand the purpose of disambiguation pages. Paul August ☎ 15:48, 29 January 2026 (UTC)

I agree with the above comment: this isn't a proper use of a disambiguation page. They are for resolving confusion about a potential article title having multiple meanings. Not including Set (mathematics) in the list of articles called Set (something or other) would be pathological. Stepwise Continuous Dysfunction (talk) 20:52, 30 January 2026 (UTC)

Participants in this discussion may be interested in this noticeboard thread I have opened: Wikipedia:Administrators'_noticeboard/Incidents#Kevincook13. --JBL (talk) 20:52, 29 January 2026 (UTC)

Would it not be more appropriate in the article, where ZF is mentioned, to refer to ZFC instead as the standard way to provide foundations? Ebony Jackson (talk) 03:29, 17 January 2026 (UTC)

The article Zermelo–Fraenkel set theory mentions both ZF and ZFC. I agree with Trovatore that we shouldn't dive too deep into the formal details here; so the lead sentence "Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations ..." is sufficient, imo (we can't mention AC already there, anyway). I agree with Kevincook13 that we should mention that set theory has a formal foundation. The lead sentence already does this, and I'd like to supplement that the notion of set, being at the bottom of the foundation, can't have a definition in terms of more primitive notions. I'd also like a footnote referring to the similar situation in Euclidean / Hilbert geometry (obtained from Ebony Jackson's post above). - Jochen Burghardt (talk) 18:59, 17 January 2026 (UTC)

That all makes sense. I have attempted an edit, with footnotes. —Ebony Jackson (talk) 19:30, 17 January 2026 (UTC)

This looks reasonable. I would prefer a different term than "governed", which suggests the axioms are in charge. Maybe "described"? --Trovatore (talk) 21:37, 17 January 2026 (UTC)

Made a tweak. --Trovatore (talk) 22:23, 17 January 2026 (UTC)