Talk:Set (mathematics)

History

Can we move the history section to the end of the article? The history section is somewhat advanced (mentioning classes, Russell's paradox, axioms of set theory), whereas the next few sections on the basic notation and concepts of set theory are more helpful for 99% of readers, I think. Ebony Jackson (talk) 02:31, 8 August 2023 (UTC)

Thank you, Ebony Jackson. By far the worst thing about Wikipedia's coverage of mathematical topics is that many of the articles are written largely by mathematicians who write as though for other mathematicians, often producing results which are incomprehensible to most ordinary readers of the encyclopaedia. It is refreshing to see someone making an attempt to reduce the extent of this problem. 👍 JBW (talk) 10:16, 4 September 2023 (UTC)

Yes, moving the history section to the end of the article, as you have now done, is a big improvement. Paul August ☎ 12:46, 4 September 2023 (UTC)

The lead of this article

Some months ago I added a "set of cows" as an example to the lead, and mentioned sets of sheep in my edit summary. It was reverted by D.Lazard with the comment "You never saw a mathematical set of sheaps, you saw a group of sheaps tha could modeled by a mathematical set (but not by a mathematicl group)". The comment above by Jochen Burghardt quoting Halmos, leads me to revisit this. I think that the notion that collections of non-mathematical objects cannot be sets but can only be modeled by sets is (1) wrong, (2) not supported by sources and contradicted by many eminent authorities such as Halmos, (3) an obstacle to the understanding of this basic concept, since it is sets of everyday objects that will be most easily understood by mathematically unsophisticated readers, (4) a source of absurdity (try to actually model a collection of cows by a set without using the notion of bijection, which is only defined between sets). So I propose to remove the assertion that the elements of sets can only be "mathematical objects". McKay (talk) 05:06, 14 March 2024 (UTC)

This article is entitled "Set (mathematics)"; so, it is about mathematical sets, not about the English meaning of the word. More precisely, the mathematical concept of a set is the abstraction of the usual concept of collection. However, intuitive examples are fundamental for understanding the concept of a set. Similarly, a line drawn on a paper sheet is not a mathematical line, although the first gives an intuition of the second, and the second is an abstraction of the first (and other examples). Confusing the physical reality with its mathematical abstraction is error prone. For example, "a set is larger than a subset" is true in everyday world, but not in the mathematical world, as soon as infinite sets are considered.

Said otherwise, if a set contains a non-mathematical object, this is not a mathematical set. D.Lazard (talk) 10:48, 14 March 2024 (UTC)

The reason for the title is to distinguish the article from other uses of the word "set", of which there are very many (see the disambiguation page set). It is not to remove from consideration some of the things that are sets. Your example doesn't work: a finite set of everyday objects behaves just the same as a finite set of mathematical objects and there is no reason to draw a line between them. Your distinction between "set" and "mathematical set" is not made by any authority that I know of. Infinite sets are a more advanced topic that is for us to explain, but there are infinite sets of everyday objects too, such as the set of all possible paragraphs. I'm waiting for others to comment, but at the moment I don't believe you have a case. McKay (talk) 00:12, 15 March 2024 (UTC)

I did pings wrong before, here is a repeat attempt: @Jochen Burghardt:. McKay (talk) 00:16, 15 March 2024 (UTC)

I seem to recall that at some point Penelope Maddy proposed an ontology whereby there were no pure sets at all; sets could contain real-world objects and other sets, but at some point the sets had to bottom out into urelements (which were, I think, supposed to be physical objects? I'm a little unsure on that point). Her work is in philosophy of mathematics, so these were definitely supposed to be mathematical sets.

In any case I think it's at least controversial to say that a set in the sense of mathematics is restricted to containing only mathematical objects. --Trovatore (talk) 01:05, 15 March 2024 (UTC)

I don't have a firm opinion yet, but here a some thoughts.

Halmos says (on the same page as my above citation) "An element of a set can be a wolf, a grape, or a pidgeon" - however, he says this on the very first page of his introduction, maybe just as an informal motivation. If I understood D.Lazard correctly, he'd have no problem using such analogies ("a set is like a herd of wolves...") in a motivation, before turning to strictly formal definitions.

Concerning the latter, I looked at the table of axioms given by Halmos at the end of his book, and they leave the question open what a set can be and what an element can be. It seems to me that the minimal model of these axioms contains only mathematical objects, more precisely: objects that can be built from the empty set (existing as a consequence of ax.2) and the infinite set (required by ax.6). However, other models may well include elements that aren't sets. Whether or not e.g. a real wolf (or the notion of it, or the reference to it, or the name of it, or whatever of it) can be such an element, seems to be a philosophical question. I feel that it can be convenient to allow real-world things as set elements, e.g. in Russell's analysis of the sentence "The present King of France is bald"; while he actually used predicate logic, one can imagine a corresponding set-theoretic argument. - Jochen Burghardt (talk) 18:57, 16 March 2024 (UTC)

More griping about the lead

While I'm at it, the lead has a more serious problem. Try to imagine an average high school student attempting to understand this. We start with "A set is the mathematical model for a collection of different things." Clicking on mathematical model, we read "A mathematical model is an abstract description of a concrete system using mathematical concepts and language." WTF? At this point our average student gives up, and yet the concept of "set" is one of the simplest to explain and an average primary school student can understand it. Now read it again carefully: this sentence says that a set of integers is not actually a collection of integers but a mathematical model of a collection of integers. Ridiculous! What the lead should start with is "A set is the mathematical concept corresponding to a collection of distinct things". Then it can continue with examples, including examples of sets of everyday objects. McKay (talk) 01:32, 15 March 2024 (UTC)

I agree to omit "mathematical model for". The article mathematical model seems to apply to e.g. sets of differential equations modelling climate, and is not too appropriate here. As for "things", this is discussed in section #The lead of this article. - Jochen Burghardt (talk) 18:01, 16 March 2024 (UTC)

I agree that, here "mathematical model" is pure pedantry for saying that, in mathematics, the concept of a set is a mathematical abstraction of the concept of a collection. Even the latter formulation must be avoided here, since the concept of abstraction is philosophy, not mathematics. So, I have changed the first sentence into

In mathematics, a set is a collection of different things; these things are called elements or members of the set and are typically mathematical objects of any kind: ...

The addition of the word "typically" allows avoiding the philosophical question of whether one can talk of a set of cows. I have revrted also the order, in the first paragraph, between infinite sets, singletons and teh empty set (set theory would not exist without infinite sets). D.Lazard (talk) 16:38, 18 March 2024 (UTC)

You do not need to say a set is a mathematical model for anything. What you can say is that a set is a collection of mathematical objects (such as numbers, points in the plane, triangles, other sets, etc.). I believe that saying using examples like "the set of colors of the rainbow" or "the set of former presidents of the United States", puts students on a path of confusion that results in them never having an understanding of sets in the way mathematicians think about them, unless and until they shake off all the sloppy things they were taught. Also, the lead definition of a "set" as a "collection of objects" is incorrect. In set theory, a collection of objects is called a *class* -- while a set is a collection of objects that is also itself an object. Some years ago, as I recall, this article said that. I have observed, however, that there has been a general decline in the technical quality of Wikipedia articles on mathematical topics, including this one. 2600:1700:3C55:F800:C88F:489A:24AC:8357 (talk) 20:28, 20 December 2024 (UTC)

Non-distinct elements

I was once chastised for thinking a set could contain duplicated elements, e.g. something like {2,2,3}. Those elements are the prime factorization of 12, and if it makes sense for them to be a set, that would be useful.
E.g., the least common denominator of two integers is simply (the product of the elements of) the intersection of their prime factorization "sets", and the product of two integers is (the product of the elements of) the union of their prime factorization "sets".

Can that be salvaged in any way?

BMJ-pdx (talk) 03:46, 28 April 2024 (UTC)

See Multiset. D.Lazard (talk) 07:49, 28 April 2024 (UTC)

Absence of interval notation

I added a section for § Interval notation. I am shocked that this article did not contain a single word about it even though I learned semantic, roster, interval, and set-builder notations in middle school. The main article about intervals even links to this article in its opening sentence that defines the term interval. Its notation deserves to be mentioned here.

I am not sure where to place the section on this page. The pre-existing sections for notations appear to be in alphabetical order, but I thought an unregistered editor like myself placing interval notation as the first in the list could be interpreted as inappropriate or disruptive. Editors who are more experienced than me would know better where to place it. 76.26.49.188 (talk) 05:23, 24 September 2024 (UTC)

I mean, I don't strictly see the problem: this article is about all sets, while Interval (mathematics) is about intervals specifically. Is there a ton of value in essentially duplicating information on that page here? Remsense ‥ 论 05:24, 24 September 2024 (UTC)

What do you say to all the other sections that have a Main article hatnote and already duplicate and summarize their relevant points? Will you be defining enumeration, or is the word interval less commonly familiar to readers? Only one sentence would have to be added to articulate a mathematical definition as other sections demonstrate. There was absolutely no mention of intervals or its notation, a notation of a type of set, on this page about sets. I, in fact, arrived here after a web search for a description of the inclusive set that uses square brackets. It directed me not to either the page that describes set notation or that describes intervals but to Bracket (mathematics), a page about about symbology, where that page gives a full 2nd-level heading to the topic of intervals that reminded me that the notation is used by a type of set called an "interval" and led me to find my way, finally, to this page that lists types of set notations. This page mentioned nothing. 76.26.49.188 (talk) 06:54, 24 September 2024 (UTC)

Intervals are only defined for sets with orderings, for example the reals. Discussion of intervals belongs in articles on mathematical order, where they are in fact discussed. --Macrakis (talk) 07:34, 24 September 2024 (UTC)

Is the empty set really a set?

If we take the definition of a set as "a collection of elements" it means that there is a contradiction, since the empty set is not a collection of elements because it has nothing. So, would it really be a set or are sets poorly defined?

A set should be: "A selection criterion for the elements that make up a collection" Greeme3 (talk) 21:19, 13 October 2024 (UTC)

@Greeme3: At the top of the article there's a note stating that the article is about "intuitive" or "naive" set theory, and contrasting that with rigorous axiomatic set theory. Rigorous axiomatic aet theory does not depend on such vague descriptions as "a set is a collection of things", and the wording you object to is a general description aimed at giving a feel for what set theory is about, rather than a definition. Unfortunately, everyday language doesn't have a word which normally refers to either a bunch of things or a bunchless absence of things, so in order to provide a description which is reasonably accessible to anyone without an advanced knowledge of the subject, it is necessary to adapt a word or expression to fit the circumstances, and extending the use of "collection" to include an empty collection is a way of doing that. Your suggestion is really no better, because if one requires the word "collection" to be used only for a collection which does actually contain some elements (as you do) then "the elements that make up a collection" can't apply to the empty set, because if there aren't any elements then there is nothing to "make up a collection". Add to that the fact that describing a set as a "selection criterion" rather than as a collection of things is less intuitive, and your suggested alternative is probably worse.

It would be perfectly possible to use some form of words describing a set as something along the lines of either a collection of things or else a nothingness, or, in the spirit of your suggestion, a selection criterion which may or may not produce elements of a collection, but for an informal introduction to the subject it would just make the description more confusing, and for a rigorous axiomatic treatment it is unnecessary, as axiomatic set theory doesn't define a set as a "collection of things" anyway. JBW (talk) 22:04, 13 October 2024 (UTC)