Talk:Set (mathematics)

Equality

Somebody removed my comment in parenthesis:

...one set equals the other, so the two sets are in fact one and the same...

about what it means for two sets to be equal. Why drop it? Personally, I think it is less than obvious what equality means in broader terms. It would be easy to imagine a world where a different term is used for the same concept in the context of sets, instead of equality, such as congruence.--Jonathan G. G. Lewis 00:28, 20 March 2021 (UTC)

Recent edits seem to have removed some of the less formal language, turning the article into what reads more like a maths textbook. I am not convinced this is an enhancement, from an ordinary, non-mathematician Wikipedia readership perspective. --Jonathan G. G. Lewis 00:28, 20 March 2021 (UTC)

Thanks for the comment. It would help other editors understand what you're talking about if you could point to the specific edit that removed this language, so that we'd have the context. I assume it's this edit. The before text reads:

Another essential property of sets is that two sets are equal (one set equals the other, so the two sets are in fact one and the same) if and only if every element of each set is an element of the other.

and the after text reads:

Two sets are equal if and only if every element of each set is an element of the other;

I agree with this edit. The old language says the same thing four times:

• two sets are equal
• one set equals the other
• the two sets are in fact one
• (the two sets are in fact) the same

Would you say "two numbers are equal, that is one equals the other, so that they are in fact one number and they are the same number, if ..."

On the other hand, I find the wording "every element of each set is an element of the other" a bit confusing and I think we can improve on it. --Macrakis (talk) 00:43, 20 March 2021 (UTC)

PS It would also help if you'd sign your comments on this page with your user name, which is apparently Jonazo, not Jonathan G. G. Lewis. --Macrakis (talk) 00:46, 20 March 2021 (UTC)

@Jonazo:/Jonathan G. G. Lewis: I think I may have been the one to make some of the edits you are talking about. I would be happy to discuss specific edits here, if you like. Regarding the definition of equality you mention above, for me it was not a matter of formal vs. informal; I was just trying to simplify the wording and eliminate some redundancy, as Macrakis said. Best wishes, Ebony Jackson (talk) 01:43, 20 March 2021 (UTC)

Actually, it was other edits I had in mind. It is a question of who is the intended readership. Most non-mathematicians seem to have a mental barrier preventing them digesting maths jargon such as 'if and only if'. Any mathematician should know all this already anyway, so what is the point, unless it is intended to reach a wider readership? As for this specific case of my more verbose explanation of what set equality means, and its removal in favour of a succinct mathematical definition, my view is neutral.--Jonathan G. G. Lewis 01:09, 28 January 2022 (UTC)

Collection

The term "collection" is not defined. Is a collection and a set the same thing? Why does Collection disambiguate to Set? Comfr (talk) 02:00, 23 February 2022 (UTC)

A set is the mathematical model of what is called a collection in common language. So, the definition of "collection" can be found in any dictionary. I agree that the first sentence of the article may be confusing, and I'll try to fix it. D.Lazard (talk) 08:48, 23 February 2022 (UTC)

Informally "defining" the most basic notion of mathematics is difficult. Halmos ("Naive set theory", German translation, 4th ed., 1976) starts his first chapter like this (my translation back to English): "A herd of wolves, a bunch of grapes, or a swarm of pidgeons are examples for sets of things. The mathematical notion of a set can be seen as the foundation of conteporary mathematics." - Maybe giving a few everyday examples is a good idea: "A set is the mathematical model of a collection of things,[1][2][3] like a herd of wolves, a bunch of grapes, or a swarm of pidgeons.[4=Halmos]"?

And maybe, later in the lead, we should mention that axiomatizations of set theory use the term "set" as a basic primitive (like "point" in geometry), that it is not defined, but its properties are fixed by the axioms (like modern geometry does no longer explain what a point is, but rather how it relates to e.g. a line). So those people who are not satisfied with the informal explanation may look at the axioms. - Jochen Burghardt (talk) 17:08, 23 February 2022 (UTC)

In practice, "collection" or "family" is typically used for higher-type objects — for example, you might have some sets, and you gather them together in a collection of sets. There's nothing technically wrong with saying "set of sets", but sometimes it's useful to keep track of the type.

As an aside, the (first-order) axioms do not actually fix the properties of sets, but only some of their properties. That's why the axiomatic method is insufficient to make the objects of discourse well-specified, whereas it is arguably possible to give a well-specified informal definition. --Trovatore (talk) 20:24, 24 February 2022 (UTC)

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)