Talk:Set (mathematics)/Archive 3

Archive 1

Archive 2

Archive 3

Definition

I just made a bold edit, and removed the term "well-defined" from the lead of the article.

The reason I removed it, is because it appears to be contentious amongst previous editors, according to much of the discussion on this talk page, and it does not seem helpful on its own in isolation to an ordinary Wikipedia reader, who is unlikely to be familiar with the term.

Furthermore, the differences between frameworks of set theory depend critically on that concept. To enshrine ""well-defined" in the definition seems to deny the possibility of alternatives from the outset. The article on set theory is a better place for that content. --Jonathan G. G. Lewis 01:18, 9 February 2021 (UTC)

First-order logic

@Jochen Burghardt:, re this diff.

ZF is a theory of first-order logic. There is no ambiguity about this whatsoever. Second-order logic allows you to quantify over sets of individuals, but in the language of set theory, sets are individuals. The second-order language of set theory is a thing; it allows you to quantify over predicates about sets, or equivalently over classes. (As a historical aside, these distinctions were not entirely clear to Zermelo at the start, but modern usage has crystallized.)

As to your second note, while full induction cannot be expressed in first-order logic, first-order Peano arithmetic uses an "induction schema", which is not a single first-order axiom, but rather one for each formula. The Gödel incompleteness theorems are most definitely about first-order logic (and in fact do not apply to second-order logic). The Gödel completeness theorem says first-order logic is "complete" in a different sense -- if something is true in every model, then there's a proof. That doesn't contradict the incompleteness theorems at all.

I really don't think we need to elaborate on these points in this article, and I would prefer that you just removed the tags you added. But if you still think there's some clarification needed, please make a proposal. --Trovatore (talk) 19:48, 10 February 2021 (UTC)

@Trovatore: Thanks for your explanation; I agree that these issues don't belong in this article, and undid my edit (except for keeping asking for a citation for the whole section).

I wasn't aware that ZF is first order until now; thanks for your explanation.

As for Gödel's theorems, I in fact would need some more clarification (which needn't appear in this article, however): Isn't Gödel's incompleteness theorem (1931) just about the sense of completeness you mentioned? He constructs a formula and argues that it cannot have a proof (in the PM or similar systems) but must hold in the natural numbers. And doesn't he need the 2nd-order induction axiom to ensure that "every model" means "the model" (categoricity)? If his formula was violated in some non-standard model of some 1st-order arithmetic, having no proof wouldn't violate completeness, I believe(d). - Jochen Burghardt (talk) 11:05, 11 February 2021 (UTC)

@Jochen Burghardt: sorry for the delay; I'm just having a little trouble following. Maybe we're talking past each other somehow? The most basic approach to the incompleteness theorem takes a first-order theory T satisfying certain conditions (ω-consistent, recursively presented, recursively axiomatized, interprets a fragment of arithmetic), and constructs a sentence G_T such that T neither proves nor refutes G_T; that is, T is "incomplete" in this sense. For the proof to work, T must be a theory of first-order logic. Theories in second-order logic can satisfy all the other conditions but be "complete" in the sense that for any sentence in the language, they logically imply either the sentence or its negation (for example, because they can be categorical).

To see that the sentence G_T is true requires reasoning that goes beyond the first-order consequences of T itself. Second-order logic would be one such approach.

Some of the things you say are true, but I have trouble reconciling them with the note you left in the "citation needed" template.

That said, I do think the language in that part of the article is a little problematic. Right now, it says it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox. But it is possible; you just have to assume more than the theory itself. There are interpretations of the claim that make sense, but I'm a little concerned about it as it stands. --Trovatore (talk) 18:52, 15 February 2021 (UTC)

My thanks to both @Jochen Burghardt: and @Trovatore: for helping with this section. My attempt to condense the results in question is admittedly "incomplete", and quite possibly incorrect, so feel free to keep improving it, and I will not take offense. --Jonathan G. G. Lewis 09:57, 28 February 2021 (UTC)

Power set is always strictly bigger than the original set?

I doubt that this claim belongs to this article. See New Foundations, where this statement does not hold. Ladislav Mecir (talk) 12:42, 15 February 2021 (UTC) Funny how the reverting editor refused to discuss this. While the claim in the article is a theorem of the Zermelo-Fraenkel set theory (ZF), the article does not discuss ZF, where the claim would be appropriate. Ladislav Mecir (talk) 14:03, 15 February 2021 (UTC)

This article is about elementary set theory. It is a consensus among mathematicians that elementary courses are based on ZFC. Some textbooks, but not all, mention results that depend on the axiom of choice. If you disagree with the implicit use of ZFC, you have to provide references to reputed elementary textbooks that do another choice for foundations of mathematics. D.Lazard (talk) 15:19, 15 February 2021 (UTC)

This article is primarily about the informal, intuitive notion of set, which ZF applies to and NF does not. --Trovatore (talk) 18:40, 15 February 2021 (UTC)

Re "If you disagree with the implicit use of ZFC, you have to provide references to reputed elementary textbooks that do another choice for foundations of mathematics." - well, I see, e.g. this book. Although it mentions "elementary set theory", perhaps you find some reason why it does not discuss elementary set theory. Ladislav Mecir (talk) 22:49, 15 February 2021 (UTC)

I may have not found the book I wanted to link to, this seems to be an "elementary set theory" book. Ladislav Mecir (talk) 23:02, 15 February 2021 (UTC)

The full text of the book you mention, Elementary Set Theory with a Universal Set, is available online at [1]. The first chapter characterizes the set theory it presents as "an alternative set theory". It also characterizes its "superficial character" as "an elementary set theory text". That is, it isn't really an elementary set theory text, but a presentation of an alternative set theory in the guise of an elementary set theory text, presumably to demonstrate how much you can do without ZFC. So it seems pretty clear that this is not a standard elementary textbook.

What's more, its author has written the "Alternative Axiomatic Set Theories" entry of the Stanford Encyclopedia of Philosophy.

The beauty of mathematics is that you can tweak assumptions (axioms) and see where that leads. Regardless of how interesting the results are, so far, it has not lead to this approach being taught as standard elementary set theory. --Macrakis (talk) 00:05, 16 February 2021 (UTC)

I was surprised the inclusion the power set result provoked such controversy. Maybe it does not belong here after all. How about someone removes it? --Jonathan G. G. Lewis 10:03, 28 February 2021 (UTC)

I think the strict inequality is an important property of the power set, and can remain. Especially given the qualification which you've now added. Paul August ☎ 16:52, 28 February 2021 (UTC)

Basic operations

Does anyone else feel that the list of set identities in the Basic operations section is excessive? Certainly the definitions of the operations (union, intersection, complement, set difference, Cartesian product) should be retained here, but I'd be inclined to leave almost all of the identities to the main pages for these operations (Union (set theory), etc.) Maybe keep De Morgan's laws. Ebony Jackson (talk) 16:44, 14 March 2021 (UTC)

Singleton sets

I just removed the paragraph I myself had added recently:

It can easily be proved that there is at most one set that contains only itself as a member. (Different frameworks of set theory vary on whether a set is allowed to contain itself as a member, or not.) However, defining other collections intensionally involving self-containment can quickly lead to paradox. (See Russell's paradox.)

The reasons I removed it are not only that the notion of a set containing itself as a member depends on the particular framework of set theory in which you work, but the uniqueness property mentioned may also be broadly wrong, and thirdly, it is unhelpful and unnecessarily abstruse for someone unfamiliar with the subject, who is the regular Wikipedia readership. Russell's paradox is covered elsewhere.

--Jonathan G. G. Lewis 01:18, 9 February 2021 (UTC)

l agree 197.189.180.155 (talk) 20:08, 7 April 2021 (UTC)