# Appropriate uses of the set-theory tag

My background and research is in mathematical logic. I have been trying, when I can, to remove the "set theory" tag from questions that are not actually about set theory. Here is my motivation. This is, of course, my opinion, but I think it's the right one.

What is set theory? It's the subject that is covered in books by that title. So it includes the study of well orderings, ordinal numbers, and cardinal numbers; axioms for set theory; formal set theories such as ZFC; and more advanced topics. It includes more basic topics such as cardinality, the uniqueness of the empty set, etc.

What is not "set theory"?

• Venn diagrams.
• "Prove that if two cosets of a subgroup have nonempty intersection then they are the same set."
• "Prove that if $f \colon A \to B$ is any function then $f^{-1}$ commutes with the union, intersection, and relative complement operations."

These things mention sets, but that does not make them set theory. The best evidence that they are not set theory is that they are primarily studied in other areas of mathematics. The second is a common exercise in abstract algebra, the third is a common exercise in real analysis or measure theory.

Similarly, not every question involving polynomials is "abstract algebra", and not every question about continuity is "topology".

If I had enough rep, I would write something like this in the "tag wiki". Since I don't, I thought I should at least start a discussion here, because the issue has come up on at least one question.

• A somewhat interesting case is math.stackexchange.com/questions/9274/… - I did not remove the set-theory tag, although this could be a problem in a real analysis book. – Carl Mummert Nov 8 '10 at 2:20
• Another example of a question that was originally tagged as set theory: "Give an example of a function such that $f \in L^2(R)$ and $f \not\in L^1(R)$." – Carl Mummert Nov 8 '10 at 2:58

This is (once again) a manifestation of the general problem that there is no difficulty- or knowledge-level rating for questions, and in the absence of such ratings, no systematic development of meta-tags to express difficulty. As long as these deficits are present, there will be a constant need for ad hoc meta-tags inside subject tags, as in the existing [algebra-precalculus] or a hypothetical [elementary-set-theory].

For tags, the better solution is to promote the modifiers such as "elementary" or "pre-calculus" to tags that can be consistently added to all subjects, e.g., [geometry] [university] and [algebra][elementary], but not [geometry-precalculus] or [abstract-algebra] that contain subordinate metatags-within-tags.

In this case, Venn diagrams and the properties of $f^{-1}(B)$ are set theory. There is no natural assignment of those problems to a different part of mathematics (Boolean algebra, measure theory, and categories don't match), and certainly none that people studying Venn diagrams or inverse functions would be expected to identify. The logic of this question most naturally leads to an [elementary-set-theory] or [pre-axiomatic-set-theory] tag, or some similar but less precise modifier such as "naive", "informal", "intuitive", "nonrigorous" or "basic". However, this type of dichotomy appears in many other subjects besides set theory, and one would equally well want to distinguish [elementary] probability questions about counting arguments for rolls of dice, from their distributional or measure-theoretic (i.e., [advanced], [research] or [university] level) cousins.

These discussions demonstrate the need for at least one of two things to happen:

1. SE platform is upgraded to allow multidimensional ratings. This would be a huge improvement, but there is no evidence so far that the SE developers intend to provide such a feature.

2. math.SE openly and completely abandons the ill-advised "death of meta-tags", a ban that may (or may not) be useful on StackOverflow but would be negative for a math site. Development and use of metatags could then include level classification as well as other (presently non-expressible) information such as [olympiad] or [task].

• I agree those would be more robust solutions, but I'm willing to accept a separate tag as a compromise. Arturo Magidin has suggested "point-set-theory" but "elementary-set-theory" would be equally fine. "Functions" and "Relations", as separate tags, would also be OK with me. I would think the same issue will also come up in number theory. – Carl Mummert Nov 8 '10 at 12:11
• On a more personal note: I believe that certain concepts, including Venn diagrams, basic properties of functions, the most basic properties of the natural and real numbers, and so forth, are parts of all areas of mathematics, which is why they are difficult to classify into any particular field. If I had to classify truly basic facts about the natural numbers (such as, they are discretely ordered and have commutative addition), I would have to say "number theory", but that's clearly not really right. Similarly, Venn diagrams are not really "set theory", they're just common heritage. – Carl Mummert Nov 8 '10 at 12:18
• It also comes up in (for example) probability, geometry, logic, number theory, algebra, statistics, and topology. We can continue improvising an inconsistent zoo of metatags-within-subject-tags, or look for a principled solution. Multidimensional rating is clearly very desirable for users, but who knows when it might happen. Adding level, task, etc tags can be done right now by the users provided that we do not have SE overlords suppressing the meta-tags, or Stackoverflow users claiming that highly contested policies on SO should determine how a math site operates. Math is different. – T.. Nov 8 '10 at 12:22
• (The comments are not threaded, and my first comment crossed with the one above it. To disambiguate: the "it" in "it comes up in [many other subjects] ..." refers to the elementary/advanced type of dichotomies, not Venn diagrams. ) – T.. Nov 8 '10 at 12:32
• That's a compelling point. I don't know why meta tags are discouraged. I suppose we could simply start using them, and see whether any moderator does anything about it. Would there be any benefit in waiting until you, or Arturo, or some others have 10,000 rep? We would need to find the right words to use, since for example "undergraduate"/"graduate" is too subjective. – Carl Mummert Nov 8 '10 at 12:44
• As a first test there are objective metatags such as [olympiad], [task], [numerical], [specified-method], [verification-request], [original-problem], [unsourced] (or its nicer but less informative equivalent, [sourced]). – T.. Nov 8 '10 at 13:44
• Carl, I see that [elementary-set-theory] has gotten some use as a result of this discussion, including original use and not only retagging. If this metatag turns out to be useful then of course that is an argument for promoting "elementary" (or "precalculus", "axiomatic", etc) to a universal tag until direct ratings of level or difficulty provide that information without tags. – T.. Nov 16 '10 at 0:26

The fact that your third example is a common exercise in real analysis and measure theory does not mean, by itself, that it is not set theory; it means that it is a result that analysis and measure theory need. By that argument, the fact that books on linear algebra that include infinite dimensional vector spaces often ask in the exercises that the student develop the basics of cardinal arithmetc means that cardinal arithmetic is not set theory, but linear algebra...

That said, I think you have a very good point that the tag would be misused in the first two, and is often misused.

I guess the question is: what tag should we use for problems that are about what one might call "point-sets"? Say, exercises along the lines of those appearing in Halmos's Naive Set Theory, or in L.E. Sigler's Exercises in Set Theory? "point-set-theory"?

• I think that Sigler's book has exercises along the lines of your third example of "not set theory", but my copy is in my office. I'll check tomorrow. – Arturo Magidin Nov 8 '10 at 2:29
• Although most real analysis books cover cardinality, most set theory books do as well, and so "of course" we will agree that's set theory. We probably also agree that general questions about cardinality shouldn't be labeled real analysis, because they aren't a particular focus of study in real analysis. The issue with the set-theory tag seems to be with questions that analysts or topologists might call "set theory", but which aren't a focus of study in set theory either. This does seem to mostly relate to "point-set" questions, that is, questions that are just about the set operations. (ctd.) – Carl Mummert Nov 8 '10 at 2:35
• Re the third example: it's very much an example of these "point set" questions. The reason I wouldn't tag it set theory is that, well, it's not set theory, it's just a basic fact. I feel bad lumping every basic fact about union and intersection into "set theory" because they are qualitatively different than the actual focus of set theorists. Since "set theory" is the name of an actual field, it seems better to reserve that tag for questions related to the field. I think that "point-set-theory" or "set operations" would be reasonable tags for these sorts of questions. – Carl Mummert Nov 8 '10 at 2:40
• For a similar phenomenon, many real analysis books have problems asking students to compute values of particular limits of sequences or series, which can be computed using just methods from calculus. I would not, in general, want to tag these questions with the "real analysis" tag. In a sense, they are "basic enough" or "computational enough" that they should get a different tag. – Carl Mummert Nov 8 '10 at 2:43
• @Carl: As I said, you have a very good point, and I can understand your frustration. I do think we need a tag for the kind of problems we are talking about (just like we have a tag for basic algebra [algebra-precalculus] to separate it from abstract algebra). – Arturo Magidin Nov 8 '10 at 2:45
• I think we are pretty much in agreement. I'd be happy to place some other tag for these. I understand "point-set-theory" by analogy to point set topology, and I don't have any objections to it. It may be a little cryptic, but I expect that the questioners who add "set theory" in the first place will not know to look for anything else. – Carl Mummert Nov 8 '10 at 2:56
• @Arturo Magidin: how about "elementary-set-theory", which T. suggested? – Carl Mummert Nov 8 '10 at 12:12
• @Carl Mummert: That's fine by me. I guess my main point is that while I sympathize with your point, I did think we need (and do not have) some tag that identifies the questions as set-theory of some kind. "elementary-set-theory" works for me. – Arturo Magidin Nov 8 '10 at 15:37

It looks like Nuno has been performing the service of more properly tagging questions "elementary-set-theory". Thanks, Nuno.

I don't have a strong opinion regarding T..'s suggestions including having separate level and subject tags, but in the meantime this seems like a helpful distinction.

• I've tagged only 7 or so, but I'll keep tagging. I'm trying to do this slowly in order to not flood the first page with old questions. Maybe in 10 days we can finish this. Also I strongly agree with Carl. The distinction between things which mention sets and set theory is ultra necessary. Especially here that there aren't many set theory questions. Also let me know if I tagged something wrong. – Nuno Nov 21 '10 at 3:13