Appropriate uses of the set-theory tag

Question

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.

Answer 1

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].

Answer 2

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"?

Answer 3

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.