If the educator decides to handle the situation by declaring that the question is beyond the scope of the course, then would it be fair to ensure that the course description and course syllabus include the name of the particular system of set theory that all theorems studied in the course are deduced from?
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11$\begingroup$ I'm not sure I get the premise of the question. Why doesn't the teacher just answer "yes" or "no" if they know the answer and "I don't know" if they don't know the answer? Or, if it's somehow subtle, say "That's a subtle issue, maybe we can discuss it after the class?" $\endgroup$– Jochen GlueckCommented Oct 9, 2023 at 4:52
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5$\begingroup$ Is the course or the study program otherwise interested in foundations and different set theories? $\endgroup$– TommiCommented Oct 9, 2023 at 6:26
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5$\begingroup$ In math courses (other than foundations), there may be no mention of any particular system of set theory. Even if the course syllabus did not say anything about such systems. We cannot assume the instructor knows anything about any system of set theory. $\endgroup$– Gerald EdgarCommented Oct 9, 2023 at 8:50
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2$\begingroup$ @Adam: I'm wondering how you arrived at this conclusion. For instance, the proofs of Tychonoff's theorem via ultrafilters in point set topology, of the Hahn-Banach extension theorem in functional analysis, and of the existence of maximal ideals in ring theory, commonly involve Zorn's lemma in a very explicit way. It's completely conceivable that a smart student interested in mathematical logic sees such a proof and wonders whether one really needs Zorn's lemma (equivalently, the axiom of choice) or whether a weaker version of it suffices to get the same theorem. $\endgroup$– Jochen GlueckCommented Oct 9, 2023 at 20:20
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4$\begingroup$ This question is impossible to answer without more details. What is the class? What theorems are we talking about? Does the instructor know the answer? (Less important, but still possibly relevant: is the OP an instructor who has been asked the question, or a student who is dissatisfied with how an instructor responded when asked this question?) $\endgroup$– mweissCommented Oct 15, 2023 at 2:10
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2 Answers
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[Personal prelude: once in a model theory course we were being presented a proof of Ramsey's theorem, which I found very similar in spirit to some proofs of Bolzano-Weierstrass, so I mentioned it and asked "Are they equivalent?" (in the non-tautological sense). The professor replied they could quite see it too, but didn't know for sure, so I rapily downloaded Simpson's Subsystems of second order arithmetic, and there it was: both statements are indeed equivalent to arithmetic comprehension, over the weaker base theory. It's nice to know there is in fact a very definite way to examine such intuitive guesses, even if one doesn't follow all the details. So:]
How should an educator answer a student who asks "Can this theorem be deduced in other systems of set theory?"
Ideally with something along the lines of "There's a lot of research about provability in set theories weaker than ZFC, the most authoritative reference for theorems of 'ordinary mathematics' certainly being Simpson's SSSO, take it from there!", but this is rather unrealistic, as we can guess most people simply haven't ever thought about such things, in which case an honest answer is the best, followed by the suggestion to look it up on the internet. Just anything that's supportive of the student's curiosity, and not dismissive/hostile, really
[...] would it be fair to ensure that the course description and course syllabus include the name of the particular system of set theory that all theorems studied in the course are deduced from?
Some texts do so - most famously Kelley, Bourbaki, and SGA4, but also, say, MacLane's CFTWM and Gabriel & Demazure -, so that one may list the same system as their chosen reference. If such questions keep popping up, that's a sign the students consider it relevant, and that maybe it's not totally beyond the scope of the course after all :)
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would it be fair to ensure that the course description and course syllabus include the name of the particular system of set theory that all theorems studied in the course are deduced from?
No, because most people simply don't care. Unless we are talking about a course on set theory, the course description or syllabus is the wrong place entirely to be bringing up such an issue.
I think you are overreacting. When a student asks a question that is beyond the scope of the course and virtually nobody else taking the course would understand or even care about the question (it wouldn't even occur to them to ask such a question), you should not try to put a response to the question in a visible part of your course materials for the future. Maybe have a comment about it in your personal notes, but leave it at that and then forget about it.
I am reminded now of an earlier question on this site here in which the OP was very concerned about using "ambiguous" terms when teaching geometry (e.g., it is technically ambiguous that the word triangle means both the boundary and plane region inside the boundary) but in practice the terms are ambiguous to nobody who would ever take the course, so all the concern was largely misplaced.
