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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?

ELM
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    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?" – Jochen Glueck Oct 09 '23 at 04:52
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    Is the course or the study program otherwise interested in foundations and different set theories? – Tommi Oct 09 '23 at 06:26
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    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. – Gerald Edgar Oct 09 '23 at 08:50
  • The title of this question/thread presupposes that some theorems are part of the course content, and -- outside of foundations courses -- ordinarily only one particular system of set theory is the deductive basis for all of the proofs of the theorems of the course. Consider, for example, a topology course. The definition of what a topological space is depends upon concepts of set theory. To be able to evaluate a student's alleged proof in an exam, the instructor will know some set theory. – ELM Oct 09 '23 at 09:45
  • You could say something like: In this course we will be making use of commonly used set-theoretic concepts and notation (e.g. those given at https://en.wikipedia.org/wiki/Set_theory#Basic_concepts_and_notation ) Though you will not be required to present formal proofs citing an axiom of any particular set theory as justification for each line of proof, you will occasionally be required to make use of a particular form of the Axiom of Choice to be specified later in the course. – Dan Christensen Oct 09 '23 at 12:25
  • None of the other comments mentions constructive mathematics, but I think this is the context of the OP's question. For example, in constructive mathematics, the classically true Extreme Value Theorem for continuous is not valid. Probably, if a student has the level of sophistication required to ask such a question, she probably belongs in a higher course :-) The traditional approach is to take a rather Platonist stance with regard to the existence of mathematical objects such as $\mathbb R$, so typically a majority of students would find comments indicating otherwise rather confusing. – Mikhail Katz Oct 09 '23 at 13:46
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    "How about you find out and report your findings to me next week in office hours?" Unless this question is relevant and not a complete distraction. Honestly, it sounds like the kid got ahead of themselves reading esoterica on the Internet. – Adam Oct 09 '23 at 14:59
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    @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. – Jochen Glueck Oct 09 '23 at 20:20
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    @JochenGlueck Sure, but sometimes you get students who ask these same questions in Calc 1. (I did say unless relevant.) The OP didn't give any context about which class this was. – Adam Oct 09 '23 at 20:29
  • @Adam: Sure, I was just under the impression that your last sentence ("Honestly, it sounds like...") made an unnecessarily strong assumption. But I agree with your reaction if this occurs in a Calculus 1 course. – Jochen Glueck Oct 09 '23 at 20:47
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    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?) – mweiss Oct 15 '23 at 02:10

2 Answers2

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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.

KCd
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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 :)

ac15
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