Mathematics Educators Stack Exchange is a question and answer site for those involved in the field of teaching mathematics. It only takes a minute to sign up.
Sign up to join this community
In teaching elementary set theory, suppose we refrain from emphasizing historical decisions that were made in theory construction.
Is there a danger that students may see the mathematical language of set theory not as a resource to help formulate a given idea, but as a framework that restricts the range of ideas that can be considered coherent and meaningful? Is there a danger that the set theory could become a bed of Procrustes for their thoughts?
This question is mixed with a lot of speculation and commentary, so I'm going to try to focus on the questions.
Should the limits of one system of set theory be the limits of a student's mathematical world?
No. The idea that a system of set theory should be a limit to mathematical ideas seems like poor pedagogy (as arbitrary restrictions usually are), and is totally at odds with mathematical practice. I've never heard of anyone teaching such a way, but if someone is, I'm skeptical that the benefits outweigh the disadvantages.
In teaching elementary set theory, while refraining from emphasizing historical decisions that were made in theory construction, is there a danger that students may see the mathematical language of set theory not as a resource providing apparatus to help formulate a given idea, but as a framework that restricts the range of ideas that can be considered coherent and meaningful?
Of course it's always conceivable that students will misunderstand something, but I'm skeptical of the value of obsessing about every conceivable misunderstanding that might happen. My experience has been that constantly trying to head off confusion before it happens just leads to students being more confused by all the digressions. Holding and then discarding misunderstandings is a natural part of the learning process; it's more useful to teach with enough dialog so that if students have this-or another-misunderstanding, there's an opportunity to learn that they think this and address the matter then.
In particular if students are taught precisely one and only one system of set theory -- with enough application of it to mathematics that it becomes like their first language, and they know of other languages of set theory primarily from hearsay -- then is there a danger that the set theory they learn could become a bed of Procrustes for the thoughts that sprout from their minds, like arms or legs growing from the body?
I suppose anything is possible, but this seems fairly unlikely. The jump from "ideas can be formalized in set theory" to "all ideas must be formalized in set theory all the time" is pretty big, and I don't see any reason an ordinary mathematics education should lead students to make such a leap. I also don't see any reason that learning other set theories is relevant; for most topics where there are issues of how to formalize them in set theory, those issues are going to persist in other systems of set theory as well.
Set theory has no important implications for 99% of normal mathematics, and the difference between, say, ZFC and NF has even less. Mathematics isn't generally written in the language of ZFC, it's written using more basic ideas and notation that are the same in other foundational systems. If I'm teaching freshman calculus and I talk about $\{x|x>0\}$, there is no implication of ZFC; the thing I'm describing would have been understandable to Euclid, if he had been willing to think of a ray as a collection of points.
But if you work in foundations of mathematics and teach set theory to upper-division math majors or math grad students, then what's stopping you from simply adopting the philosophy you seem to prefer, which is to clearly motivate the choices of axioms and definitions by contrasting them with examples from other systems of other ways that it could have been done? I think I probably would have enjoyed it if the set theory course I took had been taught that way.
