"The fact that Z denotes the set of integers is a sort of mathwide standard."
I'm pretty sure I saw that in my high school in the early 90s, but it was a one-off event where we discussed ℕ, ℝ, ℤ, and ℚ, but we never used them for anything. I'm sitting here trying to remember our high-school set theory (which is getting cognitive interference from my college training on the topic), but my memory is claiming I either never had to write {x | x ∃ ℤ} in high school, or if I ever did, we blipped over it really quickly.
High school math generally implicitly takes place in "casual ℝ". I call it casual because the only time it even gets close to really hammering on the characteristics of real numbers is in the limit discussion. I certainly never heard "Dedekind cut" in high school.
We used Z a lot when dealing with modular arithmetic and complex roots of unity, mostly just to quantify our variables. I can't recall ever using N or Q in high school, though.
Also, you don't need to mention Dedekind cuts at all when dealing with R - it can be defined by the fact that it's the smallest extension of Q that's closed under limit-taking (and I think most high school math students do understand that).
I'm pretty sure I saw that in my high school in the early 90s, but it was a one-off event where we discussed ℕ, ℝ, ℤ, and ℚ, but we never used them for anything. I'm sitting here trying to remember our high-school set theory (which is getting cognitive interference from my college training on the topic), but my memory is claiming I either never had to write {x | x ∃ ℤ} in high school, or if I ever did, we blipped over it really quickly.
High school math generally implicitly takes place in "casual ℝ". I call it casual because the only time it even gets close to really hammering on the characteristics of real numbers is in the limit discussion. I certainly never heard "Dedekind cut" in high school.