The fact that Z denotes the set of integers is a sort of mathwide standard. It comes up a lot (number theory, group theory) Did you major in some specific area of maths? Or is it just the span of years between?
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.
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).
But whatever you call it should not be too much effort to state in the beginning of a presentation or chapter or book, these are the symbols we will be using:
I = Set of Integers, ....
Sure it might be redundant but it is also easy for the reader to step over such explanations if they are familiar with them, but come back if they find some symbol they are not quite sure of. Question is are we trying to make the book easy for readers to understand, or short for the writer to write.