"I initially wrote this book with talented high-school students in mind, particularly those with math-olympiad type backgrounds. Some remnants of that cultural bias can still be felt throughout the book, particularly in assorted challenge problems which are taken from mathematical competitions. However, in general I think this would be a good reference for anyone with some amount of mathematical maturity and curiosity. Examples include but certainly not limited to: math undergraduate majors, physics/CS majors, math PhD students who want to hear a little bit about fields other than their own, high school students who like math but not math contests, and unusually intelligent kittens fluent in English."
These students would already know that Z is the set of integers, and if they didn't I don't think it would be a deterrent to pouring over the book.
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.
That's the stated motivation, but like the 40 hours mentioned around the same place, I have always assumed it was intended to be a bit tongue-in-cheek. The material covered here would span much of an undergraduate syllabus, and it would surely take several years even for an interested and hard-working mathematics student at a top university to understand and apply all of this effectively. Indeed, there are references to concepts that you wouldn't necessarily expect to have studied in detail or perhaps even encountered at all below postgraduate level.
I majored in math in college, and yet there were things which I could not decipher in the first few pages of chapter 1. For example, on page 43, what is "nonzero residues modulo p"? I guess you start with something, divide by p, and get a remainder, or residue. But what is that something? Going back to page 41, I see the hint that Z is the set of integers. I vaguely kind of remember that this was a thing that you learned once and just used forever. I had long forgotten that Z is the set of integers. I don't really see where this is clearly stated in this book.
If I was writing this for a high school student to skim through, I would make a big deal that Z is the set of integers, and Z is going to be used many times going forward, and it will always mean the same thing, the set of integers.
Someone who just learned all this stuff would be able to skim through it.
But the way it's written now, it's going to take a lot of intense work for a high school student, or someone who majored in math many years ago, to work through the whole thing.
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Edit: I see now. The prerequisites are in Appendix E. Technically, it's in there. But it's still problematic. High school books can just be read from beginning to end. For college level texts, it's OK to tuck things in Appendix E, and require the reader to go back and forth. So, no, I still would not say this is really aimed at the high school level.