In the chapter “The Reals, Axiomatically”, you describe the least upper bound axiom (p.19); if I understand correctly (it’s been a while since I took a maths course), you have the axiom right but the explanation slightly wrong.
“…there is one number which is the smallest number that’s larger than all of the members of that set.” I think “larger” should be “greater than or equal to”. There isn’t a smallest real number that’s strictly greater than a given value. (Unless you’re using “larger” to mean “greater or equal”, not “strictly greater”, which I think would be confusing.)
(Incidentally, can I suggest that the chapters should be numbered? They are in the table of contents, but not in the text or the PDF bookmark list.)