The elements of advanced mathematics. (Q2778953)

This is a textbook for a course that is offered in the USA to students who want to take upper-division mathematics classes (in which proofs, sets, and other abstract entities abound) as a transition course bridging the gap between the manipulative aspect of mathematics, which is likely to have been the only one students encountered (although they may have taken multivariable calculus and differential equations), and the formal-deductive aspect of mathematics (in historical terms, the gap between Babylonian and Greek mathematics). It contains chapters on basic logic, methods of proof, set theory (naive and ZF axiomatic), relations and functions, the real and complex numbers (including a section on nonstandard reals), some topology of the reals, and group theory and non-Euclidean geometry as examples of axiomatic theories. Students in a class attended by future high school teachers I taught from the first edition (1995; Zbl 0860.03001) would have liked to see more detail in the treatment and more solved problems. The second edition treats proof techniques somewhat more in-depth, but this class is considered so demanding that the changes will fail to satisfy the expectation that such a textbook will develop abstractness from the concrete elements with which students are familiar.