(This question could just as well go in Cafe Society, but since it’s a question about technical books I thought it might get a larger audience here. Anyway, feel free to move it if necessary).
So this fall I’m teaching a Mathematical Foundations class (college undergrads). When I taught it before (five years ago), I just used the textbook other professors were using for the class:
Logic, Sets, and Recursion, by Robert L. Causey.
When I took a similar class years ago at Virginia Tech, we used Foundations of Higher Mathematics by Fletcher and Patty. Between the two, I prefer that textbook, but since it’s retailing in the neighborhood of $300 on Amazon, I’m very reluctant to use it for my class this fall.
The reason I prefer the Fletcher/Patty book is that it includes a decent amount of “actual mathematics” (it has an introduction to group theory (definition of a group and modular arithmetic, for example), an introduction to advanced calculus (like metric spaces and epsilon-delta proofs for limits if I remember correctly; I don’t have the book in front of me), some combinatorial stuff, set theoretic stuff like countability/uncountability).
The Causey book, on the other hand, is primarily devoted to propositional calculus and predicate calculus, with a little set theory stuff thrown in the middle (like set operations, formal definition of functions, partial orders, that sort of thing. Primarily just “definitional” stuff, but not doing anything especially interesting with it). Overall, it just seemed a lot drier to me than the other book.
(I know propositional calculus and predicate calculus are important for students learning to write proofs, I would just prefer the textbook didn’t focus on that to the exclusion of “actual” mathematics, if that makes sense. If you think I’m making a big mistake and doing my students a disservice by not wanting to focus on that extensively, feel free to give me your opinion. I’m open to the idea that I’m going about this wrong and it’s possible my mind could be changed).
Anyway, bottom line: I’m looking for a “reasonably priced” Math Foundations textbook (which, at this point, I’m thinking is in the neighborhood of $150, but cheaper is always better) that, while including basic techniques and structures of proofs, also includes a good amount of mathematics like I described above for the Fletcher/Patty book.
So: Suggestions, please?