I'd like to learn formal math. Preferably, though not necessarily, starting with predicate logic/first order logic rather than higher order logic. I am trying to find resources (papers, books etc.) for doing this, but I haven't found anything I really like.
There are lots of resources for predicate and first order logic, but most do not approach the topics in a very formal way. For example, many text don't seem to try to define what they mean by "variables" or mention substitution as an important concept. Tries to explicitly describe as many of the rules of the game as it can. Many texts bring up "truth tables" without having formal rules for what you're allowed to do with those tables.
Does anyone have resources that fit these criteria?
Edit: many of the answers are good and helpful, but I feel like I should add some clarifying remarks:
Many texts mention that you can view math as merely manipulation of symbols. I don't doubt that this can be done, but I would like to see it done. A resource that explains the process of producing proofs explicitly in terms of manipulating symbols rather than in terms of functions, statements etc (at least without first defining these terms) would be helpful. I'd like to be able to pretend I was a person who didn't know any math and was just acting as a human computer, producing proofs. I'd like a resource that explains producing proofs like I was such a computer (not necessarily ONLY like that).
Best Answer
For an undergraduate-level book, none of these three can do you wrong:
They certainly go into the details, and they will leave you in a position where you could go further. If you can only look at one, try Mendelson.
An older text, which is now reprinted by Dover, is Kleene's Mathematical logic. It is also very thorough, and has the advantage of being inexpensive. (Be aware that Kleene's Introduction to metamathematics is a completely different book, which is much more advanced.)