I am interested in learning first order logic. In particular, I would like to be comfortable enough with first order logic in order to fully tackle, say, the Axioms of Zermelo-Fraenkel, as in Jech's Set Theory book (I have studied in the past other set theory books, such as Halmos' Naive Set Theory).
The main target is to eventually become comfortable enough with first order logic, and afterwards, with set theory. And then, be comfortable with the basis for the rest of mathematics. Basically, peace of mind.
However, when I try to find the right book (browsing from amazon free pages) for first order logic, I find all books I have tried so far, basically start in the first page using the word "set". The "peace of mind" breaks down then for me, because of course, I want to study first order logic in order to learn set theory later, so I do not want "sets" are a prerequisite for first order logic!
Is there a good book in first order logic that does not use sets at all?
Best Answer
Quine's Methods of Logic is a good example of a treatment that makes virtually no reference, even informal, to sets. His treatment is more to informally talk about strings and the truth functional connectives that form new strings; basically, he covers the things that go into the definition of a well-formed formula, but doesn't need to go into talk of the set of wff's.
The trade off here is that while it's easy to talk about proofs like this, the few occasions on which he discusses model theoretic ideas are very sketchy and informal. I never really found his sections on the soundness and completeness of his proof methods informative. So this "set-free" approach is good for learning how to do proofs, but isn't good for particularly deep insights into logic as a formal system; I see this as a feature as much as it's a bug, personally.
Edit: In light of Carl Mummert's comment and answer, I thought I should clarify that I use Quine as an example here; see Carl's comment below for reasons this might not be a great textbook for a mathematics student. While I do consider a treatment of logic in terms of strings and formation rules to be helpful in justifying first order logic's use in a foundational setting, I think Carl's answer also highlights why one can safely be indifferent to the use of mild set-talk.