I would like to compile a list of sources for various formal systems.$^1$ Ideally, I am looking for an explicit statements of all rules which apply to each canonical formal system (e.g. dealer's choice of Hilbert proof calculus, sequent calculus, $\lambda$-calculus, etc.) - the more, the merrier.
I ask that these references be "definitive" because I have found that most texts discussing formal systems resort to a very relaxed form of explanation that overlooks particulars of any given formal system in favour of convincing the reader that such-and-such formal system can, in principle, prove such-and-such conclusion, and that such-and-such conclusion is true. This does little to help me when I am looking for information about the system itself - in fact, it tends to act as an obstacle. Instead of finishing the book, I have to spend days playing six degrees of citation to find some obscure German paper from 1907 detailing the syntax of the system in use. I couldn't even find a written copy of the axioms of ZFC until a few months ago!$^2$
So, rather than drive myself mad searching in vain for the "SKI-combinator bible," I figured I'd just ask for gratuitous list of user manuals and get as much of it out of the way as possible.
Thusfar, I have two sources:
Principia Mathematica by Whitehead and Russell (which I will probably never consult)
Metamath: A Computer Language for Mathematical Proofs by Norman Megill (literally the documentation for a proof checker)
Still, both of these (the latter especially) devote a bit more ink to explaining why the reader should care about the system than I'd like.
$^1$ I suppose that I actually mean "various formal systems, proof systems, term rewriting systems, and logics." But I honestly don't see the purpose in distinguishing between these since, at the end of the day, each is just a means of converting sets of strings into other sets of strings.
$^2$ No, the axioms listed on the Wikipedia article on Zermelo–Fraenkel set theory, the Wolfram Mathworld entry for Zermelo-Fraenkel Axioms, and the Stanford Encyclopedia of Philosophy page on Zermelo-Fraenkel Set Theory, are not the axioms of ZFC, they are the axioms of a definitional extension of ZFC. Yes, there is a difference.
