Text books on computability
I collected the following "top eight" text books on computability (in alphabetical order):
Boolos et al., Computability and Logic
Cooper, Computability Theory
Davis, Computability and unsolvability
Hermes, Enumerability, decidability, computability
Hopcroft et al., Introduction to Automata Theory, Languages, and Computation
(thanks to Bill Province)Kleene, Introduction to Metamathematics
Minsky, Computation
Sipser, Introduction to the Theory of Computation
(thanks to Prajwal Kansakar)
I know it's opinion-based, but which important text books did I miss?
Cutland, Computability (CUP). A beautifully lucid and elegant classic text.
I'd say that [Odifreddi 1989] is still a good text to have, to use it at least as a reference book (but pretty good as a textbook too, I found).
One can also take a look at Peter Smith's "Teach yourself logic guide", where different sources are cited and commented upon. It covers much more than just computability, though there's a section exclusively on it (edit: now that I checked it again, there are actually two sections: 5.3 and 7.3 in the current version).