Books on logic, proof theory and set theory? I graduated in Computer Science at University of Bologna in Italy some years ago.
For various reasons now I am discovering a back interest in mathematic logic higher than I was a student.
I have only a pair of university books about the subject. Now I am reading that books and I am enhancing the not so developed concepts using mainly Wikipedia.
Because of this reason, maybe what I am going to write consecutively is a bit confused.
Now I am looking for the minimal list of books that cover the following areas:


*

*All kinds of logic: classical, intuitionstic, relevance, etc...

*Proof theory

*Set theory

*Philosophical implications


For each of these areas I am interested its history too.
Every help is really appreciated.
 A: You might find Graham Priest's An Introduction to Non-Classical Logic interesting. It certainly covers the non-classical logics that interest you, with some brief, but useful historical notes, and discussion of the philosophical implications by one of the leading contemporary philosophical logicians. 
A: For philosophy, I would recommend the old texts found on the Philosophy shelf of the Gutenberg.org math archives. Specifically,
http://www.gutenberg.org/wiki/Mathematics_(Bookshelf)


*

*Boole - Investigation on the Laws of Logic

*Comte - Philosophy of mathematics

*Couturat - Algebra Of Logic

*Bertrand Russel's classic on philosophy

*And A.N.Whitehead's Principia.


The books are on varied topics, and are very well reproduced. 
A: Try "Notes on logic and set theory" by P. T. Johnstone.
It is short and straight to the point.
A: Since you have a background in computer science you might appreciate some of the following books:


*

*Formal Logic by A. N. Prior

*Elements of Mathematical Logic by J. Lukasiewicz

*Aristotle's Syllogistic: From the Standpoint of Modern Logic by J. Lukasiewicz

*Polish Logic 1920-1939 Edited by Stors McCall


Prior's book has sections on propositional calculus, quantification theory, the Aristotelian syllogistic, traditional logic, modal logic, three-valued logic, and the logic of extension.  The section on propositional calculus includes a probably not very well known topic in a short of exploration of what can get variable functors. or equivalently variable truth-functions (meaning that we have at least one truth-function which qualifies as a logical variable instead of a logical constant).
Elements of Mathematical Logic starts out with a discussion of the history of logic, and then proceeds to develop two-valued propositional calculus axiomatically, using what I believe a very simple axiom set.  The axioms in words can get stated:
1) "If the first (proposition) implies the second, if the second implies the third also, then the first implies the third."
2) "If the negation of the first implies the first, then the first."
3) "If the first, then if the negation of the first, then the second."
Other sections include two-valued propositional calculus with quantifiers, problems of the independence of the axioms presented, consistency of the system, and completeness of the system.  The book has much to recommend it, in my opinion, for learning the axiomatic method, as well as realizing that many, many relationships exist or can get formed between logical laws and metatheoretical results.  Wajsberg's papers in the Polish Logic volume makes this even more apparent. 
A: There's a long and detailed annotated reading list of books on all kinds of areas of logic, at various levels, downloadable from

http://logicmatters.net/students/tyl/

