I seem to be running into topics of conversation that return to mathematical logic in some form or another a lot lately. E.g., Adam Rosi-Kessel and I got to talking about [book: Gödel, Escher, Bach]-type topics recently, namely the connection — if there is one — between consciousness (whatever that is) and self-reference in formal systems. Then there was this blog post today about programs that can print themselves and other topics.
I need to learn me some mathematical logic already, extending (let’s say) all the way from propositional logic through predicate calculus, up to Gödel’s theorem. Anyone have any recommended readings here?
Steve Reads 
Reading Godel Escher Bach gave me the same project a few years ago (I’m afraid it’s more than 15 years now). The two books I would recommend are “mathematical logic” from Kleene, and “understanding formal methods” by Jean-Francois Monin. “Mathematical Logic” covers most of what you seem to want to learn, and do it in what I would call an “autonomous way”, not requiring too much knowledge of other mathematical fields, and explaining everything. This was in contrast to current french textbooks that I could read. “Understanding formal methods” is interesting to see the connection between mathematical logic and computers (from the curry-howard correspondence), and how it can be used (ie to prove that a program is correct). It also have a presentation of mathematical logic of its own which is clear and may be easier as a first text (it is much easier to read cover to cover than Kleene).
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