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?

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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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