How much mathematics should a student of mathematical logic know? I would like to know what areas of mathematic are directly related to mathematical logic, besides the usual courses on model theory, proof theory and computability. If you suggest only one book on each subject that will be great. I prefer the books to be formal, concise and to the point( Rudin style).


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*Combinatorics and graph theory: Does a logic student need to know anything about enumeration? If so to what extent.

*Algebraic geometry: I know model theory has been applied to solve some problems in algebraic geometry, but is there any application of AB in logic? Or it only can serve as a source of inspiration?

*Number theory: Is knowing elementary number theory sufficient? 

*Toposes and sheaves: I don't know anything about sheaves and toposes. I only know some basic category theory. I know they are important for further studies in logic but they appear in different areas of math, i.e. algebraic topology , algebraic geometry and geometry. What is the best approach to learn about them?

*Geometry: I've had some courses in differential geometry and theory of manifolds. But they don't seem to be related. Maybe synthetic geometry would've been a more suitable course to take.

*Algebra(not algebraic logic ) : Paul Cohen, a logician, if I'm not mistaken have some theorems in algebra. Why!? how are logic and algebra related?
Thanks for your patience!

 A: 
How much mathematics should a student of mathematical logic know?

A student at what level? At introductory and intermediate levels, you need little specific mathematical background, just the ability to follow mathematical proofs. 
And in general, I'd say that you can get quite a long way studying various areas of mathematical logic picking up what you about other bits of mathematics on a need-to-know basis. Thereafter, it rather depends on what areas you get most interested in. For example, at one end of the spectrum, model theory gets entangled with serious algebra; while towards the other end of the spectrum, my sense is that recursion theory can be pursued a long way without requiring a rich background in other areas of mathematics. 
You mention category theory. That seems to me a special case. I like Tom Leinster's two-sentence definition of category theory: 

Category theory takes a bird’s eye view of mathematics. From high in the sky, details become invisible, but we can spot patterns that were impossible to detect from ground level.

Well, you will indeed have difficulty taking a bird's eye view and pattern-spotting if you know little mathematics at ground level! Getting seriously into topos theory (for example, by tackling Peter Johnstone's Elephant) surely requires a vastly richer background of mathematics that getting into proof theory (say). But then I'm not sure that topos theory counts as mathematical logic.
