Prereqisites for: Subsystems of second order arithmetic

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As the title suggests, im wondering what the prerequisites for Simpsons book, Subsystems of... are? Unfortunately I cant find it in the preface.

My background is a Bachelor in Philosophy and Mathematics. I heard a course on mathematical logic (completeness incompleteness) and wrote my thesis about Model Theory and Set Theory, however my background in Proof and Recusrion theory is just the very basics, needed for a first course in logic. Which would be adequate books on Proof and Recursion theory to be fit for Simpsons work?