(At the risk of being vapulated and downvoted, I'll ask this here.)
Suppose you work in a field that has nothing to do with the foundations of mathematics, but thanks to MO, you are becoming more and more interested in topics like axiomatic set theory, the different logical systems (intuitionistic, classical, finitist), category theory, type theory, etc. Thus, you want to understand all of this, and you can spend some time learning from books and articles, without any hurry. But you don't want to actually do research is this field. This is the case for me.
Question: Is there any organized way to achive this? Any book recomendatons to achive this goal? At my university there is no one working in this area, hence I have no one to ask this question directly. My background is naive set theory, naive category theory and some basic logic.
Let me explain with an example where I want to get. Suppose you saw the recent Zizek/Peterson debate, and you have read some of their books, you understood their ideas and you have an opinion. But you can't, and don't want, to sit in front of public and debate with either of them. (Of course, you would like to sit and chat with Zizek for hours :))
Thank you very much.
Best Answer
One feature of the foundations of mathematics that poses a special challenge (compared to other branches of mathematics) is that it is very easy to get confused about certain distinctions—truth versus provability, theory versus meta-theory, formal versus informal, syntax versus arithmetic, etc. One book that I think is helpful in this regard is Torkel Franzen's Inexhaustibility: A Non-Exhaustive Treatment.
Beyond that, what I would recommend depends a lot on what you're specifically interested in. A topic that comes up quite frequently on MO is reverse mathematics, and for that, I'd recommend John Stillwell's book, Reverse Mathematics: Proofs From the Inside Out. For type theory, I think that Martin-Lof's original writings are still an excellent place to start.