I'm looking for a book on set theory foundations that goes into the metamathematics of it all. I worked through Kleene's Introduction to Metamathematics. In that text he proves godels incompleteness as well as gives proofs of some reductions of classical number theory into intuitionistic number theory. I'm now left with wanting to know the next step in studying math foundations. Kleene explored some details of the how one would build a theory in first-order logic (the eliminability of definitions, etc.). I did very much enjoy that, and would love a book then constructs a theory in first-order (not necessary entirely in first-order but reducible to such).
I poked through a sample of The Foundations of Set Theory, Fraenkel, and was really intrigued by the table of contents. However, after looking a little deeper, I found that there weren't
many proofs presented in the text. Mainly references just references to other publications and such.
So I would love some recommendations on foundations that include topics like the independence of the axiom of choice, what theorems of ZFC can we reduce to ZF. Maybe even some intuitionistic set theory (I saw that was mentioned in Fraenkel) or alternate foundations. And it would be awesome if it went in to some philosophical perspectives on foundations of math. Although, this may all be too much to ask for in one book I understand.
Best Answer
I enjoyed Robert Vaught's book when I took his course at Berkeley in the late $80$'s, if you can find a copy.