In proof-theory one can prove that in, say, Peano Arithmetic one can't prove a function $f$ total. Often this seems to mean $f$ is growing too fast to be provably total.
I have some background in logic and know the very basics about formal proofs in FO, but overall I'm more familiar with finite model theory than proof-theory. Now I want to learn how to prove a function total in PA and how to prove that a function can't be proved total in PA.
Are there any books that would cover the topic with my background? Also, if there are any lecture notes near the area, I would be very happy receive pointers to them (books do cost a lot and I think I can only afford one).