I'm soon going to be a 3rd year undergrad in pure math, and next semester i will have a class about logic. I already had a logic class before so have some familiarity with first order logic. I've always thought of math as being equivalent to manipulations of finite strings, so it bothers me that most books accept infinite objects in the metatheory. For example this one is written from a formalist point of view, but is not finitist.

I'm looking for a book that like the mentioned one, that serves as an introduction to logic and that accepts only finite objects in the metatheory, as i don't believe in the existence of infinite objects.