Why can we use induction when studying metamathematics? In fact I don't understand the meaning of the word "metamathematics". I just want to know, for example, why can we use mathematical induction in the proof of logical theorems, like The Deduction Theorem, or even some more fundamental proposition like "every formula has equal numbers of left and right brackets"?
What exactly can we use when talking about metamathematics? If induction is OK, then how about axiom of choice/determincacy? Can I use axiom of choice on collection of sets of formulas?(Of course it may be meaningless. By the way I don't understand why we can talk about a "set" of formulas either)
I have asked one of my classmates about these, and he told me he had stopped thinking about this kind of stuff. I feel like giving up too......

I am reminded of this remark at the beginning of Kleene's book Mathematical Logic:
"It will be very important as we proceed to keep in mind this distinction between the logic we are studying (the object logic) and our use of logic in studying it (the observer's logic). To any student who is not ready to do so, we suggest that he close the book now, and pick some other subject instead, such as acrostics or beekeeping." 
From pages 3 and 4 of Stephen Kleene's book Mathematical Logic
