I'm currently studying A Concise Introduction to Mathematical Logic (Third Edition) by Wolfgang Rautenberg. What sparked my interest in logic is my interest in foundations in general, so I was naturally pretty scandalized when I found out how classical first-order logic is developed and how its theorems (soundness and completeness, for example) are proved. The metatheory uses the metalogical implication $\Rightarrow$, proof by contradiction, proof by induction and even Zorn's lemma! That's not the bare-bones start I though I'd get the opportunity to examine!
I don't like that, since the goal of my study of the foundations of mathematics was to find out which concepts are the most fundamental in any formal system. Let me also note that a formal system I'm after must be sufficiently expressive (not something absurd like the empty set), meaning that it must be able to encode some useful theory (e.g., a part of mathematics or some other formal theory used in philosophy; mereology, the study of parts and its wholes, comes to mind). Thank you for your insights in advance! :)