I am just beginning my study of mathematical logic (I’ve worked through the first 7 chapters of Kleene’s *Introduction to Metamathematics*) and like many others who are studying FOL for the first time, I’ve had uncertainty about the precise role and intended extent of the informal metatheory in proving results about the language we are trying to formalize. Further, I wanted to be at least somewhat economical in the assumptions I was going to have for my basis metatheory.

To placate my uneasinesss about the vagueness and strength of the informal metatheory, I chose PRA as an existing formalized system to serve as a guideline for my informal one. I heard that one could prove many syntactical theorems with just PRA, however, somewhat embarrassingly, I am not really sure what this is supposed to mean. In practice, I ended up reformulating the variable symbols in the object language in the natural iterated way to allow for a finite number of symbols and defined primitive recursive functions on these finite strings. Once I had this, I embarked on the apparently very tedious process of establishing all syntactical properties in terms of the primitive recursive functions. I constructed maps at first to perform processes like concatenation and then eventually I constructed/sketched maps that would indicate whether a string was a formula, whether a term was substitutable for a variable within a formula, whether a certain deduction was valid, etc.

Surely this could have been done more elegantly, but was I supposed to use the system in a more formal or axiomatic way, or is this what it means to use an axiomatic system as a guideline for informal reasoning? Would it have been equally valid to replace each long winded primitive recursive map with some quick induction showing some desired process is “computable” in some intuitive sense?

Further, and perhaps this should be another question, how does PRA capture finitism? Intuitively, I see that any given value of a primitive recursive map can be worked out by finite procedures, but at least superficially I appealed to countably infinite domains. Am I conflating my notions of a function in set theory with how I should be interpreting reasoning in PRA? Could I reasonably state my informal assumptions used to encapsulate my metatheory without an appeal to an infinite set?