Not every possible set. All you need is a primitive recursive procedure to decide whether a given statement is an axiom, and primitive recursive procedures to check whether each rule of inference is correctly followed in going from one step to another.

It is possible that I wasn't clear enough (my phrasing could certainly use work). I was thinking of statements such as your remark that "An automated proof-checker can identify some correct proofs - but not all of them" whereas we generally don't call something an automated proof checker unless it terminates on any proof and returns correctly if that's a valid proof or not in the axiomatic system. Sometimes we insist that this be a primitively recursive process. Sometimes this is also generalizable in a natural way such that the proofchecker can do this given some specific oracle (there are results about how these sorts of things can behave if they have access to a Halting oracle for example. I don't fully understand these except in vague generality.)