Gödel's Second Incompleteness Theorem expresses the consistency of a formal system within the system itself using a rather carefully designed proof checking predicate. The conclusion of Gödel's argument is that this formalization of consistency can't be proven in the system.
But what about other proof checking predicates? There's presumably an infinite variety of predicates which correctly check proofs. Any of which could be used to express consistency. But Gödel's argument no longer seems to apply to all of these predicates. After all, Gödel had to reason quite significantly about the internal operations of his predicate. But here the predicates are effectively black boxes.
Is it conceivable that one could prove the consistency of the system using such an alternative proof checking predicate? (Obviously it wouldn't be possible to prove that these formalizations of consistency are equivalent within the system.)