I don’t think a Gödel encoding can be done for any axiomatic system. For example, an axiomatic system with uncountably many axioms cannot necessarily be encoded in this way.
My mathematical logic is rusty, but if I’m not mistaken, the same holds for a set of axioms that’s countable, but non-computable.
JoshuaZ:
My mathematical logic is rusty, but if I’m not mistaken, the same holds for a set of axioms that’s countable, but non-computable.