I think there are some crossed wires here. "Consistency" in this sense does not talk about how two systems are related to each other[0]—it is a question of whet

I think there are some crossed wires here. "Consistency" in this sense does not talk about how two systems are related to each other[0]—it is a question of whether you can prove that a system of axioms, call it S, is not absurd (i.e., cannot produce the statement, say, 0=1). Godel's incompleteness answers the question, "can we prove S is consistent, assuming only the axioms given in S?" with a negative, assuming S is a complicated-enough set of axioms.[1] ---- [0] The usual disclaimer holds, we can talk about proving the consistency of some axiomatic system A, if some other axiomatic system B is consistent. E.g., we can show that the integers with the usual ring structure is consistent if ZFC is consistent, because we can construct integer arithmetic from ZFC. This, though, is a point I avoid for clarity since this is distinct from what I believe is being mentioned. [1] I.e., if it can embed the rules of natural-number arithmetic in some way.