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Since large cardinal axioms have arithmetic consequences not provable without them, the answer to your first question is yes. For example, let $f(n)=1$, provided that $n$ is not the Goedel code of a proof of a contradiction in ZFC. Since large cardinals imply Con(ZFC), the theory ZFC+large cardinals proves that $f$ is total. But ZFC alone does not prove this (if consistent), since it is relatively consistent with ZFC that there are proofs of contradictions from ZFC.