This is a fair point. I think the only axiom you need to rethink from ZF is powersets (since I think that's the only axiom that produces uncomputable sets from

This is a fair point. I think the only axiom you need to rethink from ZF is powersets (since I think that's the only axiom that produces uncomputable sets from computable ones (ignoring the AC, briefly)). What you'd replace it with (some sort of one based on comprehension, presumably) I couldn't say though.