A related question that's in the spirit of the previous question is, is there a version of the incompleteness theorem for set theory that guarantees an independent sentence for any theory, but with the proviso that this independent sentence has no arithmetic consequences (like CH or AC relative to ZF)?

Thanks. Do similar phenomena arise when we add the stronger generalizations of the omega-rule to set theories stronger than analysis? And is there any rule analogous to the omega-rule that makes all sentences of analysis provable or refutable?

I think the sets work for $n\ge m$. I've edited my question to fix this. Thanks for pointing it out.