Math thought:
Does there exist some set of axioms S and auxiliary statement Q such that there exists some proposition P where (S ∪ Q) → P and (S ∪ ¬Q) → P, but where S alone is not strong enough to prove P?
Can such an ensemble be constructed?
attn @Austin_Dern because I know you're a math sort of person
(sorry for the repost; corrected a mistake in the original that made the answer trivially no)
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