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cousin_it comments on AI cooperation in practice - Less Wrong
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To prove that, the proof system would need to know its own consistency, because an inconsistent proof system could prove both of these statements and indeed any statement at all. But a proof system cannot know its own consistency without actually being inconsistent (Goedel's second theorem).
For a finite set of proofs, that a given statement is not provable by a proof from that set, is provable, possibly by a proof from that set.
Technically true. You could always just evaluate the two statements and there you have your proof, with length exponential in n. I very much doubt the proof could be made shorter than n though, because the statement is equivalent to "you can't find a contradiction in these axioms using less than n steps", the difficulty of which should grow way faster than linearly in n.
Aha - very cunning.