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Vladimir_Nesov comments on AI cooperation in practice - Less Wrong
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Proof in what theory, "can't" by what definition of truth? In the extension of PA with inconsistency-of-PA axiom, it's both provable and true that PA is inconsistent.
A proof in PA that 1+1=3 would suffice. Or, if you will, the Goedel number of this proof: an integer that satisfies some equations expressible in ordinary arithmetic. I agree that there's something Platonic about the belief that a system of equations either has or doesn't have an integer solution, but I'm not willing to give up that small degree of Platonism, I guess.
You would demand that particular proof, but why? PA+~Con(PA) doesn't need such eccentricities. You already believe Con(PA), so you can't start from ~Con(PA) as an axiom. Something in your mind makes that choice.