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Penrose's Theorem of Consciousness
The idea that Gertel's theorem could have something to do with consciousness was, you know, in 1950 when Alan Turing wrote his article about the Turing test. But Penrose wants to say that even though a given formal system cannot prove its own consistency, we as humans sort of looking at it from the outside can just somehow see its consistency. The rejoinder to that has been, well, can we really? Yeah. And plausibly, we never will. There's absolutely no guarantee against an AI that repeatedly adds new axioms.