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Sewing-Machine comments on Harry Potter and the Methods of Rationality discussion thread, part 8 - Less Wrong
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Something like Scenario 1 has already happened, with Russell's paradox. People did not react by questioning their own sanity but by regarding Russell's construction as "cheating", and reconstituting the axioms so that Russell's construction was forbidden.
We're deep into insanity territory with Scenario 2, but people have speculated about such things here before.
I am fully aware of Russell's paradox. I still think some sanity checks may be worthwhile, as the number of people who have thought they achieved scenario 1 but were in fact crackpots significantly exceeds one.