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orthonormal comments on Decision Theories, Part 3.75: Hang On, I Think This Works After All - Less Wrong
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Comments (45)
To the first question, yes. That's what the top comment in this sequence proved (and Benja's reply proved much more elegantly).
To the second question, self-referential statements can't be ruled out in sufficiently powerful formal systems- that's what Gödel's Incompleteness Theorem uses!
To the third, again, if any formal system proves □A -> B and □B -> A, then it proves A and B.
Consider the statement A "□A -> B and □B -> A" and the statement B "□A -> B and □B -> A and false"
Why are those two statements not provable in every system in which they can be written?