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Benja comments on Decision Theories, Part 3.75: Hang On, I Think This Works After All - Less Wrong
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Comments (45)
Proof that a Löbian cycle works out (hat tip, Eliezer and co.):
|- □A -> B
|- □B -> A
then
|- □(□A->B)
|- □□A->□B
|- □□A->A
|- □(□□A->A)
|- □□□A->□A
|- □A->□□A
|- □□□A->□□A
(apply Löb)
|- □□A
|- A
How about just,
(simple facts about PA:)
|- □(A ∧ B) -> (□A ∧ □B)
(apply the assumptions:)
|- □(A ∧ B) -> (B ∧ A)
|- □(A ∧ B) -> (A ∧ B)
(apply Löb:)
|- A ∧ B
Heh. Nice!