Another weird one related to Gödel’s theorems is Löb’s theorem: given a sound formal system F and a sentence s, if F proves that “if s is provable in F, then s,” then F also proves s. That is:
F ⊢ (Prov_F(“s”) → s) → s
Which is strange because you might think that proving “if s is provable, then s” would be possible regardless of whether s is actually provable. But Löb’s theorem shows that such self-referential statements can only be proven when s itself is already provable.
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