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Houshalter comments on The Cartoon Guide to Löb's Theorem - Less Wrong
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Did EY post the correct answer anywhere? I've been thinking for hours and give up. The comments make no sense. The best I can tell from Larry_D'Anna's comment is that axiom 1 is wrong. But it doesn't seem wrong and I am very confused. Or maybe he's saying the way it's used in step 8 is wrong. But that seems a very straightforward inference from it. Either way, it must be correct, because Lob's theorem is an actual theorem. EY didn't just make it up to trick us.
So I went to wikipedia and it does say the same thing:
◻(◻C->C)->◻C
If you subtract the quotations (◻) from both sides, you do get (◻C->C)->C. Which obviously must be false, or 2=1. But I can't figure out what is wrong with removing the quotations. I don't even see where the quotations even came from in EYs proof. This is very frustrating.
Step 8 relies on the lemma that ◻X→◻◻X. If you try running through the proof with the ◻ removed from both sides, the corresponding lemma you would need to make step 8 work is X→◻X, but this is not true.