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Larry_D'Anna comments on The Cartoon Guide to Löb's Theorem - Less Wrong

12 Post author: Eliezer_Yudkowsky 17 August 2008 08:35PM

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Comment author: Larry_D'Anna 18 August 2008 06:47:28AM 2 points [-]

Eliezer: "Why doesn't the given proof of Lob's Theorem, the steps 1-10, show that PA + "◻C -> C" |- C?"

That's just not what it says. The hypothesis in step 2 isn't "◻C -> C" it's "◻(◻C -> C)". I suppose if the Hypothesis were "◻C -> C" we could try to find where it breaks. Step 7 is the only step that depends on 2 so it has to be there. Translating the amusing cartoons, we have

2. ◻(◻C -> C) 6. ◻(◻L -> ◻C) 7. ◻(◻L -> C)

Lets say that instead we have

2'. ◻C -> C

Well what are we supposed to do with it? We're stuck. Just because ◻C -> C doesn't mean that PA can prove it.