J_Thomas2 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: J_Thomas2 21 August 2008 11:04:25PM 0 points [-]

Let me try to say that clearer.

Suppose that A is false.

How the hell are you going to show that if PA proves A true then A will be true, when A is actually false?

If you can't prove what would happen if PA proved A true when A is actually false, then if you can prove that if PA proves A is true then A has to be true, it must be that A is true in the first place.

If this reasoning is correct then there isn't much mystery involved here.

One more time. If PA proves you are a werewolf, then you're really-and-truly a werewolf. PA never proves anything that isn't actually true. OK, say you aren't a werewolf. And I say I have a proof that if PA proves you're a werewolf then you're actually a werewolf. But in reality you aren't a werewolf and PA does not prove you are. How do I prove that PA would do that, when PA does not do that?

Once more through the mill. If PA proves that 6 is a prime number, then 6 is really a prime number. Can you prove that if PA proved 6 was a prime number then 6 would be a prime number? How would you do it?

If you definitely can't prove that, then what does it mean if I show you a proof that if PA proves 7 is a prime number then 7 must actually be prime? If you can't make that proof unless 7 is prime, and you have the proof, then 7 is actually prime.

The problem is with trying to apply material implication when it does not work.