Larry_D'Anna comments on The Cartoon Guide to Löb's Theorem - Less Wrong
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Comments (90)
J Thomas
Larry, you have not proven that 6 would be a prime number if PA proved 6 was a prime number, because PA does not prove that 6 is a prime number.
No I'm afraid not. You clearly do not understand the ordinary meaning of implications in mathematics. "if a then b" is equivalent (in boolean logic) to ((not a) or b). They mean the exact same thing.
The claim that phi must be true because if it's true then it's true
I said no such thing. If you think I did then you do not know what the symbols I used mean.
It's simply and obviously bogus, and I don't understand why there was any difficulty about seeing it.
No offense, but you have utterly no idea what you are talking about.
Similarly, if PA proved that 6 was prime, it wouldn't be PA
PA is an explicit finite list of axioms, plus one axiom schema. What PA proves or doesn't prove has absolutely nothing to do with it's definition.