Where the sentence from Curry’s paradox says “If this statement is true, then p”, Ψ says “if this statement is provable, then p”, that is, □Ψ→p.
In Curry’s paradox, if the sentence is true, that would indeed imply that p is true. And with Ψ, the situation is analogous, but with truth replaced by provability: if Ψ is provable, then p is provable. That is, □Ψ→□p.
But, unlike in Curry’s paradox, this is not what Ψ itself says! Replacing truth with provability has attenuated the sentence, destroyed its ability to cause paradox.
If only □p→p, then we would have our paradox back… and that’s Löb’s theorem.
This is all about □Ψ→□p, just about one direction of the biimplication, whereas the post proves not just that but the other direction. It seems that only this forward direction is used in the proof at the end of the post though.
I remember this by analogy to Curry’s paradox.
Where the sentence from Curry’s paradox says “If this statement is true, then p”, Ψ says “if this statement is provable, then p”, that is, □Ψ→p.
In Curry’s paradox, if the sentence is true, that would indeed imply that p is true. And with Ψ, the situation is analogous, but with truth replaced by provability: if Ψ is provable, then p is provable. That is, □Ψ→□p.
But, unlike in Curry’s paradox, this is not what Ψ itself says! Replacing truth with provability has attenuated the sentence, destroyed its ability to cause paradox.
If only □p→p, then we would have our paradox back… and that’s Löb’s theorem.
This is all about □Ψ→□p, just about one direction of the biimplication, whereas the post proves not just that but the other direction. It seems that only this forward direction is used in the proof at the end of the post though.