Let be a sentence such that
(Thus says "I am provable," in contrast to the sentence "I am unprovable" that has been found to have such interesting proberties.)
Does ?
(Notation) PA stands for "Peano arithmetic", and Prb (textbook page 266) abbreviates "Provable".
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Löb's Theorem: Assume that T is sufficiently strong recursively axiomatizable theory. If is any sentence for which , then .
Textbook says that PA is sufficiently strong theory (page 269), so I think I can apply
Löb's Theorem directly.
By hypothesis, .
We see that if , then, by Löb's Theorem.
Further, if , then we claim that . Suppose to the contrary that . Then by the choice of , which follows that .
Thus .
Could you check the above?
Thanks.