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.)
(Notation) PA stands for "Peano arithmetic", and Prb (textbook page 266) abbreviates "Provable".
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 .
Could you check the above?