I'm studying Gödel's incompleteness theorems. And I have the following slide that defines a version of Gödel's first incompleteness theorem. The point is that one can always follow the math and get the feeling that he/she grasps the idea behind the math, but Gödel's incompleteness theorem seems a bit different. It is so deep and has many consequences.
There is also the puzzle (most probably by Raymond Smullyan), and I believe that I found one unprintable statement, which is ¬PN(PN(w)). Of course, I may be wrong, but it made sense to me.
Then, I read that there is an analogy between this Gödelian puzzle and the actual proof of the Theorem GT, that is provided in the slide in my link above (I also linked it here for convenience).
The point is that I don't exactly see this analogy. If someone can explain me this analogy I would be glad. Both mathematical and informal explanation is highly welcomed.
This is correctness. A formal system $F$ is correct (with respect to an interpretation) if it proves only true sentences, i.e. if $F \vdash A$, then $A$ is true.
This is asking for completeness. A formal system $F$ is incomplete when there are sentences of the language of $F$ which can neither be proved nor disproved in $F$, i.e. there is a sentence $A$ such that neither $F \vdash A$ nor $F \vdash \lnot A$.
But one of $A$ and $\lnot A$ is true in the interpretation we are considering, and thus one of them is a true sentence that cannot be proved by $F$.