I thought Gödel’s incompleteness theorem was about statements that are “beyond truth and untruth” such as:
This statement is false.
or its more sophisticated non-self-referential cousin:
Let S be “Let S be *; the statement you get when in S you replace the asterisk by another copy of S in quote marks is false.”; the statement you get when in S you replace the asterisk by another copy of S in quote marks is false.
in the sense that there are perfectly good reasons why a system could not classify any of them as true or false without running into contradiction. (Let’s call them “trick statements”. Furthermore, it is impossible to distinguish algorithmically which statements are trick statements, so even allowing the system to answer “no comment” on trick statements would still not solve the problem.)
Maybe there is a relation between this and what you wrote, but I don’t intuitively see it.
There are two theorems. You’re correct that the first theorem (that there is an unprovable truth) is generally proved by constructing a sort of liar’s paradox, and then the second is proved by repeating the proof of the first internally.
However I chose to take the reverse route for a more epistemological flavour.
I thought Gödel’s incompleteness theorem was about statements that are “beyond truth and untruth” such as:
or its more sophisticated non-self-referential cousin:
in the sense that there are perfectly good reasons why a system could not classify any of them as true or false without running into contradiction. (Let’s call them “trick statements”. Furthermore, it is impossible to distinguish algorithmically which statements are trick statements, so even allowing the system to answer “no comment” on trick statements would still not solve the problem.)
Maybe there is a relation between this and what you wrote, but I don’t intuitively see it.
There are two theorems. You’re correct that the first theorem (that there is an unprovable truth) is generally proved by constructing a sort of liar’s paradox, and then the second is proved by repeating the proof of the first internally.
However I chose to take the reverse route for a more epistemological flavour.