I may not removed the flaw entirely, but I have definitely changed it into a less-obviously bad flaw. And also used Loeb’s Theorem to derive Goedel’s, or a close analogue.
The summary statement of Loeb is wrong. It is not the case that (◻X->X) → X, it is only the case that ◻(◻X->X) → ◻X. That is to say, that if (a proof of X implies X) is provable, then X is provable.
Using Deduction here, we get only ◻(~◻X) → ◻X, which in English is “If it is provable that X is unprovable, then X is provable”, or in other words “If PA is proved complete, it is proved inconsistent.”
I may not removed the flaw entirely, but I have definitely changed it into a less-obviously bad flaw. And also used Loeb’s Theorem to derive Goedel’s, or a close analogue.
The summary statement of Loeb is wrong. It is not the case that (◻X->X) → X, it is only the case that ◻(◻X->X) → ◻X. That is to say, that if (a proof of X implies X) is provable, then X is provable.
Using Deduction here, we get only ◻(~◻X) → ◻X, which in English is “If it is provable that X is unprovable, then X is provable”, or in other words “If PA is proved complete, it is proved inconsistent.”