The principle of non-contradiction say that p and $\neg$ p can not be both true.
- If you consider the law of excluded middle as relevant, then either p or $\neg$ p is true, and thus the other is false by the principle of non-contradiction.
- But if you consider the excluded middle as not relevant, just like intuitionistic logic, what would avoid some p and $\neg$ p to be both false ?
Actually, if true is the same than provable, then such a proposition is just a Gödel one : p is not provable and neither $\neg$ p is.
I think this is a very good interpretation and a big demystification of Gödel theorem : it is not about some true that we can not reach, it is about some absurdity that is still an absurdity when you negate it.