Godel has nothing to do with it. Godel’s Incompleteness is just the math version of the Halting Problem, which is just a fancy version of the classic paradox “This statement is false”. Godel showed that even if you outlaw self referential definitions e.g. “This statement is false”, math is rich enough in complexity to simulate itself and thus end up with self referential paradoxes anyway.

It’s not this mystical thing that people make it out to be online.