I studied mathematical logic quite extensively, and I'm pretty sure this person doesn't know what he's talking about. Godel's incompleteness theorem is about facts which are unprovable in a certain logical framework. Any sufficiently complex consistent system A has a statement G which cannot be either proven or disproven in that system. That by itself is not a paradox, there are plenty of unsolved mathematical problems that might not be solvable using the axioms we have. For example "A is/is not consistent" is such a statement.

On the contrary, any statement that can be derived within the framework A cannot contradict A because that would prove that A is inconsistent! Of course with a real sufficiently complex logical system we cannot know whether it is consistent or not, but that doesn't mean it is possible to prove a paradox in it.