Thoughts from a reflective educator.
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, a corollary of the first, shows that such a system cannot demonstrate its own consistency.
Is there a relationship between Gödel's first incompleteness theorem and the ability of humans to hold seemingly self-contradictory thoughts?
In other words, has our brain been designed in such a way as to allow us to prove as much as possible about the universe, and consequently we must hold some thoughts in a contradictory fashion; we hold an inconsistent set of axioms about the world, and so we can potentially prove anything about the world (although many proofs would be invalid, but to an inconsistent system, perhaps an invalid truth is acceptable?).