He found the incompleteness theorem

Kurt Gödel, philosopher. mathematician, logician and famous paranoid at Princeton.

No, not at all. The Incompleteness Theorem is more like, that there will always be things that can't be proven.

Further, it is impossible to find a complete and consistent set of axioms, meaning you can find an incomplete set of axioms, or an inconsistent set of axioms, but not both a complete and consistent set.

Sometimes Yes, as in Pythagoras' Theorem. Other times No, for as Godel's Incompleteness Theorem shows, there will be complete bodies of knowledge in which there will be truths that cannot be proven, and falsities which cannot be denied. [I paraphrase his theorem.]

Yes. Goedel's Incompleteness Theorem states that it's also possible to construct equations which cannot be proven to be either true or false.