Philosophy of mathematics/Gödel's Incompleteness Theorem

Austrian mathematician Kurt Gödel showed that any mathematical system complicated enough to include arithmetic must be incomplete. What does this mean? Simply that there are theorems in the system that are true but cannot be proven true from within the system. It may be possible to prove the theorem from outside the system. A lot of nonsense has been written about this topic by nonmathematicians, including assertions that we can never know anything. Some mathematical systems ARE complete, including ordinary geometry and propositional logic.

What Godel showed is that we can never have a complete knowledge of mathematics, and in the 1930's he shook the math world at its core when he proved this. Any Mathematical theory has axioms from which everything in the theory is derived. It is impossible to prove the axioms of a theory by using the theory, you have to go to a more fundamental theory to prove the axioms of the first theory. The problem is that you then have to go to an even more fundamental theroy to prove the axioms of the fundamental theory. The problem with that is you have to go to a more ...... ad infinitum. We can never prove the fundamental axioms and laws of mathematics on which everything is based and thus our knowledge will necessarily always be incomplete.