This is an amazingly awesome short description of what Godel showed about the limits of knowledge.
“In 1931, Godel’s two famous theorems seemed to prove that there are mathematical facts independent of persons and language formalisms. In essence, his theorems proved that nobody can set up a formal system and then consistently state _about_ that formal system that he or she perceives (with mathematical certainty) that its axioms and rules are correct, and that they contain all of mathematics. This is so because anyone who says they perceive the correctness of the axioms and rules must also claim to perceive their consistency. But the assertion of the consistency of those axioms is itself not provable in that formal system. Hence the person who claims to perceive the truth of something that cannot be proved in the system [i.e. the assertion of the consistency of its axioms] has to abandon the claim that the system contains all of mathematics. But then we are left with the larger problem of explaining how it is they perceive such truth. How can someone _know_ that the axioms are consistent?”
— Self-Organizing Natural Intelligence