Godel's Theorem was discovered by Kurt Godel in the 1930's at the age of 25. It is often called the incompleteness theorem. Many people make it out to prove a lot more than it actually does. All Godel's Theorem shows is that a formal system fo.r ARITHEMATIC that is both complete and consistent cannot be built. It does not show that we can't be machines. It does not show that predicate calculus or logic in general is not complete. It does not show any of these things...all it shows it that a. complete axiomatization for arithmatic is not possible.
This is basically how Godel's Theorem works:
A formal system that is powerful enough to do arithmetic is also powerful enough to capture all recursive relations. If a formal system is a system. in which one can do finite proofs of theorems, then the relation between a proof and the theorem that it proves is a recursive one. Now, all of mathematics is also a recursive (that is computable) system. So we can take a formal system and discuss it i