Kurt Gödel proved that formal axiomatic systems can be ether complete or consistent (but not both).
An interesting take on this is Jaakko Hintikka's book “The Principles of Mathematics Revisited.” He differentiates between deductive and descriptive incompleteness. Among other things, he claims that Gödel only proved the deductive incompleteness of formal axiomatic systems, but the result doesnt say anything about the descriptive completeness of systems. Hintikka's point is, that deductive completeness (the possibility of deducing all the possible sentences from given axioms), something that mathematicians had always strived for, isn't as important as system's descriptive power. “[Gödel] proved it impossible to establish the internal logical consistency of a very large class of deductive systems..” –Nagel and Newman