Illuminating the flashlight

“Gödel’s incompleteness theorem is one of the great intellectual accomplishments of the twentieth century.  Its implications are so far reaching that it is difficult to overestimate them.  Gödel’s result puts intrinsic limitations on the reach of deductive systems; that is, it shows that given any (sufficiently complex) deductive system, there are results that are beyond the reach of the system—results that are true but cannot be proved or disproved on the basis of the initial set of axioms.  The new result might be proved by adding new axioms to the system (for example, the result itself) but the new strengthened system will itself have unprovable results.” – William Byers, How Mathematicians Think