Posts: 21665
Posted: Wed Dec 31, 2008 12:28 pm
Godel showed with his Incompleteness Theorem that every (sufficiently complex) formal system is either inconsistent or incomplete. Therefore there is an inherent limitation to logic (which is a formal system): it cannot show all truths to be true, or all falseholds to be false.
Interestingly, Penrose, in his book The Emperor's New Clothes, shows that a computer could not, in principle, derive Godel's Incompleteness Theorem. So ironically, the Incompleteness Theorem that shows the inherent limitations of science, mathematics and logic, is in itself fundamentally irrational.