The limits of reason

Abstract

Kurt Gödel demonstrated that mathematics is necessarily incomplete, containing true statements that cannot be formally proved. A remarkable number known as omega reveals even greater incompleteness by providing an infinite number of theorems that cannot be proved by an finite system of axioms. Omega is perfectly well defined and has a definite value, yet it cannot be computed by any finite computer program. Omega's properties suggest that mathematics should be more willing to postulate new axioms, similar to the way that physicists must evaluate experimental results and assert basic laws that cannot be proved logically. The results related to omega are grounded in the concept of algorithmic information.