Saturday, December 19, 2009

Incomplete

clipped from www.miskatonic.org


In 1931, the Czech-born mathematician Kurt Gödel demonstrated
that within any given branch of mathematics, there would always be
some propositions that couldn't be proven either true or false using
the rules and axioms ... of that mathematical branch itself. You
might be able to prove every conceivable statement about numbers
within a system by going outside the system in order to come
up with new rules and axioms, but by doing so you'll only create a
larger system with its own unprovable statements. The implication is
that all logical system of any complexity are, by definition,
incomplete; each of them contains, at any given time, more true
statements than it can possibly prove according to its own defining
set of rules.