Wednesday, May 14, 2008
A Gödelian Puzzle
Can we know something is true but not be able to prove it? And what would that mean to how we conceive the world? Here's a puzzle I've been pondering over the last week. It's Raymond Smullyan's variation on Kurt Gödel's Incompleteness Theorem. Mull it over for at least three shakes of a cat's tail. I'll write the answer in the comments. Here goes:
Let us define a logician to be accurate if everything he can prove is true; he never proves anything false.
One day, an accurate logician visited the Island of Knights and Knaves, in which each inhabitant is either a knight or a knave, and knights make only true statements and knaves make only false ones. The logician met a native who made a statement from which it follows that the native must be a knight, but the logician can never prove he is!
Question: what statement would work?