A book of logic puzzle
s by Raymond Smullyan
. Many of the puzzles involve reasoners on the island of knight
s who always tell the truth and knave
s who always lie.
For example, if a native of the island says to a reasoner "You will never know that I am a knight", the reasoner (if she hears the statement and is sufficiently self-aware) will think the following.
"If he's a knave, then I will know he's a knight, but I can't know something false, so he must be a knight. Now I know he's a knight, which makes his statement false, so he must be a knave."
Now the reasoner believes both that the native is a knight and that he is a knave. She has become inconsistent. If the native had said "You will never believe that I am a knight" instead, things get more complicated, and it turns out that the reasoner cannot believe the native to be either a knight or a knave without becoming inconsistent. This is analogous to Godel's theorem and the statement "I am not provable in axiomatic system S".