Here are a penny and a quarter. Make a statement. If your statement is true, then I’ll give you one of these coins (not saying which). But if your statement is false, then I won’t give you either coin.
Raymond Smullyan says, “There is a statement you can make such that I would have no choice but to give you the quarter (assuming I keep my word).” What statement will accomplish that?
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One statement that works is “You will not give me the penny.” If that were false, then I would give you the penny. But I’d said that if your statement were false I’d keep both coins. So your statement can’t be false; it must be true. And if you’ve made a true statement then I owe you a coin. So I have no choice but to give you the quarter.
Smullyan related this puzzle to Gödel’s incompleteness theorem. “I thought of the penny as standing for provability and the quarter as standing for truth. Thus the statement ‘You will not give me the penny’ corresponds to Gödel’s sentence, which in effect says: ‘I am not provable.'”
(From Logical Labyrinths, 2008.)
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