O paradoxu lháře a neúplnosti

Michael Lieberman, Ph.D.

The classic Liar's Paradox is the statement "This is a lie."  If true, it is a lie and therefore false.  If false, it is a lie and, therefore, true.  The problem is one of self-reference (or diagonalization): anytime a statement can refer to itself ("This statement is blue") or its truth or falsity, this problem appears.  You might hope that mathematics is safe from this kind of paradox, but it is not!  In any system of mathematics powerful enough to do basic arithmetic, there is a mathematical statement that says, "I cannot be proven."  This fact, discovered by Kurt Gödel (born in Brno!), leads to his famous First Incompleteness Theorem.  That theorem, one of the most important in the history of mathematics, says that in any such system, there will always be statements that can neither be proven nor disproven, including the statement "I cannot be proven" (this, too, is paradoxical, though it takes a little more work to see).  This tells us that formal mathematical methods, as nice as they are, cannot possibly tell us everything we'd like to know...