Monday, May 3, 2010

Going round and round

Bumper stickers can be helpful.  I saw one yesterday that said

It's bad luck to be superstitious

What a modern thing to write!  I am reminded of Gödel and others who worked on the limits of logic and the axiomatic methods.  Gödel and others worked with recursive words, phrases, ideas and methods.  The book "Gödel, Escher, Bach: The Golden Braid" relates to similar work in art and music as well as logic.  The idea in human terms refers to the idea of a snake swallowing its own tail.  What will happen when it continues on?  How can it swallow itself completely?

There is a famous passage in the letter of St. Paul to Titus stating that a Cretan has stated that "Cretans are always liars".  Here we have a Cretan stating that Cretans lie.  If his statement is true, then it is false, since he says that every statement by a Cretan is false and he is part of that group.  If his statement is false, he is telling a lie as the statement says Cretans do, so it is true.  A puzzle!

Modern thinkers tend to be sensitive to statements such as "This statement is false" which refers to itself. Or, a pair like these:

"The statement below is false."

"The statement above is true."

There is the prize-winning "buttered cat" paradox.

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