Friday, February 16, 2007

Don't do what I tell you - a paradox.

The other day I exclaimed, "Don't do what I tell you!" "Don't do what I tell you" is a paradox along the lines of "I am a liar" (Epimenides or Cretan paradox). If you don't do what I tell you then you are doing what I told you (by not doing it), but to not do what I tell you you must do what I tell you and so it goes around.

These types of statements were the inspiration for the title of this blog, "The Honest Hypocrite". Hypocrites say one thing and do another, but if I admit that I say one thing and do another then I really am correcting what I say to what I did so I am not a hypocrite, am I. But I just said I was, and so it goes around and around.

Set theory can produce a similar contradiction, Russell's Paradox: The set of all sets that do not contain themselves as members. Russell discovered this paradox while working on Cantor's theorem which was important in understanding infinity and may have driven Cantor insane. Thus these statements are important to mathematics and can be dangerous in and of themselves.

These types of results led to the celebrated Godel's Incompleteness Theorem which says that you can never create a complete and consistent finite list of axioms, or even an infinite list that can be produced by a computer program. Each time you add a statement as an axiom, there will always be other true statements that still cannot be proved as true, even with the new axiom. Those paradoxes above will still pop into your mathematical theorems or your life, no matter how hard you try to avoid them or fix them.

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