Palo Alto Institute for Advanced Study
2007-12-18
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Palo Alto Institute for Advanced Study 2007-12-18
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FOR IMMEDIATE RELEASE The Good Shepherd’s ParadoxA New Paradox of Infinity in Set Theory2002 Feb 11
“A Golfer, a Banker, and a Man of the Cloth were playing a round of golf with a Mathematician, who told them all about the old paradoxes of infinity in set theory. At the 19th hole...” Set Theory says that “infinity + 1 equals infinity”, and that you can prove it by matching n with n + 1 for all Natural Numbers n, and then matching 0 (not strictly a natural number) with 1 . But... look at the picture above. The Golfer — a canny Scotsman — asks the others to try to imagine how they would add the lone golf ball (i.e. the + 1) to the infinity of golf balls matched strictly 1-to-1 with their “wee small glasses”. To put the ball in a glass, any glass, you must first take out the ball already there. No matter which glass you choose, no matter how many times you switch a lone ball for one in a glass, even an absolute infinity of times, (unless you cheat) you still have precisely one lone golf ball... The Banker presents “the other side of the coin” of this paradox, and the Man of the Cloth presents yet another. This is an entertaining approach the question of the inconsistency of set theory, but the math behind it is perfectly rigorous. Mathematicians usually dislike physical analogies since they can be mathematically misleading, but here the physical analogies can be quickly understood and seen to be trivially translatable into rigorous formal proofs of the inconsistency of set theory.
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