Palo Alto Institute for Advanced Study
2007-12-18
|
Palo Alto Institute for Advanced Study 2007-12-18
|
|
|
The Good Shepherd’s Paradox (cont.)The Golfer’s ParadoxAs they cooled off at the 19th hole, Willie — the Old Pro Golfer, and distinctly a Scotsman — proposed a new “paradox of infinity”. He first moved one of the many small glasses that he had been drinking from toward the center of the table. (They were wee small glasses; our Golfer, like most Scotsmen, was not much of a drinker.) He placed in it, or rather on it, a golf ball, which then rested comfortably there, almost as if the wee small glass was an oversized tee. Figure 1. One Homeless Golf Ball, One Golf Ball in One Wee Small Glass(In fact, the wee small glasses were specially shaped and colored to look somewhat like golf tees. This explains the illustrations.) He handed another ball to the Mathematician, and asked him: “Laddie, can ya put this ball with the other one so tha’ they both perch there comfortably in the one wee small glass?” “Is this a trick question, Willie? No, not if you mean that they both perch there like that one does.” “Ha about if ya switch the balls?” “You can switch the balls, but that won’t make any difference. You’ll still have a ball left over.” “Then, I can’t say much for your set theory.” “Why is that?” “Because you said that infinity plus one equals infinity, and if tha’ is so, then I ought ta be able ta get both these balls ta perch in the one wee small glass.” “One isn’t infinity.” “But... infinity starts with one! Ya said so yourself!” Willie tried again. “Let me try again. How about if ya switch the one ball for the other many times, many many times, an infinite number of times?! Aleph‑null isn’t it?” “Yes, a0, but no, Willie! Not even if you switch them a0 times.” “How about... two ta the aleph‑null, isn’t it? How about switching them two ta the aleph‑null times?” “NO! Not even if you switch them an absolute infinity of times! There is NO number of times that you can switch the two balls so that they will both biject with the one glass... both fit in the same glass!” and he added with the start of a smirk “laddie!” Our Golfer gave our Mathematician a big smile. “Here, laddie...”, and Willie handed him another wee small glass with another ball perched on top. Figure 2. One Homeless Golf Ball, Two Golf Balls in Two Wee Small Glasses“Now can ya do it? Na, I reckon ya’ll say that this second wee small glass is abstractly equivalent ta the first wee small glass — at least for purposes of wha’ we’re about here.” “That’s right!” “And any other wee small glass with its one ball that I give ya, it’ll be abstractly equivalent to the first pair, and no use switchin’?” Figure 3. Homeless Golf Ball, Aleph‑null Golf Balls in Wee Small Glasses“That’s...” and the Mathematician started to say “right”, but he paused. “And...” said Willie with a histrionic hint of sly anticipation, “how about if I give ya aleph‑null wee small glasses, each with its one-and-only-one ball?! “Ya should be able ta switch the homeless ball with the ball in the first wee small glass I give ya, then the new homeless ball with tha’ in the second, which is abstractly equivalent ta the first, then in the third, which is also abstractly equivalent ta the first ... isn’t that how ya said it goes?!” “I was afraid you were going to ask me that...” mumbled the Mathematician. “When ya said that for every n there is an n plus one that it could biject with, ya forgot tha’ n plus one was already taken and paired, until ya remove its ball; then ya are still in the same fix since ya still ’av a ball tha’s homeless. Since the set is ‘complete’, ya canna’ add another wee small glass, an’ it would do you no good if it already had its own ball that it was paired with. And it would be doubly cheatin’ if it didna’ have one.” “That last” our Banker couldn’t help interjecting “would be logically equivalent to making the set Á both non-self-equal and non-self-equivalent, wouldn’t it?!” Grins broke out, except for our Mathematician. “I was hopin’ someon’ would have a fancy way a’ saying it” responded the Scotsman. “Laddie, I call this a paradox since your hundred year-old set theory says tha’ we should be able ta put aleph‑null plus one balls in aleph‑null glasses, but the only way ya can do tha’ is ta put two golf balls in one glass! And even you canna’ do tha’!” He paused, then offered his next insight somewhat slyly. “An’ I ken tha’, since ya need one more empty wee small glass, tha’ this is also the answer to your Continuum Hypothesis question!” The others looked at Willie in astonishment, then nodded agreement among themselves. Even our Mathematician seemed to display a hint of inspiration, or was it a sudden headache? “You mathematicians eat paradoxes for breakfast? Well, ’av one for tea! You are full of paradoxes, and you say your set theory is full of paradoxes, but I think this paradox will give you and your set theory a wee bit a’ indigestion! “Laddie, perhaps your set theory is a wee bit too paradoxical. Anyway, I call this... “The Golfer’s Paradox!”
|
|
|
