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Full Idea
The problem with the Axiom of Choice is that it allows an initiate (by an ingenious train of reasoning) to cut a golf ball into a finite number of pieces and put them together again to make a globe as big as the sun.
Gist of Idea
Using Choice, you can cut up a small ball and make an enormous one from the pieces
Source
R Kaplan / E Kaplan (The Art of the Infinite [2003], 9)
Book Ref
Kaplan,R and Kaplan,E: 'The Art of the Infinite' [Penguin 2004], p.256
A Reaction
I'm not sure how this works (and I think it was proposed by the young Tarski), but it sounds like a real problem to me, for all the modern assumptions that Choice is fine.
| 15712 | 1 and 0, then add for naturals, subtract for negatives, divide for rationals, take roots for irrationals [Kaplan/Kaplan] |
| 15711 | The rationals are everywhere - the irrationals are everywhere else [Kaplan/Kaplan] |
| 15713 | The first million numbers confirm that no number is greater than a million [Kaplan/Kaplan] |
| 15714 | 'Commutative' laws say order makes no difference; 'associative' laws say groupings make no difference [Kaplan/Kaplan] |
| 15715 | 'Distributive' laws say if you add then multiply, or multiply then add, you get the same result [Kaplan/Kaplan] |
| 15717 | Using Choice, you can cut up a small ball and make an enormous one from the pieces [Kaplan/Kaplan] |