Editing Talk:804: Pumpkin Carving
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:(2) discover a compelling reason why we should not accept the axiom of choice. | :(2) discover a compelling reason why we should not accept the axiom of choice. | ||
− | :The Banach-Tarski theorem was published at the height of the debate/research, and is still frequently the first thing | + | :The Banach-Tarski theorem was published at the height of the debate/research, and is still frequently the first thing sited by someone who doesn’t accept the axiom of choice (although most working mathematicians I know do accept the axiom of choice, in part because it just seems silly to handicap yourself unnecessarily). |
:What makes the Banach-Tarski theorem seem so paradoxical is simply the fact that they show it is possible to cut a ball into a finite number of pieces (5, to be specific) and reassemble these pieces only using rotations and translations (ie, only by movements you can make with your own hands) to produce two balls, each identical in volume to the first–ie, in someways "1 [ball] = 2 [balls]", which certainly feels a bit shady. | :What makes the Banach-Tarski theorem seem so paradoxical is simply the fact that they show it is possible to cut a ball into a finite number of pieces (5, to be specific) and reassemble these pieces only using rotations and translations (ie, only by movements you can make with your own hands) to produce two balls, each identical in volume to the first–ie, in someways "1 [ball] = 2 [balls]", which certainly feels a bit shady. | ||
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:For myself, I actually *don’t* find the theorem very paradoxical at all–these 5 “pieces” into which the ball is decomposed are incredibly crazy, and nothing you could ever cut with a scalpel, or even a laser, no matter how good you are. Additionally, the theorem really follows relatively easily from a theorem everyone accepts, which is that you can similarly split up the group of rotations to four disjoint pieces and “reassemble” them via rotations into two copies of the group of rotations. It seems a little odd at first, but the thing to keep in mind that any time infinite things get involved, things are going to get a little odd (after all–what’s infinity + infinity? What about infinity/2?). Honestly, at this point, the thing which I find the most “paradoxical” about the Banach-Tarski paradox is that it can’t be done by dividing the ball into only 4 pieces, but this could be a sign I’ve been drinking the math-koolaid for too long. — Ashley | :For myself, I actually *don’t* find the theorem very paradoxical at all–these 5 “pieces” into which the ball is decomposed are incredibly crazy, and nothing you could ever cut with a scalpel, or even a laser, no matter how good you are. Additionally, the theorem really follows relatively easily from a theorem everyone accepts, which is that you can similarly split up the group of rotations to four disjoint pieces and “reassemble” them via rotations into two copies of the group of rotations. It seems a little odd at first, but the thing to keep in mind that any time infinite things get involved, things are going to get a little odd (after all–what’s infinity + infinity? What about infinity/2?). Honestly, at this point, the thing which I find the most “paradoxical” about the Banach-Tarski paradox is that it can’t be done by dividing the ball into only 4 pieces, but this could be a sign I’ve been drinking the math-koolaid for too long. — Ashley | ||
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There you have it. [[User:Lcarsos|lcarsos]] ([[User talk:Lcarsos|talk]]) 17:42, 4 September 2012 (UTC) | There you have it. [[User:Lcarsos|lcarsos]] ([[User talk:Lcarsos|talk]]) 17:42, 4 September 2012 (UTC) | ||
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