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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:To clarify something on the point above, the 5 "pieces" are described as sets of points rather than actual objects with areas, and thus cannot be created in physical space. I edited the page to accentuate this, and to remove what I believed to be a contradictory statement. The original statement "This paradox has been proven for just about anything... except objects made of atoms, which our universe is comprised of." implies that a) objects made of atoms are not considered divisible and b) that most things are considered divisible. "Just about anything" could mean the physical universe, in which case the truth is that nothing is divisible and "just about anything" is misguiding, or both things that are within the physical universe and hypothetical things, in which case it deserves further explanation. Thus I edited to explain slightly further, being a safe move to improve the article in the case of either intention.[[Special:Contributions/108.162.215.72|108.162.215.72]] 05:17, 18 May 2014 (UTC) | :To clarify something on the point above, the 5 "pieces" are described as sets of points rather than actual objects with areas, and thus cannot be created in physical space. I edited the page to accentuate this, and to remove what I believed to be a contradictory statement. The original statement "This paradox has been proven for just about anything... except objects made of atoms, which our universe is comprised of." implies that a) objects made of atoms are not considered divisible and b) that most things are considered divisible. "Just about anything" could mean the physical universe, in which case the truth is that nothing is divisible and "just about anything" is misguiding, or both things that are within the physical universe and hypothetical things, in which case it deserves further explanation. Thus I edited to explain slightly further, being a safe move to improve the article in the case of either intention.[[Special:Contributions/108.162.215.72|108.162.215.72]] 05:17, 18 May 2014 (UTC) | ||
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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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