Changes

:I think the introduction understates the motivation of the paradox.  Wasn't it to illustrate how the AoC, if embraced, can lead to absurd conclusions?--[[User:Aschlafly|Andy Schlafly]] 22:33, 5 January 2010 (EST)

:I think the introduction understates the motivation of the paradox.  Wasn't it to illustrate how the AoC, if embraced, can lead to absurd conclusions?--[[User:Aschlafly|Andy Schlafly]] 22:33, 5 January 2010 (EST)

No, it wasn't, but the Vitali set was, although I wouldn't use the word absurd, but that's a debate for another time.  The Banach-Tarski paradox was developed to investigate group representations.  Nowadays, with representation theory a hundred years more advanced, it isn't much inspiration to algebraists anymore, and is now primarily used to illustrate the strangeness you describe. It is often used to demonstrate the "weirdness" of the AoC.  Since the Vitali set WAS developed partly for the purpose you ascribe to the Banach Tarski paradox, I don't think we lose any of the strength of the original statement about the AoC by saying the Vitali paradox was developed to illustrate that weirdness.

No, it wasn't, but the Vitali set was.  The Banach-Tarski paradox was originally developed to investigate group representations.  Today, it is used to illustrate the "weirdness" of the AoC.  Since the Vitali set WAS developed partly for the purpose you ascribe to the Banach Tarski paradox, I don't think we lose any of the strength of the original statement about the AoC by saying the Vitali paradox was developed to illustrate that weirdness.

If you like, I can have a full explanation of the [[Vitali set]] in that article up tonight.  It deals more directly with why the AoC is necessary for the strange construction, and spends less time dealing with putting free groups on spheres and such things.   

If you like, I can have a full explanation of the [[Vitali set]] in that article up tonight.  It deals more directly with why the AoC is necessary for the strange construction, and spends less time dealing with putting free groups on spheres and such things.   

Summary: Although the Banach-Tarski paradox quickly and easily demonstrates the strangeness of the AoC, and I have no problem including that, that wasn't the motivation for its development. [[User:JacobB|JacobB]] 22:44, 5 January 2010 (EST)

Summary: Although the Banach-Tarski paradox quickly and easily demonstrates the strangeness of the AoC, and I have no problem including that, that wasn't the motivation for its development. [[User:JacobB|JacobB]] 22:44, 5 January 2010 (EST)