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- Thread starter eljose79
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selfAdjoint

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that is ilogical you can not make two equal balls with only one....in fact you can make two smaller balls similar to the original one but not two.....and where is the matter to make the two balls?.. i think that there is no physical sense in all that..(although maths proves to be correct)....what is the solution of the paradox?..

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why not post in the right topic :?

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Hurkyl

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Hurkyl

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selfAdjoint

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Originally posted by eljose79

that is ilogical you can not make two equal balls with only one....in fact you can make two smaller balls similar to the original one but not two.....and where is the matter to make the two balls?.. i think that there is no physical sense in all that..(although maths proves to be correct)....what is the solution of the paradox?..

That's why they call it the Banach-Tarski

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selfAdjoint

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Originally posted by Hurkyl

Hurkyl

Actually not quite right. Before you reassemble the pieces you have to perform certain special isometries on them. These are transformations that do not change the distance between points. The particular isometries push inconvenient matter up a long cusp , and you use the Axiom of Choice to ensure that there is a place to push that will accomplish the desired relationship. The actual proof is rather complicated.

Since the only "prior mathematics" that is used in the proof are basic measure theory and the Axiom of Choice, the existence of this paradox has been taken as a hit on the axiom of choice. There were several attempts to frame an axiom that does not support Banach-Tarski but still does the things that mathematicians like the A. o. C. for.

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Hurkyl

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Although I know this paradox is "evidence" against the Axiom of Choice, IMHO there's a certain inconsistency of reasoning to find it troublesome that objects outside the realm of measure theory behave differently than objects inside the realm of measure theory.

To put it in perspective, how many people complain that the axiom of infinity allows us to create a nonempty set that can be divided into two disjoint subsets both the same size as the original?

Anyways, isn't the axiom of choice used in Quantum Mechanics to assert the existance of bases for uncountably infinite dimensional vector spaces? (I've had trouble digging up the precise formulation, so I'm not sure)

Hurkyl

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HallsofIvy

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That mathematics is not physics, we are not talking about physical balls, there is no "matter" to the balls and so "physical sense" is irrelevant.

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selfAdjoint

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