Axiom of Choice and something I find to not be logical

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The discussion centers on the Banach-Tarski Paradox, which suggests that accepting the Axiom of Choice allows for the theoretical division of a sphere into parts that can be rearranged to form two spheres of greater volume. The proof involves sets that are not measurable, meaning their volume cannot be defined, leading to the counterintuitive conclusion that the original volume is not preserved. The construction notably uses a limited number of pieces, emphasizing that the intermediate steps involve disjoint subsets that collectively equal the original volume despite lacking individual measure. The mathematical complexity of the proof is acknowledged, with references to surgery theory as part of the process. Overall, the paradox highlights significant implications of set theory and the Axiom of Choice in mathematics.
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I heard something along the lines of when you accept the axiom of choice as true, you can then prove using some abstract set theory that by dividing a sphere, you can divide it and then put it together so that it is bigger than it originally was?

Is the math behind this proof difficult? And is this true?
 
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from Kuro5hin - Layman's Guide to the Banach-Tarski Paradox --->
http://www.kuro5hin.org/story/2003/5/23/134430/275

search "Banach-Tarski" for more stuff.

Feynman said phooey about B-T --->
http://www.ams.org/new-in-math/mathdigest/200112-choice.html
 
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IIRC, the proof uses surgery theory.

One of the main things to emphasize about the construction is that its intermediate steps involve sets that are not measurable. All of the clever work is done with sets for which you cannot define volume, so there isn't any reason to expect that you have the original volume when you're done.


And, incidentally, the big point about the construction is that it only uses five pieces. It's a trivial exercise to prove that you can rearrange all of the points in one sphere to form two spheres of the same size, if you do it one point at a time.
 
Last edited:
Hurkyl said:
IIRC, the proof uses surgery theory.

One of the main things to emphasize about the construction is that its intermediate steps involve sets that are not measurable. All of the clever work is done with sets for which you cannot define volume, so there isn't any reason to expect that you have the original volume when you're done.


And, incidentally, the big point about the construction is that it only uses three pieces. It's a trivial exercise to prove that you can rearrange all of the points in one sphere to form two spheres of the same size, if you do it one point at a time.


The "not measureable" subsets are disjoint and add up to the whole original ball. Therefore by linearity of measure their total measure is the original volume, even though that can't be allocated to them in any way.
 

Thread 'Formal derivation of statement from Peano Arithmetic system'

I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
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