What is the Banach-Tarsky Paradox and its connection to non-measurable sets?

[eljose79]

What is this paradox?..someone could explain me please i lost this conference given in my university.

 

[ottjes]

http://www.math.hmc.edu/funfacts/ffiles/30001.1-3-8.shtml

http://www.math.hmc.edu/funfacts/ffiles/30001.1-3-8.shtml

paradox in short:

image from the first link

 

[selfAdjoint]

in other words, if you assume basic measure theory and the axiom of choice, then you can tell how to disassemble a solid ball into five pieces and reassemble those pieces into two balls just as solid and the same size as the original. I want to emphasize that this is a mathematical prescription. You can't do it in reality because you have to use the axiom of choice to do the fitting. It says "there is a way" to fit the pieces together, but doesn't say what the way is.

 

[eljose79]

imposible...

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?..

 

[ottjes]

why not post in the right topic :?

 

[Hurkyl]

The reason you can't do it in reality is because the cuts needed to execute the paradox are very pathological and are in direct opposition to the presumed "well-behavior" of physical reality, not because of the Axiom of Choice.

Hurkyl

 

[Ben-CS]

Actually, I am not so sure that this cannot be done in reality. I know classical mechanics forbids such; but, quantum mechanics may very well allow such an operation.

 

[selfAdjoint]

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 paradox!

 

[selfAdjoint]

Originally posted by Hurkyl
The reason you can't do it in reality is because the cuts needed to execute the paradox are very pathological and are in direct opposition to the presumed "well-behavior" of physical reality, not because of the Axiom of Choice.

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.

 

[Hurkyl]

Still, there's no physical assumption that is violated by considering the Axiom of Choice valid... whether or not the Axiom of Choice is true, you're physically not allowed to cut objects up into pieces so pathological they have no measure.

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 existence of bases for uncountably infinite dimensional vector spaces? (I've had trouble digging up the precise formulation, so I'm not sure)

Hurkyl

 

[HallsofIvy]

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?..[/QUOTE}

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.

 

[selfAdjoint]

Umm, well, but good mathematics may define things that are beyond physics, but it shouldn't say things that are directly against physics. Saying that the physical impossiblility of doing this with real materials gets math of the hook misses the point. You can't construct a true circle or a length equal to pi either, but they are still a part of physics. Half the math used in the proof of the Banach-Tarski phenomenon is the same math that is used in quantum field theory - measure theory. And the axiom of choice is beloved by mathematicians because it makes impossible proofs possible.

 

[mathman]

The fundamental problem with the paradox and physics is that it is impossible to CONSTRUCT non-measurable sets, either mathematically or physically. The axiom of choice only allows their existence.