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Topic: Do you know the Banach-Tarski theorem? (Read 1923 times) |
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BNC
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Do you know the Banach-Tarski theorem?
« on: Oct 3rd, 2003, 12:29am » |
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I'm wandering if anyone here knows of this theorem. I just heard about it -- and boy, this is a strange one!
In 1924 Banach and Tarski showed that you could cut up a solid sphere into six pieces and then reassemble them into two solid spheres of exactly the same size. In fact, the sphere could be cut and reassembled into any shape or size object whatsoever.
Now, I cannot even begin to follow the froof, but it seems so counter-intuitive.... I don't know. Is it "explainable" in layman's terms?
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ThudnBlunder
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Re: Do you know the Banach-Tarski theorem?
« Reply #1 on: Oct 3rd, 2003, 12:49am » |
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Q - do you know a good anagram of "banach-tarski" ?
A - banach-tarski banach-tarski
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towr
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Re: Do you know the Banach-Tarski theorem?
« Reply #2 on: Oct 3rd, 2003, 12:51am » |
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I've heard of it.. And I'll accept it's possible, but I don't much understand it..
http://mathworld.wolfram.com/Banach-TarskiParadox.html
Someone also reduced the necessary number of pieces to 5 (and it could've been 4 if it weren't for that damn point in the center :p)..
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| « Last Edit: Oct 3rd, 2003, 12:53am by towr » |
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BNC
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Re: Do you know the Banach-Tarski theorem?
« Reply #4 on: Oct 3rd, 2003, 6:25am » |
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Thanks Barukh, that's exactly what I was looking for. I didn't realise the theorem was an "existance" rather than "constructive" one.
It still bugs me, though. I accept (after reading your link) that the theorem can never be accomplished, because actual balls are constructed from atoms, and are thus not continous. BUT -- it still seems to me to violate the conservation of mass law (although I can't think of a way to express it with infinitely dense "material" required.)
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Icarus
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Re: Do you know the Banach-Tarski theorem?
« Reply #5 on: Oct 3rd, 2003, 9:00pm » |
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I worked through the proof of the Banach-Tarski paradox in college. It is not only non-constructive, I believe it is no longer a theorem if one removes the Axiom of Choice from the set of standard axioms of set theory.
What it violates from a mathematical point of view is the additive property of measurement. One of the basic properties of counting, length, area, volume, and their higher dimensional equivalents is that if you split a set into two disjoint parts, the measure of the whole is the sum of the measures of the two parts.
In order to generalize the Cauchy-Reimann Integral, Lesbegue invented the concept of measure, which extends this property to infinite sums: if A = [bigcup]{Ai}i=1[supinfty], and all the Ai are pairwise disjoint, then [mu](A) = [sum] [mu](Ai).
Unfortunately, his method of definition does not work for all sets in [bbr]n. So some sets are left "unmeasurable".
Banach and Hausdorff explored the question of whether it was possible to extend the definition to all possible sets. They quickly established that the answer was no. But what if we discard the requirement to hold for infinite sums - i.e., is it possible for a measure to be defined for all sets so that when A & B are disjoint [mu](A [cup] B) = [mu](A) + [mu](B)?
For one and two dimensions, this turns out to be possible. But not in a natural fashion. I.e., there are infinitely many such measures, all of which restrict to the same measure on "Lesbegue measurable" sets, but which differ on unmeasurable sets. And there is no a priori reason to choose one over the others.
When Banach turned to 3 dimensions, he and Tarski (who was his student if I recall correctly) came across this result, which shows that even when restricted to finite sums, a measure cannot be defined on all sets in [bbr]3 or higher dimensions.
Though it is not a true paradox, it is often called that, because the result is so counter-intuitive.
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towr
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Re: Do you know the Banach-Tarski theorem?
« Reply #6 on: Oct 5th, 2003, 7:01am » |
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on Oct 3rd, 2003, 9:00pm, Icarus wrote:| I worked through the proof of the Banach-Tarski paradox in college. It is not only non-constructive, I believe it is no longer a theorem if one removes the Axiom of Choice from the set of standard axioms of set theory. |
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That's probably why it is also used as an argument against the Axiom of Choice.
Some find mathematics more comfortable to work/live with without it (and the 'paradox')..
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Sir Col
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Re: Do you know the Banach-Tarski theorem?
« Reply #7 on: Oct 5th, 2003, 8:17am » |
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I've always wondered about this theorem, and I've never met anyone who's worked through it before.
So, Icarus, do the two spheres have the same density?
Surely, as both towr and yourself have indicated, if this result follows from allowing the axiom of choice to be included, then it is proof that the axiom of choice leads to internal inconsistencies. I don't know what Gödel and Cohen would say, though, as I thought that they had proved that the axiom of choice is independent of the Zermelo-Fränkel set of axioms.
Once again, I find myself out-of-my-depth.
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Icarus
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Re: Do you know the Banach-Tarski theorem?
« Reply #8 on: Oct 5th, 2003, 9:04pm » |
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The Banach-Tarski theorem is surprising, but not a true paradox. I.e. it goes counter to our intuition & desires, but not to any actual mathematics. There are no internal inconsistencies. Just something that we wish were not so.
Denying the axiom of choice does not cause the problem to go away. It just means we are no longer able to decide the question either way. With the axiom of choice, we can show that there does not exist a volume measure defined on all subsets of [bbr]3. Without the axiom of choice, we still don't get such a measure. We just can no longer prove that it doesn't exist. And we know that there is no way for us to prove that it does exist. Otherwise the axiom of choice would be contradictory, and given the amount of experience we have with it, this is highly unlikely.
Quite frankly, I would not want a mathematics where the axiom of choice does not hold. Remember would imply that for some non-empty collection of non-empty sets Ai[/sup], that the set [prod][sub]i Ai is still empty.
This seems to me to be a sillier idea than the one that some subsets of [bbr]3 are so weird that they cannot be given a well-defined volume.
As for working through the theorem, I'm afraid that was better than 15 years ago. My real analysis class was broken into groups and each group assigned to research a topic and present it to the rest of the class. I got Banach-Tarski, but really would have prefered Hausdorff measures (fractals were the hot new thing at the time). Now I have forgotten how Banach-Tarski was proved, and never got around to studying up on Hausdorff measures. *Sigh*
Density is not involved in Banach-Tarski. It is about sets themselves, not material objects. And though the "double the size of the sphere" form is most often quoted, the full statement is: Given any two sets A and B with non-empty interiors, there exists a finite collection of pairwise disjoint sets {Ai} and rigid translations {ti} such that [bigcup] Ai = A, [bigcup] ti(Ai) = B, and the sets {ti(Ai)} are also pairwise disjoint. (For a set to have non-empty interior means that there is a ball somewhere of some radius > 0 that is entirely contained within the set.)
Since no density is assigned to set A or set B, there is no way to compare them.
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Sir Col
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Re: Do you know the Banach-Tarski theorem?
« Reply #9 on: Oct 6th, 2003, 12:37am » |
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If you don't mind me asking more rudimentary questions about this fascinating concept...
Is the result restricted to spheres, solids with curved surfaces, or will it apply to other solids, such as the cube?
Does it show that [bbr] are insufficient to define all 3-dimension space?
Is there any (common sense) resolve for this theorem?
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ThudnBlunder
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Re: Do you know the Banach-Tarski theorem?
« Reply #10 on: Oct 6th, 2003, 9:05am » |
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Quote:| The Banach-Tarski theorem is surprising, but not a true paradox. |
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There are several shades of meaning of the word 'paradox'.
par·a·dox (p²r“…-d¼ks”) n.
1. A seemingly contradictory statement that may nonetheless be true.
2. One exhibiting inexplicable or contradictory aspects.
3. An assertion that is essentially self-contradictory, though based on a valid deduction from acceptable premises. 4. A statement contrary to received opinion.
(American Heritage Dictionary)
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| « Last Edit: Oct 6th, 2003, 9:07am by ThudnBlunder » |
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Icarus
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Re: Do you know the Banach-Tarski theorem?
« Reply #11 on: Oct 6th, 2003, 7:43pm » |
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In mathematics, we prefer to have terms more tightly defined than is true of language in general. Mathematically speaking, something is a paradox only if both it and its negation are provable. Thus the Banach-Tarski theorem is not actually a paradox.
But even though it does not seem like it sometimes, English-speaking mathematicians are also normal speakers of the English language. So we use the looser definitions provided by normal English as well. And by that standard, Banach-Tarski is a paradox. But being mathematicians, we then have to point out that it is not a "true paradox". I.e. not a paradox by the strict mathematical meaning.
Speaking of jargon, Sir Col, if you can manage to digest the jargon in my description of the theorem in the previous post, you will see that it actually says that any two shapes can be transformed into one another, provided that they both have non-empty interior. "Non-empty interior" means that they are both solid somewhere (they do not have to be solid everywhere: A plane with a tiny ball in the middle would work).
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