Wimpy Banach-Tarski paradox - should it break my intuition?
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From: Gerry Myerson on 16 Feb 2010 17:31 In article <1523562207.202333.1266349149177.JavaMail.root(a)gallium.mathforum.org>, master1729 <tommy1729(a)gmail.com> wrote: > to quote the Banach-Tarski paradox : > > The Banach�Tarski paradox is a theorem in set theoretic geometry which > states that a solid ball in 3-dimensional space can be split into a finite > number of non-overlapping pieces, which can then be put back together in a > different way to yield two identical copies of the original ball. The > reassembly process involves only moving the pieces around and rotating them, > without changing their shape. > > THAT is constant with : > > > >Banach-Tarski > > > > > > <=> volume 3 spheres = volume 1 sphere. > > > > > > <=> 3 spheres = 1 sphere. > > > > > > <=> 3 = 1 > > > > > > <=> inconsistant. > > > > > > > > i missed nothing ! > > what did i miss ? Most of 20th century mathematics. I suspect you've missed most of 19th, 18th, and 17th century mathematics as well, and more than a little of 16th, 15th, .... -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: master1729 on 16 Feb 2010 08:18 gerry myerson wrote : > In article > <1523562207.202333.1266349149177.JavaMail.root(a)gallium > .mathforum.org>, > master1729 <tommy1729(a)gmail.com> wrote: > > > to quote the Banach-Tarski paradox : > > > > The BanachTarski paradox is a theorem in set > theoretic geometry which > > states that a solid ball in 3-dimensional space > can be split into a finite > > number of non-overlapping pieces, which can then > be put back together in a > > different way to yield two identical copies of the > original ball. The > > reassembly process involves only moving the pieces > around and rotating them, > > without changing their shape. > > > > THAT is constant with : > > > > > >Banach-Tarski > > > > > > > > <=> volume 3 spheres = volume 1 sphere. > > > > > > > > <=> 3 spheres = 1 sphere. > > > > > > > > <=> 3 = 1 > > > > > > > > <=> inconsistant. > > > > > > > > > > > > i missed nothing ! > > > > what did i miss ? > > Most of 20th century mathematics. > > I suspect you've missed most of 19th, 18th, and 17th > century > mathematics as well, and more than a little of 16th, > 15th, .... > > -- > Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for > email) nonsense , thats not even an answer , only an insult.
From: David C. Ullrich on 17 Feb 2010 06:24 On Tue, 16 Feb 2010 14:38:38 EST, master1729 <tommy1729(a)gmail.com> wrote: >> In article >> <1694796382.199908.1266322987791.JavaMail.root(a)gallium >> .mathforum.org>, >> master1729 <tommy1729(a)gmail.com> wrote: >> >> > David C Ullrich wrote : >> > >> > > On Mon, 15 Feb 2010 17:46:20 EST, master1729 >> > > <tommy1729(a)gmail.com> >> > > wrote: >> > > >> > > >> On Feb 15, 2:51 pm, Rupert >> > > <rupertmccal...(a)yahoo.com> >> > > >> wrote: >> > > >> > On Feb 16, 5:44 am, David Kevin >> > > <davidke...(a)o2.pl> >> > > >> wrote: >> > > >> > >> > > >> > >> > > >> > >> > > >> > >> > > >> > >> > > >> > > In >> > > >> > >> > > >> > >> > > >> >> > > >> >http://www.math.ucla.edu/~tao/resource/general/121.1. >> > > >> 00s/tarski.html >> > > >> > >> > > >> > > Terence Tao has proven a wimpy version of >> > > >> Banach-Tarski paradox: >> > > >> > >> > > >> > > _______________________________________ >> > > >> > > Theorem.It is possible to take a subset of >> the >> > > >> interval [0,2], cut it >> > > >> > > up into a countable number of disjoint >> pieces, >> > > >> and then translate each >> > > >> > > of these pieces so that their union is the >> > > entire >> > > >> real line. >> > > >> > >> > > >> > > The less wimpy version of this theorem >> would >> > > use >> > > >> a finite number of >> > > >> > > pieces instead >> > > >> > > of a countable number, but the proof of >> that >> > > is >> > > >> extremely technical. >> > > >> > > _______________________________________ >> > > >> > >> > > >> > > In the fact I would be really surprised by >> > > >> > > the version with finite number but I'm >> wonder >> > > >> > > about above one should look such for me. >> The >> > > >> > > reason is that there is bijection between >> > > >> > > [0; 1] and real line and bijection between >> > > >> > > [0; 1] and (0; 1] which is based on shift: >> > > >> > >> > > >> > > 1) 0->1/2->1/3->1/4->.... >> > > >> > > 2) identity for x-es which aren't in 1) >> > > >> > >> > > >> > > so I suppose that it is possible to use >> > > something >> > > >> like that to achieve >> > > >> > > a constructive proof of above theorem. So >> > > should >> > > >> wimpy Banach-Tarski >> > > >> > > paradox break my intuition? >> > > >> > >> > > >> > > I would be really grateful for the proof >> of >> > > above >> > > >> theorem in version >> > > >> > > with finite number (despite Tao's comment >> that >> > > it >> > > >> is extremely >> > > >> > > technical I don't see how to prove it) >> > > >> > >> > > >> > > Thanks in advance for responses >> > > >> > >> > > >> > Stan Wagon's book "The Banach-Tarski >> paradox" is >> > > >> good.- Hide quoted text - >> > > >> > >> > > >> > - Show quoted text - >> > > >> >> > > >> I was looked at Wagon's book after reading the >> OP. >> > > >> That book seems to >> > > >> imply that finitely additive translation >> invariant >> > > >> measures which >> > > >> normalize the unit interval and are defined on >> all >> > > >> subsets of R exist. >> > > >> If so, I don't see how it would be possible to >> get >> > > by >> > > >> with finitely >> > > >> many pieces. What am I missing? >> > > > >> > > >nothing , apart from the fact that terence tao >> > > showed - once again - that current set theory is >> > > inconsistant. >> > > >> > > Why do you say these stupid things? >> > > >> > > All that this result, or the actual Banach-Tarski >> > > "paradox" shows, >> > > is that the union of set theory and your >> intuition is >> > > inconsistent. >> > > To show set theory itself is inconsistent you >> need a >> > > _proof_ in >> > > set theory of a _contradiction_. A proof of >> something >> > > that you >> > > find hard or impossible to believe doesn't count. >> > > >> > > > >> > >> > Banach-Tarski >> > >> > <=> volume 3 spheres = volume 1 sphere. >> > >> > <=> 3 spheres = 1 sphere. >> > >> > <=> 3 = 1 >> >> Seriously: What seems more likely: >> >> (i) There's some technical detail you're overlooking >> here, >> or that you're unaware of >> >> (ii) No mathematician has every noticed this simple >> proof >> that the BT paradox shows that set theory is >> inconsistent? >> >> I mean for heaven's sake, you _really_ think nobody >> every noticed this? Jeez. This is why it's >> (informally) >> called a "paradox". > >i missed nothing and not all mathematicians are set theory mathematicians !! > >to quote the Banach-Tarski paradox : > > The Banach�Tarski paradox is a theorem in set theoretic geometry which states that a solid ball in 3-dimensional space can be split into a finite number of non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without changing their shape. > >THAT is constant with : > >> >Banach-Tarski >> > >> > <=> volume 3 spheres = volume 1 sphere. >> > >> > <=> 3 spheres = 1 sphere. >> > >> > <=> 3 = 1 >> > >> > <=> inconsistant. >> > >> > > >i missed nothing ! > >what did i miss ? The fact that it's impossible to define the volume of a subset of R^3 in such a way that (i) volume(E) is defined for _every_ set E in R^3 (ii) if you obtain F from E by a rigid motion then volume(F) = volume(E) (iii) if E and F are disjoint then the volume of the union is the sum of the volumes. I know you don't believe that that's impossible. I also know that you _cannot_ tell me what the definition of the volume of an arbitrary subset of R^3 _is_, in a way that satisfies all three conditions. > >> > > >not just for 3d spheres like banach-tarski , but >> > > already in 1 dimension. >> > > > >> > > >this might not be his intention , but he did. >> > > > >> > > >regards >> > > > >> > > >tommy1729 >> > > >> > >> > tommy1729 >> >> -- >> David C. Ullrich > >tommy1729
From: scattered on 17 Feb 2010 07:46 On Feb 16, 4:18 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > On Feb 17, 7:15 am, scattered <semiscatte...(a)gmail.com> wrote: > > > > > > > On Feb 16, 2:25 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > > > > On Feb 16, 7:02 am, scattered <semiscatte...(a)gmail.com> wrote: > > > > > On Feb 15, 2:51 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > > > > > > On Feb 16, 5:44 am, David Kevin <davidke...(a)o2.pl> wrote: > > > > > > > In > > > > > > >http://www.math.ucla.edu/~tao/resource/general/121.1.00s/tarski.html > > > > > > > Terence Tao has proven a wimpy version of Banach-Tarski paradox: > > > > > > > _______________________________________ > > > > > > Theorem.It is possible to take a subset of the interval [0,2], cut it > > > > > > up into a countable number of disjoint pieces, and then translate each > > > > > > of these pieces so that their union is the entire real line. > > > > > > > The less wimpy version of this theorem would use a finite number of > > > > > > pieces instead > > > > > > of a countable number, but the proof of that is extremely technical. > > > > > > _______________________________________ > > > > > > > In the fact I would be really surprised by > > > > > > the version with finite number but I'm wonder > > > > > > about above one should look such for me. The > > > > > > reason is that there is bijection between > > > > > > [0; 1] and real line and bijection between > > > > > > [0; 1] and (0; 1] which is based on shift: > > > > > > > 1) 0->1/2->1/3->1/4->.... > > > > > > 2) identity for x-es which aren't in 1) > > > > > > > so I suppose that it is possible to use something like that to achieve > > > > > > a constructive proof of above theorem. So should wimpy Banach-Tarski > > > > > > paradox break my intuition? > > > > > > > I would be really grateful for the proof of above theorem in version > > > > > > with finite number (despite Tao's comment that it is extremely > > > > > > technical I don't see how to prove it) > > > > > > > Thanks in advance for responses > > > > > > Stan Wagon's book "The Banach-Tarski paradox" is good.- Hide quoted text - > > > > > > - Show quoted text - > > > > > I was looked at Wagon's book after reading the OP. That book seems to > > > > imply that finitely additive translation invariant measures which > > > > normalize the unit interval and are defined on all subsets of R exist. > > > > If so, I don't see how it would be possible to get by with finitely > > > > many pieces. What am I missing?- Hide quoted text - > > > > > - Show quoted text - > > > > Terry Tao uses countably many pieces.- Hide quoted text - > > > Yes of course, but Tao says "The less wimpy version of this theorem > > would use a finite number of pieces instead of a countable number, but > > the proof of that is extremely technical." where the pronoun "this" > > *seems* to refer to the theorem he just stated (a paradoxical > > decomposition of the real line), which is impossible with finitely > > many bounded pieces. As I indicated in a subsequent post, I now think > > that Tao's "this" in "this theorem" does not refer to the theorem he > > is actually proving, but rather to the Banch-Tarski paradox itself. > > The OP evidentally also interpreted Tao's statement as I originally > > did. > > > An interesting question is if it is possible to get a "paradoxical" > > decomposition of the real line using finitely many *unbounded* pieces. > > Since there is nothing paradoxical about infinity + infinity = > > infinity, this doesn't seem as obviously impossible as getting some > > sort of paradox with bounded pieces. I suspect that it is still > > impossible, but am not 100% sure.- Hide quoted text - > > > - Show quoted text - > > You can certainly do it with countably many pieces, using the ordinary > construction of a nonmeasurable set. > > I'm not sure either. It's a good question. > > You know the Sierpinski paradox? Take the set of all points of the > form f(e^i) where f is a polynomial with nonnegative integer > coefficients, and this set is the union of a rotation and translate of > itself?- Hide quoted text - > I'm not familar with that - thanks for bringing it to my attention. So the questions is if there is a 1-dimensional analogue of that. There isn't a Banach-Tarski type paradox for the plane. The fact that Sierpinski can do something like that on the complex plane shows that such things are not automatically ruled out by the existence of finitely-additive translation-invariant measures defined on all subsets of a space. On the other hand, his construction involves rotations, so even on the plane it is not clear that translations by themselves are enough to get (weakly) paradoxical behavior.
From: Gc on 17 Feb 2010 12:43
On 17 helmi, 14:46, scattered <semiscatte...(a)gmail.com> wrote: > On Feb 16, 4:18 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > > > > > On Feb 17, 7:15 am, scattered <semiscatte...(a)gmail.com> wrote: > > > > On Feb 16, 2:25 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > > > > > On Feb 16, 7:02 am, scattered <semiscatte...(a)gmail.com> wrote: > > > > > > On Feb 15, 2:51 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > > > > > > > On Feb 16, 5:44 am, David Kevin <davidke...(a)o2.pl> wrote: > > > > > > > > In > > > > > > > >http://www.math.ucla.edu/~tao/resource/general/121.1.00s/tarski.html > > > > > > > > Terence Tao has proven a wimpy version of Banach-Tarski paradox: > > > > > > > > _______________________________________ > > > > > > > Theorem.It is possible to take a subset of the interval [0,2], cut it > > > > > > > up into a countable number of disjoint pieces, and then translate each > > > > > > > of these pieces so that their union is the entire real line. > > > > > > > > The less wimpy version of this theorem would use a finite number of > > > > > > > pieces instead > > > > > > > of a countable number, but the proof of that is extremely technical. > > > > > > > _______________________________________ > > > > > > > > In the fact I would be really surprised by > > > > > > > the version with finite number but I'm wonder > > > > > > > about above one should look such for me. The > > > > > > > reason is that there is bijection between > > > > > > > [0; 1] and real line and bijection between > > > > > > > [0; 1] and (0; 1] which is based on shift: > > > > > > > > 1) 0->1/2->1/3->1/4->.... > > > > > > > 2) identity for x-es which aren't in 1) > > > > > > > > so I suppose that it is possible to use something like that to achieve > > > > > > > a constructive proof of above theorem. So should wimpy Banach-Tarski > > > > > > > paradox break my intuition? > > > > > > > > I would be really grateful for the proof of above theorem in version > > > > > > > with finite number (despite Tao's comment that it is extremely > > > > > > > technical I don't see how to prove it) > > > > > > > > Thanks in advance for responses > > > > > > > Stan Wagon's book "The Banach-Tarski paradox" is good.- Hide quoted text - > > > > > > > - Show quoted text - > > > > > > I was looked at Wagon's book after reading the OP. That book seems to > > > > > imply that finitely additive translation invariant measures which > > > > > normalize the unit interval and are defined on all subsets of R exist. > > > > > If so, I don't see how it would be possible to get by with finitely > > > > > many pieces. What am I missing?- Hide quoted text - > > > > > > - Show quoted text - > > > > > Terry Tao uses countably many pieces.- Hide quoted text - > > > > Yes of course, but Tao says "The less wimpy version of this theorem > > > would use a finite number of pieces instead of a countable number, but > > > the proof of that is extremely technical." where the pronoun "this" > > > *seems* to refer to the theorem he just stated (a paradoxical > > > decomposition of the real line), which is impossible with finitely > > > many bounded pieces. As I indicated in a subsequent post, I now think > > > that Tao's "this" in "this theorem" does not refer to the theorem he > > > is actually proving, but rather to the Banch-Tarski paradox itself. > > > The OP evidentally also interpreted Tao's statement as I originally > > > did. > > > > An interesting question is if it is possible to get a "paradoxical" > > > decomposition of the real line using finitely many *unbounded* pieces.. > > > Since there is nothing paradoxical about infinity + infinity = > > > infinity, this doesn't seem as obviously impossible as getting some > > > sort of paradox with bounded pieces. I suspect that it is still > > > impossible, but am not 100% sure.- Hide quoted text - > > > > - Show quoted text - > > > You can certainly do it with countably many pieces, using the ordinary > > construction of a nonmeasurable set. > > > I'm not sure either. It's a good question. > > > You know the Sierpinski paradox? Take the set of all points of the > > form f(e^i) where f is a polynomial with nonnegative integer > > coefficients, and this set is the union of a rotation and translate of > > itself?- Hide quoted text - > > I'm not familar with that - thanks for bringing it to my attention. So > the questions is if there is a 1-dimensional analogue of that. > > There isn't a Banach-Tarski type paradox for the plane. The fact that > Sierpinski can do something like that on the complex plane shows that > such things are not automatically ruled out by the existence of > finitely-additive translation-invariant measures defined on all > subsets of a space. Yes, they are - if they are not null sets. Think about how many polynomials f with integer coefficients you have. Then think about the measure of set {f(e^i)| f has positive integer coefficients} >On the other hand, his construction involves > rotations, so even on the plane it is not clear that translations by > themselves are enough to get (weakly) paradoxical behavior. I makes no difference, see for example book about Lebesgue integral by Jones. |