Wimpy Banach-Tarski paradox - should it break my intuition?
in [Math]
|
From: David Kevin on 15 Feb 2010 13:44 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
From: Rupert on 15 Feb 2010 14:51 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.
From: scattered on 15 Feb 2010 15:02 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?
From: Jay Belanger on 15 Feb 2010 15:55 >> >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. >> > _______________________________________ ... > 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? Finitely many translations of subsets of [0,2] won't be able to cover all of R; I assume that whatever less wimpy version he is referring to applies to bounded sets.
From: master1729 on 15 Feb 2010 07:46
> 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. not just for 3d spheres like banach-tarski , but already in 1 dimension. this might not be his intention , but he did. regards tommy1729 |