Wimpy Banach-Tarski paradox - should it break my intuition?
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From: scattered on 16 Feb 2010 06:37 On Feb 16, 6:04 am, Gc <gcut...(a)hotmail.com> wrote: > On 15 helmi, 20:44, 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. > > There can`t be a finite version when n=1, or N=2. It was proved > someone, probably Banach, that there is a finite additive "lebesgue > measure" in P(R) and P(R^2). This famously fails when n=3. > > > > >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- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text - The wording in Tao's website isn't clear. I also originally took "the less wimpy version" to refer to another (but less) "wimpy" version in which something like the Banach-Tarski paradox is obtained on the real line. That is obviously impossible by boundedness considerations (as Jay Berlanger points out) if you are trying to get the whole line with finitely many pieces from a bounded set and less obviously impossible (but still impossible) if you are trying to rearrange a bounded piece to yield a larger bounded piece. Robert Israel is undoubtably correct in interpreting Tao to mean the full-fledged Banach-Tarski version when he alludes to a "less wimpy version".
From: Gc on 16 Feb 2010 06:51 On 16 helmi, 13:37, scattered <semiscatte...(a)gmail.com> wrote: Robert Israel is undoubtably correct > in interpreting Tao to mean the full-fledged Banach-Tarski version > when he alludes to a "less wimpy version". Yep. Also Tao has written about this (Banach-Tarski paradox) more in his online lecture notes of Real Analysis.
From: master1729 on 15 Feb 2010 21:22 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 <=> 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 > tommy1729
From: David C. Ullrich on 16 Feb 2010 11:46 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 mean duh. > <=> 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 > > > > tommy1729 -- David C. Ullrich
From: William Hughes on 16 Feb 2010 12:07
On Feb 16, 8:22 am, master1729 <tommy1...(a)gmail.com> wrote: > David C Ullrich wrote : > > > > > On Mon, 15 Feb 2010 17:46:20 EST, master1729 > > <tommy1...(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 > > <=> inconsistant. > Brilliant! Wow! Of course! Banach-Tarski shows that one sphere has the same volume as multiple spheres. No one else has ever thought of that! Shhh... Don't let this out. If you show that set theory is inconsistent then all mathematicians will lose their jobs and what will JSH be able to threaten? -William Hughes |