Wimpy Banach-Tarski paradox - should it break my intuition?
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From: Rupert on 16 Feb 2010 14:25 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.
From: master1729 on 16 Feb 2010 04:38 > 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, 2:51Â pm, Rupert > > > <rupertmccal...(a)yahoo.com> > > > >> wrote: > > > >> > On Feb 16, 5:44Â 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 ? > > > >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: Rupert on 16 Feb 2010 14:58 On Feb 17, 6:38 am, master1729 <tommy1...(a)gmail.com> wrote: > > In article > > <1694796382.199908.1266322987791.JavaMail.root(a)gallium > > .mathforum.org>, > > 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, 2:51Â pm, Rupert > > > > <rupertmccal...(a)yahoo.com> > > > > >> wrote: > > > > >> > On Feb 16, 5:44Â 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 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. > This does not follow. It would only follow if the pieces used in the paradox were Lebesgue measurable. Your later lines do not follow from this one either. > > > <=> 3 spheres = 1 sphere. > > > > <=> 3 = 1 > > > > <=> inconsistant. > > i missed nothing ! > > what did i miss ? > You missed the slightest clue of what you are talking about. > > > > >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- Hide quoted text - > > - Show quoted text -
From: scattered on 16 Feb 2010 15:15 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.
From: Rupert on 16 Feb 2010 16:18
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? |