Wimpy Banach-Tarski paradox - should it break my intuition?
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From: Aatu Koskensilta on 15 Feb 2010 18:22 master1729 <tommy1729(a)gmail.com> writes: > nothing , apart from the fact that terence tao showed - once again - > that current set theory is inconsistant. He's done it before? Of course, you're showing -- once again -- that you know pretty much nothing about set theory, whether that's your intention or not. Why, then, go on about it? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Robert Israel on 15 Feb 2010 18:44 David Kevin <davidkevin(a)o2.pl> writes: > 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. That is a rather misleading statement. The "less wimpy version", i.e. the actual Banach-Tarski paradox, has to involve a non-amenable group of motions. What Tao has here is basically a repackaging of the standard example of a nonmeasurable set. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: Aatu Koskensilta on 16 Feb 2010 10:44 master1729 <tommy1729(a)gmail.com> writes: > Banach-Tarski > > <=> volume 3 spheres = volume 1 sphere. > > <=> 3 spheres = 1 sphere. > > <=> 3 = 1 > > <=> inconsistant. Why not try and publish this wonderful stuff? The set theoretic establishment, in its stuffy stuffiness, is in need of a good kick in the knickers, what with their premice and morasses and this and what not. You have captured, in a word, the rottenness, the smelly extensive and putrescent extent, the maggoty, measly, wanting of an earlobe, corrupt core of it all, a bit like what's left of an apple after you've nibbled on it until there's nothing left. (I eat the core, myself.) -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: David C. Ullrich on 16 Feb 2010 05:43 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. > >not just for 3d spheres like banach-tarski , but already in 1 dimension. > >this might not be his intention , but he did. > >regards > >tommy1729
From: Gc on 16 Feb 2010 06:04
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 |