Wimpy Banach-Tarski paradox - should it break my intuition?
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From: Peter Webb on 18 Feb 2010 21:53 "Frederick Williams" <frederick.williams2(a)tesco.net> wrote in message news:4B7DC6B7.C7761E3A(a)tesco.net... > Gerry Myerson wrote: > >> A surface is a compact, connected topological space >> every point of which has a neighborhood homeomorphic >> to an open disk or a half-plane. > > Thank you. And given one of *those* what does it mean for it to have an > area? > A metric, ie a distance function. > -- > .... A lamprophyre containing small phenocrysts of olivine and > augite, and usually also biotite or an amphibole, in a glassy > groundmass containing analcime.
From: Rupert on 19 Feb 2010 02:07 On Feb 19, 10:01 am, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > Gerry Myerson wrote: > > A surface is a compact, connected topological space > > every point of which has a neighborhood homeomorphic > > to an open disk or a half-plane. > > Thank you. And given one of *those* what does it mean for it to have an > area? > > -- > .... A lamprophyre containing small phenocrysts of olivine and > augite, and usually also biotite or an amphibole, in a glassy > groundmass containing analcime. If it has a positive definite Riemannian metric, you can define an area easily enough for the Lebesgue measurable sets.
From: master1729 on 19 Feb 2010 08:30 compare : if you cut an apple in 2 and put those 2 pieces together again , you get an apple again with the same volume as before. it doesnt matter if you can measure the pieces or not. likewise if you cut it in 3 pieces or 4 pieces or n pieces. but banach-tarski gets a different volume. tommy1729
From: Jesse F. Hughes on 19 Feb 2010 18:50 master1729 <tommy1729(a)gmail.com> writes: > compare : > > if you cut an apple in 2 and put those 2 pieces together again , you get an apple again with the same volume as before. > > it doesnt matter if you can measure the pieces or not. > > likewise if you cut it in 3 pieces or 4 pieces or n pieces. > > but banach-tarski gets a different volume. Maybe Banach and Tarski are cutting two apples and don't know it. -- Jesse F. Hughes "So there is some sense in which your work is more akin to a work of mathematics than a banana is." -- Jim Ferry encourages James S. Harris
From: Rupert on 19 Feb 2010 20:57
On Feb 20, 10:30 am, master1729 <tommy1...(a)gmail.com> wrote: > compare : > > if you cut an apple in 2 and put those 2 pieces together again , you get an apple again with the same volume as before. > > it doesnt matter if you can measure the pieces or not. > It does matter. The Banach-Tarski paradox says that you can partition the unit ball into five pieces, not all of which are Lebesgue measurable, and rearrange them using rotational isometries to form a set of larger volume than the original unit ball (namely, the disjoint union of two balls of the same volume). Volume is only preserved under equidecomposability if the pieces are Lebesgue measurable (in dimension larger than 3, anyway). That is the whole *point* of the Banach-Tarski paradox. > likewise if you cut it in 3 pieces or 4 pieces or n pieces. > > but banach-tarski gets a different volume. > > tommy1729 |