Wimpy Banach-Tarski paradox - should it break my intuition?
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From: David C. Ullrich on 20 Feb 2010 04:43 On Fri, 19 Feb 2010 18:30:09 EST, master1729 <tommy1729(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. _Prove_ what you just said about apples, _without_ talking about the volume of the pieces. Once again: We're talking about what can be proved here, not what's intuitively obvious. Everyone agrees that the BT paradox shows that the union of set theory plus your intuittion is inconsistent - that doesn't show that set theory is inconsistent. >likewise if you cut it in 3 pieces or 4 pieces or n pieces. > >but banach-tarski gets a different volume. > >tommy1729
From: master1729 on 20 Feb 2010 09:20 > 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). but the fact that the cardinality of the disjoint union is the sum of the cardinalities of the terms in the family... ??? > > 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 >
From: master1729 on 20 Feb 2010 09:27 > On Fri, 19 Feb 2010 18:30:09 EST, master1729 > <tommy1729(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. > > _Prove_ what you just said about apples, _without_ > talking about the > volume of the pieces. > > Once again: We're talking about what can be proved > here, not > what's intuitively obvious. Everyone agrees that the > BT paradox > shows that the union of set theory plus your > intuittion is > inconsistent - that doesn't show that set theory is > inconsistent. speaking of ' intuition ' , your set theory is just an intuition of proposed axioms and further ZFC has not been proven consistant. further the conservation law of mass ( apple mass )agrees with me , so physics is on my side. > > >likewise if you cut it in 3 pieces or 4 pieces or n > pieces. > > > >but banach-tarski gets a different volume. > > > >tommy1729 >
From: Mike Terry on 20 Feb 2010 21:54 "master1729" <tommy1729(a)gmail.com> wrote in message news:1980415988.239817.1266711683586.JavaMail.root(a)gallium.mathforum.org... > > 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). > > but the fact that the cardinality of the disjoint union is the sum of the cardinalities of the terms in the family... ??? .... means that BT implies that the cardinality of the set of points in two balls equals the cardinality of the set of points in a single ball. But we knew that anyway before BT... > > > > > 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 > >
From: Aatu Koskensilta on 21 Feb 2010 05:06
master1729 <tommy1729(a)gmail.com> writes, quoting Rupert: >> 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). > > but the fact that the cardinality of the disjoint union is the sum of > the cardinalities of the terms in the family... ??? Cardinality and volume are two distinct notions. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |