Wimpy Banach-Tarski paradox - should it break my intuition?
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From: Rupert on 21 Feb 2010 05:17 On Feb 21, 11:20 am, master1729 <tommy1...(a)gmail.com> wrote: > > 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... ??? > Yes, that's fine. In the Banach-Tarski paradox, one of the pieces is countable and the other pieces have cardinality c. The sum has cardinality c. This is fine because both the unit ball and the disjoint union of two unit balls have cardinality c. > > > > > > 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- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text -
From: Rupert on 21 Feb 2010 05:19 On Feb 21, 11:27 am, master1729 <tommy1...(a)gmail.com> wrote: > > On Fri, 19 Feb 2010 18:30:09 EST, 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. > > > _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. > Physics doesn't come into it here. We are talking about arbitrary point sets. There is no reason to think that ZFC is inconsistent, and you have done nothing in particular to cast doubt on the truth of the axioms. In any event it is undeniable that the Banach-Tarski paradox is a theorem of ZFC. > > > > > > >likewise if you cut it in 3 pieces or 4 pieces or n > > pieces. > > > >but banach-tarski gets a different volume. > > > >tommy1729- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text -
From: Aatu Koskensilta on 21 Feb 2010 05:26 Rupert <rupertmccallum(a)yahoo.com> writes: > Physics doesn't come into it here. We are talking about arbitrary > point sets. From an old post by Torkel Franz�n: Some mathematical operations on mathematical balls ("solid spheres" - in mathematical terminology, a sphere is just the "shell" of a ball) correspond to physical operations on physical balls, for example cutting a ball in half. For other such mathematical operations there is no corresponding physical operation on physical balls. We simply cannot make any physical sense of the operation. Consider for example "removing from the unit ball all points with rational coordinates". What could this mean physically? In the case of the Banach-Tarski theorem, the operations considered, since they involve non-measurable sets, don't even correspond to anything in ordinary mathematical models used in physics. For the purposes of pondering why there is no physical operation corresponding to the mathematical "dissection", it is sufficient to think about the simple argument for the existence of a non-measurable subset of the interval I=[0,1]. We partition the interval by putting x and y in the same equivalence class if x-y is rational. Now let M be a set containing a representative from each of the uncountably many equivalence classes: M cannot be measurable, since I is the union of the countably many disjoint translations M+r (mod 1) of M. This definition of M has no apparent similarity to any cutting, carving or slicing that makes physical sense. (Message-ID: <vcbwug86g9c.fsf(a)beta13.sm.luth.se>) -- 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 21 Feb 2010 06:13 On Sat, 20 Feb 2010 19:27:55 EST, master1729 <tommy1729(a)gmail.com> wrote: >> 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. True. So what? Nobody's claimed that it's been proven consistent, and nobody who's familiar with Godel would make such a claim. The issue is your statement that BT shows that set theory is _inconsistent_. it doesn't. >further the conservation law of mass ( apple mass )agrees with me , so physics is on my side. In other words, you _can't_ prove it. (Which is no surprise, since it can't be proved, as the BT "paradox" shows). You should really just cut your losses, admit that you can't show that BT shows set theory is inconsistent, and stop making a fool of yourself. In case you still haven't got it: "inconsistent with intution" is not the same as "inconsistent". "Inconsistent with conservation of mass" is not the same as "inconsistent". You've (of course) given no evidence at all that BT shows that ZFC is inconsistent. Which is curious, since if ZFC _is_ inconsistent that fact has a simple proof. >> >> >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 06:19
David C. Ullrich <ullrich(a)math.okstate.edu> writes: > True. So what? Nobody's claimed that it's been proven consistent, and > nobody who's familiar with Godel would make such a claim. Whether we regard ZFC as provably consistent depends on what mathematical principles we accept. > In case you still haven't got it: "inconsistent with intution" is not > the same as "inconsistent". "Inconsistent with conservation of mass" > is not the same as "inconsistent". It makes no sense to claim ZFC is "inconsistent with conservation of mass". ZFC doesn't tell us anything about physical matters. > Which is curious, since if ZFC _is_ inconsistent that fact has a > simple proof. How so? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |