Wimpy Banach-Tarski paradox - should it break my intuition?
in [Math]
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From: Frederick Williams on 21 Feb 2010 11:36 "David C. Ullrich" wrote: > [...] if ZFC _is_ inconsistent that fact has a > simple proof. Can you expand on that, please? -- ..... A lamprophyre containing small phenocrysts of olivine and augite, and usually also biotite or an amphibole, in a glassy groundmass containing analcime.
From: master1729 on 21 Feb 2010 07:36 ullrich wrote : > 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 > >> > well , i didnt say ZFC has been proven inconsistant , i said it isnt proven consistant. thats not the same thing.
From: master1729 on 21 Feb 2010 07:37 Frederick wrote : > "David C. Ullrich" wrote: > > > [...] if ZFC _is_ inconsistent that fact has a > > simple proof. > > Can you expand on that, please? > > -- > .... A lamprophyre containing small phenocrysts of > olivine and > augite, and usually also biotite or an amphibole, in > a glassy > groundmass containing analcime. > lol
From: master1729 on 21 Feb 2010 07:39 > "master1729" <tommy1729(a)gmail.com> wrote in message > news:1980415988.239817.1266711683586.JavaMail.root(a)gal > lium.mathforum.org... > > > On Feb 20, 10:30Â 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... im talking about the finite cardinalities here. > > > > > > > > > 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: Gerry Myerson on 21 Feb 2010 18:05
In article <4B816108.37226752(a)tesco.net>, Frederick Williams <frederick.williams2(a)tesco.net> wrote: > "David C. Ullrich" wrote: > > > [...] if ZFC _is_ inconsistent that fact has a > > simple proof. > > Can you expand on that, please? If ZFC is inconsistent, then every statement (in the appropriate language) has a simple proof, no? Inconsistent means there is a staement P such that both P and not-P are theorems. For any statements Q and R, (Q and not-Q) implies R is a theorem. So if you want a simple proof of R, R follows by modus ponens from 1. (P and not-P) implies R and 2. P and not-P. -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email) |