Wimpy Banach-Tarski paradox - should it break my intuition?
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From: Gerry Myerson on 21 Feb 2010 18:06 In article <874olcg501.fsf(a)phiwumbda.org>, "Jesse F. Hughes" <jesse(a)phiwumbda.org> wrote: > 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. Has anyone pointed out that an anagram for Banach-Tarski is Banach-Tarski Banach-Tarski? -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Frederick Williams on 21 Feb 2010 19:25 "David C. Ullrich" wrote: > True. So what? Nobody's claimed that it's [ZFC] been proven consistent, and > nobody who's familiar with Godel would make such a claim. V_kappa with kappa an inaccessible cardinal is a model of ZFC, so ZFC is consistent. So ZFC has been proven consistent (by me just now). You may say that inaccessible cardinals don't exist but it's not implausible that someone somewhere thinks that they do. As for me, I am not qualified to say, so my parenthetical remark may be dismissed as a joke.
From: Rupert on 21 Feb 2010 23:27 On 2ì22ì¼, ì¤ì 9ì36ë¶, master1729 <tommy1...(a)gmail.com> wrote: > ullrich wrote : > > > > > > > On Sat, 20 Feb 2010 19:27:55 EST, 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. > > > 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. You wrote "nothing , apart from the fact that terence tao showed - once again - that current set theory is inconsistant." Current set theory is ZFC.
From: Rupert on 21 Feb 2010 23:29 On 2¿ù22ÀÏ, ¿ÀÀü11½Ã25ºÐ, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > "David C. Ullrich" wrote: > > True. So what? Nobody's claimed that it's [ZFC] been proven consistent, and > > nobody who's familiar with Godel would make such a claim. > > V_kappa with kappa an inaccessible cardinal is a model of ZFC, so ZFC is > consistent. So ZFC has been proven consistent (by me just now). > > You may say that inaccessible cardinals don't exist but it's not > implausible that someone somewhere thinks that they do. > > As for me, I am not qualified to say, so my parenthetical remark may be > dismissed as a joke. Whel David Ullrich alludes to Goedel's second incompleteness theorem he is clearly pointing out that if ZFC is consistent, then ZFC (or any weaker theory) cannot prove that ZFC is consistent. It is true that ZFC+I (where I="there is an inaccessible cardinal") can prove that ZFC is consistent, so if you accept that every theorem of ZFC+I is true then that is a perfectly good proof that ZFC is consistent. But if you had doubts about the consistency of ZFC then you would not be very likely to be happy to accept that every theorem of ZFC+I is true.
From: David C. Ullrich on 22 Feb 2010 06:53
On Mon, 22 Feb 2010 10:06:25 +1100, Gerry Myerson <gerry(a)maths.mq.edi.ai.i2u4email> wrote: >In article <874olcg501.fsf(a)phiwumbda.org>, > "Jesse F. Hughes" <jesse(a)phiwumbda.org> wrote: > >> 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. > >Has anyone pointed out that an anagram for Banach-Tarski >is Banach-Tarski Banach-Tarski? Not as far as I know. It would be nice if someone did so... |