Wimpy Banach-Tarski paradox - should it break my intuition?
in [Math]
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From: David C. Ullrich on 22 Feb 2010 06:56 On Sun, 21 Feb 2010 13:19:51 +0200, Aatu Koskensilta <aatu.koskensilta(a)uta.fi> wrote: >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. Ok. >> 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". Yes of course - I was replying to Timmy, not trying to make sense. >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? ??? If ZFC is inconsistent then there exists a proof in whatever formal system we're using that ends with P & ~ P for some P. Hmm. Ok, it may well be a very long proof - your issue is with the word "simple"?
From: David C. Ullrich on 22 Feb 2010 06:58 On Sun, 21 Feb 2010 17:36:40 EST, master1729 <tommy1729(a)gmail.com> wrote: >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 , Yes you did. Hard to decide whether you're a liar or just have bad memory, but it's very easy to look this up in the thread. > i said it isnt proven consistant. > >thats not the same thing.
From: David C. Ullrich on 22 Feb 2010 07:02 On Mon, 22 Feb 2010 00:25:09 +0000, Frederick Williams <frederick.williams2(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 No, instead I'll say that "ZFC is consistent" gives a much simpler proof of the consistency of ZFC. >but it's not >implausible that someone somewhere thinks that they do. Unless I'm totally wrong, it's not only not implausible, it's _true_ that most people who've thought about it think that ZFC is consistent. So there we have it, an ironclad proof. I stand corrected. (And by the way, keep your eyes open for my proof of the Riemann Hypothesis.) >As for me, I am not qualified to say, so my parenthetical remark may be >dismissed as a joke.
From: Aatu Koskensilta on 22 Feb 2010 10:30 David C. Ullrich <ullrich(a)math.okstate.edu> writes: > Hmm. Ok, it may well be a very long proof - your issue is with the > word "simple"? Yes. Suppose that ZFC is inconsistent but the shortest derivation of a contradiction in ZFC takes 10^10^10^10^10^10^10^10^10^10 steps. Clearly we can't prove ZFC inconsistent by presenting a derivation of contradiction in ZFC, since we can't in fact present even the shortest derivation. Rather, any proof we can present that ZFC is inconsistent must consist in mathematical reasoning establishing the existence of the derivation. (Indeed, in logic we very rarely prove the formal derivability of anything in formal theories by exhibiting formal derivations.) There is no necessity to this reasoning being simple. It may well involve tortuously complex mathematics, far beyond anything we've seen to this day in its staggering ingenuity, beyond even the dreams of Groethendieck. Here I'm using the word "proof" in its usual sense, to mean a piece of compelling mathematical reasoning intended for human consumption. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 22 Feb 2010 10:57
Rupert <rupertmccallum(a)yahoo.com> writes: > 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. No need to drag in the soundness of ZFC + I. If you accept (in addition to the usual set theoretic principles) that an inaccessible exists, you get a proof of consistency of ZFC without invoking any reflection principle. An inaccessible is an overkill, of course; a logically weaker but very natural consistency proof proceeds by way of a truth predicate. > 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. There's no reason to connect any doubts or their alleviation to consistency statements. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |