Wimpy Banach-Tarski paradox - should it break my intuition?
in [Math]
|
From: A on 22 Feb 2010 11:12 On Feb 22, 10:30 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > David C. Ullrich <ullr...(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. > Which dreams of Grothendieck are you referring to?
From: Aatu Koskensilta on 22 Feb 2010 11:20 A <anonymous.rubbertube(a)yahoo.com> writes: > Which dreams of Grothendieck are you referring to? Well, take for one his dream of proving the standard conjectures, to provide a general cohomology theory by way of a correct definition of mixed motive. Or any of his crazy dreams about God. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Gerry Myerson on 22 Feb 2010 17:12 In article <b0s4o5li069i42b6oq46u93359g52r4cpb(a)4ax.com>, David C. Ullrich <ullrich(a)math.okstate.edu> wrote: > 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... Looks like a job for the Department of Redundancy Department. -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Rupert on 24 Feb 2010 14:25 On Feb 23, 2:57 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Rupert <rupertmccal...(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. You are right, it might have been better if I had just said "if you accept 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. > Indeed. > > 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. > I'm not sure if I know what you mean by this. People certainly do have doubts about the consistency of this theory or that theory, no? I don't know if I understand what point you wanted to make.
From: Frederick Williams on 24 Feb 2010 16:32
Gerry Myerson wrote: > > 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? If simple means short, then no. Your simple proof of R below needs to be preceded by a simple proof of P and not-P _from_ ZFC. As Aatu points out (and as is obvious to you, me and David) such a proof from a (let's say) inconsistent ZFC might be very long. The last sentence of David's reply to Aatu suggests to me that he might wish to withdraw the word "simple", but I have no wish to get into an argument with him about it. > 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. |