Wimpy Banach-Tarski paradox - should it break my intuition?
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From: David C. Ullrich on 25 Feb 2010 05:41 On Wed, 24 Feb 2010 21:32:02 +0000, Frederick Williams <frederick.williams2(a)tesco.net> wrote: >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. The word "simple" is really not well-defined. It may well be a bit silly to call a proof that takes 10^10^10^10 lines "simple", but in fact when I said there would be a simple proof I _was_ meaning to include that possibility - that astronomically long proof is simple in that it uses nothing but axioms and modus ponens. _Finding_ the proof is simple in the same sense; enumerate all the finite strings of symbols and check each one to see whether it's a proof of a contradiction in ZFC. No need to point out again that that may not seem so "simple", I've agreed to that. I do think that, for example, an analogous use of the word "trivial" is useful - playing perfect chess is trivial since chess is a finite tree. That will strike many as a perverse notion of "trivial", but it _is_ true that chess _is_ trivial in a sense in which some other things are not, since there _is_ a "simple" algorithm. That may or may not be a silly notion of "trivial". But it does seem to me that we want a word that means what I use "trivial" for, in particular a word for the concept that does not entail "easy". It happens all the time that I tell students this or that is trivial, and then I feel I should assure them that this does not mean it's easy. An example came up just the other day. A student was stuck on proving something (say X and Y are Banach spaces, T : X -> Y is a bounded linear operator. If we identify X with its image in the second dual X** under the natural injection, and similarly for Y, then the second adjoint T**, restricted to X, is equal to T). I predicted that this would be trivial. It took me a few minutes to sort out all the definitions and get all the stars straight. But it _was_ completely trivial, even though it took me a few minutes... >> 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.
From: Gc on 25 Feb 2010 08:31 On 25 helmi, 12:41, David C. Ullrich <ullr...(a)math.okstate.edu> wrote: > > No need to point out again that that may not seem so > "simple", I've agreed to that. I do think that, for example, > an analogous use of the word "trivial" is useful - playing > perfect chess is trivial since chess is a finite tree. > That will strike many as a perverse notion of "trivial", > but it _is_ true that chess _is_ trivial in a sense in which > some other things are not, since there _is_ a "simple" > algorithm. I think it`s common to see chess trivial from mathematics point of view, especially because Hardy makes this point in his famous book. IMHO In some sense all finite things are trivial or simple, (I understand this point because I am interested in analysis :)). Maybe you don`t wan`t to use this with respect to proofs, because as usually seen proofs are finite, but if you just look proofs syntactically they could be called terrible simple if you compare them to some infinite strings like (i think at least most) decimals expansions of irrationals.
From: William Hughes on 25 Feb 2010 10:29 On Feb 25, 6:41 am, David C. Ullrich <ullr...(a)math.okstate.edu> wrote: > ... But > it does seem to me that we want a word that means > what I use "trivial" for, in particular a word for the > concept that does not entail "easy". It happens all the > time that I tell students this or that is trivial, and then > I feel I should assure them that this does not mean > it's easy. > The closest I can find in English is "straightforward" or maybe "very straightforward". I disagree a little, to me "trivial" should mean easy (though not necessarily practical), e.g. there is a trivial algorithm to play perfect chess, but there is no (known) practical trivial algorithm to play good chess. > An example came up just the other day. A student was > stuck on proving something (say X and Y are Banach > spaces, T : X -> Y is a bounded linear operator. If > we identify X with its image in the second dual > X** under the natural injection, and similarly for > Y, then the second adjoint T**, restricted to X, > is equal to T). I predicted that this would be trivial. > It took me a few minutes to sort out all the definitions > and get all the stars straight. But it _was_ completely > trivial, even though it took me a few minutes... Well, definition chasing is certainly straightforward. It is trivial? I am undecided. - William Hughes P.S. Should we now substitute Ullrich for Von Neumann in the famous anecdote. Professor : It's trivial Student: Are you sure? Professor: <works at the blackboard for half an hour> Yes, it's trivial.
From: master1729 on 25 Feb 2010 06:06 Frederick Williams 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. that is indeed a joke.
From: Gerry Myerson on 25 Feb 2010 16:46
In article <4298268c-76cc-435c-89b7-aef6b611347a(a)m37g2000yqf.googlegroups.com>, William Hughes <wpihughes(a)hotmail.com> wrote: > On Feb 25, 6:41�am, David C. Ullrich <ullr...(a)math.okstate.edu> wrote: > > > ... But > > it does seem to me that we want a word that means > > what I use "trivial" for, in particular a word for the > > concept that does not entail "easy". It happens all the > > time that I tell students this or that is trivial, and then > > I feel I should assure them that this does not mean > > it's easy. > > > > The closest I can find in English is "straightforward" > or maybe "very straightforward". How about "mechanical"? -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email) |