Why can no one in sci.math understand my simple point?
in [Logic]
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From: Virgil on 20 Jun 2010 16:01 In article <ccb88bb1-d8a6-4f04-bc6a-ca3346f7c8ad(a)c10g2000yqi.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 20 Jun., 02:04, "Mike Terry" > > > > No, you're misunderstanding the meaning of computable. > > > > Hopefully you will be OK with the following definition: > > > > � � A real number r is computable if there is a TM (Turing machine) > > � � T which given n as input, will produce as output > > � � the n'th digit of r. > > Whatever might be the true meaning: The Turing machine need a finite > definition. Therefore the computable number has a finite definition. > > There are only countable many finite definitions. And every diagonal > of a defined Cantor list has also a finite definition. Cantor's argument does not require that any member of a list of reals be computable beyond a finite number of decimal places, so enough of each can be finitely defined for the proof to work.
From: Virgil on 20 Jun 2010 16:08 In article <d55211a9-4dbf-48b3-8d8d-88d1fd859ab3(a)c33g2000yqm.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 20 Jun., 16:33, Newberry <newberr...(a)gmail.com> wrote: > > On Jun 19, 6:06 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > > > > Newberry <newberr...(a)gmail.com> writes: > > > >> Because every infinite sequence of digits represents a real number? > > > >> And > > > >> the antidiagonal is one such sequence? > > > > > > If it does not exist then it does not represent anything let alone a > > > > number. > > > > > > Now it is clear that it does not exist. Since all the reals are on the > > > > list and the anti-diagonal would differ from any of them. This > > > > violates the assumption. Hence the anti-diagonal does not exist. > > > > > Wow. Are you saying that the *sequence of digits* specified by the > > > anti-diagonal does not exist? > > > > That's right. There is no formula or algorithm to construct the list. > > It means that you would have to flip each and every digit one by one. > > And that is impossible. > > Well, there are some lists defined by finite definitions like > 0.1 > 0.11 > 0.111 > ... > > But the number of these lists and the number of their diagonals is > countable. > > > > > Anyway, in a sense, you're right. If we assume that every real is > > > represented by a sequence on the list, then we can prove that every > > > sequence occurs on the list (ignoring the issue of multiple > > > representations). And yet, we can also show that a particular sequence > > > is not on the list. > > > > Which sequence is that? > > O, don't be unfair. Matheolgicians "prove" their claims. They are not > used to give examples. Remember Zermelo who "proved" that every set > can be well ordered. Although every sober mind today knows that > hisproof is wrong, set theory is built upon this lie. Given the axiom of choice, every set is in principle well orderable. But without the axiom of choice, it is not always possible. So WM must be rejecting the axiom of choice for himself. But he does not have either the right, or, more importantly, the power to impose that rejection on everyone else.
From: Virgil on 20 Jun 2010 16:18 In article <f92c169d-ee85-40c2-aa82-c8bdf06f7b55(a)j4g2000yqh.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 20 Jun., 17:51, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > > > Cantor was this utterly insane freak who chose not to accept Newberry's > > word for it, and instead *prove* that there was no list of all real > > numbers. �Obviously, his proof is nonsense, because, after all, Newberry > > said there was no list. > > His proof is nonsense because it proves that a countable set, namely > the set of all reals of a Cantor-list and all diagonal numbers to be > constructed from it by a given rule an to be added to this list, > cannot be listed, hence, that this indisputably countable set is > uncountable. That is a deliberate misrepresentation of the so called "diagonal proof". What that proof (not actually by by Cantor himself) ays is that any listing of reals is necessarily incomplete because given such a listing one can always construct a real not listed in it. What Cantor himself proved was that for any list of infinite binary sequences there are binary sequences not listed in that list. Cantor had previously proved by quite a different method, which WM seems unable or unwilling to criticize, that the set of reals was not countable. > > Even completely blinded matheologicians should be able, perhaps after > some contemplation, to recognize that. What completely blinded matheologicians are able, perhaps after some contemplation,to recognize is that WM is not working in the same world as mathematicians do.
From: Jesse F. Hughes on 20 Jun 2010 18:13 "Mike Terry" <news.dead.person.stones(a)darjeeling.plus.com> writes: > "Peter Webb" <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote in message > news:4c1e3743$0$1028$afc38c87(a)news.optusnet.com.au... >> No, to calculate the first n terms of the anti-diagonal you require only > te >> first n items on the list to n places of accuracy. > > So you require as input: > - the 1st digit to 1 place of accuracy > - the 2nd digit to 2 places of accuracy > - the 3rd digit to 3 places of accuracy > - ... > > The ... above is the give away! For your algorithm to work its going to > require an INFINITE amount of input data. (Remember, you are trying to find > a SINGLE stand-alone FINITE algorithm that produces ALL the required digits > of the antidiagonal.) > > So Tim was quite right, and you haven't answered his point correctly > yet... No, you fail to understand the brilliance that Peter brings to this conversation. Every real is computable. You just have to produce a finite amount of data to compute the real to n digits, namely, the first n digits of the real you want to compute. All of you naysayers are just holding back inevitable progress. -- "And yes, I will be darkening the doors of some of you, sooner than you think, even if it is going to be a couple of years, and when you look in my eyes on that last day of work at your school, then maybe you'll understand mathematics." -- James S. Harris on Judgment Day
From: Newberry on 20 Jun 2010 20:18
On Jun 20, 1:18 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <f92c169d-ee85-40c2-aa82-c8bdf06f7...(a)j4g2000yqh.googlegroups.com>, > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > On 20 Jun., 17:51, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > > > Cantor was this utterly insane freak who chose not to accept Newberry's > > > word for it, and instead *prove* that there was no list of all real > > > numbers. Obviously, his proof is nonsense, because, after all, Newberry > > > said there was no list. > > > His proof is nonsense because it proves that a countable set, namely > > the set of all reals of a Cantor-list and all diagonal numbers to be > > constructed from it by a given rule an to be added to this list, > > cannot be listed, hence, that this indisputably countable set is > > uncountable. > > That is a deliberate misrepresentation of the so called "diagonal proof". > > What that proof (not actually by by Cantor himself) ays is that any > listing of reals is necessarily incomplete because given such a listing > one can always construct a real not listed in it. Please CONSTRUCT the anti-diagonal in the space provided below. Virgil's anti-diagonal construction BEGIN END > What Cantor himself proved was that for any list of infinite binary > sequences there are binary sequences not listed in that list. > > Cantor had previously proved by quite a different method, which WM seems > unable or unwilling to criticize, that the set of reals was not > countable. > > > > > Even completely blinded matheologicians should be able, perhaps after > > some contemplation, to recognize that. > > What completely blinded matheologicians are able, perhaps after > some contemplation,to recognize is that WM is not working in the same > world as mathematicians do. |