Cantor Confusion
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From: William Hughes on 11 Dec 2006 09:45 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > > > Recall this post from Dec 1 > > > > > > > > We extend this to potentially infinite sets: > > > > > > > > A function from the potentially infinite set A to the > > > > potentially infinite set B is a potentially infinite set of > > > > ordered pairs (a,b) such that a is an element of A and b is > > > > an element of B. > > > > > > A function, according to modern mathematics, is a set, actually fixed > > > and complete > > > > Yes, but according to modern mathematics the natural numbers > > are a set, actually fixed and complete. You cannot pick and choose the > > bits of modern mathematics you want to use. > > But I can point to those bits which are selfcontradictory and refuse to > accept them. Your assumption of an actually potentially infinite set is > such a piece. > Do you now claim the natural numbers are not a potentially infinite set? the natural numbers do not exist? one can't define functions on the natural number?. Not that all of the above is completely independent of whethe the natural numbers actually exist, we just need that the natural numbers exist. I do not claim that a potentially infinite set actually exists, I merely claim that a potentailly infinite set exists. That is all I need. - William Hughes
From: mueckenh on 11 Dec 2006 09:49 William Hughes schrieb: > > Given that you find the words function, and bijection so > distasteful, I have reworked my points to avoid using > them in conection with potentially infinite sets. > Please stop me at the first point you disagree with. > [...] > > -given two sets of natural numbers E and F where E is a > potentially > infinite set, and F has a largest element. there does > not exist an equitransform between E and F > Until here all is correct, as far as I see. > -the diagonal is the potentially infinite set of natural > numbers. > > -every line L has a largest element > > -there is no equitransform between the diagonal and a line L And here we have the contradiction: 1) The set of lines is also the potentially infinite set of natural numbers. 2) Every element of the diagonal is in at least one line. 3) Every initial segment of the diagonal is (in) a line. 4) The limit (n -->oo, for the number n of elements) of the diagonal is oo. 5) The limit (m -->oo, for the number m of elements of a line) is less than infinite. 6) Therefore not every initial segment of the diagonal can be (in) a line. 7) Contradiction between (3) an (6). Regards, WM
From: stephen on 11 Dec 2006 09:50 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > stephen(a)nomail.com wrote: >> Nobody has said that time is a set. But time can be modelled using >> sets. > Modelling time as implied with functions results in a vicious circle, > as I have demonstrated in a previous poster. No you did not demonstrate any such thing. >> Apples are not sets either, nor are they numbers, but you >> can reason about apples using sets and numbers. Your arguments are >> becoming increasingly irrational. You really do not appear to have >> any rational opposition to set theory, just emotional ones, >> and your arguments boil down to Communist style propaganda slogans. > Well, you're quite close, after all. But I can also say that _your_ > arguments boil down to Capitalist style propaganda slogans. Who is > the hypocrite here? At least _I_ will not deny that my ideas _are_ > influenced by the world outside. While I don't expect _you_ to admit > that Set Theory is only "fundamental" because of its setting within > the Capitalist Economic System. You think you are objective and free > of emotion, but you are not. Your propaganda is as bad as mine. > Han de Bruijn You were the one complaining about Set Theory restricting your freedom within mathematics. You are now the one denying people people the freedom to pursue mathematics you do not like. That is hypocrisy, plain and simple. And one again, you demonstrate that you do not know what the word "foundation" means. Set Theory is fundamental in the sense that mathematics can be described using set theory. That is all anyone is saying. You have turned this simple statement into some political/spiritual/philosophical nonsense. You are just making up positions to argue against, instead of actually responding to what people say. I suppose you think the fact that Turing machines are a foundation of computing is just propaganda as well. Stephen
From: mueckenh on 11 Dec 2006 09:59 Virgil schrieb: > In article <1165614850.907256.254500(a)j72g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > [concerning Cantor's first proof of uncountability of the real numbers] > > > > > > In the reals, any subset which has a real upper bound has a real least > > > upper bound and, similarly , any subset which has a real lower bound has > > > a real greatest lower bound. > > > > > > The only subsets of the reals for which there is a similar property are > > > real intervals. > > > > > > Thus it is only for the set of all reals, or for real intervals, that > > > the proof appplies. > > > > Yes. And it does not apply if only one single element of the > > investigated real interval is missing. As the uncontability property of > > this interval cannot depend on this single element, the whole proof > > fails. > > The proof does not fail. It merely does not apply to the whole any more, > but it does apply separately to each of the pieces separated by that > removed point. Invalid arguing, because the proof would fail as well for the remainig sets after removing one element. > > Or is WM arguing that removing a single element from a set can make an > uncountable set countable, This change is highly improbable. *Therefore* the proof is invalid. > even though its removal real partitions the > remaining reals into two equally uncountable sets. In particular, the proof would fail for the whole set of reals after removing the set of rationals (as I explained in my paper) - and there would not remain any sets which are provably uncountable by this or Cantor's second proof. Regards, WM
From: mueckenh on 11 Dec 2006 10:03
Virgil schrieb: > In article <1165615065.285043.115990(a)73g2000cwn.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > Franziska Neugebauer schrieb: > > > >> mueckenh(a)rz.fh-augsburg.de wrote: > > > >> > > > >> > Dik T. Winter schrieb: > > > >> >> > Everybody knows what the number of ther EC states is. > > > >> [...] > > > >> > The number of EC states is "the number of EC states". > > > >> > > > >> This is hardly a definition. > > > >> > > > >> > It is simply a notion which can be equal to a natural number. > > > >> > > > >> Which may _evaluate_ to a number. > > > > > > > > No. It evaluates to a number as little as 6 evaluates to a number. It > > > > *is* a number, though not a fixed number. > > > > > > Mathematically one modells such "not-fixed numbers" as functions. > > > Conclusively this function has value 6 at 1968. > > > > > > > That is a matter of definition of the word "number". > > > > > > Provide one. Don't forget to provide a definition of "not-fixed" number > > > and "not-fixed" set. And please show that one gains advantage over the > > > function concept. > > > > > A function is a set of ordered pairs and as such it is not variable. > > So sets of ordered pairs cannot be 'variable' but other sets can? Unfortunately you misunderstood (again): IF a set cannot be variable, THEN also a function cannot be variable. It is useless to distinguish between set and function. Regards, WM |