Cantor Confusion
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From: Lester Zick on 28 Oct 2006 16:18 On Sat, 28 Oct 2006 00:36:01 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: >In article <h4n4k25f4lnlsrdfqb6mku1v7eg6gj8v0b(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > > On Fri, 27 Oct 2006 00:36:24 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > > wrote: > > >You need transfinity when you want to show that something that holds in > > >the finite case also is valid in the infinite case. Induction will not > > >show that 0.111... is rational, it can only show that all the finite > > >initial parts are rational. And I again note that the notation 0.111... > > >(in the decimals) has only meaning due to the definition of that notation. > > > > However one can certainly show the square root of 2 without > > transfinity through rac construction even though its decimal expansion > > is infinite. > >You need not tell that to me. You should tell that to Wolfgang >Mueckenheim who insists that sqrt(2) does not exist because it is >impossible to know all the decimals in its decimal expansion. Well you know I talked to WM about this a year or two ago and of course I disagree with him in general terms. But an interesting aspect of the problem popped up just a few days ago when another person suggested that mathematical definitions have to designate a "domain of discourse" in what appear as particular terms such as card(x) = . . . Now without debating the issue of mathematical definition in general I will say that if mathematikers insist on definitions drawn exclusively in particular terms then WM may have some basis for his conclusions in that the infinity of mathematical expressions cannot be accommodated in a physically limited universe. (Of course I don't agree that the universe is physically limited but that's another issue.) In other words if mathematics insists its definitions can only be valid if cast in particular terms of specific domains to which they apply and there can be no infinite domains in a finite physical universe then math definitions cannot apply to infinity. At least that's how I would argue the issue. Now my solution to the problem would be to adopt a method of general definition in mathematics instead of the particular method math seems to prefer. Then if properly drawn mathematics definitions could certainly apply to infinity provided mathematics definitions don't need to specify any specific domain of discourse to establish the scope and validity of its definition. ~v~~
From: mueckenh on 28 Oct 2006 16:37 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > Ah, but none of them actually get the same conclusion you do. So, you > > > and they can't both be right. > > > > There are several mathematicians understanding my argument, but the > > conclusions are no in question here. Some have understood my argument > > but as it does not fit their expectation and conviction over many > > years, they try to find a fault. That is a legitimate process of > > finding the truth by discussion. Even Virgil has understood, although > > he does not try to find any error but plainly refuses the result > > because there is a contrary proof by Cantor. And I know from many of my > > students that they understood me, although they do not study > > mathematics but various other topics. > > Please name the mathematicians that agree that your argument is correct. > (Han doesn't count, since he says he is a physicist.) Every mathematician whose mind has not yet suffered the drill of set theory you probably were exposed to knows that my argument is correct. (And there are many mathematicians who never studied set theory but every mathematician knows the geometric series.) However due to the strong pressure of the orthodox fraction it may be disadvantageous for a mathematician to see his name printed here. Further it is my general principle not to publish names of private correspondents. Regards, WM
From: mueckenh on 28 Oct 2006 16:40 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Dik T. Winter schrieb: > > > > > > > Within the *real* numbers the limit does exist. And a decimal number is > > > nothing more nor less than a representative of an equivalence classes. > > > > So we are agian at this point: The real numbers do exist. > > Not according to you (at least not by the usual definition). > According to > your reasoning, there can only be a finite number of real numbers ever > defined. This makes limits kind of weird, since you cannot get > arbitrarily close. You cannot get arbitrarily close either. Can you calculate sqrt(2) to 10^100 digits? No. But we can both get as close as necessary to obtain correct results. > > > > > > For the real > > numbers we have LIM 10^(-n) = 0. Therefore, i ther limit n--> oo tere > > is no application of Cantor's argument. > > > > > Given that there are only a finite number of integers, what > do you mean by n--> oo ? > In these discussions I understand the usual by n-->oo, because I do not believe that my private opinions are of great interest for the majority. But if you are interested: I understand by, e.g., n --> oo 1/n a function the argument of which gets as large as necessary in order to make 1/n as small as desired, such that we cannot distinguish it from 0 in any calculation. This is possible because we have the tools to construct natural numbers as large as we want. At least I do not see a limit. This is what many mathematicians do not understand: You can create and know a number M as large as you want although you cannot know all the numbers between 1 and M. The information (the bits) required to know [pi*10^10^100] is not available in the whole universe. The information to know 10^10^10^10^10^10^10 is available in a pea nut shell. Regards, WM
From: mueckenh on 28 Oct 2006 16:46 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > > > > > > Cantor laid the foundations of sets theory. If modern set theory would > > > > not accept his definition, then it should not call itself set theory. > > > > > > It's called Z set theory. If you don't want it to be called 'set > > > theory' just because it has refinements not in Cantor's work, then I > > > guess we could call it 'zet zery' or whatever. But that's missing the > > > point. Upon Skolem's refinement of Zermelo's set theory, we have a > > > formal set theory. > > > > Why did Skolem not like it? > > Skolem wrote a very strong crtique of set theory. It's in the Van > Heijenoort anthology 'From Frege To Godel'. Why do you think did he that? > But Skolem made an > important suggestion. In the axiom schema of separation, Zermelo had > used a notion of a 'definite property', which is not rigorous (what in > heck is a definite property?). So Skolem suggested dumping 'definite > property' and instead using well formed formulas (in a certain way). > And that alteration makes it possible for us to fully formalize Zermelo > set theory. As you think, until someone asks: what in heck is a wff ? Like: What in heck is a not defined well order? Is it a not very well order? > > > I proved that there are not more real numbers than a countable set has > > elements. I proved it in a manner which everybody with moderate > > mathematical knowledge can understand. And, what is important, in a > > manner completely independent of your special language. > > Then yours is a proof in an informal context of your own mathematical > understanding (and what CLAIM to be a context shared by anyone with > moderate mathematical knowledge). And, so, as I've said already about a > half a dozen times by now, that is not a proof that set theory is > inconsistent. To prove set theory is inconsistent, you must provide > prove, IN set theory, a sentence P and a sentence ~P that are sentences > in the language of set theory. > > > It is clear and proven *from outside* that their results > > are wrong and, therefore, uninteresting for me. > > It matters little to me that set theory is uninteresting to you on > account of your having convinced yourself that it conflicts with > certain ideas of yours that are outside set theory. > > I view set theory (in any of a number of different formulations) Therefore I don't see any necessity to use a special language like ZFC or NBG, but use mathematics which can be understood by any freshman. > as one > among many possible formal axiomatizations to provide the usual > theorems of real analysis. Obviously it does not provide the recursion f(n+1) = 1*E + f(n)/2 with f(1) = 1*E. But that is the standard construction of the number of E related to a segment of a path. And the limit for n --> oo is the number of E related to an infinite path. It is ridiculous to require transfinite induction to prove that. > If you have a proof that set theory is > inconsistent, then a lot of people, including me, would like to see the > proof. But what you've given is just an argument as to why you think > set theory is wrong. Take the tree as a proof of consistency of ZFC, satisfying Skolem's theorem: It is impossible from inside this model of ZFC, i.e. the real numbers, to proof its countability, because the function required to do that is not available there. Outside this model the function is available which relates fractions of edges to paths. Therefore, the real numbers show ZFC is consistent, because it has a countable model. Only those sitting inside the cage of this model do not get to know it, because they refuse to use fractions. Regards, WM
From: mueckenh on 28 Oct 2006 16:48
William Hughes schrieb: > And once again WM deliberately confuses, "all positions are > finite", with "there are a finite number of positions". There is no confusing! Every finite position belongs to a finite segment of positions (indexes). If you don't believe that and assert the contrary, then try to find a finite position which dos not belong to a finite segment (of indexes). It is simply purest nonsense, to believe that "all positions are finite" if "there are an infinite number of positions". Regards, WM |