Cantor Confusion
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From: mueckenh on 14 Oct 2006 09:52 Alan Morgan schrieb: > In article <990aa$452e542e$82a1e228$16180(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >Dik T. Winter wrote: > > > >> In article <1160646886.830639.308620(a)c28g2000cwb.googlegroups.com> > >> mueckenh(a)rz.fh-augsburg.de writes: > >> ... > >> > If every digit position is well defined, then 0.111... is covered "up > >> > to every position" by the list numbers, which are simply the natural > >> > indizes. I claim that covering "up to every" implies covering "every". > >> > >> Yes, you claim. Without proof. You state it is true for each finite > >> sequence, so it is also true for the infinite sequence. That conclusion > >> is simply wrong. > > > >That conclusion is simply right. And yours is wrong. Completed infinity > >does not exist. > > sqrt(-1) doesn't exist either. Frankly, I have a much harder time > believing in "imaginary" numbers than I do believing in infinite > sets. But sqrt(-1) does not yield contradictions, as far as I know. Regards, WM
From: mueckenh on 14 Oct 2006 09:54 Virgil schrieb: > In article <1160676113.344404.246370(a)h48g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > > > Cantor uses Lim{n} n = omega witout much ado. > > > > > > That is not a limit of "sets". > > > > It is the limit of the natural numbers. The limit is omega, an ordinal > > number. Meanwhile we know that every number is a set. Hence, it is a > > limit of sets. > > N ( or omega) is only a limit in the sense of being a union of its > members, and is the first non-empty ordinal to be equal to the union of > its members. No other form of 'limit of a sequence' of sets is defined > in ZF. This union just gives Lim {n-->oo} {1,2,3,..,n} = N. > > > > > > > In his first paper he uses even Wallis' symbol oo. What should there > > > > require a definition, if all natural did exist? > > > > This is what I use and write in modern form: Lim {n-->oo} {1,2,3,..,n} > > > > = N. > > > > > > Yes, and that fits my definition. On the other hand, how would you > > > define lim{n --> oo} {n, n+1, ...}? > > > > I would not attempt to define that. > > It is obviously the empty set, i.e., the intersection of all the sets of > form {n, n+1, ...}. Above you wrote that only the *union* and "no other form of 'limit of a sequence' of sets is defined in ZF". Now you accept the intersection. Regards, WM
From: mueckenh on 14 Oct 2006 09:55 David Marcus schrieb: > Virgil wrote: > > In article <1160650371.242557.284430(a)h48g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Dik T. Winter schrieb: > > > > > > > In article <1160578706.221013.145300(a)c28g2000cwb.googlegroups.com> > > > > mueckenh(a)rz.fh-augsburg.de writes: > > > > > Dik T. Winter schrieb: > > > > > > So the definition I gave for a limit of a sequence of sets you agree > > > > > > with? Or not? I am seriously confused. With the definition I gave, > > > > > > lim{n = 1 .. oo} {n + 1, ..., 10n} = {}. > > > > > > > > > > Sorry, I don't understand your definition. > > > > > > > > What part of the definition do you not understand? I will repeat it here: > > > > > What *might* be a sensible definition of a limit for a sequence of sets > > > > > of > > > > > naturals is, that (given each A_n is a set of naturals), the limit > > > > > lim{n = 1 ... oo} A_n = A > > > > > exists if and only if for every p in N, there is an n0, such that either > > > > > (1) p in A_n for n > n0 > > > > > or > > > > > (2) p !in A_n for n > n0. > > > > > In the first case p is in A, in the second case p !in A. > > > > Pray, read the complete definition before you give comments. > > > > > > I do not believe that definition (2) is of any relevance. > > > Cantor uses Lim{n} n = omega witout much ado. > > > omega is simply defined as the limit of the increasing natural numbers. > > > In his first paper he uses even Wallis' symbol oo. What should there > > > require a definition, if all natural did exist? > > > This is what I use and write in modern form: Lim {n-->oo} {1,2,3,..,n} > > > = N. > > > > Where in ZFC or NBG does "Mueckenh"find any definition of any such limit? > > Or, in what book does mueckenh find this? My name is Wolfgang Mueckenheim, briefly known as WM. "mueckenh" is only used by Virgil due to his bad education or behaviour. The book I recommend is the collected works of Cantor. But sometimes I dare to write things not yet included in books. Regards, WM
From: mueckenh on 14 Oct 2006 09:56 Tony Orlow schrieb: > > > > Why shouldn't it? If every digit position of 0.111... is a finite > > position then exactly this is implied. Your reluctance to accept it > > shows only that you do not understand how an infinite set can consist > > of finite numbers. In fact, nobody can understand it, because it is > > impossible. > > > > Regards, WM > > > > But Wolfgang, surely that consideration does not impact, say, the set of > reals in (0,1], which are all finite, yet whose number is infinite. It > is not a requirement that a set of all finite values be finite. That > conclusion follows from the combination of that fact with the fact there > is a constant positive unit difference between consecutive elements. Of course, Tony, you are right! Regards, WM
From: mueckenh on 14 Oct 2006 09:58
David R Tribble schrieb: > mueckenh wrote: > >> Yes, but the assertion of Fraenkel and Levy was: "but if he lived > >> forever then no part of his biography would remain unwritten". That is > >> wrong, because the major part remains unwritten. > > > > David R Tribble wrote: > >> What part? > > > > mueckenh wrote: > >> That part accumulated to year t, i.e., 364*t. > > > > David R Tribble schrieb: > >> It's stated that he lives forever, so what value of t you are using? > > > > mueckenh wrote: > > You can use any positive value of t and prove that the unwritten part > > n(t) for t > t_0 is larger than the unwritten part for t_0. You can > > even use the formal convergence criterion for the convergent function > > 1/n(t). There is no room for he assumption that the written part could > > ever surpass the unwritten part. > > If I use any positive value for t, then there is still the positive > value t+1 (and 2t, t^2, and all the rest), none of which satisfies the > "lives forever" part. So I can't use any positive value of t. > Obtain the limit of the balls in the vase for t --> noon like the limit of n from Lim {n --> oo} (1/n) = 0. > > mueckenh wrote: > >> If you think Lim {t-->oo} 364*t = 0, we need not continue to discuss. > > > > David R Tribble schrieb: > >> I don't think anyone has said that. I merely asked which pages (days) > >> in the "major part" of the book don't get written. Do you have a > >> certain t in mind? > > > > mueckenh wrote: > > I merely answer that it is completely irrelevant to speak of certain t. > > Then why did you say "use any positive value of t"? > > > The paradox is raised only by the asumption that the set of all t did > > exist. > > What paradox? The result Lim{n-->oo} 9n = 0 where mathematics leads to Lim{n-->oo} 1/9n = 0. Regards, WM |