Cantor Confusion
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From: Virgil on 24 Dec 2006 13:15 In article <1166952651.859923.213150(a)i12g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > stephen(a)nomail.com schrieb: > > > Newberry <newberry(a)ureach.com> wrote: > > > > > > What is supposed to follow from all this? > > > > That what is true "in the limit" is not necessarily true for the infinite > > case. > > For instance the diagonal number of Cantor's proof differs from any > list entry of a finite list. But it certainly does not differ from any > list entry of an infinite list. (Compare the case 1.000...1 = 0.999...) The "diagonal" for any list differs from every number in /that/list. > > > For instance the bijection {1,2,3,...,n} <--> {2,4,6,...,2n} holds for > every finite sequence, but it does not hold for an infinite sequence The mapping f: N <--> E: n |--> 2*n is a bijection of infinite sets. > (obviously, because there are more natural numbers than even atural > numbers). It is a pity that mathematicians have not known your theorem > earlier. It would have spared us much confusion. "Knowing" that has not prevented WM from being terminally confused. Not knowing what is not so has prevented much confusion for the rest of us.
From: Virgil on 24 Dec 2006 13:22 In article <1166952847.829226.324240(a)79g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > As not all natural numbers do exist, the set is potentially infinite, > > > i.e., it is finite. It has a maximum. L_D. Taking this maximum and > > > adding 1 or takin L_D ^ L_D or so yields another maximum. In any cae > > > there is a maximum. > > > > Which is always exceeded by a larger maximum. > > Which then is the maximum. As the creation of your "maximum" simultaneously creates its successor, the time span for any natural to be maximal is zero. > > > > > > > The proof that L_D does not exist > > > > makes no assumption about whether all natural numbers > > > > exist. All we need is that the natural numbers are not > > > > bounded. This is true whether or not all the natural numbers > > > > exist. > > > > > > Here is anothe proof: > > > The set of all even natural numbers contains at least one number larger > > > than its cardinality. > > > > Can WM name that number? If not, how can he prove its existence? > > Nobody can name it, because the set is not fixed. In what set theory are sets not fixed. Wm's theories of volatile sets do not work in mathematics. > > Which "largest existing natural number" is that? > > You name one, and we will find a larger one. > > That is it, then. > > > > > > > > Assume it exists. Call it N_L. > > > > > > > > It is easy to see that the set > > > > A={1,2,3,...,N_L} exists. > > > > > > If we do not take into accoun that there are physical constraints, yes. > > > > What physical constraints limit non-physical existence? > > The absence of more than 10^100 bits limits the existence of objects > which require more than 10^100 bits to exist In physical existence one may need physixcal bits, but not otherwise.
From: Virgil on 24 Dec 2006 13:50 In article <1166952651.859923.213150(a)i12g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > stephen(a)nomail.com schrieb: > > > Newberry <newberry(a)ureach.com> wrote: > > > > > > What is supposed to follow from all this? > > > > That what is true "in the limit" is not necessarily true for the infinite > > case. > > For instance the diagonal number of Cantor's proof differs from any > list entry of a finite list. But it certainly does not differ from any > list entry of an infinite list. (Compare the case 1.000...1 = 0.999...) The Cantor construction rule produces for any countably infinite list a number which necessarily differs from every number in the list.
From: Virgil on 24 Dec 2006 13:52 In article <1166962110.414621.9660(a)n51g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > You know the Cauchy criterion. For every epsilon > 0 you must be able > to find an n_0 such that for n > n_0 we have |a_n - pi| < epsilon. > > If you have less than 10^100 bits available, then you cannot realise an > a_n which reproduces more than the first 10^100 bits of a number like > pi. (Unless pi was a number showing some pattern, like the number > 0.111.., which would make it possible to express every a_n using less > than 10^100 bits.) > > You cannot find such a number a_n without having pi already. But this > number shall be used to establish the existence of pi. This is > impossible. pi does not exist *as a number* (it exists as an idea and > in form of close approximations). Which idea, in a non-physical world populated only by ideas, suffices.
From: Virgil on 24 Dec 2006 13:59
In article <1166962877.448330.134980(a)73g2000cwn.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > cbrown(a)cbrownsystems.com schrieb: > > > > It is when you try to argue "lim n-> oo E(n)/P(n) = 2 contradicts > > > > E(n)/P(n) < 1 in the infinite case" that your confusing notation > > > > "looks" like a contradiction (to you; to me it looks like nonsense). > > > > > > Then you really do accept 1 + 1/2 + 1/4 + ... < 1 ? > > > > Of course not. How do you infer that from my statements? > > >From your statement that the same calculation, in case of edges, yields > less than edge per path. On the contrary, it is WM's claim that the limit of a quotient must always equal the quotient of limits that is the problem. WM claims lim n -> oo E(n)/P(n) = (lim n -> oo E(n))/(lim n -> oo P(n)), but it does not hold for any form of limit definition in standard mathematics. And until WM has some explicit axiom system in which he can prove it, he cannot legitimately claim it. |