Cantor Confusion
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From: Bob Kolker on 4 Dec 2006 16:02 Eckard Blumschein wrote: > > For my feeling, Dedekind and Cantor were lacking power of abstraction. Actually they had more such power than Kronecker was comfortable with. Bob Kolker
From: Virgil on 4 Dec 2006 18:21 In article <1165241048.445660.143440(a)73g2000cwn.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1164982199.959381.134510(a)j72g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Eckard Blumschein schrieb: > > > > > > > > > > I recall being a little boy wondering when I was told that while there > > > > is no evidence proving the existence of god there is also no evidence > > > > showing his non-existence. Are those crippled who don't believer in CH? > > > > I consider the background of CH given in the difference between number > > > > and continuum. This might be crippled down to the truth? Do you agree? > > > > > > > No, I am sorry, I do not. The continuum is nothing but our failure to > > > look closely enough. In physics it lasted 2000 years to settle the idea > > > of the atom and to supplement and complete it by the uncertainty > > > relations. The majority of matematicians is not yet far sighted enough > > > to recognize the same situation in their realm. > > > > > > We do not yet /know/ that the physical universe is not continuous, so > > why should we reject a mathematically continuous real number system? > > Ever heard of the Uncertainty Relations? They guarantee a grainy > structure. But in physics we know even more. We know that the shortest > distance is given by shortest distance one can measure. This is given > by the shortest wavelength one can generate. This is roughly given by > the photon wavelength > > lambda = h/mc with m = 5*10^55 g, the mass of the accessible part of > the universe. What limits thought to only the accessible part of the universe? That is a limitation of the physical world that need not affect the worlds of imagination.
From: Virgil on 4 Dec 2006 18:27 In article <1165241975.343837.131650(a)j44g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > cbrown(a)cbrownsystems.com schrieb: > > > > > So your relation is supposed to be a function mapping edges -> sets of > > > > paths, correct? > > > > > > Yes. But the mapping is not the usual one (one edge --> one path). The > > > edges are subdivided in shares. > > > > Then your eventual claim that you are creating a bijection between > > edges and paths (or sets of paths) is false; you are instead showing > > that there is some mapping between "subdivisions of edges" and paths. > > That's fine; but it says nothing about whether there are equal > > cardinalities of edges and paths. > > I think, nobody would oppose to dividing the edges merely in two halves > each. If the series 1 + 1/2 + 1/4 + ... yields 2, then we can extend > this knowledge to bijections too. In order to have a bijection from edges to paths one must have a rule which assigns each edge ,in its entirety, to some path. WM does not do this, ergo has no bijection. > > If you dislike the fractions only, then let us map the edges on the > paths by random choice. We know that there are enough edges, because > when two paths split, there is always an edge which can be mapped onto > that path not yet carrying an edge. WM may think he knows something, but one never has just two paths splitting in an infinite tree. At each split, infinitely many paths go each way from that node, in fact, as many in each "half" as in the entire tree.
From: Virgil on 4 Dec 2006 18:29 In article <1165242365.580949.112500(a)l12g2000cwl.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > To have a big mouth is not enough to create a world, not even a notion. WM certainly has tried the big mouth approach and can verify its lack of success.
From: Virgil on 4 Dec 2006 18:31
In article <1165244124.997659.179710(a)f1g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > The natural numbers form a potentially > > infinite set. The diagonal contains the potentially infinite set > > of natural numbers. > > There is nothing to contain! Then where did all those natural numbers disappear to? Don't nobody leave the room! |