Cantor Confusion
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From: mueckenh on 6 Dec 2006 05:57 Virgil schrieb: > 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? The fact that any brain consists of not more than this limited part of the universe. And the fact that the power set of your neurons and ideas and other contents of our brain is a finite set. And the power set of this set is a finite set too. And so on, in infinity ... (potential infinity , of course) Regards, WM
From: mueckenh on 6 Dec 2006 06:01 Virgil schrieb: > > 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. In order to see that a ball which is contained in a closed box cannot escape, we need not have a rule determining its position. So is the ratio between paths and edges. The number of paths is irrevocably bounded by the number of edges. This is so for any finite part of the tree. In order to reverse this ratio, there must be parts of paths which do not consist of edges. > > 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. And each of them owns the shares of two edges. In fact there are not less edges than paths. Regards, WM
From: mueckenh on 6 Dec 2006 06:04 Dik T. Winter schrieb: > > > I have difficulty with the tree because your explanations are confused > > > and sometimes contradictionary. > > > > Which one? > > Many. Care to name one? Please do not mistake your misunderstandings for errors of mine. For instance I never stated that nodes represent numbers, as you erroneously believed. > > > > > No. Paths are only another notation for the reals in usual > > > > representation and usual definition. > > > > > > You state that you are using limits with your infinite paths? > > > > Of course. The paths are nothing else but another way of denoting a > > real number in binary representation. > > But in that case you are doing something non-standard. Not at all! I represent numbers by standard binary notations. Regards, WM
From: mueckenh on 6 Dec 2006 06:06 Dik T. Winter schrieb: > In article <1165242365.580949.112500(a)l12g2000cwl.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > By the axiom it is not assumed that it exists, it is stated that it > > > exists. It is similar to the parallel postulate from Euclid that > > > does not assume that there is one line going through a point not on > > > another line and parallel to that other line. It is stated as fact. > > > > To have a big mouth is not enough to create a world, not even a notion. > > You would see that if you tried to say where the assumed object > > existed. > > In my mind. I can reasonably think about the set of all natural numbers. > Honest, I have no problem with it. You believe you could. That is a difference to "can". Other people think they can reasonably think of having an immortal soul. Regards, WM
From: mueckenh on 6 Dec 2006 06:09
Dik T. Winter schrieb: > > Everybody knows what the number of ther EC states is. > > That is *not* what I did ask you. You state that it is simply a matter > of definition how one interprets "to grow" and "number", sure. > and I asked you > to provide definitions. Moreover, the number of the EC states is not > fixed, so you can only state what the number of the EC states is at a > particular point in time. The number of EC states is "the number of EC states". It is simply a notion which can be equal to a natural number. The set of prime numbers does not contain the number 1. But once upon a time it did contain it. And 2000 years ago it did not contain it (because at that time 1 was not considered to be a number). > > > > > > That is not "the set of states". You can talk about "the current > > > > > set of states" or about "the set of states in 1957" or whatever. > > > > > At least mathematically. In mathematics, by definition, a set > > > > > can not grow. > > > > > > > > Wrong. Read my explanation above. > > > > > > Wrong. Read my explanation above. I am talking mathematics. > > > > Do you really think so? > > Yes. > That is a matter of definition. You may say: Where I am, there is the top. > > > > > You are, of course, entitled to use another definition, but that will > > > > > not clarify the discussion at all (and you are not using standard set > > > > > theory). > > > > > > > > Hrbacek and Jech teach standard set theory including the fact that in > > > > ZF everything is a set. > > > > > > Yes? > > > > I remember that you opposed. Now you agree because you have learnt? > > No. I still do not agree. The universe in ZF is *not* a set. Then there is no universe in ZF. Regards, WM |