Cantor Confusion
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From: mueckenh on 4 Dec 2006 09:04 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. In mathematics, we have an "uncertainty relation" between numbers and their digits: numbers * digits < 10^100 Regards, WM
From: mueckenh on 4 Dec 2006 09:19 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. > > We have a function f mapping the > > paths onto the real numbers of he interval [0,1]. The domain of f is > > the set of all infinite paths. The function is not injective because > > some real numbers have two representations. But the function is > > surjective. > > Fine; and with a little tweaking, we can make f a bijection. We also > have that there is a bijection h (edges -> N). You then define g so > that for every path p, the limit of the sum over all edges e of g(p, e) > = 2. None of this is in dispute. Fine. That is a big advantage for me. Usually my arguing is disputed already at an earlier stage. > Hmmm. So instead of g : (paths X edges) -> Q, you have that g : (paths > X edges X N) -> R, and that for all paths p, lim n->oo sum (over all > edges e) g(p, e, n) = 2. > > Whatever. It is not disputed that this limit is 2. What is disputed is > that /therefore/ there exists a bijection (or surjection) T : (edges -> > paths). How do you propose to use g to construct T? 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. Regards, WM
From: mueckenh on 4 Dec 2006 09:26 Dik T. Winter schrieb: > In article <1164975696.538746.93710(a)j44g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <456EEA86.20001(a)et.uni-magdeburg.de> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > ... > > > > So far I recall "the set IN of naturals is countable". > > > > I understood set theory means a hypothetical (alias fictitious) set off > > > > all natural numbers. > > > > > > Oh, for once, try to talk mathematics. By the axiom of infinity the > > > set of all naturals is neither hypothetical nor fictitious. > > > > You mean that by stating the axiom this fictitious set gains some > > reality? It is assumed to exist. That's all. "Ficititious" is an > > adjective describing its state extremely well. > > 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. Regards, WM
From: mueckenh on 4 Dec 2006 09:44 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > [...] > > Everybody knows what the number of the EC states is. > > Everybody except you knows that EC states are not part of any set > theory. Oh, there are two of us. You forgot Cantor.Beiträge zur Begründung der transfiniten Mengenlehre. (That *is* a set theory). "Unter einer "Menge" verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten in unsrer Anschauung oder unseres Denkens (welche die "Elemente" von M genannt werden) zu einem Ganzen." Or do you insist on living creatures? Further I am in accordance with he sentence: "Sets are not objects of the real world: they are created by our mind, not by our hands." Of course I understand by EC states the mind-created set of EC states. > > "To grow" colloquially means "1. To increase in size by a natural > process." > "Process" means "1. A series of actions, changes, or functions bringing > about a result" > (source: thefreedictionary) in der Analysis haben wir es nur mit dem Unendlichkleinen und dem Unendlichgroßen als Limesbegriff, als etwas Werdendem, Entstehendem, Erzeugtem, d.h., wie man sagt, mit dem potentiell Unendlichen zu tun. [David Hilbert: Über das Unendliche, Math. Ann. 95 (1925) p. 167] > > Usually the "series of action" means a succession in _time_. Since > mathematics does not comprise a (physical) concept of time we have to > define what "to grow" shall mean mathematically. It is not unusual to say f(x) grows with x (if it does so). > Set theoretically numbers are sets. Only sets which are functions can > grow. To model the "number of states of the EC" set theoretically > we must define the number-of-states-of-the-ec-function n. This function > should contain the ordered pairs (1958, 6) and (2006, 25). According to > my definition n _does_ grow. Correct. So the EC-states, after all, are in ZFC. I knew it. > NB the numbers 6, 25, 1958 and 2006 do not > grow. That is not a surprise. I agree completely with you. Regards, WM
From: Eckard Blumschein on 4 Dec 2006 09:47
On 12/2/2006 2:56 AM, Dik T. Winter wrote: > In article <1164975696.538746.93710(a)j44g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <456EEA86.20001(a)et.uni-magdeburg.de> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > ... > > > > So far I recall "the set IN of naturals is countable". > > > > I understood set theory means a hypothetical (alias fictitious) set of > > > > all natural numbers. > > > > > > Oh, for once, try to talk mathematics. By the axiom of infinity the > > > set of all naturals is neither hypothetical nor fictitious. > > > > You mean that by stating the axiom this fictitious set gains some > > reality? It is assumed to exist. That's all. "Ficititious" is an > > adjective describing its state extremely well. > > 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. > And that gives us Euclidean geometry. In the same way, the axiom of > infinity gives us ZF set theory where the set of naturals does exist > as a reality. The Latin word factum means something which has been done. Wo created the factum of set theory? I do not refer to the axiom of infinity. This is just an abused version of the Archimedean original. I refer to the idea that there are more countables than fictitious elements of the uncountable continuum. Yes, more than 100 years of fruitless confusion in mathematics go back to Dedekind and Cantor. |