Cantor Confusion
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From: mueckenh on 5 Dec 2006 08:14 cbrown(a)cbrownsystems.com 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. > > If I say that the sets {a} and {p,q} have the same cardinality, I do not follow this "path". Only for 1/2 edge together with another 1/2 edge together I assert to have 1 edge with cardinal number 1. And that is correct. > > Furthermore, many sets of things (for example, the set of all finite > simple abelian groups) have elements which cannot be "divided" in the > way you seem to imply. What is "half" of the group of order 7? But a unit can be divided. You know what 1/2 is. And you know what 1/4 is. You know tat 1/2 + 1/2 = 1. > > If you dislike the fractions only... > > I don't dislike fractions; some of my best friends are fractions. > > I simply don't see that you have produced, from the functions f, g, and > h, a surjective function T (edges -> paths). You appear to have no > response to the request that you produce one; and instead change the > problem. As those parts of an edge which are mapped on a path are not mapped on any other path, there is obviously a bijection, though not from undivided edges but from the shares of divided edges onto paths. So I take it you agree that your original argument is flawed? No. The necessity of as many edges as path is so obvious that this fact is impossible to overlook - once one has discovered it. I add an appendix to one of my papers, where this is underlined I (here the arguing is based on nodes instead of edges, but that doesn't matter much): The bijection of paths and nodes of the binary tree proposed in this paper has been accused to cover only those paths which represent rational numbers (ending with an infinite sequence of ones: 111..). But obviously no path can split into two paths other than at a node. To assert the existence of more path than nodes is to assert the existence of more split positions than nodes - and that is equivalent to claim the existence of more nodes than nodes. Therefore, if the objection is justified, then it is only because there are no other than rational numbers and, hence, there are no other than the corresponding paths. Nevertheless this objection can be met either by using a random mapping or by a fractional allocation of nodes. In the first case we know that there is one more node with one more split (because a split position is a node). Therefore it does not matter which path is declared to be that path carrying a node already and which path is in need of a node. In the latter case, the first node is divided into two equal shares. One share is inherited by all the path beginning with 0.0 while the other share is inherited by all the paths beginning with 0.1. Similarly, half of the first node 0 (on level 1) is inherited by all the paths beginning with 0.00 while the other half is inherited by all the paths beginning with 0.01. Half of the first node 1 (on level 1) is inherited by all the paths beginning with 0.10 while the other half is inherited by all the paths beginning with 0.11. Continuation of this process leads to the following sequence of shares of nodes which are mapped on the finite segments of paths: 1/2 node up to level 1 1/2 + 1/4 nodes up to level 2 1/2 + 1/4 + 1/8 nodes up to level 3 briefly 1 - 1/2^n nodes up to level n Continuing this calculation we obtain for the complete infinite path covering every level 1/2 + 1/4 + 1/8 + ... = 1 node. As the number of nodes is countable, the number of paths is countable too. But the number of paths is not less than the number of real numbers in the interval [0, 1]. This is a contradiction to Cantor's theorem. Regards, WM
From: mueckenh on 5 Dec 2006 08:20 Virgil schrieb: > In article <1165237393.288598.129130(a)j44g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > > > > Informally we have that a potentially infinite set is a set > > > which is always finite, but to which we can add an element > > > whenever we want. We say that x is an element of > > > the potentially infinite set if we can add enough elements > > > to get to x. > > > > Yes. In particular this method of adding elements guarantees that such > > a set can never be uncountable. > > But having "added an element to it" produces a different set according > to the axiom of extensionality. Please learn: Common set theory is not possible with potentially infinite sets. > > > > > The set N produced by induction is potentially infinite. It does not > > have a largest element because it is not a fixed set but a set which > > can grow. > > A set to which one can add an element and have the same set is not a set > at all in any reasonable set theory. If WM wants such things he will > have to create a separate "set" theory in which they are allowed as they > are not allowed in and standard set theory/ > > > It has at most a temporarily largest element. > > For how long? Until a larger one is constructed. > > > > How does the definition I am using differ from the > > > 'commonly accepted meaning of "potentially infinite"'? > > > > You consider complete sets. Potential infinity is an unending process. > > {(n-1)/n:n in N} is , by WM;s standards, and endless process, but is > completely contained in rational interval [0,1] intersection Q. > > > > > > The set of numbers is potentially infinite. So the real numbers > > > are potentially infinite. > > > > So you can construct the set of all real numbers (of the interval [0, > > 1] in binary representation) by: > > > > 0.0 > > 0.1 > > 0.01 > > 0.11 > > ... > > > > This set is countable. > > Then it appears that what WM actually means is that > > WM's "potentially infinite" = our "countable" > and > WM's "actually infinite" = our "uncountable" No. Your set N is countable but actually infinite. The digit indexes of 1/3 in decimal representation form a countable set. Nevertheless there are actually infinitely many. Regards, WM
From: mueckenh on 5 Dec 2006 08:22 Eckard Blumschein schrieb: > On 12/1/2006 3:10 PM, 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. > Why should we abandon the old and proven concepts number and continuum? > To my understanding, they may or may not ideally fit the reality. Even > after I know that solids consist of molecules, atoms, and smaller > particles, there is no reason to start at this insight when designing > let's say a building. There is no reason to give up continuity for applied mathematics. Ony those who want to learn the real truth may bother. Regards, WM
From: Franziska Neugebauer on 5 Dec 2006 08:37 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > 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.Beitr�ge zur Begr�ndung >> > 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? >> >> Since "Cantor" is still present in the subject I have to ask you >> whether you want to discuss anachronisms or if you want to learn how >> contemporary set theory works. > > Contemporary set theory *is* an anachronsm. Compare, for instance, P. > Lorenzen: Die endlichen Weltmodelle der gegenw�rtigen > Naturwissenschaft zeigen deutlich, wie diese Herrschaft eines > Gedankens einer aktualen Unendlichkeit mit der klassischen > (neuzeitlichen) Physik zu Ende gegangen ist. Befremdlich wirkt dem > gegen�ber die Einbeziehung des Aktual-Unendlichen in die Mathematik, > die explizit erst gegen Ende des vorigen Jahrhunderts mit G. Cantor > begann. Set theory does not claim to rule in science (physics) or politics at all. Hence my argument prevails that EC states are not part of any set theory. ,----[ <45742128$0$97220$892e7fe2(a)authen.yellow.readfreenews.net> ] | Not everything is "in" ZF. Neither the EC nor their states are "in" | ZF. Hence not everything you know is necessarily a set. `---- >> > 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. >> >> I will not discuss this anachronisms but modern concepts instead: > > After you will have learned how it works, you wil see that it is but > an anachronism. Since I already know how it works I can clearly see its timeless elegance. >> I am especially interested in the growing number and the growing set. > > The number of your contributions has increased by 1 with your post I > just answer. The same holds for the set of your contributions. Which Z-set theoretically can only be modelled by having a set which is a function. You remember: Z set theory does not comprise a concept of time. ,----[ <1165228714.543673.34570(a)j72g2000cwa.googlegroups.com> ] | No. X and Y do not grow, they remain "X" and "Y". The set they denote | does grow. The number of EC states may be n. "n" does not grow. The | number denoted by "n" does grow. `---- Do you agree that X and Y are functions? F. N. -- xyz
From: Bob Kolker on 5 Dec 2006 08:40
mueckenh(a)rz.fh-augsburg.de wrote: > > There is no reason to give up continuity for applied mathematics. Ony > those who want to learn the real truth may bother. And what is the "real truth"? Pontius Pilate |