Cantor Confusion
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From: cbrown on 5 Dec 2006 14:15 mueckenh(a)rz.fh-augsburg.de wrote: > 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. > I know that 1/2 of the length of a unit edge + 1/2 of the length of a unit = 1 unit edge in length; but I don't know that 1/2 of an edge + 1/2 of an edge = 1 edge; because I don't know what "1/2 of" an edge is. Some things (unit lengths) can be divided in a sensible manner; and some things cannot. Likewise, some things, such as /lengths/, can be added so that 1/2 + 1/2 = 1; but 1/2 of the group of order 7 + 1/2 of the group of order 7 is not the group of order 7; because "1/2 of" is meaningless here. Your argument seems to say something about lengths of paths and lengths of edges; but not about number of paths and number of edges (where by "number" I mean "cardinality"). > > > 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. > Your original argument was: "Edges are countable. Paths are uncountable. There exists a surjection T of edges onto paths; therefore there countable >= uncountable; contradiction". Now you say that T is /not/ a surjection of edges onto paths, but some other thing. I have two questions: (1) What is the range and domain of the bijection T you claim to have provided? More exactly, when you say T maps "shares of divided edges" one-to-one onto "paths", how do you characterize the set of "shares of divided edges"? What exactly are the elements of this set? (2) How does the existence of T then lead to a contradiction? (Prediction: you are somehow conflating the fact that lim n->oo sum (over all edges e) g(p,e,n) = 2 with the (false) assertion that sum (over all edges e) lim n->oo g(p,e,n) = 2.) > > 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. > If it is so obvious, you should, as a professor, also be able to prove it; otherwise, it is simply your firmly held conviction. > 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 4> much): > Your appendix fails to address the key question: what is the domain of the function T? If e is an edge, what is the set of "shares of the divided edge e"? <snip repetition of the undisputed definition of g: (paths X edges X N) -> R with lim n->oo sum (over all edges e) g(p,e,n) = 2> Cheers - Chas
From: Virgil on 5 Dec 2006 14:45 In article <1165321642.871472.309360(a)73g2000cwn.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > William Hughes schrieb: > > > > > > > > > > > > > > > > We extend this to potentially infinite sets: > > > > > > > > > > > > A function from the set potentially infinite set A to the > > > > > > potentially infinite set B is a potentially infinite set of > > > > > > ordered pairs (a,b) such that a is an element of A and b is > > > > > > an element of B. > > > > > > > > > > > > We can now define bijections on potentially infinite sets > > > > > > and extend the bijection equivalence relation to include > > > > > > potentially infinite sets. Thus we can define > > > > > > equivalence classes under bijection of potentially infinite sets. > > > > > > Thus we can define "cardinal numbers" of potentially > > > > > > infinite sets. > > > > > > > > > > > There is only one "cardinal number". In order to apply any of Cantor's > > > > > proofs of higher cardinal numbers, a set of aleph_0 must be complete. > > > > > But it cannot be complete in potential infinity. > > > > > > > > You now agree that a potentially infinite set can have > > > > a cardinal number and that this cardinal is not > > > > a natural number. > > > > > > > I wrote: a "cardinal number". oo is not a cardinal number in the sense > > > of set theory. > > > > > > > We have: there exists a bijection between sets or potentially infinite > > > > sets > > > > A and B iff the cardinal number of A is the same as > > > > the cardinal number of B. > > > > > > > > Now apply this. > > > > > > > > The natural numbers form a potentially > > > > infinite set. The diagonal contains the potentially infinite set > > > > of natural numbers. > > > > > > There is nothing to contain! You are too much caught in the terms of > > > set theory. You cannot have the complete set because then it would be > > > complete, i.e., actually existing, i.e., actually infinite. > > > > > > If you want to avoid the word contain, reword > > "The diagonal contains the potentially infinite set > > of natural numbers." as "if x is an element of the potentially > > infinite set of natural numbers, then x is an element of > > the potentially infinite set of elements of the diagonal". > > And as well: If x is an element of the potentially infinite set of > natural numbers, then x is an element of a line (and all natural > numbers y < x are also elements of that very line). > > And there is no natural number x, which is no element of a line. Every natural but one is not an element of some line. There is no line containing every natural. The diagonal contains every natural if, as defined, it contains the end of every line.
From: Virgil on 5 Dec 2006 14:52 In article <1165322064.705072.182240(a)80g2000cwy.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > The number of your contributions has increased by 1 with your post I > just answer. The same holds for the set of your contributions. Such time dependent "sets" are not the same as sets under the rubric of set theories, as they do not, for example, obey the axiom of extensionality.
From: Virgil on 5 Dec 2006 14:54 In article <1165322199.723733.167650(a)79g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > 1/3 is not a sequence at all. It is a rational number. > > > > Some correspondents try to think themselves. I encourage you to join > them. WM rekes not his own rede.
From: Virgil on 5 Dec 2006 15:02
In article <1165322444.264500.206440(a)80g2000cwy.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > stephen(a)nomail.com schrieb: > > > Sets, like numbers, do not > > grow. You, like many other people who do not understand set theory, > > think of sets as mutable objects, that change as we perform operations > > on them. This is akin to thinking that numbers change when we perform > > addition. If I add 3 to 7, neither 3 or 7 changes. > > Please read before answering. > > > > It is only a matter of definition and in principle no reason for > > > quarrel. But it is amusing to see ho set theorists insist on the > > > complete and actual existence of the sets of numbers. Of course 25 will > > > not switch to 27 but the number of states will switch from 25 to 27. > > > That's all. Only by this notion we can talk of growing sets and > > > introduce the notion of potential infinity. One can speak of a growing "family" of sets to which more sets can be added without becoming a different "family" in the same way that the addition of a child to an everyday family does not make it a different family. But such "families" of sets are not mere sets, they are /functions/ whose domain is time and whose values are the different sets which occur at different times. WM should read some Korzybski to help him get his head straightened out. |