Cantor Confusion
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From: Virgil on 18 Oct 2006 15:08 In article <1161158340.546454.133620(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > David Marcus schreef: > > > > > > > Han de Bruijn wrote: > > > > > > > > > > How can we know, heh? Can things in the real world be true AND false > > > > > (: definition of inconsistency) at the same time? > > > > > > > > That is not the definition of "inconsistency" in Mathematics. On the > > > > other hand, I don't know of any statements in Mathematics that are both > > > > true and false. If you have one, please state it. > > > > > > What then is the precise definition of "inconsistency" in Mathematics? > > > > How many times does it have to be posted? > > > > G is inconsistent <-> G is a set of formulas such that there exists a > > formula P such that P and its negation are both members of G. > > Like: The vase is empty a noon and the vase is not empty at noon. However concluding that the vase is not empty requires a breach of the conditions set by the gedankenexperiment, so is irrelevant to that gedankenexperiment.
From: Virgil on 18 Oct 2006 15:14 In article <1161158540.655217.49600(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > The assumption |{2,4,6,...}| = alpheh_0 > n forall n e N is false. The assumption that |{2,4,6,...}| > n forall n e N is TRUE, because |{2,4,6,...}| >= |{2,4,6,..., 2(n+1)}| = n+1 for all n e N is TRUE.
From: Virgil on 18 Oct 2006 15:17 In article <1161158657.883924.152580(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > Right. No matter which balls you pick you are going to remove an > > infinite > > number of balls. So the number of balls you remove does not matter. > > Which balls you remove does matter. > > Fine. The question is, however, is this set "all balls" or how many > balls remain in the vase at noon? According to the original gedankenexperiment description, each ball inserted is removed before noon. Which, to those who can reason, would suggest that all of them are removed leaving the vase empty.
From: William Hughes on 18 Oct 2006 15:18 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > In theory every possible way of describing a number is known > > (this includes descrbing the representation). > > > I am afraid, here you are in error. Proof: If you were right, then also > every impossible way of doing mathematics should be known. That implies > that set theory should be known as inconsistent. Most of your party > don't know that yet. Piffle. Every possible string is known in theory. Yes, this includes every crank theory that set theory is inconsistent. However, this does not make set theory inconsistent. > > > > > > Ingenuity may or may not be limited, but there is a limited amount > > > > of time to communicate the results of ingenuity. > > > > > > Yes. It can be communicated, however, in highly compressed form. Also > > > the compression depends on ingenuity. > > > > There are only a limited number of messages that can be communicated > > during the lifetime of the universe. This incudes descriptions > > and use of compression methods. > > But all you argue does not limit the size of a numbe There are only a limited number of messages that it is possible to communicate. One of these messages describes a larger number than any of the other messages (description includes description of any compression method used). > > > > > > > > > > > > > > > > > > > > > > > That is true too. And it is easy to see: If we define Lim [n-->oo] > > > > > {1,2,3,...,n} = N, then we can see it easily: > > > > > > > > > > For all n e N we have {2,4,6,...,2n} contains larger natural numbers > > > > > than |{2,4,6,...,2n}| = n. > > > > > > > > So we have something that is true for finite sets. > > > > > > It is true for finite numbers and sets of finite numbers. Should it not > > > be true for all sequences of finite even numbers, then there must be > > > some of even finite numbers, X, in {2,4,6,...,2n, X} which care to push > > > the cardinality without increasing the sizes. That is obviously > > > impossible. > > > > Nope. Adding a single element, or a finite > > number of elements cannot take you > > from a finite set to an infinite set. > > You have to add an infinite number of elements. > > So your set X above must be infinite. Adding an > > infinite set to a finite set certainly changes the cardinality. > > Of course, but necessarily it also changes the maximum sizes of > elements. As long as the sizes all are finite, the cardinality is > finite too. Nope. It is possible to have a set with infinite cardinality composed of element all of which have finite size (consider the real numbers in [0,1]). >You assume that only the one is changed, the other is not. > But you seem not to be aware that in natural numbers size and > cardinality are strictly the same I take it you mean that for a set of natural numbers of the form {1,2,3,...,n}, the cardinality is equal to the size of the maximum element. This is only true if the set has a maximum element >while with the even natural numbers > size grows faster than cardinality. Therefore, for every set X we have > again the same problem but in increased form: X contains by far larger > elements than expressed by its cardinality (because the smaller ones > are already used up). > And since an infinite set does not contain a largest element there is no problem. > > > > > > > > > > > > There is no larger natural number than aleph_0 = |{2,4,6,...}|. > > > > > Contradiction, because there are only natural numbers in {2,4,6,...}. > > > > > > > Wiithout any justification whatsoever you state something > > > > about infinite sets. > > > > > > I state something about finite numbers. > > > > You state something about the set {2,4,6,...} > > That are finite numbers. That is also an infinite set. > > > > > > > > > > Something that is true about finite sets does not have to > > > > be true about infinite sets. > > > > > > That is your standard excuse. It seems that nothing can be true for > > > infinite sets. > > > > Piffle. If I were to say "Something that is true about sets of > > integers > > does not have to be true about sets of real numbers" would > > you say "It seems that nothing can be true for sets of real numbers"? > > That is a wrong example. The set N consists o finite numbers. If it > contained infinite numbers, you could be right, but that were > uninteresting. Piffle. Numbers are not infinite, sets are infinite. It is possible to have an infinite set composed of finite numbers (consider the real numbers in [0,1]). > > > > > > > The brain is contstrained by physical laws. The concepts produced by > > > > the brain are not contrained by physical laws. > > > > > > The puppet hangs on the string. The feet of the puppet hang on the > > > puppet, but not on the string? They do not fall down when the string is > > > cut? > > > > The beauty of the puppet cannot exist without the puppet. If the > > string > > is cut the beauty of the puppet does not fall down. > > If the puppet falls down and if it is of porcelain, the beauty is gone. > So? The beauty may be gone, but the beauty does not fall down. The beauty of the puppet depends on the puppet, but the beauty of the puppet is not a physcial entity. - William Hughes
From: Virgil on 18 Oct 2006 15:22
In article <1161158964.561827.87020(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > I warned you that this point is new: The edges are split in shares of > > > 1/2, 1/4, and so on. But when fractions were introduced in mathematics, > > > most people may have had the same problems as you today. I am sure you > > > can understand it from the written text above. (Many others have > > > already understood it.) > > If a fraction of an edge is related to a path, which is really new, as > far as I know, then the whole edge is related to a set of paths. > Otherwise it would be meaningless to use fractions. This new technique > was exactly described in my original text. > > Regards, WM In attempting to establish bijections, there is no point in fractionating the members of either set as it is only one-to-one correspondences that are of any interest. And if edges are to be split, so can paths be split, and into as many pieces. |