An uncountable countable set
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From: mueckenh on 15 Sep 2006 05:15 Dik T. Winter schrieb: > In article <1158133724.305764.245170(a)d34g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > > You keep confusing the situation where edges do terminate (and so all paths > > > do terminate) with the situation where edges do not terminate (and so all > > > paths do not terminate). > > > > What do you imagine? Every edge terminates in a node. > > Yes, so every path made up of a sequence of edges terminates. Does the path of 0.111... terminate? Does the path of pi terminate? Each of their edges terminates. > > > > This example says: Countability can be proven without a bijection. > > A misapprehention. Countability is *by definition* the presence of a > bijection with (a subset) of the set of natural numbers. Aleph_0 is defined by the presence of a bijection with N. But if people assert that aleph_0 is larger than any natural number and smaller than 2^aleph_0, i.e., that aleph_0 has properties of a number, then one can test that property with more general means like the rational relation. This unavoidably and unambiguously leads to the result that aleph_0 is the number of edges is not less than the number of paths is 2^aleph_0. > > Oh yes, it does. My paths are as long as the diagonal of Cantor's list. > > Every number which can in principle be the diagonal of Cantor's list > > can be represented by a path of my tree. > > > > Why do you accept the axiom of infinity for Cantor's list, but deny it > > for my tree? > > I do not deny it, I only state that in the "completed" tree either there > are edges that do not terminate, or all paths do terminate. How could an edge "not terminate"? Do you expect non-terminating digits in 0.111...? What is a non-terminating edge? What would a non-terminating digit be? I can't make any sense of your remark that non-terminating edges should exist. > > > > It is well known that the set of *finite* sequences of digits is countable. > > > You are telling nothing new here. > > > > The paths of my tree are not finite paths. If you insist, that no > > infinite paths do exist, then you must also deny the existence of an > > infinite diagonal of Cantor's list. > > You should reread what I have written, I have never written that infinite > paths do not exist. I have only stated that the paths either terminate > at a node (when all edges terminate at a node) or do *not* terminate at > a node (when all edges do not terminate at a node). If paths terminate > at a node, 1/3 is not in your tree, because that is not a node in your > tree. 1/3 is not a node but a path. No path i the tree does terminate. The nodes in the tree-representation are an image of the digits in binary representation. Regards, WM
From: mueckenh on 15 Sep 2006 05:19 Dik T. Winter schrieb: > Still you have not answered my question: "what law is he using"? I improved this translation already to "truth" > On the > other hand, you should familiarise yourself with the meaning of the > English word "axiom". One of the meanings from Merriam-Webster: > an established rule or principle or a self-evident truth > Note the latter part, which (I think) entirely conforms with Cantor's > "Grundsatz". The latter part does conform with Cantor's but not with the meaning of "axiom" in modern mathematics. Webster obviously was not a modern mathematician. > > > > Although I disagree that > > > changing them leads to rubbish. > > > > because you have not yet understood what Cantor's truths are. > > Educate me. What are they? Pray provide sources. But your rubbish > is not my rubbish... Cantor's truths are self-evident truths which cannot be changed arbitrarily in contrast to the axioms of modern set theory. Regards, WM
From: mueckenh on 15 Sep 2006 05:29 Dik T. Winter schrieb: > > > Yes, it is not in the list. So what is the problem? All possible indexes > > > are in the list, so: > > > (1) A{p = digit position} E{q = list item} {such that q indexes p} > > > which you deny. > > > > That is not an argument. > > *Why* is it not an argument? You state: "all possible indexes are in the > list". What is *wrong* about the argument? You always state "it is wrong", > but never state what part of the argument is wrong. It is wrong because not for all {p = digit position} there exists {q = list item} {such that q indexes p} , but only for those p, which can be indexed, i.e., which are present in the list. As 0.111... is not in the list, there must be a digit position, which is not in the list. (As such a position cannot be identified, 0.111... cannot exist.) > No. There are "numbers" (exactly one) that can be indexed by that list > but that is not in that list. That is a false belief. And why should there be only one of those numbers? Why couldn't it be two at least or even ten? If you answer this question, you have a good chance to recognize that not even one exists that is not in that list but can be indexed by the numbers of that list. > > > But there is a single "number" that can be indexed by your list but not > covered by your list. 0.111... is one (as I defined it above). But it > is *not* a natural number. Therefore it cannot be indexed by natural numbers. Try to find out, why you insist on only one of those strange numbers. There could be infinitely many. Could they all be indexed by natural numbers?. > > > Do you agree that the non-terminating list can index non-terminating > numbers? If the answer is no, why? Because it should index at least two different of those infinite numbers if it could index one of them. Regards, WM
From: mueckenh on 15 Sep 2006 05:42 Dik T. Winter schrieb: > > That does not exist. At least it is not described by n. III is a > > representation of 3. > > Now you shift back to representation, pray remain consistent. Representation is number. There is no difference. Numerals have no "soul". > > > If you know the positions by heart, then you need no addition actually. > > You had already used it or the person who devised that technique had > > used it. But earlier or later your knowing by heart will end and you > > will have to count +1. > > Nope. I will only to need to know successors. Anyhow, you read much more > in the successor of the Peano axioms than is present. The successor is > defined without even any knowledge of addition at all. So > succ(George V) = Edward VIII, succ(Edward VIII) = George VI and > succ(George VI) = Elizabeth II within the set of British kings and queens. > I do not think what way of addition you would propose for that. Of > course, this successor function does not satisfy all of Peano's axioms, > but I hope you get the idea. I was talking of *counting* (remember: Zahl and zaehlen) which requires natural numbers which require the ability to add 1 after you have run out of the numbers known by heart. > > > > Counting without numbers would be nonsense. A boy like Virgil may count > > > > 1, 2, 3, 3, 3, ... > > > > > > a, b, c, d, e, f, ... > > > > The old Greek and othe cultures have just used their letters as numbers > > too. > > Yes, every culture had their representation of numbers. Sometimes base 10, > sometimes other bases. Base 20 is quite common in Europe. Mixed bases > are also used. Sometimes they were used to count from 1. Sometimes from > 0. Who did so before Cantor? > I may note that the decimal system you appear to like most is, in > Europe, mostly due to Simon Stevin, and at that time it included 0 as > a digit. Did he count from 0? Regards, WM
From: Han de Bruijn on 15 Sep 2006 05:47
mueckenh(a)rz.fh-augsburg.de wrote: > David R Tribble schrieb: > >>Tony Orlow wrote: >> >>>Wolfgang's and my position is that N is unbounded but finite, >> >>"Unbounded but finite" is a contradiction, meaning "not finite but >>finite". I'm sure you and Wolfgang think this double-think makes >>sense, but the rest of us don't. > > Your position only reflects the miseducation in mathematics during the > last decades. > > Actual or finished infinity is a contradicton. Surpassed infinity is a > contradiction. > > Unbounded but finite is mathematical reality. Think of the set of all > natural numbers which have been realized by writing down these numbers. > Think of the set of known prime numbers. Think of the set of written > novels. Think of the set of postings. > > These sets are unbounded because they can be extended without end. > Nevertheless they are always finite. Sorry for jumping in so late. But VM is quite right, of course. We have encoutered utterly absurd consequences of thinking otherwise, like the mainstream "theorem" that the probability of a natural being a multiple of 3 doesn't exist. While the obvious truth is that it is equal to 1/3 . This topic has been discussed at length in a thread called "Calculus XOR Probability". Let Google be your friend, eventually. Han de Bruijn |