Cantor Confusion
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From: mueckenh on 25 Nov 2006 04:27 Virgil schrieb: > In article <1164307852.961046.140060(a)f16g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > > I will accept that it can be connected if the "right" uncountable set is > > > > > removed, but I can think of uncountable sets of points whose removal at > > > > > least appears to totally disconnect the set of those remaining. > > > > > > > > Of course, for example if you remove all points. > > > > > > The remaining empty set is trivially connected. > > > > That is nonsense. If there is nothing to be connected then there is > > nothing connected! > > Topologically, a set is connected unless it is a union of two or more > non-empty disjoint open sets. > > See http://en.wikipedia.org/wiki/Connected_space > > According to this definition, an empty set is connected. I prefer to think myself. > > What definition of connectedness does WM use which disconnects the empty > set? A set is connected, if it has at least two elements and if its elements are all connected. A set with only one element cannot be connected or disconnected. A set with less than one element is, strictly speaking, not a set at all. So it is neither connected nor disconnected.(Sometimes it is considered a set for formal reasons. Like the number 1 not considered a prime number in order to maintain unique prime factorization.) > > > > > Cantor wanted to suggest that the removal of a countable set leaves the > > > > plane connected while the removal of an uncountable set does not. > > > > > > Those who try to read Cantors mind often deceive themselves. > > > > I read his words. > > Not very well. > > What did I misread, according to your opinion? > > With other words, you did not understand them. But our question > > concerned the appendix only. The proof given there is very simple, so > > simple that even Mr. Bader asserted to have understood it (it seemed so > > to him, at least). You cannot confirm that my proof is correct? > > In reading the beginning of that paper, I found enough errors to > dissuade me from bothering. So it should be possible to name one, unless you prefer to join the slanderer, Mr. Bader. > > > > Regards, WM > > > > PS: You really did not see that Cantor's arguing in his first proof is > > valid for rational numbers as well as for transfinite numbers? > > If WM thinks this, he has not understood the first proof. > > There is an increasing sequence of rationals whose LUB is sqrt(2), > e.g., f(n) = floor( 2^n * sqrt(2) ) / 2^n. > > And a decreasing sequence of rationals whose GLB is sqrt(2), > e.g., g(n) = Ceiling( 2^n * sqrt(2) ) / 2^n > > If Cantor's first proof held for the rationals, as WM claims, then WM > would have proved that sqrt(2) is rational. You misunderstood completely. I claim that Cantor's proof is "symmetric". It can be applied to the algebraic numbers, showing that the limit is not algebraic. It can as well be applied to the transfinite numbers, showing that the limit is not transfinite. This has nothing to do with rational numbers and sqrt(2) because that would not prove any uncountability at all (sqrt(2) is algebraic and as such belongs to a countable set). Regards, WM
From: mueckenh on 25 Nov 2006 04:31 Virgil schrieb: > > > > For finite indexes it is correct. > > > > > > Infinite things are not constrained to behave in all respects like > > > finite ones. That is because "infinite" means "not finite". > > > > But *the lines are all finite*. > > The *set* of lines in WM's so-called EIT is not finite. The number of elements of every line is finite. And only that is relevant. > > > That's just why I chose the EIT as > > example. > > > So your magic belief is easily > > destroyed. > > Not by those who cannot keep their quantifiers straight. In dealing with linearly ordered finite sets there is no quantifier magic. Regards, WM PS: >WM is stuttering now. He posted the identical article 4 times in a row. Sorry, don't know the reason.
From: mueckenh on 25 Nov 2006 04:34 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Why did it take so long to switch from "I don't need any advice from > > you" and from "slightly misleading" to "erratum"? > > It took "so long" because you made my wording an issue and because you > prefered to ride this dead horse insistently claiming that it was not > dead. > So I am guilty again! Small wonder. Horse killed. Have a nice Weekend. Regards, WM
From: Franziska Neugebauer on 25 Nov 2006 04:57 mueckenh(a)rz.fh-augsburg.de wrote: > If the set 1,2,3,...,n is finite, then the set 1,2,3,..., n+1 is > finite too. > Therefore, by induction we (that is: those who can think logically) > prove that the generated set is always finite. Any set {0, 1, 2, ..., n} n e omega is finite. You have not proved that the set of all such sets { {}, {0}, {0, 1}, {0, 1, 2}, ... } is finite. F. N. -- xyz
From: Ross A. Finlayson on 25 Nov 2006 05:42
mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > Oh, perhaps. But now you are entering the field of philosphy end exiting > > the field of mathematics. > > Philosophy of mathematics belongs to mathematics. It is its foundation. > The most solid one is MatheRealism. No, the only complete and consistent theory could be the null axiom theory. Obviously you're probably heard of Goedel and the notion that no adequate theory to found mathematics could be complete, that is not necessarily so, with some arguing the notion of a maximal ordinal, for limited cases of arithmetic, Goedel's incompleteness doesn't apply to the null axiom theory. It was a notion of Hilbert and has long been a goal of the practice of formalism of mathematics to unify all mathematics. To the ancient Greeks mathematics was geometry, the postulates named after Euclid serve well and today there are even partially-non- and super-Euclidean geometries. That's digression, influential Goedel closed the door on the Hilbert program, or correspondingly schools to unify, unitize, the theory of mathematics. The reason that is so is because Goedel's are results of mathematics, though to sophisticates it may be seen that he worked under some false assumptions, false axioms, that do not have to be seen to hold. They're convincing to many, particularly those with interests in the application of mathematics who never encounter any reason to stringently peruse the foundations, in the blur that is application, they don't care. They don't care, because their answers are complete and consistent, in the perfect clarity that is application. Dang, I was all coasting on having my words compared to Hegel's. Ross |