Cantor Confusion
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From: Virgil on 23 Nov 2006 16:35 In article <1164285635.946311.260210(a)j44g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: WM is stuttering now. He posted the identical article 4 times in a row. As I have already answered it once, this is merely to note his repetition.
From: Virgil on 23 Nov 2006 16:40 In article <45659C9B.8000800(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/22/2006 8:39 PM, Virgil wrote: > > > Something which is potentially, but not actually, > > infinite cannot be a set at all in any set theory yet seen here. > > Isn't there consensus that the set IN of natural numbers is countable? > > How do you imagine bijection with a actually infinite set? > Notice: Actual infinity is a "Gedankending" something fictitious. Al numbers are "Gedankendingens" (if the is the right plural). > I do not say it is unconceivable or nonsense. It is just unapproachable. I t is conceivable as it has been conceived. > > Moreove, potentially and actually infinite are mutually excluding points > of view. So one can round file one of them. Mathematicians, by and large, round file the first, anti-mathematicians sometimes round file both.
From: Virgil on 23 Nov 2006 16:43 In article <4565D4D5.2030509(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/23/2006 9:14 AM, Virgil wrote: > > >> Eckard>> That little ambiguity? Doesn't it make a categorical difference > >> if a set > >> >> has been a priori set for good? > >> > > >> Virgil> When one speaks of the sequence {1,2,3,...} and another speaks > >> of the > >> > set {1,2,3,...}, the notational ambiguity should not override the words > >> > "sequence" and "set". > >> > >> Perhaps you did not get or at least did not accept my point: > >> Does it not make a categorical difference whether a set is something > >> unchanging beeing already perfect for good or something that just > >> formally includes the vital property of the series of nagtural numbers > >> to have no end at all? > >> The notation {1, 2, 3, ...} is ambiguous with this respect. It depends > >> on the decision whether ... denote potential or actual infinity. > > > > Are the labels "sequence" and "set" ambiguous to EB? If so perhaps he is > > just muddled. > > The ambiguity resides within the three points "..." > They may denote either actual or potential infinity. > > > >> Meanwhile I see mounting evidence for my suspicion that set theory is > >> some sort of (self?)-deception. > > > > EB sees what he tells himself he should see, regardless of whether there > > is anything there to see of not. > > Those who follow the discussion will not immediately change their > opinion but should have a chance for doing so. If one has a choice between a red hat and a green hat, merely saying "hat" does not avoid ambiguity, but prefixing "hat" with "red" or "green" does. Thus "sequence {1,2,3...}" or "set [1,2,3...}" are no more ambiguous that "red hat" or "green hat".
From: Virgil on 23 Nov 2006 16:47 In article <4565D569.9080200(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/23/2006 9:16 AM, Virgil wrote: > > In article <456556CD.4030107(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > > > >> Virgil> While all sequences are sets ( as functions from N to some set of > >> > values) not all sets are sequences. > >> > >> So you could be correct if referring to sets that are no sequences? > >> Look above: I referred to something going on forever. Which "forever"? In time, in space, or in something else. As neither time not space, in any physical sense, exists in mathematics, it must be in something, as yet unexplained, else.
From: Virgil on 23 Nov 2006 17:05
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. What definition of connectedness does WM use which disconnects the empty set? > > > 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. > > > > As an > > > example he chose the algebraic numbers (obviously in order to > > > distinguish them from the transcendental numbers). > > > > > > I proved that there is by far a simpler proof for the algebraic > > > numbers. > > > I proved that this proof can also be applied to the transcendental > > > numbers. > > > See the appendix of http://arxiv.org/pdf/math.GM/0306200 > > > > I have seen it. No serious mathematician would dare present such a > > sloppy paper for publication. Nor would any well-referreed mathematical > > journal publish it. > > The alleged proofs, at least as far as I bothered > > to read, are fatally flawed, > > In fact, this was the opinion of the mathematicians at Cantor's time > already. Therefore he had problems to get his papers published. > Therefore I wrote my paper. > > > and do not establish their claimed results. > > 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. > > 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. |