Cantor Confusion
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From: William Hughes on 24 Nov 2006 10:28 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: > > > > > > > > > > > > > If there is no bijection with one line, then there must be an element > > > > > of the diagonal outside of every line. > > > > > > > > No. Not unless the one line > > > > contains every element from every line. > > > > > > This holds for every finite line. Every finite line contains every > > > element from every preceding line. There are only finite lines. > > > > > > I claim > > > > There does not exists a line L_1 such that L_1 contains every > > element from every line. > > > > You counter with > > > > For every line L_1, L_1 contains every element from every line > > preceding L_1. > > > > However, these two statments are not contradictory. > > > > "every element from every line" > > > > is not the same thing as > > > > "every element from every line preceding L_1" > > > > > > Yes it is true that > > > > For any element of the diagonal, d_nn, there exists a line > > L_2, such that L_2 contains d_nn. > > > > However, the statement > > > > There exists a line L_1, such that > > L_1 contains every element from every line > > preceding L_1. > > > > is not the same as the statement > > > > There exists a line L_1, such that > > L_1 contains every element from every line > > preceding L_2. > > > > So we cannot use > > > > There exists a line L_1, such that > > L_1 contains every element from every line > > preceding L_1. > > > > to show that > > > > There exists a line L_1, such that L_1 > > contains every element from every line > > For the lines, each of with has a finite number of elements, this *is > the same*. > It is very important to keep the two concepts, the set of elements of an arbitray line l the set of all elements from all lines L distinct. Both are composed of finite elements. l has a finite number of elements. L has an infinite number of elements. > If you have a linearly ordered set with a finite number of elements, > then there is a maximum. Yes > Every line has a finite number of elements. Further the elements are > linearly ordered. >Therefore the above requirement is satisfied. Yes, for any set of elements of an arbitrary line l, l has a maximum element. Now consider L. L is linearly ordered, but does not have a finite number of elements > There > is a line which contains all the elements off all lines - unless there > were an infinite number of elements. But that would not represent a > natural number. Here you confuse an arbitrary line with the set of all lines. Recall l and L are not the same thing .. > > Therefore you cannot counter with the usual argument that the elements > of any line are all finite but that there are infinitely many of them. Your phasing makes the antecedent of "them", "the elements of any line". However, I do not claim that this set is infinite. I do claim that both the set of "lines" and the set of "every element of every line"are infinite. Rephrasing The elements of any line are all finite, futher there are only a finite number of elements in any one line. However, there are an infinite number of lines. If we let L be the set of every element from every line, then the set L is inifinite. - William Hughes
From: Franziska Neugebauer on 24 Nov 2006 10:29 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. F. N. -- xyz
From: Eckard Blumschein on 24 Nov 2006 12:37 On 11/23/2006 10:47 PM, Virgil wrote: > 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. Neither nor. Counting is just a repetitious operation. > > As neither time not space, in any physical sense, exists in mathematics, > it must be in something, as yet unexplained, else. Not unexplained but best imagined via the application on time or space.
From: Eckard Blumschein on 24 Nov 2006 12:43 On 11/23/2006 10:43 PM, Virgil wrote: >> 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". As ambiguous as "hat" in the sense of as many red as you like and "hat" in the sense the quality green includes something irrational: all of indefinitely much.
From: Eckard Blumschein on 24 Nov 2006 12:57
On 11/23/2006 10:40 PM, Virgil wrote: > In article <45659C9B.8000800(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> 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). The correct plural would be Gedankendinge. Fraenkel (p. 6) referiert Cantor: das als reines Gedankending offenbar nichts widerspruchsvolles in sich birgt. Indeed, the actually infinite set is selfcontradictory. > >> I do not say it is unconceivable or nonsense. It is just unapproachable. > > I t is conceivable as it has been conceived. >> >> Moreover, 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. I do not understand your idioms "round file" and "by and large". Notice. Mathematics has to do with potential infinity when dealing with genuine numbers. But strictly speaking it deals with the actual infinity when considering real numbers and the continuum. |