Cantor Confusion
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From: Eckard Blumschein on 7 Dec 2006 06:15 On 12/7/2006 4:41 AM, Virgil wrote: > In article <4576DFFC.1000807(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/5/2006 11:04 PM, Virgil wrote: >> > In article <4575B9F3.7080107(a)et.uni-magdeburg.de>, >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> >> >> >> Reals according to DA2 are fictitious >> > >> > No one mathematically competent who is at all familiar with Cantor's 2nd >> > proof finds any such thing falsehoods in it. >> > >> > It is EB who is fictitious. >> >> Set theorist may wish this. No my arguments are real and unrefuted. > > In mathematics they are neither real nor unrefuted. In some sort of EB > wonderland EB may be able to dictate what is or is not allowable, but in > mathematics, he is outside the pale, and his dictates are of no > consequence.. Just try and refute: Reals according to DA2 are fictitious Reals according to DA2 are fictitious With fictitious I meant: They must not have a directly approachable numerical address. This was the basis for the 2nd DA by Cantor afer an idea by Emil du Bois-Raymond. Good luck
From: Franziska Neugebauer on 7 Dec 2006 06:17 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > Franziska Neugebauer schrieb: >> >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > Bob Kolker schrieb: >> >> [...] >> >> >> Rational numbers are non-genuine. Nowhere in the physical world >> >> >> outside of our nervouse systems do they exist. >> >> > >> >> > The nervous system is a part of the physical world. >> >> >> >> Hence this issue is to be disussed in a physics or neuro sciences >> >> newsgroup. >> >> >> >> > The integers belong to the nervous system. >> >> >> >> The nervous system is generally not an issue in mathematics. >> > >> > That is why many mathematicians overestimate their capabilities so >> > grossly. >> >> Which mathematician overestimates her or his capabilities? > > Those who pretend to be able of knowing every integer and, therefore, > to imagine the whole actually infinite set. I don't know any mathematician who "pretends" an ability of "knowing" (what does this mean?) every integer. Who "pretends" so? F. N. -- xyz
From: mueckenh on 7 Dec 2006 06:23 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Please clarify a point of nomenclature. > Do you consider a potentially infinite > set containing only finite elements > (e.g. the natural numbers) to be: > > 1. a set ? > 2. a finite set? It is neither an actually infinite set nor is it a set in the sense of set theory. I call it a set, because "set" is a handsome word. I call it an infinte set, because it is not a finite set. But if I talk to you about that, you cannot understand, because you can only think in the notions of set theory. > > > Note that the collection of all lines is > a potentially infinite set. So if we have > a statement involving the collection of > all lines we cannot exchange quantifiers. We know that every line consists of a finite number of elements. That is the point! That shows that for each line the quantifier can be reversed. > > > > > Recall my simple example: Every set of even natural numbers like > > {2,4,6,...,2n} has a cardinal number which is less than some number in > > the set. This theorem does not become invalid if n grows infinitely. > > No. Something that is true for finite n may or may not be > true in the limit. Handwaving! Belief in the miracle of infinity. Something may be changing. But if we know that the difference between n_k and n_0 is more than k for all k then we can conclude that in the limit for k --> oo the difference will not change its sign. n_k - n_0 > k ==> lim [k-->oo] (n_k - n_0) > 0. That can be concluded with absolute certainty. And just this describes our question. > > > A > > set of even atural numbers cannot have an infinite cardial number. > > Piffle. The collection of all even natural numbers is a potentially > infinite > set. Every potentially infinite set has an infinite cardinal number. A potentially infinite set has no cardinal number. I said the set may have the "cardinal number" oo. But that is not a number. Therefore, I used the quotation marks. > > > The potentially infinite set of natural numbers > > > exists. > > > > That does not involve the existence of every natural number. (Don't ask > > me which is missing. Those which can be named can be incorporated.) > > I have never claimed nor used the existence of every natural number. > It is enough to claim the existence of the potentially infinite sets > to define bijections between potentially infinite sets. You do not > have to claim the existence of every element of a potentially > infinite set. A bijection is a set. If it is not complete, then you cannot obtain your conclusions. > > > > infinite set. > > > > Not a complete bijection. A bijection is but a set of ordered pairs. If > > the domain is incomplete (and the range too), then the bijection cannot > > be complete. > > Recall this post from Dec 1 > > We extend this to potentially infinite sets: > > A function from the potentially infinite set A to the > potentially infinite set B is a potentially infinite set of > ordered pairs (a,b) such that a is an element of A and b is > an element of B. > > We can now define bijections on potentially infinite sets > > A bijection on potentially infinite sets is not a set of ordered pairs > it > is a potentially infintite set of ordered pairs. It is not necessary > to show that a bijection is "complete" to show that it exists. What means to exist in this case? > > > > > But it cannot be false for a finite line, because finite sets allow > > quantifier reversal. Now there is no infinite line. > > > > However, the statement is not about the elements of a line, > but about the elements of the diagonal. > The set of elements of the diagonal is potentially infinite. > Potentially > infinite sets do not allow quantifier reversal. Here is the main point of misunderstanding: The statement is about a line, because the diagonal consists only of the elements of lines. And all lines are finite and, therefore, can be treated as I do.. > > > If the diagonal would actually exist, then a line of same length would > > actually exist. > > It is customary to define a term (e.g. "actually exist") before you use > it. Below you clarify that for a set or potentially infinite set > to actually exist, it must both exist and all of its elements must > exist. > > It is irrelevent whether the diagonal "actually exists" or not. It is > enough to know that the diagonal exists to define a bijection > involving the diagonal. Now any line L actually exists, actually? > so it > is possible to ask the question: Does a bijection between > the diagonal and L exist? > > Do you intend to keep claiming that a bijection can exist > between the diagonal and a single line?. Of course. But it is not an actual or complete bijection, because neither the diagonal nor any line L actually exists. The contradiction you allude to would only occur in case of complete existence. I devised this example n order to disprove this case. > > > > > > > > > > > > > > > If the "infinite set of finite numbers" existed, then we would have a > > > > bijection between the diagonal and a line. > > > > > > > > > > We have agreed to disagree on whether the "infinite set of finite > > > numbers" > > > exists. However, we have agreed that the "potentially infinite set > > > of finite numbers" exists. > > > > But we have not agreed that it exists actually (i.e., is complete). > > Since I have never used the property that the set > of natural numbers is complete, this is irrelevent. > > [Note by the way you claim that the set of natural > numbers exists, Please understand: The set of natural numbers does *not* exist in the sense of set theory. > but does not actually exist. This > doesn't sound any better in German. You need to > work on your nomenclature.] > > > > > > > > > > > (A diagonal cannot exist without being in a line.) > > > > > > False. Every element of the diagonal must be in some > > > line. However, there is no line that contains all elements of the > > > diagonal. So the diagonal is not in a line. > > > The elements of the diagonal form the potentially infinite > > > set of natural numbers. This potentially infinite set exists, > > > so the diagonal exists. So the diagonal can exist without > > > being in a line. > > > > Why don't you directly say that the potentially infinite diagonal > > actually exists? > > Because it is not necessary to say that the potentially infinite > diagonal is complete, either to show that it exists, or to > define a bijection on it. And what do you mean by "it exists"? Regards, WM
From: Eckard Blumschein on 7 Dec 2006 06:34 On 12/7/2006 4:38 AM, Virgil wrote: > In article <4576DF19.7070005(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/5/2006 11:01 PM, Virgil wrote: >> > In article <4575B727.6070006(a)et.uni-magdeburg.de>, >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> > >> >> On 12/5/2006 3:36 PM, Georg Kreyerhoff wrote: >> >> > Eckard Blumschein schrieb: >> >> > >> >> >> Do not confuse Cantor's virtue of belief in god given sets with my power >> >> >> of abstraction. >> >> > >> >> > Your power of abstraction is nonexistant. You're not even able to >> >> > distinguish >> >> > between representations of numbers and the abstract concept of numbers. >> >> > >> >> > Georg >> >> >> >> Really? >> > >> > Really! >> >> No. Maybe some mistakable wording gave rise for a false impression. >> More likely, you and Georg do not understand that there is no number >> outside an appropriate representation. > > Who gave you the power to dictate what a "number" is, EB? I just realized how brutally Cantor raped the old and correct notion of number and feel safe with Gauss all the others. > > A "number" in mathematics is what the majority of mathematicians agree > it is, regardless of what anti-mathematicians like EB try to dictate. The majority of really important mathematics perhaps lived before Cantor or did not take issue towards his at best somewhat strange and absolutely unfounded violation of the notion number. >> The abstract concept of numbers >> must not be misused as to declare rationals and embeded rationals >> likewise existent. > > The "abstract concept of number" can be used in any way that > mathematicians choose to use it, If there was really general agreement among mathematicians, then there would be an acceptable printed definition. Since Cantor's definition of set has been declared untennable without substitute, I do not expect a clean definition of number either. and anti-mathematical pipsqueeks like > EB have no power to dictate what mathematicians are or are not allowed > to do. Insult always indicates poor personality and lacking arguments.
From: mueckenh on 7 Dec 2006 06:37
Virgil schrieb: > > But WM's construction requires an infinite sequence of branches for each > object to be paired with a path. And we already know that there are > uncountably many such infinite sequences constructable from countably > many branches(or nodes), so WM's pairing scheme is phony. I will try to make it simpler for you: Divide the root node of the binary tree by 2^n and let n --> oo. The number of pieces then becomes infinite, but it remains countable. Map one piece on each path. Their number is 2^n with n --> oo. This number is countable too, by the fact that a finite number can never become actually uncountable by multiplication with two. The clue of the tree is merely its continuity. Every path is connected with all the others. Therefore uncountability is excluded. Regards, WM |