Cantor Confusion
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From: David Marcus on 7 Dec 2006 02:18 Eckard Blumschein wrote: > On 12/5/2006 2:13 PM, Bob Kolker wrote: > > For the latest time. Uncountability is a property of sets, not > > individual numbers. > > I know this widespread view. So you claim. However, last time I asked you to give the standard definitions, you failed. Care to try again? Define "countable" and "uncountable". > > There is no such thing as an uncountable real > > number. > > Real numbers according to DA2 are uncountable altogether. People like > you will not grasp that. And, people like you don't listen when we point out that "DA2" does not define/construct/characterize the real numbers. Of course, if you actually bothered to read a modern book, you could learn this for yourself. > > Countability > > /Uncountability are properties of -sets-, not individuals. > > Do not reiterate what I know but deny. How come you get to deny things, but we don't? Doesn't seem fair. -- David Marcus
From: David Marcus on 7 Dec 2006 02:20 Ralf Bader wrote: > Bob Kolker wrote: > > > Eckard Blumschein wrote: > >> an exact numerical representation available. Kronecker said, they are no > >> numbers at all. Since the properties of the reals have to be the same as > >> these of the irrationals, all reals must necessarily also be uncountable > >> fictions. > > > > For the latest time. Uncountability is a property of sets, not > > individual numbers. There is no such thing as an uncountable real > > number. Nor is there any such thing as a countable integer. Countability > > /Uncountability are properties of -sets-, not individuals. > > > > You have been told this on several occassions and you apparently are too > > stupid to learn it. > > These people (Blumschein and Mückenheim) don't know how words and notions > are used in mathematics. Blumschein takes a word, e.g. "uncountable", and > attempts to infer from its everyday usage its mathematical meaning. That > such-and-such is uncountable means for Blumschein that it doesn't have the > nature of a number, or something like that. So integers are "countable" for > Blumschein, whereas irrationals are swimming in that sauce Weyl spoke about > (but not meaning what Blumschein thinks) and therefore are "uncountable" in > Blumscheins weird view. For Blumschein, your explanations are just your > prejudices. It is pointless to repeat them. He can't understand you. Undoubtedly correct. On the other hand, not everything in life needs to have a point. -- David Marcus
From: David Marcus on 7 Dec 2006 02:24 Eckard Blumschein wrote: > On 12/6/2006 12:55 AM, Ralf Bader wrote: > > These people (Blumschein and Mückenheim) don't know how words and notions > > are used in mathematics. Blumschein takes a word, e.g. "uncountable", and > > attempts to infer from its everyday usage its mathematical meaning. That > > such-and-such is uncountable means for Blumschein that it doesn't have the > > nature of a number, or something like that. So integers are "countable" for > > Blumschein, whereas irrationals are swimming in that sauce Weyl spoke about > > (but not meaning what Blumschein thinks) and therefore are "uncountable" in > > Blumscheins weird view. For Blumschein, your explanations are just your > > prejudices. It is pointless to repeat them. He can't understand you. > > According to the axiom of extensionality, a set has been determined by > its elements. "is" not "has been". > Why do you not admit the possibility that countability of > a set requires countable numbers. Doesn't it make sense? Of course, it doesn't make sense. The reasons are: 1. You haven't defined/explained what a "countable number" is. 2. There is already a standard definition of "countability of a set" and it seems unlikely that you can define "countable number" in such a way to make your statement true. > I didn't find a single counter-example. That doesn't prove anything. In mathematics, we prove things. -- David Marcus
From: Virgil on 7 Dec 2006 02:38 In article <MPG.1fe18bc534a955549899ed(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Eckard Blumschein wrote: > > On 12/6/2006 12:55 AM, Ralf Bader wrote: > > > These people (Blumschein and Mückenheim) don't know how words and > > > notions > > > are used in mathematics. Blumschein takes a word, e.g. "uncountable", and > > > attempts to infer from its everyday usage its mathematical meaning. That > > > such-and-such is uncountable means for Blumschein that it doesn't have > > > the > > > nature of a number, or something like that. So integers are "countable" > > > for > > > Blumschein, whereas irrationals are swimming in that sauce Weyl spoke > > > about > > > (but not meaning what Blumschein thinks) and therefore are "uncountable" > > > in > > > Blumscheins weird view. For Blumschein, your explanations are just your > > > prejudices. It is pointless to repeat them. He can't understand you. > > > > According to the axiom of extensionality, a set has been determined by > > its elements. > > "is" not "has been". > > > Why do you not admit the possibility that countability of > > a set requires countable numbers. Doesn't it make sense? > > Of course, it doesn't make sense. The reasons are: > > 1. You haven't defined/explained what a "countable number" is. > > 2. There is already a standard definition of "countability of a set" and > it seems unlikely that you can define "countable number" in such a way > to make your statement true. > > > I didn't find a single counter-example. > > That doesn't prove anything. In mathematics, we prove things. And just how hard do you suppose EB looked for counter-examples to his own pet theories?
From: David Marcus on 7 Dec 2006 02:44
Han de Bruijn wrote: > Franziska Neugebauer wrote: > > You still have not yet understood the concept of inifinite sets. > > There may be no person in the world who understands them better than > Wolfgang Mueckenheim. Yeah, sure. And, it may be that the sun won't come up tomorrow. -- David Marcus |