Cantor Confusion
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From: David Marcus on 9 Dec 2006 03:32 mueckenh(a)rz.fh-augsburg.de wrote: > > Virgil schrieb: > > [concerning Cantor's first proof of uncountability of the real numbers] > > > > In the reals, any subset which has a real upper bound has a real least > > upper bound and, similarly , any subset which has a real lower bound has > > a real greatest lower bound. > > > > The only subsets of the reals for which there is a similar property are > > real intervals. > > > > Thus it is only for the set of all reals, or for real intervals, that > > the proof appplies. > > Yes. And it does not apply if only one single element of the > investigated real interval is missing. As the uncontability property of > this interval cannot depend on this single element, the whole proof > fails. Are you really saying that the proof is wrong because it doesn't prove a different theorem? -- David Marcus
From: Virgil on 9 Dec 2006 03:41 In article <MPG.1fe436bb3bd6003c989a18(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Eckard Blumschein wrote: > > On 12/8/2006 1:05 AM, David Marcus wrote: > > > Eckard Blumschein wrote: > > >> On 12/5/2006 9:23 PM, Virgil wrote: > > >> > > >> >> Do not confuse Cantor's virtue of belief in god given sets with my > > >> >> power > > >> >> of abstraction. > > >> > > > >> > Cantor's religious beliefs are as irrelevant as EB's beliefs in his > > >> > own > > >> > infallibility. > > >> > > >> I am not infallible. Show me my errors, and I will express my gratitude. > > > > > > Show you your errors or convince you that they really are errors? The > > > former is simple, but the latter appears to be impossible. We can't > > > force you to learn, if you don't wish to. > > > > I can force you to either refute e.g. my hint that Cantor's definition > > of a set has been declared untenable or tacitly accept this fact. > > Please restate the definition that you say is untenable. Let's take a > look. And tell us by whom it has been declared untenable, and why we should take his word for anything.
From: Virgil on 9 Dec 2006 03:44 In article <MPG.1fe43a42ec04d3c1989a1c(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Eckard Blumschein wrote: > > On 12/7/2006 7:06 PM, Lester Zick wrote: > > > On Thu, 7 Dec 2006 02:04:34 -0500, David Marcus > > > <DavidMarcus(a)alumdotmit.edu> wrote: > > >>Bob Kolker wrote: > > >>> Eckard Blumschein wrote: > > >>> > > >>> > Roughly speaking, it just claims that a set is unambiguously > > >>> > determined > > >>> > by its elements. If i recall correctly A=B<-->(A in B and B in A) > > >>> > > > >>> > Perhaps the Delphi oracle provided less possibilities of tweaked > > >>> > interpretation betwixed and between potential and actual infinity. > > >>> > > >>> What is "potential" infinity. Can you define it rigorously? > > >> > > >>Even a non-rigorous defintion would be a start. > > > > > > Well since according to David a definition is "only an abbreviation" > > > how about "X"? > > > > Strictly speaking there is no potential infinity. Infinity is a > > fictitious quality. The series of natural numbers is potentially > > infinite. Aristotele wrote: Infinity exists potentially. There is no > > actual infinity. I was under the impression that Aristoteles wrote in Greek, as neither German nor English were available to him. > > > > Marcus is quite right. We should better explain such basic terminology. > > You appear to have just said that there is no "potential infinity" and > no "actual infinity". Is this correct? If so, why do you keep using the > phrases??? And, what in the world have you been talking about all this > time? Nothing? Sheesh. At least make an effort.
From: David Marcus on 9 Dec 2006 04:29 Ralf Bader wrote: > Eckard Blumschein wrote: > > > On 12/6/2006 12:55 AM, 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. > > > > According to the axiom of extensionality, a set has been determined by > > its elements. Why do you not admit the possibility that countability of > > a set requires countable numbers. Doesn't it make sense? I didn't find a > > single counter-example. > > And I didn't find any counterexample to the fact that anything you write is > stupid gibberish. Nonetheless I will give you a "proof" of the > inconsistency of sert theory: There are different definitions of real > numbers which, however, are said to be equivalent. One is with Dedekind > cuts, by which a real number is a set of rational numbers, x = {q e Q | q < > x}. This is a countable set (so you may belabor this by rambling about > "countable numbers"). Another definition of real number is as a class of > Cauchy sequences. A diagonal argument easily reveals the fact that those > classes are uncountable. So in this view a real number is a certain > uncountable set, and we have a "contradiction": A real number is a > countable set and an uncountable set at the same time. Of course, this is > utter nonsense. But I think it fits well into your (and Mückenheim's) > follies. Actually, it is too close to being sense to fit with the follies. -- David Marcus
From: David Marcus on 9 Dec 2006 04:33
Virgil wrote: > In article <MPG.1fe436bb3bd6003c989a18(a)news.rcn.com>, > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > > Eckard Blumschein wrote: > > > On 12/8/2006 1:05 AM, David Marcus wrote: > > > > Eckard Blumschein wrote: > > > >> On 12/5/2006 9:23 PM, Virgil wrote: > > > >> > > > >> >> Do not confuse Cantor's virtue of belief in god given sets with my > > > >> >> power > > > >> >> of abstraction. > > > >> > > > > >> > Cantor's religious beliefs are as irrelevant as EB's beliefs in his > > > >> > own > > > >> > infallibility. > > > >> > > > >> I am not infallible. Show me my errors, and I will express my gratitude. > > > > > > > > Show you your errors or convince you that they really are errors? The > > > > former is simple, but the latter appears to be impossible. We can't > > > > force you to learn, if you don't wish to. > > > > > > I can force you to either refute e.g. my hint that Cantor's definition > > > of a set has been declared untenable or tacitly accept this fact. > > > > Please restate the definition that you say is untenable. Let's take a > > look. > > And tell us by whom it has been declared untenable, and why we should > take his word for anything. I fear that if we ask for too many things at once, we are unlikely to get any of them. Although admittedly, asking for one thing at a time hasn't generally produced much. -- David Marcus |