Cantor Confusion
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From: David Marcus on 21 Nov 2006 00:49 Randy Poe wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > Randy Poe schrieb: > > > > > > I know that at most 10^100 digits of sqrt(2) can be determined, in > > > > principle. > > > > > > In principle, if a is the sqrt(2) to 10^100 digits, then > > > 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. > > > > > > What "in principle" prevents me from calculating 2/a, > > > or adding it to 1, or taking 0.5*(a + 2/a)? > > > > Lack of bits to represent 2/a if a already requires all bits available. > > That means it's hard in practice. > > But what prevents 2/a from existing in principle? > > > If you don't believe me, simply try it. By about 330 calculations you > > should be able to produce the first 10^100 digits. If you have a slow > > computer you will need one second per calculation. So let it run for 5 > > minutes and you will know what prevents you from calculating 2/a. > > What difference does the speed of an actual computer make? We > aren't talking about implementation, we're talking about "in > principle". > > How can a represent "all bits available" IN PRINCIPLE? In principle, > there is no limit on the available bits. Just a guess, but I think that for WM, the only things that are possible in principle are those that are possible in practice. It reminds me of the original Star Trek series: There was a barrier at the edge of the galaxy. So, if you got to the edge of the galaxy, you couldn't go any further (except in those episodes where they figured out how to get through the barrier). -- David Marcus
From: Ralf Bader on 21 Nov 2006 01:24 Dik T. Winter wrote: > In article <MPG.1fc5e10844be8cf2989911(a)news.rcn.com> David Marcus > <DavidMarcus(a)alumdotmit.edu> writes: > > Dik T. Winter wrote: > > > In article <1163469888.057589.117040(a)k70g2000cwa.googlegroups.com> > > > "William Hughes" <wpihughes(a)hotmail.com> writes: > > > > So lets count the set of all natural numbers {1,2,3,...} > > > > There are no natural number left. So we stop > > > > using natural numbers and use ordinals > > > > (and to nobody's surprise a few things change). > > > > > > This is wrong. There is no ordinal needed to count the elements of > > > the > > > set of all natural numbers. You can count until you weigh an ounce > > > (;-)) > > > but you will never finish. Neither the elements you wish to count > > > will > > > be exhausted nor the numbers with which you count. I think this is > > > potential infinity. On the other hand, when you ask "how many" > > > elements there are in N, you need an infinity (and this is, I think, > > > actual > > > infinity). But all this hinges quite closely on the semantics of the > > > word "count". If seen as a process, you do not need an infinity; > > > when > > > seen as the result of a process, you do need an infinity. In many > > > languages (German and Dutch amongst others) there are different words > > > for the two meanings, but the meanings are conflated in English. > > > > Are you discussing languages or math? What would a mathematical meaning > > for "count the set of all natural numbers" be? > > I am discussing both language and maths. In German (as in Dutch) > there is a clear distinction between "Abzählen" and "Zählen". And as > you are in discussion with somebody from German origin, it is important > to keep this in mind. The first word means the process of counting, the > second means counting to get a result. In English for both the verb > "to count" is used (and is given as translation in dictionaries). I > think "to enumerate" is a better translation of "Abzählen". As a native speaker of German, I can't confirm this. There is a difference between "zählen" and "abzählen", because one can't use these words interchangeably, and this difference seems to have something to do with what you wrote, but it is far less clear-cut than your remarks seem to say. However this may be, in discussions with Mückenheim it doesn't matter anyway, because Mückenheim plainly doesn't understand German, or, more precisely, Cantor's original papers. Mückenheim gives ample proof of this fact in his publications on the arxiv. In his paper "On a severe inconsistency..." Mückenheims ascribes some assertions to Cantor, and confirms this by quoting from Cantor's papers, which are obviously wrong. If one checks this quotation, one sees that Mückenheim ignored assumptions which Cantor had made previously in his text. In another paper whose title I don't remember Mückenheim asserts to give a simpler proof of a theorem of Cantor's, namely that the real plane stays connected if all points whose coordinates are rational are left out. Comparing with Cantor's original, one notices that Cantor had proved a much more general result, namely that the real plane stays connected if an arbitrary countable set of points is left out. The set of points with rational coordinates he later mentions as an example, and Mückenheim took that example for the main issue, ascribing some other weird opinions to Cantor on the way. Although, surprisingly, Mückenheims simplified proof of a restricted assertion is in principle correct, it is also telling that he seems to consider it worth to be published what actually is a mildly interesting exercise in basic point set topology. It simply doesn't make any sense, and doesn't lead anywhere, to discuss mathematics with Mückenheim, neither in English nor in German. Ralf
From: David Marcus on 21 Nov 2006 01:52 Ralf Bader wrote: > Dik T. Winter wrote: > > In article <MPG.1fc5e10844be8cf2989911(a)news.rcn.com> David Marcus > > <DavidMarcus(a)alumdotmit.edu> writes: > > > Dik T. Winter wrote: > > > > In article <1163469888.057589.117040(a)k70g2000cwa.googlegroups.com> > > > > "William Hughes" <wpihughes(a)hotmail.com> writes: > > > > > So lets count the set of all natural numbers {1,2,3,...} > > > > > There are no natural number left. So we stop > > > > > using natural numbers and use ordinals > > > > > (and to nobody's surprise a few things change). > > > > > > > > This is wrong. There is no ordinal needed to count the elements of > > > > the > > > > set of all natural numbers. You can count until you weigh an ounce > > > > (;-)) > > > > but you will never finish. Neither the elements you wish to count > > > > will > > > > be exhausted nor the numbers with which you count. I think this is > > > > potential infinity. On the other hand, when you ask "how many" > > > > elements there are in N, you need an infinity (and this is, I think, > > > > actual > > > > infinity). But all this hinges quite closely on the semantics of the > > > > word "count". If seen as a process, you do not need an infinity; > > > > when > > > > seen as the result of a process, you do need an infinity. In many > > > > languages (German and Dutch amongst others) there are different words > > > > for the two meanings, but the meanings are conflated in English. > > > > > > Are you discussing languages or math? What would a mathematical meaning > > > for "count the set of all natural numbers" be? > > > > I am discussing both language and maths. In German (as in Dutch) > > there is a clear distinction between "Abz?hlen" and "Z?hlen". And as > > you are in discussion with somebody from German origin, it is important > > to keep this in mind. The first word means the process of counting, the > > second means counting to get a result. In English for both the verb > > "to count" is used (and is given as translation in dictionaries). I > > think "to enumerate" is a better translation of "Abz?hlen". > > As a native speaker of German, I can't confirm this. There is a difference > between "z?hlen" and "abz?hlen", because one can't use these words > interchangeably, and this difference seems to have something to do with > what you wrote, but it is far less clear-cut than your remarks seem to say. > > However this may be, in discussions with M?ckenheim it doesn't matter > anyway, because M?ckenheim plainly doesn't understand German, or, more > precisely, Cantor's original papers. M?ckenheim gives ample proof of this > fact in his publications on the arxiv. In his paper "On a severe > inconsistency..." M?ckenheim ascribes some assertions to Cantor, and > confirms this by quoting from Cantor's papers, which are obviously wrong. > If one checks this quotation, one sees that M?ckenheim ignored assumptions > which Cantor had made previously in his text. In another paper whose title > I don't remember M?ckenheim asserts to give a simpler proof of a theorem of > Cantor's, namely that the real plane stays connected if all points whose > coordinates are rational are left out. Comparing with Cantor's original, > one notices that Cantor had proved a much more general result, namely that > the real plane stays connected if an arbitrary countable set of points is > left out. The set of points with rational coordinates he later mentions as > an example, and M?ckenheim took that example for the main issue, ascribing > some other weird opinions to Cantor on the way. Although, surprisingly, > M?ckenheim's simplified proof of a restricted assertion is in principle > correct, Quite surprising. > it is also telling that he seems to consider it worth to be > published what actually is a mildly interesting exercise in basic point set > topology. > It simply doesn't make any sense, and doesn't lead anywhere, to discuss > mathematics with M?ckenheim, neither in English nor in German. I'm pretty sure we all are well aware of this. For that matter, I don't think we've been "discussing mathematics" with WM. Thank you for your very interesting comments. -- David Marcus
From: Eckard Blumschein on 21 Nov 2006 03:24 On 11/13/2006 10:21 PM, Virgil wrote: > mueckenh(a)rz.fh-augsburg.de wrote: >> Franziska Neugebauer schrieb: >> > mueckenh(a)rz.fh-augsburg.de wrote: >> > > Yes. But, sorry to see, it is the fundament of modern mathematics. >> > >> > "Finished infinities" is your wording. >> >> Precisely describing the fundament of modern mathematics. > > It may describe WM's fundament, but need not describe anyone elses'. The style of discussion seems to correspond to the subject: Cantor Confusion. Confusion due to Cantor. Confusion how to interpret Cantor. Confusion of Cantor himself .... Cantor and Confusion are almost synonymous: Infinite confusion on infinity. To what extent criticism by WM might be justified? Let me tell you my position: WM is most likely wrong if he sees the finished infinities the fundament of modern mathematics. At first, modern mathematics is not at all really based on set theory but on valuable precantorian tradition and independent continuation of it, combined with a rather pragmatic ignorance of principles. Secondly, nothing would be really wrong with the apparent logical contradiction between the infinite divisibility of continuum and the finite amount of attributed numbers available. I wrote "would" because I revealed an either overlooked or more likely carefully hidden categorical gap between the primary notion of continuum and the so called Hausdorff continuum of real numbers. Mathematics learned from Cantor logically splitted thinking. On one hand, geometry requires continuity. As made obvious by DA2, this would demand a definition of the reals like fictitious limits ore like perfectly endless continued decimals or the like, i.e. fictions. On the other hand, algebra requires genuine numbers. So all set theory may claim to include infinity. It only nurtures a pertaining illusion. Real numbers as defined e.g. by nested intervals are strictly speaking just very fine grained rationals based on the potential infinity. In so far, WM is absolutely correct. Isn't he? Ich ?bersetze frei: In welchem Umfang hat WM Recht? Ich meine: WM irrt sich h?chstwahrscheinlich, wenn er das beendete Nieendende ironisch als das Fundament moderner Mathematik benzeichnet. Erstens basiert die moderne Mathematik ?berhaupt nicht wirklich auf der Mengenlehre sondern auf wertvoller Vorcantor-Tradition und deren unabh?ngigem Ausbau, kombiniert mit einer ziemlich pragmatischen Mi?achtung von Grunds?tzen. Zweitens w?re nichts einzuwenden gegen dem scheinbaren logischen Widerspruch zwischen unendlicher Teilbarkeit des Kontinuums und der begrenzten Menge von zurechenbaren verf?gbaren Zahlen. Ich schrieb "w?re" weil ich einen entweder ?bersehenen oder, was wahrscheinlicher ist, sorgf?ltig versteckten kategorischen Unterschied zwischen dem originalen Kontinuumsbegriff und dem sogenannten Hausdorff-Kontinuum der reellen Zahlen aufgedeckt habe. Die Mathematik hat von Cantor gelernt logisch zweigleisig zu denken. Einerseits braucht die Geometrie die Stetigkeit. Wie das DA2 verdeutlicht, w?rde dies eine Definition der reellen Zahlen als fiktive Grenzwerte oder mit perfekt endlosen Nachkommastellen erfordern, also Fiktionen. Andererseits erfordert die Algebra echte Zahlen. Somit mag alle Mengenlehre behaupten das Unendliche einzuschlie?en. Dies n?hrt nur entsprechende Illusionen. Reelle Zahlen, so wie man sie beispielsweise mittels Intervallschachtelung definiert, sind genau genommen nur sehr hochaufl?sende Rationalzahlen, beruhend auf potentieller Unendlichkeit. Insoweit hat WM v?llig Recht. Oder etwa nicht? Eckard Blumschein
From: Heinz Mau on 21 Nov 2006 03:45
Am Tue, 21 Nov 2006 09:24:54 +0100 schrieb Eckard Blumschein: > Ich schrieb "w?re" weil ich einen entweder ?bersehenen oder, was > wahrscheinlicher ist, sorgf?ltig versteckten kategorischen Unterschied > zwischen dem originalen Kontinuumsbegriff und dem sogenannten > Hausdorff-Kontinuum der reellen Zahlen aufgedeckt habe. You are only a stupid little girl from germany. Du hast nur gezeigt, wie dumm Du bist. Sach mal, an der Uni Sommerpause oder wieso kannst du auf einmal Deine ganze Zeit wieder hier verplempern? |