An uncountable countable set
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From: mueckenh on 15 Aug 2006 13:47 Virgil schrieb: > In article <1155485742.529974.141050(a)i3g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > Of course you can index each n. But your "each n" stems from the true > > list. And we know that 0.111... is not in the true list, because it is > > distinguished from any element of the true list. > > It can be indexed by the first 1, and each 1 indexed by the next 1 and > everything is then indexed. Everything which is in the true list 0.1 0.11 0.111 .... can be indexed. But nothing which is not in the true list. Regards, WM
From: mueckenh on 15 Aug 2006 13:49 Virgil schrieb: > In article <1155487078.836213.291820(a)p79g2000cwp.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > In article <1155314675.845888.174190(a)i3g2000cwc.googlegroups.com>, > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > Virgil schrieb: > > > > > > > > So we have infinitely many finite triangles, but without that "final > > > > > edge", no infinite triangle. > > > > > > > > Cantor's list has no final line. Is it finite? > > > > > > Since a triangle requires 3 edges, without all 3 it is not a triangle. > > > An endless list does not require an end, so is "complete" without one. > > > > A symmetric rectangular triangle is completely determined by one edge > > next to the right angle. > > But edges, as sides of triangles, being segments, must have two ends, > and that "edge" does not. The edge has a definite size, it is a well defined quantity (according to Cantor). Regards, WM
From: mueckenh on 15 Aug 2006 13:51 Virgil schrieb: > > > > > Thus "mueckenh" is claiming to be able to inject the power set of the > > > > > naturals into the set of naturals. > > > > > > > > > > We should be interested in seeing his attempts to perform this > > > > > impossibility. > > > > > > > > Where does *my arguing* fail? > > It assumes without proof that what holds in finite cases must hold in an > infinite case. So does Cantor and all his followers in his diagonal argument. According to Hilbert operating in the infinite can only be secured by the finite. "das Operieren mit dem Unend¬lichen kann nur durch das Endliche gesichert werden" > > What is true up to every level n of the tree is true for the whole > > tree. > > Then "Mueckenh" claims the infinite tree must be finite as every finite > level of it is finite. Of course. There is no actual infinity. > > > > Else: What is true for a finite segment of Cantor's list need not be > > true for the whole infinite list. > > The Cantor statement is not about finite segments of the list but about > individual members of the list, that the constructed number is not equal > to the number in place n in place n. The constructed number has infinitely many finite segments and is, itself, actually infinite. > > > > What is the difference (in concluding from finity to infinity) between > > list and tree? > > Lists don't branch. That is a pity. Had Cantor seen a branching example he would have withdrawn his set theory. Regards, WM
From: mueckenh on 15 Aug 2006 13:53 Virgil schrieb: > > > What Cantor does say that is that no list can be complete, but does not > > > say that there is any number which cannot be listed. > > > > We know that the set of *all* those numbers which can ever appear in > > lists, be it as original entries or as diagonal numbers, is countable. > > Even you know it! Why do conclude from Cantor's idea such incoherent > > nonsense? > > Cantor never gets into the issue of whether there are inaccessible > numbers. Inaccessible numbers have nothing to do with this issue. > That issue did not come up until later. He recognized that, if this issue was correct (which he did not believe), the continuum was countable: Wäre KÖNIGs Satz, daß alle 'endlich defi¬nierbaren' reellen Zahlen einen Inbegriff von der Mächtigkeit aleph_0 ausmachen, richtig, so hieße dies, das ganze Zahlenkontinuum sei abzählbar ... (Letter to Hilbert, 8. 8. 1906) . > He merely says that no > list can include all real numbers. Which is true. So there are countable sets which can be enumerated and such which cannot? Larger countable sets and smaller countable sets? Countable sets which cannot be counted? Another great idea of set theory. Be sure, even the natural numbers cannot be enumerated, because a natural number can be added to every list of natural numbers. Regards, WM
From: mueckenh on 15 Aug 2006 13:54
Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > An infinite sum of 1's is not infinite? > > n > lim sum 1 = lim n =def L > n -> oo i = 1 n -> oo > > There is no such L in N. Correct. Therefore there are not infinitely many difference of 1 between natural numbers. Regards, WM |