An uncountable countable set
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From: David Hartley on 7 Jul 2006 14:39 In message <vmhjr2-E08407.11521207072006(a)news.usenetmonster.com>, Virgil <vmhjr2(a)comcast.net> writes >In article <1152282125.757193.28320(a)m73g2000cwd.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > >> There is no limit other than in Canrtor's diagonal. EVERY set initially >> indexed by natural numbers will remain indexed by natural numbers. And >> it will unavoidably become ordered by size too. > >At best you have proved that a countable subset of the rationals have >ben restored to their usual ordering, but you have nowhere shown that >your process ever re-orders ALL of them. > >After each of your countably many transpositions, there is still a >countably infinite subset of the rationals left which are not in natural >order. That is a necesssary consequence of the entire set being >originally well ordered. Suppose x, y are any two rationals. Suppose they are indexed by m and n in the first enumeration, and let k = max(m,n). From the k-th stage on, x and y are in their natural order. Under any reasonable definition of the limit of a sequence of orderings, here it will be the natural ordering. (WM's fallacy is his claim that the (order-preserving) indexation by the naturals carries over to the limit.) -- David Hartley
From: Virgil on 7 Jul 2006 14:42 In article <1152283137.725274.18470(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1152183139.346593.62140(a)m73g2000cwd.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > > > > > Anyone is quite free to reject any axiom set, but no one is free to > > > > impose any prohibition of any axiom set on others. > > > > > > Unless there appears a conradiction. > > > > Not even then. Though if there were any contradiction WITHIN an axiom > > system there would be little point in pursuing that system further. > > What axiom is not obeyed by my transpositions? What system of axioms are you claiming your transpositions are operating in? And it may transpire that it is some theorem deduced from the axioms of that system as a whole that you are violating, which theorem requires a complex of axioms, not merely a single axiom. But as far s I can see, in any appropriate axiom system, say ZF or ZFC or even NBG, you merely show that you can natually order some finite initial segment of your list, not the entire list.
From: Virgil on 7 Jul 2006 14:44 In article <1152283261.085370.83060(a)s16g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1152183314.431125.225030(a)m79g2000cwm.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > > > > To say n --> oo does not require that there exist any "oo". > > > > > > > > It is merely an abbreviation for "as n increases unboundedly beyond any > > > > given natural". > > > > > > But always remaining a natural itself, yeah. And the number of naturals > > > < n is always a finite number, so that it never does reach infinity > > > too. > > > > But it reaches past every finite value!!! > > There is always a finite value past every finite value. So we are and > remain sufficiently save within the domain of finit values. So one never reaches an end. That is what "infinite" for a sequence like the naturals means, "without end".
From: Virgil on 7 Jul 2006 14:59 In article <1152283547.949104.56780(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > This case would invalidate Cantor's diagonal proof. > > > And after than, Mueckenheim must show why Cantor's first proof is invalid > > No problem, that is easy to do. Thousands have tried, but none have succeeded, just as tens of thousands have tried to invalidate his second proof but none have succeeded. > > > and also why Cantor's proof that no set surjects to its power set is > > invalid. > > That has been done already. By whom? Certainly not by anyone so dim as "mueckenh". > > > And then Mueckenheim can take on all those many other proofs of > > the uncountability of the reals. > > > > Only when he has finished all of that can Mueckenheim claim that the > > Cantor theorem is invalid. > > There are no further proofs but only some variants of three proofs > mentioned. Even the second and the third are fairly similar. "Mueckenh" is quite wrong about the existence of other proofs. Though AFAIK Cantor only produced 2, others have produced others. > > Regards, WM
From: Dik T. Winter on 7 Jul 2006 21:48
In article <pTsU8Z33pqrEFw3G(a)212648.invalid> David Hartley <me9(a)privacy.net> writes: .... > (WM's fallacy is his claim that the (order-preserving) indexation by the > naturals carries over to the limit.) Indeed. The most basic standard fallacy he uses. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |