An uncountable countable set
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From: Virgil on 6 Jul 2006 14:41 In article <1152191768.157605.15960(a)b68g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Now, the complete > set of transpositions is of order type omega. And it maintains the > well-order by index while it achieves well-order by size. If this is the set of rationals "mueckenh" claims is being reordered, "mueckenh" is claiming to have found a smallest rational number!!! But to see why his scheme fails: Suppose we have a well-ordering of the rationals, "mueckenh"'s scheme of reordering by size the first two, then the first 3, then the first 4 and so on overlook the fact that the "number" which are left unreordered after each stage never diminishes, so his iterations make no progress at all.
From: Virgil on 6 Jul 2006 14:45 In article <1152191944.782522.136800(a)j8g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > Since any line and the following line both act on the same elements, > > > > they cannot be applied simultaneously, and they do not "commute", but > > > > must be applied sequentially. > > > > > > What has commutation to do with this proof? > > > > Absence of commutativity, which is the case with certain transpositions > > and sequences of transpostions, means that they must be applied in > > sequence and not simultaneously as "mueckenh"'s theory requires. > > They are not "applied" at all but are given in zero time. If transpositions are not applied sequentially, then their effect is often undefined, since altering that sequence can alter their effect. Learn some group theory before pontificating on how it works.
From: Virgil on 6 Jul 2006 14:53 In article <1152192189.464866.10470(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > Cantor's "diagonal has no problems of validity, only problems of > > comprehesibility. "Mueckenh", for example, hasn't a clue. > > Are there in fact people who can believe that anybody was unable to > understand Cantor's arguing? Was it such a problem for you to > comprehend it? I have no problem with Cantor's first proof, nor his second, of the theorem that the reals are uncountable, nor his proof that there are no surjections from any set to its power set. I can easily believe that there are people who have trouble with each of these proofs, as I have seen evidence to that effect. "Mueckenh" has not addressed Cantor's first proof, so I do not know if he understands it, though considering the way he misunderstands the second, I would doubt it.
From: Dik T. Winter on 6 Jul 2006 19:05 In article <vmhjr2-4E845C.11135406072006(a)news.usenetmonster.com> Virgil <vmhjr2(a)comcast.net> writes: > In article <1152180620.790783.87220(a)75g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > Virgil schrieb: > > > > > Also, in ZF, ZFC and NBG, any countable union of countable sets can be > > > proved countable, so "mueckenh"'s attempts to claim otherwise must fail. > > > > Feferman and Levy showed that one cannot prove that there is any > > non-denumerable set of real numbers which can be well ordered. > > In ZFC and NBG, every set is in theory well-orderable. > > > Moreover, they also showed that the statement that the set of all real > > numbers is the union of a denumerable set of denumerable sets cannot be > > refuted. Yes, that was already written by Herman Rubin. It is consistent that the reals are a countable union of countable sets. And I think they even had a model where that was the case. > Where did they show this, and for what system did they show this??? > Absent any such reference, I doubt that it is valid in ZFC, and ZFC is > demonstably as consistent as ZF. You need choice (and I think countable choice is sufficient). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 6 Jul 2006 19:13
In article <vmhjr2-085B41.11335706072006(a)news.usenetmonster.com> Virgil <vmhjr2(a)comcast.net> writes: > In article <1152182749.926627.310460(a)a14g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: .... > > Of course. But that does not exclude that these transposition can be > > executed and finished (if Cantor's list can be finished). > > But it prohibits them from being executed out of their prescribed order. > Since the order of execution induces a well ordering of the set of > transpositions, there would have to be a first transpostion producing > any given effect. This is (as has been noted alread) wrong. Consider the sequence of transpositions on N (where I use ordinal numbers for the elements): (1, 2)(2, 3)(3, 4)(4, 5)... This sequence places the number 0 further in the sequence of numbers at each step. If we were to define something like a limit on it, number 0 would be greater than any other natural number (in the imposed ordering), and so the ordered set would become: (1, 2, 3, ..., 0) so the ordinal number of the set is changed from w to w+1, but there is no first transposition that performs that change. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |