An uncountable countable set
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From: mueckenh on 5 Jul 2006 17:03 Virgil schrieb: > > > > 0.1 > > > > 0.11 > > > > 0.111 > > > > ... > > > > 0.111...1 > > > > ... > > > > > > > > But in this list the number 0.111... is not contained. Hence not all of > > > > its digits can be identified. > > > > > > You have just proved that your example supports Cantor's theorem by > > > producing a number, 0.111..., not in your own list. > > > > All digits of a number must be indexed by natural numbers. Otherwise > > they cannot be identified. All digits which can be identified are > > pesent in the list which contains all unary representations of naturlal > > numbers. > > > That presumes that the set of naturalas is finite, which in ZFC and NBG > is false. It presumes only that *every* natural is finite. 0.111... is not member of the sequence 0.1 0.11 0.111 .... That is a fact, proven by analysis, and the only possible reason for a difference is the number of digits. > > And "mueckenh" has not presented any consistent system of axioms in > which it is true. > > > > 0.111... does not belong to this set. > > It does in ZFC and NBG in both of which N is infinite. So in ZFC and NBG the limit of a strictly monotonically increasing sequence is a term of that sequence? Interesting. Regards, WM
From: mueckenh on 5 Jul 2006 17:05 Virgil schrieb: > > Try to map R on the set {1, 2, 3, K} and you will see it. > > R surjects to {1,2,3,L} with no problem, when L is not "mueckenh"'s > circularly defined K. > > When one restricts the functions one is allowed to use to functions > which cannot exist, as "mueckenh" keeps doing, one canot do much math. Therefore one should not use such functions and sets to prove fundamental theorems. Regards, WM
From: mueckenh on 5 Jul 2006 17:07 Virgil schrieb: > In article <1151961303.406453.116760(a)75g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > Consider a surjective mapping from N on the set of even positive > > numbers & K. > > Only if we consider a function g and K = K(f). > > > Or consider a surjective mapping from R on the set of rational numbers > > & K. > > Only if we consider a function g and K = K(f). > > > Or Consider a surjective mapping from R on the set {1, 2, 3, K}. > > Only if we consider a function g and K = K(f). > > > > > > But without readjusting f by g. That is forbidden in case of Cantor's > > diagonal as well as in our case. > > Then "mueckenh" does not understand either 'Cantor's diagonal' or your > case. > > In the Cantor "diagonal" on has a choice of uncountably many diagonals > rules to choose from, any one of which is sufficient to prove the > incompleteness of the list. Wrong. There are no uncountable sets. Compare the binary tree. Every element shows it. | o / \ o o / \ / \ Regards, WM
From: mueckenh on 5 Jul 2006 17:09 Virgil schrieb: > > Wrong. The scheme is a fixed matrix and each line can be applied > > simultaneously. > > > > (1,2), (3,4), (5,6), ... > > 1, (2,3), (4,5), (6,7), ... > > (1,2), (3,4), (5,6), ... > > 1, (2,3), (4,5), (6,7), ... > > (1,2), (3,4), (5,6), ... > > 1, (2,3), (4,5), (6,7), ... > > 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? Regards, WM
From: mueckenh on 5 Jul 2006 17:11
Virgil schrieb: > > > > 0.111.. is not in the list, then it must have more digits than can be > > > > indexed (and hence, can exist). > > > > > > They are satisfactorily indexed by the infinite set of finite natural > > > numbers, N. > > > > You just proved that there are infinitely digit positions which are not > > indexed by natural numbers (*all* of which are given in the list). > Having infinitely many does not require that any one of them be > infinitely large. > > And each of the infinitely many naturals is only finitely large. Either the diagonal number 0.111... is not distinguished from all finitely large numbers of the list 0. 0.1 0.11 0.111 .... then Cantor's proof fails. Or 0.111... is distinguished from all finitely large numbers of the list 0.1 0.11 0.111 .... then the digits of 0.111... cannot all be indexed by natural numbers. Either or! That means there is no actual infinity - neither nor. Regards, WM |