An uncountable countable set
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From: mueckenh on 16 Aug 2006 06:43 Dik T. Winter schrieb: > In article <1155640559.355146.166090(a)74g2000cwt.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > Says who? You state that an infinite union of finite sets is finite. > > > I ask you for a quote or a proof, and you refrain to give some. Are you > > > not able to either prove that or give a quote? > > > > It is the definition of a natural number that it is a (positive) finite > > number. > > What is the relation with your statement that "an infinite union of > finite sets is finit"? Where is the *proof* of that statement? Are > you not able to either prove that or give a quote? The stair case is my proof that a union of infinitely many 1's gives an infinite set. The representation of infinitely many natural numbers by the stairs requires infinitely many stairs. Infinitely many stairs require infinite height. You disagree. You state there is no stair of infinite height, so you state that an infinite union of finite sets (1's) is not infinite. That is the proof. Regards, WM
From: mueckenh on 16 Aug 2006 06:45 mike4ty4(a)yahoo.com schrieb: > "mueck...(a)rz.fh-augsburg.de " wrote: > > An uncountable countable set > > > > There is no bijective mapping f : |N --> M, > > where M contains the set of all finite subsets of |N > > and, in addition, the set K = {k e |N : k /e f(k)} of all natural > > numbers k which are mapped on subsets not containing k. > > > > > > This shows M to be uncountable. > > > > > > Regards, WM > > A set can't be both uncountable and countable at the same time Are you sure? The set of all constructible numbers including all real numbers in Cantor-lists and all of their diagonal numbers is countable. Nevertheless most people assert that the construction of a diagonal number would show the uncountability of this countable set of constructible numbers. Regards, WM
From: Dik T. Winter on 16 Aug 2006 08:49 In article <1155724907.014576.23510(a)74g2000cwt.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > 1 0.1 > > > 2 0.11 > > > 3 0.111 > > > ... > > > > > > which contains all digit positions which can be indexed - by > > > definition. > > > > > > 0.111... is not in the list. > > > > Indeed, it is not in the list. But K can be indexed. The problem you > > appear to have is that K does not have a largest index position. > > The list does not have a largest number either. There is no largest > index position which could be indexed by the numbers of the list. > Nevertheless the number 0.111... has positions which are larger than > all positions of numbers in the list. Otherwise 0.111... would be in > the list. I do not think I can parse this. You state "nevertheless the number 0.111... has positions which are larger than all positions of numbers in the list". If we define 0.111... = K that number such that for each p in N, the p-th digit is 1, and there are no other digit positions, how can your statement be true? Which position of K is larger than all positions in the list (note that the positions in the list *also* correspond to the natural number). -- 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 16 Aug 2006 08:51 In article <1155725007.690845.21360(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1155640559.355146.166090(a)74g2000cwt.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > > Says who? You state that an infinite union of finite sets is finite. > > > > I ask you for a quote or a proof, and you refrain to give some. Are you > > > > not able to either prove that or give a quote? > > > > > > It is the definition of a natural number that it is a (positive) finite > > > number. > > > > What is the relation with your statement that "an infinite union of > > finite sets is finit"? Where is the *proof* of that statement? Are > > you not able to either prove that or give a quote? > > The stair case is my proof that a union of infinitely many 1's gives an > infinite set. The representation of infinitely many natural numbers by > the stairs requires infinitely many stairs. Infinitely many stairs > require infinite height. > > You disagree. You state there is no stair of infinite height, so you > state that an infinite union of finite sets (1's) is not infinite. That > is the proof. If you always misrepresent what I state. No, that is *not* what I have stated. The stair has infinite height and width, but there is no element with infinite width. -- 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 16 Aug 2006 08:55
In article <1155725133.570350.44280(a)74g2000cwt.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > mike4ty4(a)yahoo.com schrieb: .... > The set of all constructible numbers including all real numbers in > Cantor-lists and all of their diagonal numbers is countable. > Nevertheless most people assert that the construction of a diagonal > number would show the uncountability of this countable set of > constructible numbers. But can the list of constructable numbers be constructed? That they are countable comes from other considerations. You use Turing machines and they are countable, but not all of them deliver a constructible numbers, namely those that do not halt do not deliver such a number. So constructing a list of constructible numbers is equivalent to solving the halting problem. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |