An uncountable countable set
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From: mueckenh on 12 Jul 2006 11:38 Virgil schrieb: > In article <1152638895.480011.23630(a)h48g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > 0,111... is a sequence which does not represent a natural. > > > > And it cannot be completely indexed by natural numbers. > > If such an obviously well ordered set has digits not indexed by members > of N it must have a first such digit > > Which digit is that first non-indexed digit, "mueckenh"? > > Can't answer? It is the first 1 behind the zeros at the rights hand side. 0.111... - 0.111...1 = 0.000...0111... Regards, WM
From: mueckenh on 12 Jul 2006 11:40 Virgil schrieb: > In article <1152638595.246620.165700(a)p79g2000cwp.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > > > > > > > > It can only be proved > > > > after having counted the digits from 1 to n without leaving out a > > > > single one. > > > > > > Nonsense. The Cantor rule generates a digit to go in the nth decimal > > > place of the number being created without any reference to any other > > > decimal place. > > > > > You cannot identify any place without counting from 1 to that place. > > The axiom of infinity guarantees us all those "places" without counting > anything. But it does not guarantee what is existing there. In order to find out you must count. Regards, WM
From: mueckenh on 12 Jul 2006 11:43 Franziska Neugebauer schrieb: > > You are using "to index" simultaneously for two different meanings. The > first is "to index a specific position" the second is undefined. What > does "0.1111 is indexed by the list number 4 = 0.1111" exaclty mean? The unary number n indexes every position 1,2,3,...,n. > > >> > I say: What can be indexed by different list numbers can also be > >> > indexed by one alone. > >> > >> Meaning? Proof? > > > > The first 1 of 0.1111 can be indexed by 0.1, the second 1 of 0.1111 > > can be indexed by 2 = 0.11 etc. But all 1's of 0.1111 can be indexed > > by 4 = 0.1111 simultaneously. > > What does "being index simultaneously" mean? Can't see that. The unary number n indexes every position 1,2,3,...,n. > > >> > If you do not believe that, > >> > >> I do not know what you mean. > >> > >> > then you should be able to show an example where more > >> > than one list number is required. Of course it must be a finite > >> > example, because there are only finite list numbers. And note: all > >> > list numbers are unary representations of natural numbers. > >> > >> But not all unary sequences are representations of natural numbers. > >> 0,111... is a sequence which does not represent a natural. > > > > And it cannot be completely indexed by natural numbers. > > What does "completely indexed" mean? All digits are indexed. Regards, WM
From: Franziska Neugebauer on 12 Jul 2006 13:07 mueckenh(a)rz.fh-augsburg.de wrote: > > Franziska Neugebauer schrieb: >> >> You are using "to index" simultaneously for two different meanings. >> The first is "to index a specific position" the second is undefined. >> What does "0.1111 is indexed by the list number 4 = 0.1111" exaclty >> mean? > > The unary number n indexes every position 1,2,3,...,n. How exactly is that achieved? >> >> > I say: What can be indexed by different list numbers can also be >> >> > indexed by one alone. >> >> >> >> Meaning? Proof? >> > >> > The first 1 of 0.1111 can be indexed by 0.1, the second 1 of 0.1111 >> > can be indexed by 2 = 0.11 etc. But all 1's of 0.1111 can be >> > indexed by 4 = 0.1111 simultaneously. >> >> What does "being index simultaneously" mean? Can't see that. > > The unary number n indexes every position 1,2,3,...,n. How exactly is that achieved? >> >> > If you do not believe that, >> >> >> >> I do not know what you mean. >> >> >> >> > then you should be able to show an example where more >> >> > than one list number is required. Of course it must be a finite >> >> > example, because there are only finite list numbers. And note: >> >> > all list numbers are unary representations of natural numbers. >> >> >> >> But not all unary sequences are representations of natural >> >> numbers. 0,111... is a sequence which does not represent a >> >> natural. >> > >> > And it cannot be completely indexed by natural numbers. >> >> What does "completely indexed" mean? > > All digits are indexed. All positions *are* indexed. There is a bijection between the index set N and the figures a_i: ... i ... ^ | v ... a_i ... F. N. -- xyz
From: Virgil on 12 Jul 2006 13:20
In article <1152707645.284464.200140(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > you are arguing that 1/9 does not exist. > > 1/9 does exist. What does not exist is its representation 0.111... Odd how "mueckenh" just used what he says does not exist in order to say it does not exist. |