Re: An uncountable countable set

In article <1152133914.661066.91950@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

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.

OR, as is actually the case, the endless sequence of 1's fraction
0.111... is distinct from every finite truncation of it AND every digit
of it CAN be indexed by a natural number.

So that the actuality is that both of "mueckenh"'s alternatives are
wrong simultaneoulsy.
.

Relevant Pages

  • Re: An uncountable countable set
    ... Indexing the digit number n is equivalent to covering the ... string up to digit number n. ... Even a finite string of an infinite set of finite ... naturals does exist, and by Cantor, the cardinality is aleph-0. ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... > Okay, I don;t think I understood what you were saying. ... > naturals, you might as well call it something, I suppose. ... In TO's system of "whole numbers", there is a most significant digit and ... > infinite unending string of bits, even if most are generally ignored. ...
    (sci.math)
  • Re: Galileos Paradox and the Project of the Reals
    ... the positive integers which I defined the other day. ... A finite real, then, may be defined as any finite natural, or any number between any two finite naturals on the real line, by subdivion of the unit interval. ... We can also construct a linear enumeration of the reals using powers as I suggested with the H-riffic numbers. ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... >>> rightmost zero in an unending sequence of ever more rightward ... >> naturals, you might as well call it something, I suppose. ... > In TO's system of "whole numbers", there is a most significant digit and ... >> infinite unending string of bits, even if most are generally ignored. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, with each increment in the domain, the range increases by some amount. ... you had said that the existence ... Like it's the number of unit intervals, and the number of reals in the unit interval. ... You are using a form of infinite induction, making a claim for an infinite set based on all finite initial segments of it. ...
    (sci.math)