Re: An uncountable countable set

In article <1152193390.830398.326300@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
....
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.

If we assume that that list contains natural numbers in some unary notation
(as I think you do) than:

Or 0.111... is distinguished from all finitely large numbers of the

0.111... is not a natural number in unary notation. So it is inherently
different from all elements of the list.

"To be different" means for all unary representations of n
An : 0.111... - n =/= 0

How do you propose to define that subtraction when 0.111... is not a
natural number?

Every digit can be indexed by a natural number means that there is no
digit, which is different from all unary representations of natural
numbers.

Right.

By construction of 0.111... this means further that every
digit at position n and all digits at smaller positions can be indexed.
This means there does not exist any digit which differs from all unary
representations.
not forall n: 0.111... - n =/= 0

But that is a wrong conclusion. Let's call 0.111... (as a sequence of
digits) K. And let's define K[i] is the i'th digit of K and An[i] the
i'th digit of An. The following statement is correct:
There is no i such that for all n K[i] != An[i] (1)

En : 0.111... - n = 0.

And that is wrong, because that means:
There is an n such that for all i K[i] = An[i]. (2)
pray tell us under what logical reasoning you transform (1) to (2).

And to simplify it, take A0 = 0.10, A1 = 0.01, K = 0.11. (1) is
satisfied, (2) is not satisfied.
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