Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 12.12.2024 20:06, FromTheRafters wrote:

WM was thinking very hard :

On 12.12.2024 12:30, FromTheRafters wrote:

WM wrote :

But if Cantor can apply all natural numbers as indices for his bijections, then all must leave the sequence of endsegments.  Then the sequence (E(k)) must end up empty.

 Impossible, since k and k+1 etcetera are natural numbers.

Then Cantor cannot get them out of the endsegments either. He claims to use all natural numbers as indices in mappings. For instance the famous Cantor-list would not prove anything uncountable iunless all natural numbers with no exception were indices of the lines.

 

And there must be a continuous staircase from E(k) to the empty set.

 A limit set or element does not have to be completely within a set or interval.

But all numbers remaining in endsegments are not usable for mappings.
Regards, WM