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

On 28.12.2024 19:48, Jim Burns wrote:

On 12/28/2024 11:50 AM, WM wrote:

A good example is the set of FISONs.
Every FISON contains only
a negligible quantity of natural numbers.

 Yes.
 

A generous estimation is:
Every FISON contains
less than 1 % of all natural numbers.
There is no FISON that contains more than 1 %.

 Yes.
 

Therefore the union of all FISONs contains less than 1 % of all natural numbers.

 No.
The union of FISONS contains
100% of all members of any FISON.

Of course. But that is less than 1 % of all natural numbers.

Still,
each FISON contains
less than 1% of the union of FISONs.

That is nonsense. Note that when two or 10 or many FISONs miss 99 %, then their union misses 99 %.

 

Outside of the union of FISONs are almost all natural numbers.

 No.

You violate straight mathematics and logic like with your Bob. There is a upper threshold for all FISONs (99 %). This cannot be surpassed by their union.
Regards, WM