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

On 12/29/24 5:53 AM, WM wrote:

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 %.

But the union of ALL the FISONs, the whole INFINITE set, gets all of them.

>

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.

NBo, you "Naive" (which you call straight) mathematics just can't handle this sort of math, and explodes into smithereens when you try to use it for this.
The problem is the upper limit for the ratio of a FINITE set of fusions to the whole set of Natural Numbers is *0*, as the finite sets is immesurably small compared to the infinite set.
But, once you allow the unioning of an INFINITE set of FUSIONs, that lets you get to 100%.

 Regards, WM