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

On 05.01.2025 18:39, Richard Damon wrote:

On 1/5/25 12:26 PM, WM wrote:

On 05.01.2025 13:47, Richard Damon wrote:

On 1/5/25 5:31 AM, WM wrote:

On 04.01.2025 11:59, joes wrote:

Am Sat, 04 Jan 2025 09:52:08 +0100 schrieb WM:

On 04.01.2025 05:06, Richard Damon wrote:

On 1/3/25 12:15 PM, WM wrote:

>

Every union of FISONs which stay below a certain threshold stays below
that threshold.

Every union of a finite number of FISONs is just an admssion that you
can't do the actual union of *ALL* FISONs.

For all FISONs:|ℕ \ {1, 2, 3, ..., n}| = ℵo.

But not for the union.

>
What should make the union larger than all FISONs?
>
Every union of FISONs which stay below a certain threshold stays belown that threshold.

>
Because you never actually USED *ALL* FISONs,

>
All FISONs are smaller than 1 % of |ℕ|.
Find a FISON {1, 2, 3, ..., n} such that {1, 2, 3, ..., 100n} is a superset of ℕ.

I never said there was,

You said that I never used all FISONs. But I do. All are insufficient.

but your claim doesn't match your conclusion, as the union of *ALL* the FISIONs will reach the size of the Natural Numbers, even though no finite subset reaches a measurable percentage of it.
 

How do they do it? Do one or more FISONs grow during the union process? (One would be sufficient.)
Regards, WM