Re: Simple enough for every reader?

On 19.05.2025 15:57, Mikko wrote:

On 2025-05-18 12:20:47 +0000, WM said:
 

On 18.05.2025 12:30, Mikko wrote:

On 2025-05-17 15:00:33 +0000, WM said:
>

Are you aware of the fact that in
>
{1}
{1, 2}
{1, 2, 3}
...
{1, 2, 3, ..., n}
...
>
up to every n infinitely many natural numbers of the whole set
>
{1, 2, 3, ...}
>
are missing? Infinitely many of them will never be mentioned individually. They are dark.

>
For example, if we pick 5 for n we have
>
{1}
{1, 2}
{1, 2, 3}
{1, 2, 3, 4}
{1, 2, 3, 4, 5}
>
then 6 and infinitely many other numbers are missing. So numbers
6, and 7 are dark as are ingfinitely many other numbers.

 

Maybe for a 3-year old child. Doves can count to 7. Earthworms may
fail at 1 already.

 Many animals can differentiate quantities up to about 7. As far as
we know most of them needn't and can't count. They just see the
difference. Accurate determination of larger quantities may require
counting.
 None of which is relevant to may observation that if n = 5 then your
definition makes 6 dark.

If you have no idea of 6, it is dark for you. I you arbitrarily stop at 5 although you know 6, 5 is not dark for you.

 

It depends on the system. But important is that no system can get over the infinite gap of dark numbers.

 Why not? Cantor quite obviously gets over quite large infinities.

Yes, but not by counting them one by one. He jumps to ω and can from there go on. He could also go back ω-1, ω-2, ... . But there is no finite initial segment reaching to ω/n for every n that has a finite initial segment. Let alone reaching to ω-n.
Regards, WM