Re: Simple enough for every reader?

On 05/18/2025 03:30 AM, 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.
>

If you start with finite ordinals as "sets that don't contain
themselves", then collect them as the "set of all sets that
don't contain themselves", yet not containing itself, then
Russell's paradox says that it would contain itself.
So, it would be extra-ordinary in the sense of Mirimanoff,
and reflect on Skolem's extensions and fragments, and
Cohen's forcing, quite simply.
Infinity, you know, it kind of is "infinity plus one".