Re: collective and individual removal

On 04/28/2025 07:04 AM, FromTheRafters wrote:

WM explained on 4/28/2025 :

We can remove collectively all terms from the harmonic series. Nothing
remains.
>
If we restrict the removal to individually definable terms, then an
infinity remains.
>
Is this remainder caused by the impossibility to define infinitely
many terms individually? No! It is sufficient to define one term,
namely the last one definable.

>
But there is no last one definable since they all are.
>

This is an irrefutable proof of the existence of dark numbers.

>
For some values of irrefutable.

It seems a reverie on "pair-wise union" versus "infinite union",
also called "the illative", or sometimes, "univalent in HoTT".
See, ZF only has pair-wise union, otherwise various antinomies
are readily derived, yet, completions are claimed to follow as
if they were.
Then, since there are various models of integers, it results
that the inductive set as axiomatized may be seen as a sort
of negative axiom as much as a positive axiom, since naive
comprehension would arrive at either and both of fragments
and extensions as existing, models of integers.
So, it's a non-standard setting, naturally, say.