Re: universal quantifiers, because g⤨(g⁻¹(x)) = g(y) [1/2] Re: how

On 05/06/2024 01:16 PM, Jim Burns wrote:

On 5/6/2024 2:59 PM, WM wrote:

Le 05/05/2024 à 19:59, Jim Burns a écrit :

On 5/3/2024 4:31 PM, WM wrote:

>

If all could be counted to,
they would not remain after every counted number.

>
Of those which CAN be counted.to,
each CAN be counted to.
>

From each number n which CAN be counted.to
for each number k which CAN be counted.to
more.than.k numbers which CAN be counted.to
can be reached immediately
from n to n+k+1
>
None of these immediate more.than.k numbers remain
after every number n which CAN be counted.to.

>
Nevertheless almost all, namely ℵo, remain.

>
Of those which CAN be counted.to,
each CAN be counted to,
and
none remain after all which CAN be counted.to,  and
each has its own immediate followers  such that
  each follower
    CAN be counted.to  and
    does NOT remain after all which CAN be counted.to
  and
  they are more.than.1.many
>
    "Infinite" does not mean "humongous".
>
Of those which CAN be counted.to,
each CAN be counted to,
and
none remain after all which CAN be counted.to,  and
each has its own immediate followers  such that
  each follower
    CAN be counted.to  and
    does NOT remain after all which CAN be counted.to
  and
  they are more.than.2.many
>
Of those which CAN be counted.to,
each CAN be counted to,
and
none remain after all which CAN be counted.to,  and
each has its own immediate followers  such that
  each follower
    CAN be counted.to  and
    does NOT remain after all which CAN be counted.to
  and
  they are more.than.3.many
>
...
>
Of those which CAN be counted.to,
each CAN be counted to,
and
none remain after all which CAN be counted.to,  and
each has its own immediate followers  such that
  each follower
    CAN be counted.to  and
    does NOT remain after all which CAN be counted.to
  and
  they are more.than.k.many
>
...
>
Of those which CAN be counted.to,
each CAN be counted to,
and
none remain after all which CAN be counted.to,  and
each has its own immediate followers  such that
  each follower
    CAN be counted.to  and
    does NOT remain after all which CAN be counted.to
  and
  they are more.than.any.countable.to
  they are ℵ₀.many
>
>

Well, first of all, it's after pondering that there
is quantifier comprehension artifacts of the extra sort,
as of a set of all sets, order type of ordinals, a universe,
set of sets that don't contain themself, sets that contain
themselves, and so on.
Then, English affords "any, "each, "every, "all".
The -any means for example that "it's always a fragment".
So in this sense the usual universal quantifier is for-each.
Then, for-each, means usual comprehension, as if an enumeration,
or a choice function, each.
Then, for-every, means as a sort of comprehension, where it
so establishes itself again, any differently than -each,
when -each and -every implies both none missing and all gained.
Then, "for-all", sort of is for that what is so "for-each"
and "for-every" is so, "for-all", as for the multitude as
for the individual.
Then, I sort of ran out of words, "any", "each", "every", "all",
then that seems their sort of ordering, about comprehension,
in quantification, in the universals, of each particular.
About sums it up, ....