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

On 05/05/2024 10:59 AM, Jim Burns wrote:

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

Le 03/05/2024 à 21:32, Jim Burns a écrit :

On 5/3/2024 9:57 AM, WM wrote:

>

It is the idea that infinite sets are fixed
such that no element can be added or removed.
It is the idea that
the natural numbers reach immediately from 0 until ω

>

Each number which CAN be counted.to from 0
is before
ℵ₀.many numbers which CAN be counted.to from 0

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

>
1.
 From each number n which CAN be counted.to
the numbers which CAN be counted.to reach immediately
more.than.1 from n to n+2
>
None of these immediate more.than.1 numbers remain
after every number n which CAN be counted.to.
>
>
The meaning of (1.) depends upon
'n' NOT being
the true name of any number which CAN be counted.to,
the way that "Rumpelstiltskin" is the true name of
a certain straw.into.gold.spinner.
>
An analogy better than "name" is "pronoun"
'n' is more like a pronoun than a name.
"It is a natural number", etc.
>
Variable.names are a big improvement over pronouns
because, in every natural language I'm aware of,
there are no more than a handful of pronouns,
used with many handfuls of referents, and
their distinct referents are kept distinct
by context, AKA, figuring.it.out.
Even if the figuring.out doesn't fail, a lot of work.
>
The expression
| x < y and y < z implies x < z
|
is a big improvement in clarity over
a paragraph of muddle with three pronouns.
x y z act like pronouns, though.
>
2.
 From each number n which CAN be counted.to
the numbers which CAN be counted.to reach immediately
more.than.2 from n to n+3
>
None of these immediate more.than.3 numbers remain
after every number n which CAN be counted.to.
>
3.
 From each number n which CAN be counted.to
the numbers which CAN be counted.to reach immediately
more.than.3 from n to n+4
>
None of these immediate more.than.3 numbers remain
after every number n which CAN be counted.to.
>
...
>
k.
 From each number n which CAN be counted.to
the numbers which CAN be counted.to reach immediately
more.than.k from n to n+k+1
>
None of these immediate more.than.k numbers remain
after every number n which CAN be counted.to.
>
...
>

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

>
 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.
>
>

How about disambiguating quantifiers so that something
like the universal quantifier gets disambiguated to
reflect a for-any/for-each/for-every/for-all when it's
so that things like the Sorites/Heap or transfer principle
apply.
Similarly the existential quantifier is often to be
disambiguated "exists", "exists-unique", "exists-plural",
these kinds of things.
English as a language has a rich variety of copulas.