Re: Replacement of Cardinality

On 8/4/2024 11:44 AM, WM wrote:

Le 03/08/2024 à 19:42, joes a écrit :

[...]

>
The reversed quantification is nonsense

...because from ∀x∃u to ∃u∀x is unreliable.
 From ∃u∀x to ∀x∃u  is reliable.
 From ∃u∃x to ∃x∃u  is reliable.
 From ∀u∀x to ∀x∀u  is reliable.
However,
from ∀x∃u to ∃u∀x is unreliable.
In some cases,
from ∀x∃u to ∃u∀x is not.only.unreliable,
it is incorrect.
'<' is a case in which it is incorrect.
 From ∀x∃u≠x:x<u  to  ¬∃u∀x≠u:x<u is reliable.

because
ℵ₀ unit fractions need ℵ₀*2^ℵ₀ points above zero.

Sets ordered so that all non.{}.subsets are 2.ended
are finite sets.
Sets ordered so that any non.{}.subset is non.2.ended
are infinite sets.
For each unit.fraction,
the set of unit.fractions below it is non.2.ended.
It is an infinite set.
For each unit.fraction,
each non.{}.subset of unit.fractions above it is 2.ended.
The unit.fractions above it are finitely.many.