Re: Replacement of Cardinality

On 8/21/2024 6:43 AM, WM wrote:

Le 20/08/2024 à 22:05, Jim Burns a écrit :

No min.⅟ℕᵈᵉᶠ exists.

>
The reason is potential infinity.

Is potentialᵂᴹ.infinity mathematicsᵂᴹ?
Judging from your edicts on this topic,
the answer looks to be both 'yes' and 'no'.
We are agreed, though, that
no min.⅟ℕᵈᵉᶠ exists.
It follows that ⅟ℕᵈᵉᶠ has properties that
the unit.fractionsᵈᵉᶠ > x > 0
don't have.
Each non.empty subset S ⊆ ⅟ℕᵈᵉᶠ(x,1]
holds two ends, max.S min.S
Contrary to that,
⅟ℕᵈᵉᶠ as a whole has
a non.empty subset which does not hold two ends,
itself, ⅟ℕᵈᵉᶠ

But dark unit fractions are assumed to be
actually infinite.

As we move along, I find out more about
what darkᵂᴹ unit.fractions and darkᵂᴹ numbers are NOT.
We agree that darkᵂᴹ unit.fractions are NOT
positive lower.bounds of the visibleᵂᴹ unit.fractions,
NOT zero, NOT negative, NOT larger than
any visibleᵂᴹ unit.fraction.
NOT lower.bound and NOT not.lower.bound.
I don't see anything left for the darkᵂᴹ to NOT not.be.

[...] actually infinite.

Consider finiteⁿᵒᵗᐧᵂᴹ sets as
sets orderable with each non.{}.subset 2.ended.
Each subset of a finiteⁿᵒᵗᐧᵂᴹ set is finiteⁿᵒᵗᐧᵂᴹ.
Each superset of an infiniteⁿᵒᵗᐧᵂᴹ set is infiniteⁿᵒᵗᐧᵂᴹ.
In order to know those, we only need to know
what is a subset, what is a superset.
⎛ Each non.{}.subset of
⎜ a non.{}.subset of a finiteⁿᵒᵗᐧᵂᴹ set
⎜  is
⎜ a non.{}.subset of a finiteⁿᵒᵗᐧᵂᴹ set
⎜  and thus is
⎜ 2.ended.
⎜
⎜ Each non.{}.subset of a finiteⁿᵒᵗᐧᵂᴹ set
⎜  is
⎝ finiteⁿᵒᵗᐧᵂᴹ.
⎛ Each superset of
⎜ a set orderable with a non.2.ended subset
⎜  is
⎜ a set orderable with a non.2.ended subset
⎜
⎜ Each superset of an infiniteⁿᵒᵗᐧᵂᴹ set
⎜  is
⎝ infiniteⁿᵒᵗᐧᵂᴹ.
It seems to be that
potentiallyᵂᴹ.infinite sets and
infiniteⁿᵒᵗᐧᵂᴹ sets are the same.
Only potentiallyᵂᴹ.infinite sets have
potentiallyᵂᴹ.infinite subsets.