Re: Replacement of Cardinality

On 8/25/2024 3:41 PM, WM wrote:

Le 24/08/2024 à 22:16, Jim Burns a écrit :

Many of us are more familiar with necklaces which
have some minimum size of bead (or of whatever).

>
The unit fractions have no minimum distance
but more than nothing.

Positive real numbers have no minimum positive
but are each positive.
A minimum is one of what it's a minimum of.
0 is not the minimum of the positive real numbers.
There aren't ever two greatest.lower.bounds.
If
a greatest.lower.bound is NOT
one of what it's greatest.lower.bound of,
then
there is NO OTHER greatest.lower.bound which IS
one of what it's greatest.lower.bound of,
and
there is no minimum of them:
no minimum exists, visibleᵂᴹ or darkᵂᴹ.

For rationals,
there is a greatest lower bound of distances
between different rationals,
but,
for a minimum distance to exist,
the greatest lower bound needs to be a distance.

>
For a distance to exist in every case
mathematics is sufficicient:
∀n ∈ ℕ: 1/n - 1/(n+1) > 0 .

For each n ∈ ℕ:
1/(n+1) is a counter.example disproving the claim
that 1/n is the smallest unit.fraction:
1/n - 1/(n+1) > 0

The greatest lower bound of distances is 0,

>
The distances between all unit fractions are positive.
The GLB is to comfort blind persons who cannot think.

Some sets have a greatest.lower.bound not.in it.
A set with a greatest.lower.bound not.in it
doesn't have a second greatest.lower.bound in it.
doesn't have a minimum.
doesn't have a lower end.
doesn't have all its non.empty subsets 2.ended.
is an infinite set.
'Bye, Bob.