Re: How many different unit fractions are lessorequal than all unit fractions?

On 9/24/2024 4:37 PM, WM wrote:

On 24.09.2024 22:19, Jim Burns wrote:

On 9/24/2024 3:28 PM, WM wrote:

On 23.09.2024 19:58, Jim Burns wrote:

On 9/23/2024 8:57 AM, WM wrote:

a smallest unit.fraction, visibleᵂᴹ or darkᵂᴹ,
is gibberish.

>
But
the increase of NUF(x) from 0 to infinity
without intermediate steps is not gibberish?

 Anything which
can be reached by intermediate steps
is not infinite.
(You (WM) apparently mean something different.)
 Thus
increasing NUF(x) from 0 to infinity
WITH intermediate steps
is gibberish,

>
The only alternative would by
infinitely many unit fractions at one point.

The correct (different) alternative is:
infinitely.many unit.fractions
at one point per unit.fraction,
with infinitely.many points in all.
Although
no more than finitely.many points
can be stepped.through end.to.end
we don't require these points to do more than _exist_
More.than.finitely.many can _exist_

The only alternative would by
infinitely many unit fractions at one point.
That is not gibberish but wrong.

Each positive point δ, visibleᵂᴹ or darkᵂᴹ,
is undercut by a visibleᵂᴹ.unit.fraction
The only alternative is gibberish.
⎛ Assume δ is positive and not.undercut.
⎜
⎜ β is between
⎜ points.undercut > β and points.not.undercut < β
⎜
⎜ β ≥ δ > 0
⎜
⎜ 2⋅β > β
⎜ 2⋅β is undercut
⎜ 2⋅β > ⅟k ∈ ⅟ℕdef
⎜ ½⋅β > ¼⋅⅟k ∈ ⅟ℕdef
⎜ ½⋅β is undercut.
⎜
⎜ However,
⎜ ½⋅β < β
⎜ ½⋅β is not.undercut
⎜
⎜ "½⋅β is undercut and not.undercut"
⎝ is gibberish.

The only alternative would by
infinitely many unit fractions at one point.
That is not gibberish but wrong.

Each positive.point, visibleᵂᴹ or darkᵂᴹ,
is undercut by a visibleᵂᴹ.unit fraction
Each visibleᵂᴹ.unit.fraction
is undercut by more than.k unit.fractions,
at one point per unit.fraction,
where k is a countable.to number.
Each positive point, visibleᵂᴹ or darkᵂᴹ,
is undercut by more.than.finitely.many
(infinitely.many)
visibleᵂᴹ unit.fractions.
The only alternative is gibberish.

Of many suitable definitions of natural numbers,
one is:
they are well.ordered (subsets minimummed or empty)
they continue (have successors)
they are reached by a step (≠0 have predecessors)
>
The natural numbers are our Paradigm of Finite.
>
There is no first unreachable natural number.
By that and by its well.order,
there is no unreachable natural number.

>
The natural numbers n belonging to
the first infinitely many unit fractions 1/n, i.e.
there where NUF(x) increases at one point
from 0 to infinity,
cannot be distinguished in your opinion.
Thus they are unreachable.

1. Describe 'reachable'.
2. Observe that infinitely.many
(our 'infinitely.many', not what you think it is)
is not what that is, is not reachable.
⎛ A finite ordered set begins and ends, or it is empty.
⎝ Each of its subsets begins and ends, or it is empty.
In a finite ordered set
for each split,
the foresplit ends and, in one step more,
the hindsplit begins.
In a finite ordered set,
for each split,
the hindsplit _can be reached_
from the foresplit.
To insist that /1 can be reached from 0
unit.fraction by unit.fraction
is
to insist that the unit fractions are
finitely.many.
To insist they are is gibberish, and
it can be (has been) shown to be gibberish.

By that and by its well.order,
there is no unreachable natural number.
>
ω is not a natural number.
⎛ Each before ω can be reached.
⎝ Each which can be reached is before ω.
>
If ω-1 existed such that (ω-1)+1 = ω
then ω could be reached

 If ω-1 could be seen. But it cannot.

ω-1 is gibberish, seen or unseen.