Re: Replacement of Cardinality

On 8/13/2024 10:21 AM, WM wrote:

Le 12/08/2024 à 19:44, Jim Burns a écrit :

On 8/12/2024 9:50 AM, WM wrote:

Le 11/08/2024 à 19:56, Jim Burns a écrit :

 

What causes an exception: nₓ ∈ ℕ:
⅟nₓ > 0 without ⅟(nₓ+1) > 0 ?

>
The end of the positive axis.

>
There is no ⅟nₓ before the end of the positive axis
without ⅟(nₓ+1) before the end of the positive axis.

>
You cannot see it. It is dark.

 

The function NUF(x) is a step-function. It can increase from 0 at x = 0 to greater values,

 0 isn't a unit.fraction.
 

either in a step of size 1
or in a step of size more than 1.
But increase by more than 1 is excluded by
the gaps between unit fractions.

 0 isn't a unit.fraction.
 

(Note the universal quantifier there,

 quantified over unit.fractions
 

according to which never –
in no limit and in no accumulation point –
two unit fractions occupy the same point x.)

 Each unit fraction has GLB β > 0
which other unit.fractions are at least as far as.
 0 is not a unit.fraction.
0 does not have GLB β > 0
which unit fractions are at least as far as.
Otherwise,
½⋅β is lower.bound and not.lower.bound.
 

Therefore
the step size can only be 1,

 ...at a unit.fraction.
0 isn't a unit.fraction.

 

resulting in a real coordinate x with NUF(x) = 1.

 INVNUF(1) > ⅟ ⌊⅟INVNUF(1) +1⌋ > ⅟ ⌊⅟INVNUF(1) +2⌋
 NUF(INVNUF(1)) > 1
Contradiction.