On 5/20/2022 6:03 AM, WM wrote:
> Jim Burns schrieb
> am Donnerstag, 19. Mai 2022 um 21:45:49 UTC+2:
>> On 5/19/2022 2:32 PM, WM wrote:
>>> All definable fractions get indexed.
>>> Most fractions don 't get indexed.
>>> Dark fractions.
>>
>> All definable fractions get indexed ==
>> all fractions get indexed.
>> No dark fractions.
>
> Wo you are not willing to analyse the facts.
"Calculemus." -- Gottfried Leibniz
> Enumerate all positive fractions.
These fractions:
m/n such thatm and n are _definable_ ==
only finite-many '+1' from 0 ==
for each BEFORE and AFTER =< m
and each BEFORE and AFTER =< n,
some j ends BEFORE and j+1 begins AFTER.
> 1/1, 1/2, 1/3, 1/4, ...
> 2/1, 2/2, 2/3, 2/4, ...
> 3/1, 3/2, 3/3, 3/4, ...
> 4/1, 4/2, 4/3, 4/4, ...
> 5/1, 5/2, 5/3, 5/4, ...
> ...
>
> We check the number of indexes by bijecting them
> with the fractions of the first column (we could use
> every other column or line as well). When applying
> the indexes for indexing fractions such that
> m/n gets the index
> k = (m + n - 1)(m + n - 2)/2 + m
Calculemus.
For k = (m + n - 1)(m + n - 2)/2 + m
for each BEFORE and AFTER =< k,
some j ends BEFORE and j+1 begins AFTER,
because...
For each BEFORE and AFTER =< m
and each BEFORE and AFTER =< n,
some j ends BEFORE and j+1 begins AFTER.
...etc, etc, etc...
...and so,
for each BEFORE and AFTER =< (m+n-1)
and each BEFORE and AFTER =< (m+n-2),
some j ends BEFORE and j+1 begins AFTER.
For each BEFORE and AFTER =< (m+n-1)*(m+n-2),
some j ends BEFORE and j+1 begins AFTER.
because
for each BEFORE and AFTER =< (m+n-1)
and each BEFORE and AFTER =< (m+n-2),
some j ends BEFORE and j+1 begins AFTER.
and,
otherwise, there are contradictions.
| Assume OTHERWISE[1].
| Assume,
| for each BEFORE and AFTER =< (m+n-1)
| and each BEFORE and AFTER =< (m+n-2),
| some j ends BEFORE and j+1 begins AFTER,
| but
| NOT, for each BEFORE and AFTER =< (m+n-1)*(m+n-2),
| some j ends BEFORE and j+1 begins AFTER.
|
| There is a _first_ p₁+1 =< (m+n-2) such that
| NOT, for each BEFORE and AFTER =< (m+n-1)*(p₁+1),
| some j ends BEFORE and j+1 begins AFTER,
| AND it is
| TRUE, for each BEFORE and AFTER =< (m+n-1)*p₁,
| some j ends BEFORE and j+1 begins AFTER.
|
| By definition of '*',
| (m+n-1)*(p₁+1) = (m+n-1)*p₁+(m+n-1)
|
| However,
| for each BEFORE and AFTER =< (m+n-1)*p₁+(m+n-1),
| some j ends BEFORE and j+1 begins AFTER,
| because
| for each BEFORE and AFTER =< (m+n-1)*p₁,
| for each BEFORE and AFTER =< (m+n-1),
| some j ends BEFORE and j+1 begins AFTER,
| and,
| otherwise, there are contradictions.
|
|| Assume OTHERWISE[2].
|| Assume,
|| for each BEFORE and AFTER =< (m+n-1)*p₁
|| and each BEFORE and AFTER =< (m+n-1),
|| some j ends BEFORE and j+1 begins AFTER,
|| but
|| NOT, for each BEFORE and AFTER =< (m+n-1)*p₁+(m+n-1),
|| some j ends BEFORE and j+1 begins AFTER.
||
|| There is a _first_ s₁+1 =< (m+n-1) such that
|| NOT, for each BEFORE and AFTER =< (m+n-1)*p₁+(s₁+1),
|| some j ends BEFORE and j+1 begins AFTER,
|| AND it is
|| TRUE, for each BEFORE and AFTER =< (m+n-1)*p₁+s₁,
|| some j ends BEFORE and j+1 begins AFTER.
||
|| By definition of '+',
|| (m+n-1)*p₁+(s₁+1) = ((m+n-1)*p₁+s₁)+1
||
|| However,
|| for each BEFORE and AFTER =< ((m+n-1)*p₁+s₁)+1
|| some j ends BEFORE and j+1 begins AFTER,
|| because
|| for each BEFOREless and AFTERless =< (m+n-1)*p₁+s₁,
|| some j ends BEFOREless and j+1 begins AFTERless
||
|| For each BEFORE and AFTER =< ((m+n-1)*p₁+s₁)+1
||
|| either (i)
|| BEFORE = BEFOREless
|| AFTER = AFTERless∪{((m+n-1)*p₁+s₁)+1}
|| and the j which ends BEFOREless ends BEFORE
|| and the j+1 which begins AFTERless begins AFTER
||
|| or (ii)
|| BEFORE = {all =< (m+n-1)*p₁+s₁}
|| AFTER = {((m+n-1)*p₁+s₁)+1}
|| and the j which ends BEFORE = (m+n-1)*p₁+s₁
|| and the j+1 which begins AFTER = ((m+n-1)*p₁+s₁)+1
||
|| CONTRADICTION[2]:
|| NOT, for each BEFORE and AFTER =< (m+n-1)*p₁+(s₁+1),
|| some j ends BEFORE and j+1 begins AFTER,
|| AND it is
|| TRUE, for each BEFORE and AFTER =< (m+n-1)*p₁+(s₁+1),
|| some j ends BEFORE and j+1 begins AFTER.
|
| Therefore,
| for each BEFORE and AFTER =< (m+n-1)*p₁+(m+n-1),
| some j ends BEFORE and j+1 begins AFTER.
| and,
| for each BEFORE and AFTER =< (m+n-1)*(p₁+1),
| some j ends BEFORE and j+1 begins AFTER.
|
| CONTRADICTION[1]:
| NOT, for each BEFORE and AFTER =< (m+n-1)*(p₁+1),
| some j ends BEFORE and j+1 begins AFTER,
| AND it is
| TRUE, for each BEFORE and AFTER =< (m+n-1)*(p₁+1),
| some j ends BEFORE and j+1 begins AFTER.
Therefore,
for each BEFORE and AFTER =< (m+n-1)*(m+n-2),
some j ends BEFORE and j+1 begins AFTER.
...etc, etc, etc...
Therefore,
for k = (m + n - 1)(m + n - 2)/2 + m
for each BEFORE and AFTER =< k,
some j ends BEFORE and j+1 begins AFTER.
Therefore,
k/1 is one of "these fractions",
k is _definable_
k is only finite-many '+1' from 0
> with the resulting sequence
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> then the integer fractions have to supply these indexes.
...which we just now saw that they do.
> They are stripped off these indexes.
> The number of not indexed fractions remains constant
For each indexed fraction,
the number of _indexed_ fractions after it
remains constant and infinite.
For k = (m + n - 1)(m + n - 2)/2 + m
there are no _not-indexed_ fractions.
Infinity is not a reallyreallyreallyreallyreallyreally
large number. It is a different kind of thing.
> although all definable fractions get indexes.
All definable fractions == all fractions.
All fractions get indexes.
> It remains constant in all infinitely many cases
> because every applied index is taken from
> an indexed fraction.
>
> What is difficult to understand here?
My best guess is that you don't understand that
infinity is not a reallyreallyreallyreallyreallyreally
large number. It is a different kind of thing.