Re: Does the number of nines increase?

On 6/30/2024 11:15 AM, WM wrote:

Le 29/06/2024 à 23:18, joes a écrit :

Am Sat, 29 Jun 2024 17:01:10 +0000 schrieb WM:

Like "0.111..." it is a formula.
Formulas determine sequences.
The other way round is not possible.

>

Then all real numbers are formulas.

>
Yes, you got it!

Then there are more formulas than there are formulas.
Formulas which are
finite strings from a finite alphabet
can be assigned finite indices.
Each nonempty set of formulas holds
  a first.indexed member.
For each formula,
  there is a last.before indexed.formula and
  a first.after indexed.formula,
   except the very.first, which has only a first.after.
There is a sequence of nested intervals of
the real number line
such that
each interval contains later intervals,  and
its end.points are indexed.formulas, and
any interior indexed formulas are
  later than its end.points.
Any formula interior to all the intervals
is later than infinitely.many endpoints.
But there are no indexed.formulas
later than infinitely.many end.points
There is no formula interior to all the intervals.
However,
there is a point interior to all the intervals,
or else
the Intermediate Value Theorem fails, and
there exist functions 'continuous' at all points
which nevertheless jump.
So, there is a point there.
So, there is a formula there.
But there isn't an indexed.formula there.
So, all the indexed.formulas
(finite strings, finite alphabet)
aren't all the formulas.
Somehow.