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

On 10/5/2024 5:43 AM, WM wrote:

On 05.10.2024 10:46, Moebius wrote:

a quantifier shift is NOT reliable
und wird daher in der Mathematik tunlichst vermieden (und nicht nur dort).

⎛ a quantifier shift is NOT reliable
⎜ and is therefore avoided in mathematics
⎝ (and not only there).

[In] many cases it is correct.

"Many cases" is insufficient when
the argument requires "all cases".
Finite beings can learn _some facts_ true of
each one of infinitely.many.
The technique developed uses
finitely.many finite.length claims which are
true in all cases of concern
(and silent in cases not.of.concern).
It isn't _known_ which
  individual an all.cases claim refers to.
It probably could be said that asking which
  is a question for which no answer exists.
Nonetheless,
it _is_ known that such a claim must be true.
There exists no reading in which it is false.
For a claim true in no more than _many_ cases,
the all.cases.claims technique doesn't work.
If someone has a technique using many.cases.claims,
they should explain _why_ their technique works.
This has already been done,
by literally centuries of work,
for the all.cases.claims technique.

For instance
if every definable natural number has
 ℵo natural successors,
then there are ℵo natural numbers larger than
 all definable natural numbers.

All.cases.claims:
⎛ A set of (definable) natural numbers has
⎜ a minimum or is empty
⎜
⎜ A (definable) natural number has
⎜ predecessor.natural or is 0
⎜
⎜ A (definable) natural number has
⎜ a successor.natural.
⎜
⎜ You: definable natural number
⎝ We: natural number
In all cases _of concern_ those are true claims.
There are other cases, in some of which they're false,
but we aren't concerned with those here and now.
When we are concerned, we'll make different claims.

if every definable natural number has
 ℵo natural successors,
then there are ℵo natural numbers larger than
 all definable natural numbers.

Each (definable) natural number has
ℵ₀.many (definable) natural numbers after (>) it.
There are 0.many (definable) natural numbers which
bounds (≥) all (definable) natural numbers.
Of many cases in which quantifier shift is correct,
here and now is NOT one of them.
Because quantifier shift is sometimes wrong,
quantifier shift is never used.
It is reliability which gives finite beings
the ability to know about infinity.

They are dark however and cannot be specified.

Either they are
well.ordered, successored and (≠0)predecessored.
or they are
not of concern here and now.
That is how finite beings explore infinity.