Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 12/15/2024 7:00 AM, WM wrote:

On 14.12.2024 23:04, Jim Burns wrote:

On 12/14/2024 5:26 AM, WM wrote:

the set of what remains unused, i.e.,
 of intersections of endsegments
 (1) E(1), E(1)∩E(2), E(1)∩E(2)∩E(3), ...
loses all content.
Then,
by the law
(2) ∀k ∈ ℕ :
∩{E(1),E(2),...,E(k+1)} =
∩{E(1),E(2),...,E(k)}\{k}
the content must become finite.

>

Explain your vision of the problem:

>
A finite member ⟦0,ψ⦆ of the (well.ordered) ordinals
is smaller.than its successor ⟦0,ψ⦆∪{ψ}
>
If ⟦0,ψ⦆ is smaller than its successor ⟦0,ψ⦆∪{ψ}
then ⟦0,ψ+1⦆ = ⟦0,ψ⦆∪{ψ} is smaller.than
its successor ⟦0,ψ+1⦆∪{ψ+1}
  which means
If ψ is finite, then ψ+1 is finite.
If ψ+1 is finite, then ψ+2 is finite.

>
Yes, that is
the potentially infinite collection of
definable numbers. But it explains nothing.

Unless you have changed whatᵂᴹ you (WM) mean,
an actuallyᵂᴹ infinite set is smaller.than
a fuller.by.one set, but
it contains a potentiallyᵂᴹ infinite subset, meaning
a subset not.smaller.than a fuller.by.one set.
Unless you have changed whatᵂᴹ you (WM) mean,
to completeᵂᴹ a potentiallyᵂᴹ infinite set means
to insert an epilogue (presumably darkᵂᴹ) so that
set+epilogue is actuallyᵂᴹ infinite.
Your (earlier, possibly.current) explanation of how
a smaller.than.fuller.by.one set can contain
a not.smaller.than.fuller.by.one subset
is that
the (undeniably one.to.one) identity map x ↦ x
doesn't work somehowᵂᴹ in the darkᵂᴹ appendix.
Georg Cantor and matheologians do not use
sets in which x ↦ x doesn't work somehowᵂᴹ.
They (we) use sets which do not change.
I include among our sets
sets which you (WM) say change, of which
you often also say that
they're only potentiallyᵂᴹ infinite.
But, no, all of our sets do not change.
Some of our (unchanging) sets are
not.smaller.than a fuller.by.one set.

Explain your vision of the problem:

Some of our (unchanging) sets are
not.smaller.than a fuller.by.one set.
That's not the problem.
The problem is that you (WM) don't accept that.
In order to avoid accepting that,
you (WM) try various work.arounds, for example,
by declaring that
x ↦ x doesn't work when that'd be inconvenient.
My attempts to resolve your problem (not our problem)
have tried to explain why our way is what it is,
and have pointed out internal conflicts in your way.
Your attempts justify
_your_ problem being _our_ problem
have mostly.to.all referred to
whatever you're talking about as though it's
whatever we're talking about.
That will never, ever, ever be successful because,
although you might be confused about what we mean,
we are not confused about what we mean.
⎛ But then, Wolfgang Mückenheim,
⎜ there is your evangelization of your students at
⎜ the Augsburg University of Applied Sciences,
⎜ students who have been raised to _trust_ their teachers.
⎜ You might have a small success there.
⎜ When you allow them to assume that
⎜ you and we are talking about the same thing,
⎜ you are lying, and betraying that trust.
⎜ Unfortunately, I doubt that lying to one's students is
⎝ _technically_ a crime. Technically.

ω is the first upper bound of finite ordinals.
If ψ < ω, then ψ < ψ+1 < ψ+2 ≤ ω
>
If ω-1 exists
then
ω-1 is last.before.ω
α < β < ω  ⇒  α ≠ ω-1
>
If ω-1 exists
then
ω-1 < (ω-1)+1 < (ω-1)+2 ≤ ω
ω-1 ≠ ω-1
>
Therefore,
ω-1 doesn't exist

>
Not as a definable number.

Not as any of the (well.ordered) ordinals.

That is common knowledge.
But you should not only say what not exists.

When I make claims about the (well.ordered) ordinals,
they aren't claims about not.the.(well.ordered).ordinals.

(1) E(1), E(1)∩E(2), E(1)∩E(2)∩E(3), ...
loses all content.

For end.segments of the finite.ordinals,
each finite.ordinal is lost,
no finite.ordinal is last,
no end.segment is finite,
the set of common finite.ordinals is empty.

By the law
(2) ∀k ∈ ℕ :
∩{E(1),E(2),...,E(k+1)} =
∩{E(1),E(2),...,E(k)}\{k}
the sequence gets empty one by one.

The sequence gets emptier one.by.one.
Each end.segment is larger.than
each set larger.than an emptier.by.one set.
Each end.segment isn't
a set larger.than an emptier.by.one set.
Each end.segment
⎛ can get emptier.by.one.
⎝ cannot get smaller.by.one.
The limit set {}
⎛ holds all common finite.ordinals.
⎝ isn't in the sequence.