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

On 12/2/2024 9:28 AM, WM wrote:

On 02.12.2024 12:53, FromTheRafters wrote:

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[...]

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Infinite endsegments contain an infinite set each,
infinitely many elements of which
 are in the intersection.

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Yes to:
⎛ regarding finite.cardinals,
⎜ for  each  end.segment   E(k)
⎜ there is a subset S such that
⎝ for each finite cardinal j, j < |S| ≤ |E(k)|
>
No to:
⛔⎛ regarding finite.cardinals,
⛔⎜ ⮣ there is a subset S such that ⮧
⛔⎜ ⮤ for  each  end.segment   E(k) ⮠
⛔⎝ for each finite cardinal j, j < |S| ≤ |E(k)|
>
A quantifier shift tells you (WM) what you (WM) _expect_
  but a quantifier shift is untrustworthy.
>

An empty intersection cannot come before
an empty endsegment has been produced by
losing one element at every step.

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No.
Because see below.
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E(1), E(2), E(3), ...
and
E(1), E(1)∩E(2), E(1)∩E(2)∩E(3), ...
are identical for every n and in the limit
because
E(1)∩E(2)∩...∩E(n) = E(n).

>
They are
identical COUNTER.EXAMPLES to what you expect.
>
----

An empty intersection cannot come before
an empty endsegment has been produced by
losing one element at every step.

>
No.
For the set of finite cardinals,
EVEN IF NO END.SEGMENT IS EMPTY,
  the intersection of all end segments is empty.
>
⎛ The set of finite.cardinals holds
⎜  only finite.cardinals.
⎜
⎜ Each finite.cardinal is finite.
⎜
⎜ For each finite.cardinal,
⎜  only finitely.many finite.cardinals are ≤ it.
⎜
⎜ For each finite.cardinal,
⎜  only end.segments which start ≤ it
⎜ hold it.
⎜
⎜ For each finite.cardinal,
⎝  only finitely.many end.segments hold it.
>
⎛ The set of finite cardinals holds
⎜  all finite.cardinals.
⎜
⎜ Each finite.cardinal is followed by
⎜  another finite.cardinal.
⎜
⎜ No finite.cardinal is last.
⎜
⎜ The set of finite cardinals has
⎜  a subset (itself) which is not two.ended.
⎜
⎜ The set of finite cardinals is infinite.
⎜
⎜ Each finite.cardinal starts an end.segment.
⎜
⎝ There are infinitely.many end.segments.
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⎛ For each finite.cardinal,
⎜  only finitely.many end.segments hold it.
⎜
⎜ There are infinitely.many end.segments.
⎜
⎜ For each finite.cardinal,
⎜  not all end.segments hold it.
⎜
⎜ For each finite.cardinal,
⎜  the intersection doesn't hold it
⎜
⎜ EVEN IF NO END.SEGMENT IS EMPTY,
⎜⎛ For each finite.cardinal,
⎜⎜  only finitely.many end.segments hold it.
⎜⎜ For each finite.cardinal,
⎜⎜  only finitely.many end.segments hold it.
⎜⎜ For each finite.cardinal,
⎜⎝  the intersection doesn't hold it
⎜
⎜ EVEN IF NO END.SEGMENT IS EMPTY,
⎝  the intersection of all end segments is empty.
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