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

On 05.12.2024 23:20, Jim Burns wrote:

On 12/5/2024 2:30 PM, WM wrote:

Your following exposition is lucid and clear. It was a pleasure for me to discuss it!

>
And it is the empty endsegment.

 Depending upon how 'end.segment' is defined,
{} either is or isn't an end.segment.
Consider the options.
 ⎛ With {} as an end.segment,
⎜ there are more.than.finite.many end.segments,
⎜ too many for any finite.cardinal to be
⎜ in common with all end segments.
⎜ And therefore,
⎜ the intersection of all holds no finite cardinal.

Yes.

⎜
⎜ With {} NOT as an end.segment,

all endsegments hold content.

⎜ there STILL are more.than.finite.many end.segments,

Not actually infinitely many however. If all endsegments have content, then not all natnumbers are indices, then the indices have an upper bound.

⎜ too many for any finite.cardinal to be
⎜ in common with all end segments.

Too many for all definable natnumbers. But by assumption of content not all natnumbers have become indices.

⎜ And therefore,
⎝ the intersection of all STILL holds no finite cardinal.

No definable finite cardinal.

The intersection of all non.empty.end.segments
of the finite.cardinals,
which are each infinite non.empty.end.segments,
is empty.

That is a wrong conclusion because inclusion monotony prevents an empty intersection of non-empty endsegments:
∀n ∈ ℕ: E(1)∩E(2)∩...∩E(n) = E(n)
Identical sequences have identical limits.

 Because,
⎛ for each finite.cardinal,
⎜ there are fewer finite.cardinals before it
⎝ than there are finite.cardinals after it.

That is true only for definable or accessible cardinals.
All cardinals however can be exhausted with no remainder:
"The infinite sequence thus defined has the peculiar property to contain the positive rational numbers completely, and each of them only once at a determined place." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]
None is missing,let alone a natural number.

Not because
( an end.segment is empty.

This argument is wrong if infinite bijections are assumed to exist. However "fewer finite.cardinals before it than there are finite.cardinals after" is the your only argument.
Regards, WM