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

On 1/9/2025 3:23 PM, WM wrote:

On 09.01.2025 20:46, Jim Burns wrote:

On 1/9/2025 1:25 PM, WM wrote:

On 09.01.2025 18:52, Jim Burns wrote:

Sets do not change.

>
But the terms (E(n))
differ from their successors by one number.

>
Each end.segment is larger than
any ordinal smaller.than fuller.by.one sets.

>
What makes it so?

ℕ holds each
ordinal smaller.than a fuller.by.one ordinal,
thus
ℕ is larger than
any ordinal smaller.than fuller.by.one sets.
The ordinals are well.ordered.
  If
ANY ordinal begins an end.segment which
  is NOT larger than
  any ordinal smaller.than fuller.by.one sets,
  then
a FIRST ordinal begins an end.segment which
  is NOT larger than
  any ordinal smaller.than fuller.by.one sets.
  and
its NEXT.BEFORE.FIRST ordinal begins an end.segment which
  IS larger than
  any ordinal smaller.than fuller.by.one sets.
However,
that situation cannot arise.
For each finite ordinal,
there is a fuller.by.one ordinal which is finite.
The NEXT.BEFORE.FIRST is larger.than 1,2,3,...
The (supposed) FIRST is larger than 0,1,2,3,...
But that contradicts FIRST being FIRST.
Thus, because it's contradictory, there is no FIRST.
Thus, because ordinals are well.ordered,
there are NONE AT ALL which begin
an end.segment which is NOT larger than
  any ordinal smaller.than fuller.by.one sets.