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

On 1/3/2025 12:00 PM, WM wrote:

On 03.01.2025 17:51, Jim Burns wrote:

On 1/3/2025 3:39 AM, WM wrote:

all finite ordinals can be subtracted from ℕ
without infinitely many remaining.

>
If

>
No if.

If set A is smaller.than set B
then A∪{a} ≠ A is smaller.than B∪{b} ≠ B
Thus,
if ⟦0,j⦆ is smaller than ⟦0,j+1⦆
then ⟦0,j⦆∪⦃j⦄ is smaller than ⟦0,j+1⦆∪⦃j+1⦄
⟦0,j+1⦆ = ⟦0,j⦆∪⦃j⦄
⟦0,j+2⦆ = ⟦0,j+1⦆∪⦃j+1⦄
If #⟦0,j⦆ < #⟦0,j+1⦆
then #⟦0,j⦆ < #⟦0,j+1⦆ < #⟦0,j+2⦆
If ⟦0,j⦆ is
smaller.than fuller.by.one (finite)
then ⟦0,j⦆ is not.largest
smaller.than fuller.by.one (finite).
There is no largest
smaller.than fuller.by.one (finite).
Darknessᵂᴹ and visibilityᵂᴹ do not affect
the existence of claims in a finite sequence,
each claim of which is true.or.not.first.false.

Is it true or false?

All finite.ordinals removed from
the set of each and only finite.ordinals
leaves the empty set.
There is no largest finite.ordinal.