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

On 1/3/2025 2:48 PM, WM wrote:

On 03.01.2025 19:46, Jim Burns wrote:

There is no largest finite.ordinal.

>
There is no definable largest finite ordinal.

There is no largest
ordinal smaller.than fuller.by.one ordinals.
The ordinals smaller.than fuller.by.one ordinals
are finiteⁿᵒᵗᐧᵂᴹ.
There is no largest finiteⁿᵒᵗᐧᵂᴹ.ordinal.
----
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 (finiteⁿᵒᵗᐧᵂᴹ)
then ⟦0,j⦆ is not.largest
smaller.than fuller.by.one (finiteⁿᵒᵗᐧᵂᴹ).
There is no largest
smaller.than fuller.by.one (finiteⁿᵒᵗᐧᵂᴹ).