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

On 02.01.2025 19:19, Jim Burns wrote:

On 1/2/2025 6:31 AM, WM wrote:

On 31.12.2024 21:26, Jim Burns wrote:

On 12/31/2024 1:20 PM, WM wrote:

 

Every union of FISONs which stay below a certain threshold stays
below that threshold.
Note: Every FISON stays below 1 % of ℕ.

For each finite ⟦0,j⦆, there are more.than.#⟦0,j⦆.many finite ⟦0,k⦆

So it is for all definable natural numbers.

So it is for each finite.ordinal.

That is impossible, because all finite ordinals can be subtracted from ℕ
without infinitely many remaining. If you don't understand that, it is
useless to go on.
>

natural numbers which have only few successors because when they are
removed nothing remains ℕ \ {1, 2, 3, ...} = { } .

... the non.finite.ordinal natural.numbers.
We define natural.numbers to be only finite.ordinals, of which there
are more.than.any.finite.many.

Infinitely many can be removed without remainder. But only finitely many
can be defined by FISONs.