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

On 15.12.2024 19:53, Jim Burns wrote:

On 12/15/2024 7:00 AM, WM wrote:

Unless you have changed whatᵂᴹ you (WM) mean,
to completeᵂᴹ a potentiallyᵂᴹ infinite set means
to insert an epilogue (presumably darkᵂᴹ) so that
set+epilogue is actuallyᵂᴹ infinite.

This epilogue is required to empty ℕ by |ℕ \ {1, 2, 3, ...}| = 0. All definable numbers fail: ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

(1) E(1), E(1)∩E(2), E(1)∩E(2)∩E(3), ...
loses all content.

 the set of common finite.ordinals is empty.

Fine.

 

By the law
(2) ∀k ∈ ℕ :
∩{E(1),E(2),...,E(k+1)} =
∩{E(1),E(2),...,E(k)}\{k}
the sequence gets empty one by one.

The limit set {}
⎛ holds all common finite.ordinals.
⎝ isn't in the sequence.

If all natural numbers are individually available for indexing then they are available for individually leaving the intersection.
Regards, WM