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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Sun, 15 Dec 2024 20:16:29 +0100
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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