Sujet : Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
De : wolfgang.mueckenheim (at) *nospam* tha.de (WM)
Groupes : sci.logicDate : 23. Nov 2024, 09:54:36
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <vhs58b$1krl6$2@dont-email.me>
References : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
User-Agent : Mozilla Thunderbird
On 22.11.2024 23:56, Jim Burns wrote:
On 11/22/2024 4:30 PM, WM wrote:
Remember:
The intersection of all endsegments is empty,
but the intersection of
endsegments which can be counted to
is infinite.
No one should "remember" that.
It is incorrect.
∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k), |E(k)| = ℵ₀
∩{E(1), E(2), ...} = { }.
Note that every endsegment loses only one number.
Finite cardinalities can lose one number.
For all endsegments:
∀k ∈ ℕ: |E(k+1)| = |E(k)| - 1
Therefore there must
exist infinitely many finite endsegments.
Regards, WM
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