Sujet : Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
De : wolfgang.mueckenheim (at) *nospam* tha.de (WM)
Groupes : sci.logicDate : 25. Nov 2024, 14:52:24
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <vi1vep$2pjuo$1@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
User-Agent : Mozilla Thunderbird
On 24.11.2024 22:33, Jim Burns wrote:
On 11/24/2024 3:56 PM, WM wrote:
Endsegments can change by 1 element.
Therefore their number of elements can change by 1.
Yes,
each end.segment.set can change by 1 element
However,
for each end.segment.set E(k)
for each finite cardinality j
there is a larger.than.j subset E(k)\E(k+j+1)
Finite cardinalities belong to dark endsegments. Not every dark endsegment has a successor.
For each end.segment.set E(k)
for each finite cardinality j
j is not the cardinality of E(k)
The endsegments only can have an empty intersection if there are endsegments with 3, 2, 1, 0 elements.
|∩{E(k) : k ∈ ℕ_def}| = ℵ₀
∀k ∈ ℕ : E(k+1) = E(k) \ {k}
∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
∩{E(k) : k ∈ ℕ} = { }
∀k ∈ ℕ: |E(k+1)| = |E(k)| - 1
Regards, WM