Sujet : Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
De : james.g.burns (at) *nospam* att.net (Jim Burns)
Groupes : sci.logicDate : 24. Nov 2024, 22:33:52
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <67d9867b-2614-4475-975c-938bafca5c00@att.net>
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 11/24/2024 3:56 PM, WM wrote:
On 24.11.2024 21:17, Jim Burns wrote:
On 11/24/2024 2:42 PM, WM wrote:
On 24.11.2024 20:26, Jim Burns wrote:
What we mean by
|E(k)| ≤ |E(k+1)|
is that
there is a one.to.one function
from E(k) to E(k+1)
The successor operation, for example.
>
What I mean is the fact that
∀k ∈ ℕ: |E(k+1)| = |E(k)| - 1
whereas Cantor's ℵo is a very unsharp measure.
>
Finite cardinalities can change by 1.
>
Endsegmentes 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)
j < j+1 = |E(k)\E(k+j+1)|
For each end.segment.set E(k)
for each finite cardinality j
j is not the cardinality of E(k)
j < |E(k)\E(k+j+1)| ≤ E(k)
j ≠ |E(k)|
That contradicts |E(k)| being any finite cardinal j
any cardinal j which can change by 1
Which contradicts |E(k)| changing by 1
…