Sujet : Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
De : mikko.levanto (at) *nospam* iki.fi (Mikko)
Groupes : sci.logicDate : 12. Nov 2024, 14:45:53
Autres entêtes
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Message-ID : <vgvm6h$1k8co$1@dont-email.me>
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On 2024-11-11 11:33:52 +0000, WM said:
On 11.11.2024 12:15, Mikko wrote:
On 2024-11-10 10:54:02 +0000, WM said:
The measure of all intervals J(n) = [n - √2/10, n + √2/10] is smaller than 3.
Maybe, maybe not, depending on what is all n.
It is, as usual, all natural numbers.
The measure of the interval J(n) is √2/5, which is roghly 0,28.
Agreed, I said smaller than 3.
The measure of the set of all those intervals is infinite.
The density or relative measure is √2/5. By shifting intervals this density cannot grow. Therefore the intervals cannot cover the real axis, let alone infinitely often.
Between the intervals J(n) and (Jn+1) there are infinitely many rational
and irrational numbers but no hatural numbers.
Therefore infinitely many natural numbers must become centres of intervals, if Cantor was right. But that is impossible.
Where did Cantor say otherwise?
-- Mikko
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