Sujet : Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
De : wolfgang.mueckenheim (at) *nospam* tha.de (WM)
Groupes : sci.logicDate : 21. Nov 2024, 11:21:40
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <vhn1jk$jf6v$1@dont-email.me>
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User-Agent : Mozilla Thunderbird
On 21.11.2024 10:16, Mikko wrote:
On 2024-11-20 11:42:15 +0000, WM said:
The intervals before and after shifting are not different. Only their positions are.
The intervals are different. A shifted interval contains a different
set of numbers.
Consider this simplified argument. Let every unit interval after a natural number n which is divisible by 10 be coloured black: (10n, 10n+1]. All others are white. Is it possible to shift the black intervals so that the whole real axis becomes black?
No. Although there are infinitely many black intervals, the white intervals will remain in the majority. For every finite distance (0, 10n) the relative covering is precisely 1/10, whether or not the intervals have been moved or remain at their original sites. That means the function decribing this, 1/10, 1/10, 1/10, ... has limit 1/10. That is the quotient of the infinity of black intervals and the infinity of all intervals.
Regards, WM
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