Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 17.11.2024 09:55, Mikko wrote:

On 2024-11-16 19:42:22 +0000, WM said:
 

On 16.11.2024 10:21, Mikko wrote:

On 2024-11-15 12:00:43 +0000, WM said:
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On 15.11.2024 11:43, Mikko wrote:

On 2024-11-14 10:34:52 +0000, WM said:

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No. Covering by intervals is completely independent of their individuality and therefore of their order.

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Translated intervals are not the same as the original ones. Not only their
order but also their positions can be different as demonstrated by your
example and mine, too.

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If they do not cover the whole figure in their initial order, then they cannot do so in any other order.

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So you want to retract your claims that involve another order?

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My claim is the obvious truth that the intervals [n - 1/10, n + 1/10] in every order do not cover the positive real line, let alone infinitely often.
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So you regard invalid what you said on 2024-11-13 16:14:02 +0000:
 

The intervals J(n) = [n - 1/10, n + 1/10] cover the relative measure 1/5 of ℝ+. By translating them to match Cantor's intervals they cover ℝ+ infinitely often.

That is the claim of set theory.

This is impossible.

This is my claim.

Therefore set theorists must discard geometry.

 There you translate them so that not only their order but also their
positions change. But You also rejected my J' intervals without giving
any reason that does not reject your "translated" intervals.

Your J'(n) = (n/100 - 1/10, n/100 + 1/10) are 100 times more than mine.
For every reordering of a finite subset of my intervals J(n) the relative covering remains constant, namely 1/5.
The analytical limit proves that the constant sequence 1/5, 1/5, 1/5, ... has limit 1/5. This is the relative covering of the infinite set and of every reordering.
Regards, WM