Re: Incompleteness of Cantor's enumeration of the rational numbers

On 24.11.2024 03:22, Richard Damon wrote:

On 11/23/24 4:11 PM, WM wrote:

Cover the unit intervals of prime numbers by red hats. Then shift the red hats so that all unit intervals of the positive real axis get red hats.
>

And you can, as
the red hat on the number 2, can be moved to the number 1
the red hat on the number 3, can be moved to the number 2
the red hat on the number 5, can be moved to the number 3

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A very naive recipe.

 But it works.

It fails in every step to cover the interval (0, n] with hats taken from this interval.

Yes, for every n that belongs to a tiny initial segment.

 No, for EVERY n.
 Show one that it doesn't work for!

The complete covering fails in every interval (0, n] with hats taken from this interval.

 

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so all the numbers get covered.

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No.

 WHich one doesn't.

Almost all. The reason is simple mathematics. For every interval (0, n] the relative covering is 1/10, independent of how the hats are shifted. This cannot be remedied in the infinite limit because outside of all finite intervals (0, n] there are no further hats available.
On the other hand, we cannot find a first n that cannot be covered by a hat. This dilemma cannot be resolved by negating one of the two facts. It can only be solved by dark numbers.
Regards, WM