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

On 11/24/24 5:31 AM, WM wrote:

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.
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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.

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But it works.

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

But that isn't the requirement.
The requirement it to map from ALL Prime Natural Numbers to ALL Natural Numbers

 

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

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

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

Which isn't the interval in question.
Your funny-mental fallacy is that you think an infinite set can be thought of as just some finite set allowed to keep growning until it reaches infinity,
That is just the wrong model.

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so all the numbers get covered.

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No.

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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.

But finite sets aren't infinite sets, and don't act the same as them.

 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.
 

Nope, your idea of mathematics has contradictions in it that have exploded your mind and left just a "dark hole" behind. You can not just use finite mathematics on infinite sets.

Regards, WM