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

on 11/20/2024, Ross Finlayson supposed :

On 11/20/2024 12:05 PM, FromTheRafters wrote:

WM wrote on 11/20/2024 :

On 20.11.2024 15:14, FromTheRafters wrote:

WM formulated on Wednesday :

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It does not make it wrong, but it unmasks it at imprecise. That's
why I don't like it. We can do better.

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It works well enough.

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Really? Then you can answer the following questions:
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Let every unit interval after a natural number on the real axis be
coloured white with exception of the powers of 2 which are coloured
black. Is it possible to shift the black intervals so that the whole
real axis becomes black?

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No, of course not.
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Or: Let every unit interval after a natural number on the real axis be
coloured as above with exception of the intervals after the odd prime
numbers which are coloured red. Is it possible to shift the red
intervals so that the whole real axis becomes red?

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No, of course not.
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What colour has the real axis after you have solved both tasks?

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Depending on the order of the tasks. I think half red or half black.

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Well you have to reference academic reference and describe "supertask"
besides "asymptotics" about where "the asymptotic density of black or
red respectively is 1 in the limit", that you point to "supertask"
instead of mumbling like it's not already considered by proper minds,
not just ketonic neck-flap gaspers of having failures altogether
of any sort of related-rates problems.
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This is mathematics: humor is irrelevant, and so is what
anybody "thinks", or, "feels".
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It only matters what "is", and there's a language of it,
so use it. (Or lose it.)
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Good sir

If painted black and then red, it will be red. If painted red and then black, it will be black. These are real intervals, and as such I assume real powers of two. In both scenarios, none of the negative real axis is at all affected.