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

On 07.11.2024 01:51, Richard Damon wrote:

On 11/6/24 1:22 PM, WM wrote:

These other intervals also have irrational endpoints. Every point outside of an interval is next to some endpoint which is irrational.

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Or rational endpoints.

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No, the rationals are centres of their intervals.

 They can also be endpoints of intervals.

Not of their own intervals.

 

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There is no "next to" on the dense line

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Every positive point is nearer to zero than to any negative point.
Of -x and 0 the latter is next to any positive x.
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 But that isn't "next to".

It is next to when between a point and the interval no further point exists, like here:
Use the intervals J(n) = [n - 1/10, n + 1/10]. Without splitting or modifying them they can be translated and reordered, to cover the whole positive axis and every rational as a midpoint - if Cantor was right.
Regards, WM