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

On 11/7/24 4:28 AM, WM wrote:

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.

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They can also be endpoints of intervals.

 Not of their own intervals.

But they can define intervals that way.
You don't seem to understand what thosse intervals do

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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:

But that doesn't exist for closed intervals (intervals define with a point on their end).

 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.

I don't think you understand what Cantor was doing or trying to show.
But of course, you never seemed to have known what you were talking about.
IF you are talking about the thing I think you are, Cantor was just showing that between any two irrational numbers will be a rational number. (Not at its MID-point, just between them).

 Regards, WM