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

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

On 06.11.2024 12:46, Richard Damon wrote:

On 11/5/24 7:34 AM, WM wrote:

On 05.11.2024 13:03, Richard Damon wrote:

On 11/5/24 3:45 AM, WM wrote:

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I describe that a point between two finite intervals has two finite intervals around it. Even this simple conclusion must be denied by the believers in matheology.

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If you mean that between to finite length intervals, which include there endpoints, and a point that is between those two intervals, then YES there exist two other intervals between the point and those two intervals.

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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.

 No, the rationals are centres of their intervals.

They can also be endpoints of intervals.

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There is no "next to" on the dense line

 Every positive point is nearer to zero than to any negative point.
Of -x and 0 the latter is next to any positive x.
 

But that isn't "next to".
There is no point that is NEXT TO any other point, as if X and Y were "next to" each other, then the point (X+Y)/2 would be between them, and thus they are not next to each other.
By the rules of arithmetic of Rational and Reals, that expresion is ALWAYS a value, and will ALWAYS be between the two numbers.

Regards, WM