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

On 17.12.2024 00:55, Richard Damon wrote:

On 12/16/24 3:55 AM, WM wrote:

On 15.12.2024 22:14, Richard Damon wrote:

On 12/15/24 2:29 PM, WM wrote:

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Next is a geometric property, in particular since the average distance of intervals is infinitely larger than their sizes.

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Not sure where you get that the "average" distance of intervals is infinitely larger than ther sizes.

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The accumulated size of all intervals is less than 3 over the infinite length. Hence there is at least one location with a ratio oo between distance to the interval and length of the interval. Start there with the cursor. It will hit one next interval. Crash.

 Since none of the gaps are infinte, and none of the intervals are of 0 size, there is no "infinite" ratio of any gap to any interval.

There is no upper bound for the ratio between distance and size of intervals. This excludes the density of intervals. This excludes covering of all rationals by intervals. This excludes a bijection between natural numbers and rational numbers.
Regards, WM