Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 15.11.2024 11:43, Mikko wrote:

On 2024-11-14 10:34:52 +0000, WM said:

No. Covering by intervals is completely independent of their individuality and therefore of their order.

 Translated intervals are not the same as the original ones. Not only their
order but also their positions can be different as demonstrated by your
example and mine, too.

If the do not cover the whole figure in their initial order, then they cannot do so in any other order.

 

Therefore you can either believe in set theory or in geometry. Both contradict each other.

 Geometry cannot contradict set theory because there is nothing both
could say. But this discussion is about set theory so geometry is not
relevant.

There is something both could say: Set theory claims that the intervals are enough to cover every rational number by a midpoint.. i.e., to cover ℝ+ infinitely often. Geometry denies this.

By shuffling the same set of intervals which do not cover ℝ+ in the initial configuration, it is impossible to cover more. That's geometry.

 So what part of ℝ+ is not covered by my J'?

Since according to Cantor's formula the smaller parts of ℝ+ are frequently covered, in the larger parts much gets uncovered. Every definable rational is covered. That is called potential infinity.
Regards, WM