On Friday, February 4, 2022 at 5:16:31 AM UTC-5, WM wrote:
> zelos...@gmail.com schrieb am Donnerstag, 3. Februar 2022 um 08:25:38 UTC+1:
> > onsdag 2 februari 2022 kl. 10:27:37 UTC+1 skrev WM:
>
> > > 1, O, O, O, ...
> > > 2, O, O, O, ...
> > > 3, O, O, O, ...
> > > 4, O, O, O, ...
> > > ...
> > > 1/1, 1/2, 1/3, 1/4, ...
> > > 2/1, 2/2, 2/3, 2/4, ...
> > > 3/1, 3/2, 3/3, 3/4, ...
> > > 4/1, 4/2, 4/3, 4/4, ...
> > > ...
> No fraction can settle at a definable index in the first column and cause an empty place in the matrix. But the matrix must get empty, when all fractions will settle in the first column.
> > It is all meaningless garbage
> Try to understand it. Find a mistake.
>
> Regards, WM

For a free thinker such as you WM, I would think that you could see that the rational value as fundamental is a fraud.
Your choice to stay with the natural numbers and dodge real values seems quite reasonable to me now that I have been through the analysis.
To rely upon the division of natural values in order to arrive upon the continuum is far away from fundamental integrity. You expose the two dimensional nature of the rational number system above. No discussion of this dimensional quagmire exists within standard mathematics does it?

You at least expose that the rational numbers are too large; they are not efficient. They are wasting too much in their redundancies.
Worst of all, the idea that you could divide two natural values and suddenly, voila, you land in a continuum... all the while on a 2D grid. Yes, there is something to it, but fundamentally the continuum is not a grid like that. We know from working upon the continuum that numbers take a gray status. When we issue forth a serious value such as 2.01, and I suppose the issue of units here, say Volts, we see that the mathematician versus the engineer will take this value differently. As to who has the more accurate version: it is the engineer. The act of chasing digits is epsilon/delta thinking. Close enough is good enough is not the mathy way. Yet an integral over a hundred samples versus a thousand: either will do the job. The seeking of perfection in the numbers is acceptable and worthy, but has limits too. Clean theory ought to have superseded clean values. Instead two-signed morons win the day.

fredag 4 februari 2022 kl. 11:16:31 UTC+1 skrev WM:
> zelos...@gmail.com schrieb am Donnerstag, 3. Februar 2022 um 08:25:38 UTC+1:
> > onsdag 2 februari 2022 kl. 10:27:37 UTC+1 skrev WM:
>
> > > 1, O, O, O, ...
> > > 2, O, O, O, ...
> > > 3, O, O, O, ...
> > > 4, O, O, O, ...
> > > ...
> > > 1/1, 1/2, 1/3, 1/4, ...
> > > 2/1, 2/2, 2/3, 2/4, ...
> > > 3/1, 3/2, 3/3, 3/4, ...
> > > 4/1, 4/2, 4/3, 4/4, ...
> > > ...
> No fraction can settle at a definable index in the first column and cause an empty place in the matrix. But the matrix must get empty, when all fractions will settle in the first column.
> > It is all meaningless garbage
> Try to understand it. Find a mistake.
>
> Regards, WM
There are mistakes, we've pointed out it out you just refuse to understand