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

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

On 05.11.2024 04:12, Richard Damon wrote:

On 11/4/24 11:55 AM, WM wrote:

 

Cantor showed how to count the Rationals in a countable infinity.

 No. Then the real axis would have measure zero.

No, because the real axis is made of Real Numbers, not Rational Numbers.
Yes, the proportion of Real Numbers that are Rational is 0

>
He showed that the Reals Could not be counted, not even a finite length line of them.

 That is the nonsense believed by matheologians. He assumes a fixed set ℕ. But if Hilbert's hotel was real, then always a diagonal number could be enumerated by inserting it into the list at line no. 1.

A Fixed but COUNTABLY INFINITE set.
I don't think you understand what a diagonal proof is (I know you don't as you have shown you don't).
The fact that Hilbert shows we can "add" new entries into the infinite set of Natural Numbers by transforming them shows some of the properties of infinite sets that finite sets do not have.
This isn't showing a "contradiction", it is showing that infinite sets are diffferent than finite sets, and some of the propreties you are insisting on, just don't work, so if you insist on them, you can't have the infinite sets.
PERIOD.
Saying that infinite sets exist, and your properties still exist, is like claiming that 1 == 2.
Your logic system just blows up and creates your "darkness" in the void it leaves behind.

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Therefore the
point has no nearest interval.

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That is an unfounded assertions and therefore not accepted.

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No, it is a PROVEN statement, therefor true.

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It is proven in an inconsistent theory. I describe the true mathematics.

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No, you describe

 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.
 Regards, WM
 

You seem to be just babbling. 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.
The key here is you need to be using CLOSED intervals that contain their end points, and that just follows from the fact that the Real number line is dense, and between any two points is an infinite number of other points (and thus a finite interval).
Your concept of "adjacent" points is the problem case which you just showed can't exist, so apparently yours is the "mathology" that doesn't have a firm foundation.