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

On 2024-11-04 18:12:55 +0000, WM said:

On 04.11.2024 18:49, Mikko wrote:

On 2024-11-04 10:47:19 +0000, WM said:
 

On 04.11.2024 11:31, Mikko wrote:

On 2024-11-04 09:55:24 +0000, WM said:
 

On 03.11.2024 23:18, Jim Burns wrote:
 

There aren't any neighboring intervals.
Any two intervals have intervals between them.

 That is wrong. The measure outside of the intervals is infinite. Hence there exists a point outside. This point has two nearest intervals

 No, it hasn't.

 In geometry it has.

 This discussion is about numbers, not geometry.

 Geometry is only another language for the same thing.

Another language is an unnecessary complication that only reeasls
an intent to deceive.

Between that point an an interval there are rational
numbers and therefore other intervals

 I said the nearest one. There is no interval nearer than the nearest one.

 There is no nearesst one. There is always a nearer one.

 Nonsense.

No, the meaning is clear. Of course, because some intevals overlap,
you should have specified what exacly you mean by "nearer". But as
ε shriks the overlappings disappear and the distance between any
two intevals approaches the distance between their centers we may
define distance between the intervals as the distance between their
endpoints even wne ε > 0.

Therefore the
point has no nearest interval.

 That is an unfounded assertions and therefore not accepted.

 It is not unfounded.

 Of course it is. It is the purest nonsense.

That you don't even try to support your clam to support your claim
indicates that you don't really believe it. Cantor's results are
conclusions of proofs and you have not shown any error in the proofs.
You are free to deny one of more of the assumptions that constitue
the foudations of the results but you havn't. Even if you will that
will not make the results unfounded. It only means that you want to
use a different foundation. Whether you can find one that you like
is your problem.
--
Mikko