On Mon Oct 2 12:38:06 2023 bassam karzeddin wrote:

On Monday, October 2, 2023 at 8:09:44 PM UTC+3, Jim Burns wrote:

On 10/1/2023 10:07 PM, bassam karzeddin wrote:

On Saturday, August 8, 2020
at 11:27:08 AM UTC+3, bassam karzeddin wrote:

the discontinuity of the so-called
real number in modern mathematics
1) What is the greatest real number that is
strictly less than sqrt(3)?
it doesn't exist

2) What is the least real number that is
strictly greater than sqrt(3)
It doesn't exist,

hence real numbers are
isolated and discontinuous
and they are certainly discrete numbers

You are using "isolated" as though it means
the opposite of what it actually means.

| In mathematics,
| a point x is called
| an isolated point of a subset S
| (in a topological space X)
| if
| x is an element of S and
| there exists a neighborhood of x that
| does not contain any other points of S.
|
https://en.wikipedia.org/wiki/Isolated_point

However,
the real numbers are only described as
"constructible" numbers as
distinct existing distances on
the real number line

Each real number is distinct.
No real number is isolated.

If a real-number function jumps,
then at least one point exists at which
that function is not continuous.

That is why,
for each split F H of the rationals
an irrational u \ is between F and H
F < u < H

Consider two irrationals u v
between two splits F H F H of
F < u < H F < v < H

One of (i) (ii) (iii) is true.

(i)
H and F overlap
F < u < H F < v < H
H F
p/q H F
u < p/q < v
u < v

(ii)
H and F overlap
F < v < H F < u < H
H F
p/q H F
v < p/q < u

v < u

(iii)
Neither H and F nor H and F overlap
H F = H F =
F = F < u < H = H
F = F < v < H = H
(|u-v| > 0)
u = v

Points between splits of rationals
are ordered according to
whether a rational is between them, or not.

[...] FOR SURE

The entire confusions actually are sourced from the so-called set theory where it was refuted by my modest self & few other distinguished members of sci.math where the mainstream academic sheeples usually don't accept anything rigorously proven &

globally published freely from public disgusting sources like sci.math

However, the same confusion comes basically from the unlimited density of "constructible " numbers

Where between any distinct locations on the real number line & no matter however close to each others or far from each others one might think, there are always & perpetually an endless number of only other constructible numbers where it is absolutely

impossible to determine all of them

Where no other types of real numbers other than the constructible numbers ever exist, FOR SURE

Any bigginer in number theory can immediately well-understand what I simply announce as perpetual truth about real numbers

Bassam karzeddin

*
Bassam: I believe that you know much more about mathematics than Archimedes Pluotonium.

earle
*

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