That escalated quickly now ChatGPT has also an argument
in favor that the intersection all end-segments is non-empty.
I guess the creativity of ChatGPT spans also truth. The
error comes possibly from a wrong interpretation of
end-segment, leaning towards segment, i.e. a closed
interval [a,b]. Woa! Thats a splendid blooper:
Q: The intersection of all end-segments cannot be empty,
which shows the existence of dark numbers. Is that right?
A: It sounds like you may be referring to a specific mathematical
concept or proof, but without more information, it is
difficult to know for certain.
In general, an "end-segment" can refer to a subset of the real numbers
that have an upper and lower bound. If the intersection of all end-segments
is non-empty, it would mean that there is at least one number that
is common to all end-segments.
It's important to note that intersection of all end-segments cannot be empty.
Since the set of real numbers are a union of all end-segments, so there must
be an overlap, otherwise the union will not be the real numbers set.
The intersection of all end-segments is the set of all its accumulation
points, which are the limit points of the end-segments.
The concept of "dark numbers" is not typically used in mathematics and
I'm not aware of any proof that refers to dark numbers in the way that
you described. If you could provide more information or context about
the specific proof or concept you're referring to, I would be happy to try
to help you understand it better.
I made a screenshot.