The problem was so important that German mathematician David Hilbert
placed it at the head of his famous list of 23 outstanding problems for the next
century, presented to the International Mathematical Congress in Paris in 1900.
Gödel emphatically believed the hypothesis to be false, but he did not prove it.
He did prove (in 1938) that the hypothesis was compatible with the Zermelo–
Fraenkel axioms for set theory. A quarter of a century later, Paul Cohen startled
Gödel and the logicians by proving that the continuum hypothesis could not be
deduced from the Zermelo–Fraenkel axioms. This is equivalent to showing the
axioms and the negation of the hypothesis is consistent. Combined with Gödel’s
1938 result, Cohen had shown that the continuum hypothesis was independent
of the rest of the axioms for set theory.
This state of affairs is similar in nature to the way the parallel postulate in
geometry (see page 108) is independent of Euclid’s other axioms. That discovery
resulted in a flowering of the non-Euclidean geometries which, amongst other
things, made possible the advancement of relativity theory by Einstein. In a
similar way, the continuum hypothesis can be accepted or rejected without
disturbing the other axioms for set theory. After Cohen’s pioneering result a
whole new field was created which attracted generations of mathematicians who
adopted the techniques he used in proving the independence of the continuum
hypothesis.

the condensed idea

Many treated as one