continuum hypothesis

It was proved by Georg Cantor that the power set of the natural numbers is cardinally larger than the set of natural numbers themselves, i.e., there is no surjection of the natural numbers onto their power set. Cantor hypothesized that there is no ‘intermediate’ cardinality between that of the natural numbers and that of their power set— that every infinite set is either the ‘size’ of the natural numbers, or at least as big as the continuum. (Note: the power set of the natural numbers has the same cardinality as the set of real numbers)

This hypothesis, which Cantor believed was true but which he couldn't prove, became known as the Continuum Hypothesis (often abbreviated CH)

The statement of the hypothesis can be generalized to the statement that there is no intermediate cardinality between any set and its power set. This is called the Generalized Continuum Hypothesis (GCH).

In the 20th century Kurt Gödel and Paul Cohen proved results that together show that the GCH is independent of the other common axioms of set theory. That is, our common formalization of set theory is not strong enough to decide this question.