The continuum hypothesis is a hypothesis that there is no set that is both bigger than that of the natural numbers and smaller than that of the real numbers. Georg Cantor stated this hypothesis in 1877.
There are infinitely many natural numbers, the cardinality of the set of natural numbers is infinite. This is also true for the set of real numbers, but there are more real numbers than natural numbers. We say that the natural numbers have infinite cardinality and the real numbers have infinite cardinality, but the cardinality of the real numbers is greater than the cardinality of the natural numbers.
This hypothesis is the first problem on the list of 23 problems David Hilbert published in 1900. Kurt Gödel showed in 1939, that the hypothesis cannot be falsified using Zermelo–Fraenkel set theory. The Zermelo–Fraenkel set theory is the set theory commonly used in mathematics. Paul Cohen showed in the 1960s that the Zermelo-Fraenkel set theory cannot be use to prove the continuum hypothesis, either. For this, Cohen was awarded the Fields Medal.