Continuum hypothesis
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 used to prove the continuum hypothesis, either. For this, Cohen was awarded the Fields Medal.