 Gödel hoped that on the basis of such large cardinal axioms, the most important open question of set theory could eventually be settled. This is the continuum problem. The continuum hypothesis was proposed by Cantor in the late nineteenth century. It states that there are no sets S which are too large for there to be a one-to-one correspondence between S and the natural numbers, but too small for there to exist a one-to-one correspondence between S and the real numbers. Despite strenuous efforts, all attempts to settle the continuum problem failed. Gödel came to suspect that the continuum hypothesis is independent of the accepted principles of set theory (ZFC). Around 1940, he managed to show that the continuum hypothesis is consistent with ZFC. A few decades later, Paul Cohen proved that the negation of the continuum hypothesis is also consistent with ZFC. Thus Gödel’s conjecture of the independence of the continuum hypothesis was eventually confirmed.