Is there an infinity between the whole numbers and the points on a line?

Nobody knows, and the strange thing is that the standard rules of mathematics cannot tell us. The question is the continuum hypothesis: is there a set whose cardinality is strictly between that of the integers and the real numbers, the points of a line? Cantor advanced the hypothesis in 1878, saying there is no such set. Settling it was the first of Hilbert's problems presented in 1900.

The answer turned out to be neither yes nor no. The continuum hypothesis is independent of ZFC, the standard axioms of set theory: those axioms can neither prove nor disprove it, so either the hypothesis or its negation can be added as a new axiom, with the resulting theory consistent if and only if ZFC is consistent. The independence was proved in 1963 by Paul Cohen, complementing work by Kurt Gödel in 1940.

The proof came in two halves. Gödel showed that CH cannot be disproved from ZFC. Cohen showed it cannot be proven from the ZFC axioms, and to do it he developed the method of forcing, which starts from a model of ZF in which CH holds and builds a larger one in which it fails. Cohen was awarded the Fields Medal in 1966 for his proof.

What does that mean for the truth of CH? Mathematicians disagree. Gödel was a Platonist, so he believed the question was meaningful and that CH is false. Shelah pictures many possible set theories, all conforming to ZFC. Skolem argued as early as 1923 that the conception of set is not specific enough to decide. The hypothesis remains an active topic of research, not a closed case.