My understanding is this: Gödel proved his earth shattering result in the 1930s. Then the dust settled and mathematicians thought it might be “academic” and that mathematics might not come tumbling down after all. Then in the 1960s Paul Cohen proves that the continuum hypothesis is undecidable. That is, we will never know whether an infinite set can have a cardinality between the natural numbers and the real numbers.

My understanding comes from Keith Devlin’s wonderful book “Mathematics: A new golden age”.