Full Idea
Gödel proved that (if set theory is consistent) we cannot refute the continuum hypothesis, and Cohen proved that (if set theory is consistent) we cannot prove it either.
Clarification
The hypothesis is that the numbers contain gaps
Gist of Idea
If set theory is consistent, we cannot refute or prove the Continuum Hypothesis
Source
report of Kurt Gödel (What is Cantor's Continuum Problem? [1964]) by William D. Hart - The Evolution of Logic 10
Book Reference
Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.273