Full Idea
In 1938, Gödel showed that ZF plus the General Continuum Hypothesis is consistent if ZF is. Cohen showed that ZF and not-GCH is also consistent if ZF is, which finally shows that neither GCH nor ¬GCH can be proved from ZF itself.
Gist of Idea
The General Continuum Hypothesis and its negation are both consistent with ZF
Source
Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1217)
Book Reference
'From Kant to Hilbert: sourcebook Vol. 2', ed/tr. Ewald,William [OUP 1996], p.1217