Kurt Gödel showed that GCH is a consequence of ZF + V=L (the axiom that every set is constructible relative to the ordinals), and is therefore consistent with ZFC.
Gödel's work shows that GCH (a specific mathematical concept) follows from the axioms ZF + V=L, which asserts that all sets are constructible within the ordinal system. Therefore, GCH aligns with ZFC.