The independence proof just described shows that CH is independent of ZFC. Further research has shown that CH is independent of all known large cardinal axioms in the context of ZFC. Moreover, it has been shown that the cardinality of the continuum can be any cardinal consistent with König's theorem. A result of Solovay, proved shortly after Cohen's result on the independence of the continuum hypothesis, shows that in any model of ZFC, if is a cardinal of uncountable cofinality, then there is a forcing extension in which . However, per König's theorem, it is not consistent to assume is or or any cardinal with cofinality .
The independence proof just described shows that CH is independent of ZFC. Further research has shown that CH is independent of all known large cardinal axioms in the context of ZFC. Moreover it has been shown that the cardinality of the continuum can be any cardinal consistent with König's theorem. A result of Solovay, proved shortly after Cohen's result on the independence of the continuum hypothesis, shows that in any model of ZFC, if is a cardinal of uncountable cofinality then there is a forcing extension in which . However, per König's theorem, it is not consistent to assume is or or any cardinal with cofinality .