Range of values for ##2^{\aleph_0}##

[WWGD]

TL;DR Summary

Possible values of cardinality in different models of ZFC+ ~CH

Ok, so assume we have a model for ZFC where CH does not hold. What values may $2^{\aleph_0}$ assume over said models?

 

[stevendaryl]

According to Wikipedia,

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 $\kappa$ is a cardinal of uncountable cofinality, then there is a forcing extension in which $2^{\aleph_0} = \kappa$. However, per König's theorem, it is not consistent to assume $2^{\aleph _{0}}$ is $\aleph _{\omega }$ or $\aleph _{\omega _{1}+\omega }$ or any cardinal with cofinality $\omega$.

 

[WWGD]

Excellent, Steven Daryl, thank you.