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

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In models of ZFC where the Continuum Hypothesis (CH) does not hold, the value of 2^{\aleph_0} can vary significantly. Solovay's result indicates that for any uncountable cardinal \kappa, there exists a forcing extension where 2^{\aleph_0} equals \kappa. However, König's theorem restricts the values of 2^{\aleph_0}, stating it cannot be \aleph_{\omega}, \aleph_{\omega_1 + \omega}, or any cardinal with cofinality \omega. This highlights the complexity of cardinality in set theory under different models. Understanding these relationships is crucial for exploring the implications of the Continuum Hypothesis.
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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?
 
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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##.
 
Excellent, Steven Daryl, thank you.
 

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