It's now well known that Continuum Hypothesis (CH) is independent from standard axioms of set theory: one can assume that $c=\aleph_1$ or assume that $c \neq \aleph_1$. Let us assume the second case-then the natural queation rise:

What 'values' $c$ can take? I know that it's impossible that $c=\aleph_{\omega}$? Is it possible to give simple criterium for a cardinal $\alpha$ to have a property: $c=\alpha$ in some model of set theory? In particular, is there a cardinal $\alpha$ defined not using the term $c$ for which we have $c<\alpha$? And the last question, is it possible for $c$ to be weakly inaccessible?