People living in M cannot construct a G which is P-generic over M. They may believe on faith that there exists a being to whom their universe, M, is countable. Such a being will have a generic G and an f_G=\cup G. The people in M do not know what G and f_G are but they have names for them, \Gamma and \Phi. They may also read the preceding few paragraphs and thus figure out certain properties of G and f_G…
I could only say that Paul Cohen’s idea on the Continuum Hypothesis is amazing. And I am quite curious about his religious belief.