Why does forcing seem to be so vacuously true?
It seems like you are just reversing the subset containment of the model of ZFC + CH to be the other way in the poset. So, why is this valid? Why are you allowed to just put the empty set at the top of the partial order*?
*I have just started learning set theory so I have only seen one example of forcing and it was the forcing of the set of reals to be equal to the second uncountable cardinal.
Edit: It seems to just be a beautiful fact of set theory that one must accept the continuum hypothesis, for example, to prove that there is a universe where the continuum hypothesis is false.