- Do I understand correctly that Cohen's technique forces sets from V which are not in L into the gap between the naturals and its power set? If so, does one use the implicit assumption that V is not equal to L?
Cohen's forcing essentially takes Gödel constructible universe L and adds sets from the larger universe of sets V until it provides a set having a cardinality between that of the naturals N and its power set. But in order to do that, you must assume that V has sets to add that are not in L, because if V=L, the continuum hypothesis CH holds. If this is right so far, then does one assume something like the existence of Ramsey cardinals to do the proof? I suspect I am missing something here, as I have never seen such an assumption in expositions of forcing.