Cohen forcing

Cohen forcing consists of finite partial approximations to a new subset or function, ordered by reverse inclusion. Meeting the ground-model dense sets makes the union a total generic object.

Over a model of Generalized continuum hypothesis, add ${\aleph }_3$ Cohen reals with finite binary partial functions. The Delta-system lemma proves the countable chain condition for forcing. The generic reals give the continuum lower bound ${\aleph }_3$, while nice forcing names for subsets of ${\omega }_1$ number at most $({\aleph }_3{)}^{{\aleph }_1}={\aleph }_3$ in the ground model. Cardinal preservation then gives $2^{{\aleph }_0}=2^{{\aleph }_1}={\aleph }_3$.