Cohen forcing

To add one binary real number, Cohen forcing usesStronger conditions decide more bits. For a generic filter $G$, the compatible conditions have a union $c=\bigcup G$. For each $n$, the set of conditions deciding bit $n$ is a dense subset of a forcing order, so $c$ is total.

For each ground-model binary sequence $r$, the conditions disagreeing with $r$ at some coordinate are dense: extend any finite condition at an unused coordinate with the opposite bit. Genericity therefore gives $c\ne r$ for every such $r$. This explains precisely why the union is a new real. Restricting to conditions that decide an initial segment gives an equivalent dense suborder of finite binary strings.

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 $2^{{\aleph }_0}=2^{{\aleph }_1}={\aleph }_3$.