 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.