= Cohen forcing
{c}
{wiki}
To add one binary <real number>, <Cohen forcing> uses
$$
\mathbb P=\{p:p\text{ is a finite partial function }\omega\to\{0,1\}\},\qquad q\leq p\ \Longleftrightarrow\ q\supseteq p.
$$
Stronger 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.
Back to article page
