Countably closed forcing (${\omega }_1\text{-closed}$)

Every descending countable sequence of conditions has a common stronger condition. This is ${\omega }_1$-closed forcing and adds no countable ordinal sequences.

Conditions are countable successor-length sequences $⟨A_{\xi }\subseteq \xi ⟩$, ordered by end extension. A countable fusion construction decides a named subset below a fresh limit index and writes that trace at the index inside any named club. The generic sequence satisfies the diamond principle.

Countable partial functions from ${\omega }_1$ to a nonempty ground-model set $A$, ordered by extension, add a surjection ${\omega }_1\to A$. Countable closure preserves ${\omega }_1$ and adds no reals.