Closed forcing ($\kappa$-closed)

A forcing order is $\kappa$-closed when every decreasing sequence of conditions of length below $\kappa$ has a lower bound. Countably closed forcing adds no new countable sequences of ground-model elements and in particular adds no new real numbers.

If a forcing is $\kappa$-closed in the ground model, then it adds no functions from a ground ordinal $\alpha <\kappa$ into a ground set $B$. Recursively decide each value inside the ground model and take common stronger bounds at limit stages and after the final step. Conditions deciding a whole ground function are dense below any condition asserting this type of function. The dense-below generic meeting lemma ensures the generic filter meets that dense subset of a forcing order. One arbitrary bound need not be in the filter in an ordered set.

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.

If a ground-model order is $\lambda$-closed, it adds no ordinal-valued functions with domain $\mu <\lambda$. Recursively decide each name value inside the ground model, using closure at intermediate limit stages and once more at the end. Below each starting condition, conditions deciding the entire function are dense. A generic filter meets them by the dense-below generic meeting lemma. The recursion must be internal to the ground model, since its closure property only covers its own sequences.

A $\lambda$-closed forcing order preserves a ground-model cardinal number $\lambda$, even when it is singular. A proposed surjection from an ordinal below $\lambda$ to $\lambda$ would be an old function because closed forcing adds no short ordinal sequences, contradicting old cardinalhood. This concerns $\lambda$ and smaller cardinals; closure alone does not supply preservation of all larger cardinals.