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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.

The forcing is $\kappa$-closed in $M$ when every descending sequence of conditions of length less than $\kappa$, belonging to $M$, has a lower bound in $M$:We use the convention that $q\le p$ means that $q$ is stronger. In particular, ${\omega }_1$-closed forcing is countably closed. Internal quantification over the sequences is essential when $M$ is externally countable.

Use standard notation for forcing: $q\le p$ means that $q$ is stronger than $p$. A closed forcing order is $\lambda$-closed when every decreasing sequence $⟨p_{\xi }:\xi <\delta ⟩$ with $\delta <\lambda$ has a common lower bound:The chain condition for forcing isAn antichain in a forcing order consists of pairwise incompatible forcing conditions; incompatibility means having no common stronger extension. In particular, the ${\aleph }_1$-chain condition is the countable chain condition for forcing. These properties and their sequences or antichains are computed inside the ground model $M$.

If $\mathbb{P}$ is ${\aleph }_1$-closed in $M$, a descending sequence deciding successively all entries of a proposed function $\mathbb{N}\to M$ has a common lower bound. Thus the extension contains no new countable sequences of ground-model elements and in particularIt follows that ${\aleph }_1^M$ remains uncountable and hence is preserved. More generally such closure preserves cardinals at most ${\aleph }_1$, but closure alone need not preserve larger cardinals.

The forcingis ${\aleph }_1^M$-closed, so by closed forcing it adds no new real numbers. Its generic union can be viewed as a sequenceof old reals. For every $r\in \wp (\mathbb{N}{)}^M$, conditions asserting that some unused row equals $r$ are dense: assigning all countably many values of that row is a legitimate condition. Thus the generic sequence surjects ${\aleph }_1^M$ onto the old set of reals.

In $M$, that set had cardinality ${\aleph }_2^M$. The forcing therefore collapses ${\aleph }_2^M$ to ${\aleph }_1^M$, while adding no reals and preserving ${\aleph }_1^M$. Consequently