Countable forcing

A forcing order that is countable inside the ground model. It has the countable chain condition and preserves ${\omega }_1$. Internal countability is distinct from external countability of the entire model.

Over a Generalized continuum hypothesis ground model, adding one Cohen real preserves every infinite power $2^{\lambda }={\lambda }^{+}$. For the countable forcing, nice subset forcing names for $\lambda$ number at most $(2^{{\aleph }_0}{)}^{\lambda }=2^{\lambda }$ in the ground model. Old subsets give the matching lower bound and the countable chain condition for forcing preserves cardinals. Starting from the constructible universe, the new real is nonconstructible while the constructible levels remain unchanged.

A ground-model Suslin tree remains Suslin after countable forcing. A new uncountable chain or antichain would contain an uncountable ground-model subset by the ground-model uncountable subset lemma for countable forcing.

An uncountable set of ground-model elements in a generic extension by countable forcing contains an uncountable ground-model subset. Partition membership witnesses according to the countably many forcing conditions.