Let $\mathbb{P}$ and $\mathbb{Q}$ be two forcing notions. Recall that we say $\mathbb{Q}$ is a subforcing of $\mathbb{P}$ if there exists a regular embedding $\mathbb{Q} \to \text{r.o.}(\mathbb{P}).$

Question. Let $\mathbb{P} \in \{Add(\omega, \kappa), Col(\omega_1, \kappa) \}.$

(1) Is there a subforcing of $\mathbb{P}$ which is not Proper?

(2) Is there a subforcing of $\mathbb{P}$ which is not semi-Proper?

Remark. In Subalgebras of Cohen algebras need not be Cohen, Koppelberg and Shelah show that there is a subalgebra of Cohen forcing which is not Cohen.


No. A subforcing of a c.c.c. forcing is c.c.c. A subforcing of a countably closed forcing is countably-strategically-closed, which implies proper. (This is easy to see via countable elementary submodels. Use the strategy to construct a generic condition.)

Furthermore, every subforcing of a proper forcing is proper. Properness is equivalent to preserving stationary subsets of $[\kappa]^\omega$ for all $\kappa$. The stationarity of $X \subseteq [ \kappa ]^\omega$ cannot be restored once killed, since killing it is just adding some $f : [\kappa]^{<\omega} \to \kappa$ such that no $x \in X$ is closed under $f$. (See Abraham's chapter of the Handbook.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.