Iterating Generalised Perfect Set Forcing Along Well-Founded Orders

Abstract

The technique of geometric forcing iteration was developed by Kanovei \cite{zbMATH01335192} and used to prove that the perfect set forcing can be iterated with countable supports along any partial order, while preserving ${\aleph }_1$. In \cite{Property-B} we considered a generalised perfect set forcing with respect to a filter on a cardinal $\hat{I}º$ satisfying $\hat{I}{º}^{<\hat{I}º}=\hat{I}º$, which we denoted $\mathbb{P}(\mathcal{F})$, and we proved that its iteration with supports of size $\le \hat{I}º$ along any ordinal preserves cardinals up to and including $\hat{I}{º}^{+}$. We show that there is a version of the geometric iteration technique that applies to $\mathbb{P}(\mathcal{F})$ and yields that for $\hat{I}º$ satisfying $\hat{I}{º}^{<\hat{I}º}=\hat{I}º$ and for appropriate filters $\mathcal{F}$, the forcing ${\mathbb P} (\FF)$ can be iterated with supports of size $\le \hat{I}º$ along any well-founded partial order and preserve cardinals up to and including $\hat{I}{º}^{+}$. As an application of our technique we obtain that common notions of arboreal forcings on $\ddot{I}‰$ can be iterated with countable supports along any well-founded partial order and that such iterations preserve ${\aleph }_1$.

The version submitted to the Proceedings of the Steklov Institute of Mathematics