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 $\kappa$ satisfying $\kappa^{<\kappa}=\kappa$, which we denoted ${\mathbb P} (\mathcal F)$, and we proved that its iteration with supports of size $\le\kappa$ along any ordinal preserves cardinals up to and including $\kappa^+$. We show that there is a version of the geometric iteration technique that applies to ${\mathbb P} (\mathcal F)$ and yields that for $\kappa$ satisfying $\kappa^{<\kappa}=\kappa$ and for appropriate filters $\mathcal F$, the forcing ${\mathbb P} (\FF)$ can be iterated with supports of size $\le\kappa$ along any well-founded partial order and preserve cardinals up to and including $\kappa^+$. As an application of our technique we obtain that common notions of arboreal forcings on $\omega$ can be iterated with countable supports along any well-founded partial order and that such iterations preserve $\aleph_1$.