Iterating Generalised Perfect Set Forcing Along Well-Founded Orders

TL;DR

This paper extends the geometric iteration technique to a generalized perfect set forcing along well-founded orders, ensuring cardinal preservation up to ^+ for certain cardinals.

Contribution

It demonstrates that the generalized perfect set forcing (\u0192) can be iterated with supports of size  along any well-founded partial order, preserving cardinals.

Findings

The iteration preserves cardinals up to ^+ for  satisfying ^{<}}=.

The geometric iteration technique applies to () forcing along well-founded orders.

The method generalizes previous results on iteration along partial orders.

Abstract

Vladimir Kanovei \cite{zbMATH01335192} developed the technique of geometric iteration and used it 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 proved that its iteration with supports of size $\le\kappa$ along any ordinal preserves cardinals up and including $\kappa^+$. We show that there is a version of the geometric iteration technique that applies to ${\mathbb P} (\mathcal F)$, to yield that for $\kappa$ satisfying $\kappa^{<\kappa}=\kappa$, the forcing ${\mathbb P} (\FF)$ can be iterated with supports of size $\le\kappa$ along any well-founded partial order, while preserving cardinals up and including $\kappa^+$.