Adding cofinal countable sequences through multiple regular cardinals by ssp forcing
Ben De Bondt, Boban Velickovic
TL;DR
The article develops a direct stationary-set preserving forcing scheme to realize $\omega$-cofinal sequences for every regular cardinal in a fixed set $${\mathcal{K}}$, while keeping other uncountable regular cardinals with their cofinalities intact. The construction uses two layers of side-condition forcing built from Foreman–Magidor style games and a family of multiple Namba games, yielding forcings $\mathbb{P}_0^{\mathbf{Nm}}({\mathcal{K}})$ and $\mathbb{P}_1^{\mathbf{Nm}}({\mathcal{K}})$ that are strongly ssp and semiproper. The first forcing adds $\omega$-cofinal sequences through each $\kappa \in {\mathcal{K}}$, while the second forces the remaining regular cardinals in a regulated way so that their cofinalities become $\omega_1$; together they shape the global cofinality structure while preserving projective-stationary models. An application to the countable-cofinality-constructible model $C^*$ demonstrates coding of larger objects and preservation of stationarity in the constructible hierarchy, illustrating the practical impact of the method. The framework also provides a thorough semiproperness analysis, including the role of measurable cardinals in ensuring stronger preservation properties.
Abstract
We present a direct construction of stationary set preserving forcings that make $ω$-cofinal all the members of some arbitrary set $\mathcal{K}$ of regular cardinals $κ> ω_1$. In addition, it is made possible to ensure that no other uncountable regular cardinals from the ground model acquire countable cofinality in the forcing extension. Our method is elementary, being based on a combinatorial argument by Foreman and Magidor together with generalizations of typical side-condition arguments and needs no assumptions beyond $\mathsf{ZFC}$.