Adding cofinal countable sequences through multiple regular cardinals by ssp forcing

TL;DR

This paper introduces a new forcing technique that preserves stationarity and manipulates cofinalities of regular cardinals, ensuring specific cardinals become countably cofinal without affecting others, all within ZFC.

Contribution

It provides an elementary, combinatorial forcing construction to cofinally manipulate multiple regular cardinals simultaneously, extending previous methods without extra assumptions.

Findings

Successfully makes selected regular cardinals $ ext{cofinal}( ext{countable})$

Preserves stationarity of certain sets during forcing

Ensures no unintended regular cardinals change cofinality

Abstract

We present a direct construction of stationary set preserving forcings that make $\omega$-cofinal all the members of some arbitrary set $\mathcal{K}$ of regular cardinals $\kappa > \omega_1$. 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 $\mathsf{ZFC}$.