Tree Properties at Successors of Singulars of Many Cofinalities
William Adkisson
TL;DR
This work addresses forcing TP and the strong tree property at successors of singular cardinals across multiple cofinalities. It introduces a master forcing based on a sequence of indestructibly supercompact cardinals, combined with side forcings $\mathbb{H}$ and $\mathbb{L}_s$, and leverages absorption and $\kappa$-approximation techniques to preserve branches. The main results establish TP at $\aleph_{\omega+\omega+1}$ and at $\aleph_{\omega_n+1}$ for all $0<n<\omega$, and achieve the strong tree property at the same cardinals, with extensions to uncountably many cofinalities. This provides a flexible framework for distributing TP and STP across a broad spectrum of successor cardinals of singulars, advancing Magidor-style questions about small-cardinal instances of these properties.
Abstract
From many supercompact cardinals, we show that it is consistent for the tree property to hold at many small successors of singular cardinals, each with a different cofinality. In particular, we construct a model in which the tree property holds at $\aleph_{ω+ω+1}$ and at $\aleph_{ω_n+1}$ for all $0<n<ω$. We show that this can be done for the strong tree property as well, and extend the technique to large uncountable sequences of desired cofinalities.