Tree Properties at Successors of Singulars of Many Cofinalities

TL;DR

This paper demonstrates the consistency of the tree property at many small successors of singular cardinals with different cofinalities, using models derived from supercompact cardinals.

Contribution

It introduces a method to establish the tree property at multiple successors of singular cardinals with varying cofinalities, extending to the strong tree property and large sequences.

Findings

Tree property holds at $eth_{ ext{omega}+ ext{omega}+1}$ and $eth_{ ext{omega}_n+1}$ for all $n< ext{omega}$.

The technique extends to the strong tree property and large uncountable sequences.

Models are constructed from many supercompact cardinals.

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 $\aleph_{\omega+\omega+1}$ and at $\aleph_{\omega_n+1}$ for all $0<n<\omega$. 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.