The tree property at double successors of singular cardinals of uncountable cofinality

TL;DR

This paper demonstrates how to force a model where a singular strong limit cardinal of uncountable cofinality has the tree property at its double successor, assuming large cardinal hypotheses.

Contribution

It introduces a forcing construction that achieves the tree property at the double successor of a singular cardinal with uncountable cofinality under strong large cardinal assumptions.

Findings

Tree property holds at b2 of a singular strong limit cardinal.

Constructs models with prescribed cofinality for singular cardinals.

Uses large cardinal assumptions to achieve the result.

Abstract

Assuming the existence of a strong cardinal $\kappa$ and a measurable cardinal above it, we force a generic extension in which $\kappa$ is a singular strong limit cardinal of any prescribed cofinality, and such that the tree property holds at $\kappa^{++}$.