The tree property at double successors of singular cardinals of uncountable cofinality with infinite gaps

TL;DR

This paper demonstrates how to force a model where a singular strong limit cardinal has the tree property at its double successor, with specific cofinality and continuum size, assuming large cardinals.

Contribution

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

Findings

The existence of a model with the tree property at b2 of a singular cardinal.

Control over the cofinality and continuum size of the singular cardinal.

Consistency results assuming strong and weakly compact cardinals.

Abstract

Assuming the existence of a strong cardinal $\kappa$, a weakly compact cardinal $\lambda$ above it and $\gamma > \lambda,$ we force a generic extension in which $\kappa$ is a singular strong limit cardinal of any given cofinality $\delta$, $2^\kappa\geq \gamma$ and such that the tree property holds at $\kappa^{++}$.