The tree property at first and double successors of singular cardinals with an arbitrary gap

Abstract

Let $\mathrm{cof}(\mu )=\mu$ and $\kappa$ be a supercompact cardinal with $\mu <\kappa$. Assume that there is an increasing and continuous sequence of cardinals $⟨{\kappa }_{\xi }\mid \xi <\mu ⟩$ with ${\kappa }_0:=\kappa$ and such that, for each $\xi <\mu$, ${\kappa }_{\xi +1}$ is supercompact. Besides, assume that $\lambda$ is a weakly compact cardinal with ${\sup }_{\xi <\mu }{\kappa }_{\xi }<\lambda$. Let $\Theta \ge \lambda$ be a cardinal with $\mathrm{cof}(\Theta )>\kappa$. Assuming the ${\mathrm{GCH}}_{\ge \kappa }$, we construct a generic extension where $\kappa$ is strong limit, $\mathrm{cof}(\kappa )=\mu$, $2^{\kappa }=\Theta$ and both $\mathrm{TP}({\kappa }^{+})$ and $\mathrm{TP}({\kappa }^{++})$ hold. Further, in this model there is a very good and a bad scale at $\kappa$. This generalizes the main results of [Sin16a] and [FHS18].