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

## TL;DR

This paper constructs a model in set theory where the tree property holds at the first and double successors of a singular cardinal with an arbitrary gap, under certain large cardinal assumptions and the GCH.

## Contribution

It generalizes previous results by establishing the tree property at both  and  successors of a singular cardinal with an arbitrary gap, using advanced forcing techniques.

## Key findings

- Tree property holds at  and  successors of a singular cardinal.
- Constructs a model with specified cardinal characteristics and the GCH.
- Existence of both a very good and a bad scale at .

## 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 $\langle\kappa_\xi\mid \xi<\mu\rangle$ 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 $\Theta\geq\lambda$ be a cardinal with $\mathrm{cof}(\Theta)>\kappa$. Assuming the $\mathrm{GCH}_{\geq\kappa}$, 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].