Abstract
Let $\kappa$ be an infinite regular cardinal. The tree property at $\kappa$ is a compactness principle which says that every $\kappa$-tree has a cofinal branch.
Obtaining the tree property at the double successor of an infinite regular cardinal $\kappa$ is relatively easy and only a weakly compact cardinal is required (''Mitchell forcing''). The situation is more complex when we wish to get this result at the double successor of a singular strong limit cardinal $\kappa$ since we need to ensure the failure of SCH at $\kappa$.
In this talk we will discuss the important case of $\aleph_\omega$ and show that if $\kappa$ is a certain large cardinal (not too large), and $1 < n<\omega$ is fixed, then there is a forcing $\P$ such that the following hold in $V^\P$: \begin{itemize} \item $\kappa = \aleph_\omega$ is strong limit, \item $2^{\aleph\omega}=\aleph_{\omega+n}$, and \item The tree property holds at $\aleph_{\omega+2}$. \end{itemize} The forcing $\P$ is a combination of several subforcings which first prepare the universe and then use a combination of the Mitchell forcing and the Prikry forcing with collapses to force the tree property.