We prove various theorems about the preservation and destruction of the tree property at ω2. Working in a model of Mitchell [9] where the tree property holds at ω2, we prove that ω2 still has the tree property after ccc forcing of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\aleph_1}$$\end{document} or adding an arbitrary number of Cohen reals. We show that there is a relatively mild forcing in this same model which destroys the tree property. Finally we (...) prove from a supercompact cardinal that the tree property at ω2 can be indestructible under ω2-directed closed forcing. (shrink)

Assuming some large cardinals, a model of ZFC is obtained in which $\aleph_{\omega+1}$ carries no Aronszajn trees. It is also shown that if $\lambda$ is a singular limit of strongly compact cardinals, then $\lambda^+$ carries no Aronszajn trees.