We prove various theorems about the preservation and destruction of the tree property at ω 2. Working in a model of Mitchell  where the tree property holds at ω 2, we prove that ω 2 still has the tree property after ccc forcing of size א 1 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 ω 2 can be indestructible under ω 2-directed closed forcing.
- Large cardinals
- Tree property