UC Irvine

The existence of a saturated ideal on $\aleph_2$ and the $\aleph_2$-tree property are interesting properties in their own right, with the literature filled with results involving one or the other. However, it seems that all known techniques for producing a model with one of these two properties destroys the other property. The question of constructing a model with both properties has appeared in the literature.

In Chapter 1, we cover the technical background needed throughout the rest of the dissertation. In Chapter 2, we present a variant of the Kunen-Magidor construction and use it to construct a model that has a saturated ideal on $\aleph_2$ and satisfies $\square_{\omega_1}$. In Chapter 3, we prove that $\square_{\omega_1, \omega}$ and $\square(\omega_2)$ fail in the original Kunen-Magidor model, and we also show that the original Kunen-Magidor model has simultaneous stationary set reflection. The question of constructing a model that has a presaturated/strong ideal---instead of a saturated ideal---on $\aleph_2$ and has the $\aleph_2$-tree property may be more approachable because it seems that some variant of the Kunen-Magidor construction \textit{may} work in the presaturated/strong case. Thus, in Chapter 4, we present another variant of the Kunen-Magidor construction and use it to prove a partial result towards the goal of constructing a model that has a presaturated/strong ideal on $\aleph_2$ and has the $\aleph_2$-tree property. We then discuss a potential strategy for using the variant to construct such a model.