Stationary reflection principles and two cardinal tree properties

We study consequences of stationary and semi-stationary set reflection. We show that the semi stationary reflection principle implies the Singular Cardinal Hypothesis, the failure of weak square principle, etc. We also consider two cardinal tree properties introduced recently by Weiss and prove that they follow from stationary and semi stationary set reflection augmented with a weak form of Martin's Axiom. We also show that there are some differences between the two reflection principles which suggest that stationary set reflection is analogous to supercompactness whereas semi-stationary set reflection is analogous to strong compactness.Comment: 19 page

On Indestructible Strongly Guessing Models

In \cite{MV} we defined and proved the consistency of the principle ${\rm GM}^+(\omega_3,\omega_1)$ which implies that many consequences of strong forcing axioms hold simultaneously at $\omega_2$ and $\omega_3$. In this paper we formulate a strengthening of ${\rm GM}^+(\omega_3,\omega_1)$ that we call ${\rm SGM}^+(\omega_3,\omega_1)$. We also prove, modulo the consistency of two supercompact cardinals, that ${\rm SGM}^+(\omega_3,\omega_1)$ is consistent with ZFC. In addition to all the consequences of ${\rm GM}^+(\omega_3,\omega_1)$, the principle ${\rm SGM}^+(\omega_3,\omega_1)$, together with some mild cardinal arithmetic assumptions that hold in our model, implies that any forcing that adds a new subset of $\omega_2$ either adds a real or collapses some cardinal. This gives a partial answer to a question of Abraham \cite{AvrahamPhD} and extends a previous result of Todor\v{c}evi\'{c} \cite{Todorcevic82} in this direction

Proper forcing remastered

In these notes we present the method introduced by Neeman of generalized side conditions with two types of models. We then discuss some applications: the Friedman-Mitchell poset for adding a club in \omega_2 with finite conditions, Koszmider's forcing construction of a strong chain of length \omega_2 of functions from \omega_1 to \omega_1, and the Baumgartner-Shelah forcing construction of a thin very tall superatomic Boolean algebra.Comment: 29 page