Abstract A diagonal version of the strong reflection principle is introduced, along with fragments of this principle associated with arbitrary forcing classes. The relationships between the resulting principles and related principles, such as the corresponding forcing axioms and the corresponding fragments of the strong reflection principle, are analyzed, and consequences are presented. Some of these consequences are “exact” versions of diagonal stationary reflection principles of sets of ordinals. We also separate some of these diagonal strong reflection principles from related axioms.

Abstract We prove a variety of theorems about stationary set reflection and concepts related to internal approachability. We prove that an implication of Fuchino–Usuba relating stationary reflection to a version of Strong Chang’s Conjecture cannot be reversed; strengthen and simplify some results of Krueger about forcing axioms and approachability; and prove that some other related results of Krueger are sharp. We also adapt some ideas of Woodin to simplify and unify many arguments in the literature involving preservation of forcing axioms.

Abstract We prove a variation of Easton’s lemma for strongly proper forcings, and use it to prove that, unlike the stronger principle IGMP, GMP together with 2ω ≤ ω2 is consistent with the existence of an ω1-distributive nowhere c.c.c. forcing poset of size ω1. We introduce the idea of a weakly guessing model, and prove that many of the strong consequences of the principle GMP follow from the existence of stationarily many weakly guessing models. Using Namba forcing, we construct a model in which there are stationarily many indestructibly weakly guessing models which have a bounded countable subset not covered by any countable set in the model.

Abstract We prove that a wide class of strongly proper forcing posets have quotients with strong properties. Specifically, we prove that quotients of forcing posets which have universal strongly generic conditions on a stationary set of models by certain nice regular suborders satisfy the ω1-approximation property. We prove that the existence of stationarily many ω1-guessing models in Pω2(H(θ)), for sufficiently large cardinals θ, is consistent with the continuum being arbitrarily large, solving a problem of Viale and Weiss [13].

Abstract It is well known that saturation of ideals is closely related to the “antichain-catching” phenomenon from Foreman–Magidor–Shelah [10]. We consider several antichain-catching properties that are weaker than saturation, and prove: (1) If ${\cal I}$ is a normal ideal on $\omega _2 $ which satisfiesstationary antichain catching, then there is an inner model with a Woodin cardinal; (2) For any $n \in \omega $ , it is consistent relative to large cardinals that there is a normal ideal ${\cal I}$ on $\omega _n $ which satisfiesprojective antichain catching, yet ${\cal I}$ is not saturated (or even strong). This provides a negative answer to Open Question number 13 from Foreman’s chapter in the Handbook of Set Theory ([7]).

Abstract We obtain lower bounds for the consistency strength of fully nonregular ultrafilters on ω2.