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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 I is a normal ideal on w2 which satisfies stationary antichain catching, then there is an inner model with aWoodin cardinal; (2) For any n ∈ w, it is consistent relative to large cardinals that there is a normal ideal I on wn which satisfies projective antichain catching, yet 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]).

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 FROM AUTHOR]

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 2 ω ≤ ω 2 is consistent with the existence of an ω 1-distributive nowhere c.c.c. forcing poset of size ω 1. 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 FROM AUTHOR]

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 FROM PUBLISHER]

We obtain lower bounds for the consistency strength of fully nonregular ultrafilters on ω2. [ABSTRACT FROM AUTHOR]

We obtain lower bounds for the consistency strength of fully nonregular ultrafilters on ω2. [ABSTRACT FROM AUTHOR]