Sean Cox

QUOTIENTS OF STRONGLY PROPER FORCINGS AND GUESSING MODELS

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 simple universal strongly generic conditions on a stationary set of models by certain nice regular suborders satisfy the

Ideal Projections and Forcing Projections

\omega_1

Characterizing large cardinals in terms of layered posets

-approximation property. We prove that the existence of stationarily many

Chang’s conjecture and semiproperness of nonreasonable posets

\omega_1

Lower consistency bounds for mutual stationarity with divergent uncountable cofinalities

-guessing models in

Prevalence of Generic Laver Diamond

P_{\omega_2}(H(\theta))

Indestructible Guessing Models and the Continuum

, for sufficiently large cardinals

Martin’s Maximum and tower forcing

\theta

Indestructibility of generically strong cardinals

, is consistent with the continuum being arbitrarily large, solving a problem of Viale and Weiss.

Layered posets and Kunen's universal collapse

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