Utilizing Brief Spiritual Assessments with Clients who belong to the Church of Jesus Christ of Latter-day Saints:

In recent years social work has increasingly focused on spirituality and religion as key elements of cultural competency. The Joint Commission—the nation's largest health care accrediting organization—as well as many other accrediting bodies require spiritual assessments in hospitals and many other mental health settings. Consequently, specific intervention strategies have been fostered in order to provide the most appropriate interventions for religious clients. The Church of Jesus Christ of Latter-day Saints is the fourth largest and one of the faster growing churches in the United States. In an effort to facilitate cultural competence with clients who are members of the Church of Jesus Christ, a brief spiritual assessment instrument was developed. This mixed-method study asked experts in Church culture (N = 100) to identify the degree of cultural consistency, strengths, and limitations of the brief spiritual assessment instrument. Results indicate that the framework is consistent with Church culture and a number of practice-oriented implications are offered.

SQUARES, ASCENT PATHS, AND CHAIN CONDITIONS

With the help of various square principles, we obtain results concerning the consistency strength of several statements about trees containing ascent paths, special trees, and strong chain conditions. Building on a result that shows that Todorčević’s principle $\square \left( {\kappa ,\lambda } \right)$ implies an indexed version of $\square \left( {\kappa ,\lambda } \right)$, we show that for all infinite, regular cardinals $\lambda < \kappa$, the principle $\square \left( \kappa \right)$ implies the existence of a κ-Aronszajn tree containing a λ-ascent path. We then provide a complete picture of the consistency strengths of statements relating the interactions of trees with ascent paths and special trees. As a part of this analysis, we construct a model of set theory in which ${\aleph _2}$-Aronszajn trees exist and all such trees contain ${\aleph _0}$-ascent paths. Finally, we use our techniques to show that the assumption that the κ-Knaster property is countably productive and the assumption that every κ-Knaster partial order is κ-stationarily layered both imply the failure of $\square \left( \kappa \right)$.

HIERARCHIES OF (VIRTUAL) RESURRECTION AXIOMS

I analyze the hierarchies of the bounded resurrection axioms and their “virtual” versions, the virtual bounded resurrection axioms, for several classes of forcings (the emphasis being on the subcomplete forcings). I analyze these axioms in terms of implications and consistency strengths. For the virtual hierarchies, I provide level-by-level equiconsistencies with an appropriate hierarchy of virtual partially super-extendible cardinals. I show that the boldface resurrection axioms for subcomplete or countably closed forcing imply the failure of Todorčević’s square at the appropriate level. I also establish connections between these hierarchies and the hierarchies of bounded and weak bounded forcing axioms.

HIERARCHIES OF FORCING AXIOMS, THE CONTINUUM HYPOTHESIS AND SQUARE PRINCIPLES

I analyze the hierarchies of the bounded and the weak bounded forcing axioms, with a focus on their versions for the class of subcomplete forcings, in terms of implications and consistency strengths. For the weak hierarchy, I provide level-by-level equiconsistencies with an appropriate hierarchy of partially remarkable cardinals. I also show that the subcomplete forcing axiom implies Larson’s ordinal reflection principle atω2, and that its effect on the failure of weak squares is very similar to that of Martin’s Maximum.

The consistency strength of choiceless failures of SCH

We determine exact consistency strengths for various failures of the Singular Cardinals Hypothesis (SCH) in the setting of the Zermelo-Fraenkel axiom system ZF without the Axiom of Choice (AC). By the new notion of parallel Prikry forcing that we introduce, we obtain surjective failures of SCH using only one measurable cardinal, including a surjective failure of Shelah's pcf theorem about the size of the power set of ℵω. Using symmetric collapses to ℵω, , or , we show that injective failures at ℵω, , or can have relatively mild consistency strengths in terms of Mitchell orders of measurable cardinals. Injective failures of both the aforementioned theorem of Shelah and Silver's theorem that GCH cannot first fail at a singular strong limit cardinal of uncountable cofinality are also obtained. Lower bounds are shown by core model techniques and methods due to Gitik and Mitchell.

Hierarchies of forcing axioms I

We prove new upper bound theorems on the consistency strengths of SPFA(θ), SPFA(θ-linked) and SPFA(θ+ -cc). Our results are in terms of (θ, Γ)-subcompactness, which is a new large cardinal notion that combines the ideas behind subcompactness and Γ-indescribability. Our upper bound for SPFA(ϲ-linked) has a corresponding lower bound, which is due to Neeman and appears in his follow-up to this paper. As a corollary, SPFA(ϲ-linked) and PFA(ϲ-linked) are each equiconsistent with the existence of a -indescribable cardinal. Our upper bound for SPFA(ϲ-c.c) is a -indescribable cardinal, which is consistent with V = L. Our upper bound for SPFA(ϲ+-linked) is a cardinals κ that is (κ+,)-subcompact, which is strictly weaker than κ+-supercompact. The axiom MM(ϲ) is a consequence of SPFA(ϲ+-linked) by a slight refinement of a theorem of Shelah. Our upper bound for SPFA(ϲ++-c.c.) is a cardinal κ that is (κ+, )-subcompact, which is also strictly weaker than κ+-supercompact.