We establish an equiconsistency between (1) weak indestructibility for all $\kappa +2$-degrees of strength for cardinals $\kappa $ in the presence of a proper class of strong cardinals, and (2) a proper class of cardinals that are strong reflecting strongs. We in fact get weak indestructibility for degrees of strength far beyond $\kappa +2$, well beyond the next inaccessible limit of measurables (of the ground model). One direction is proven using forcing and the other using core model techniques from inner model theory. Additionally, connections between weak indestructibility and the reflection properties associated with Woodin cardinals are discussed. This work is a part of my upcoming thesis [7].

After reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove that second-order reflection with the abundant atom axiom is bi-interpretable and hence also equiconsistent with the existence of a supercompact cardinal. The proof relies on a reflection characterization of supercompactness, namely, a cardinal $\kappa $ is supercompact if and only if every $\Pi ^1_1$ sentence true in a structure M (of any size) containing $\kappa $ in a language of size less than $\kappa $ is also true in a substructure $m\prec M$ of size less than $\kappa $ with $m\cap \kappa \in \kappa $.

We show the consistency, relative to the appropriate supercompactness or strong compactness assumptions, of the existence of a non-supercompact strongly compact cardinal $\kappa _0$ (the least measurable cardinal) exhibiting properties which are impossible when $\kappa _0$ is supercompact. In particular, we construct models in which $\square _{\kappa ^+}$ holds for every inaccessible cardinal $\kappa $ except $\kappa _0$, GCH fails at every inaccessible cardinal except $\kappa _0$, and $\kappa _0$ is less than the least Woodin cardinal.

We prove two theorems concerning indestructibility properties of the first two strongly compact cardinals when these cardinals are in addition the first two measurable cardinals. Starting from two supercompact cardinals $\kappa _1 < \kappa _2$ , we force and construct a model in which $\kappa _1$ and $\kappa _2$ are both the first two strongly compact and first two measurable cardinals, $\kappa _1$ ’s strong compactness is fully indestructible (i.e., $\kappa _1$ ’s strong compactness is indestructible under arbitrary $\kappa _1$ -directed closed forcing), and $\kappa _2$ ’s strong compactness is indestructible under $\mathrm {Add}(\kappa _2, \delta )$ for any ordinal $\delta $ . This provides an answer to a strengthened version of a question of Sargsyan found in [17, Question 5]. We also investigate indestructibility properties that may occur when the first two strongly compact cardinals are not only the first two measurable cardinals, but also exhibit nontrivial degrees of supercompactness.

Extending the work of Larson and Todorcevic, we show that there is a model of set theory in which normal spaces are collectionwise Hausdorff if they are either first countable or locally compact, and yet there are no first countable $L$-spaces or compact $S$-spaces. The model is one of the form $\text{PFA}\left( S \right)\left[ S \right]$, where $S$ is a coherent Souslin tree.

We obtain an equiconsistency for a weak form of universal indestructibility for strongness. The equiconsistency is relative to a cardinal weaker in consistency strength than a Woodin cardinal, Stewart Baldwin's notion of hyperstrong cardinal. We also briefly indicate how our methods are applicable to universal indestructibility for supercompactness and strong compactness.

We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve and unify Theorems 1 and 2 of [5], due to the first author.

Can a supercompact cardinal κ be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness? In this article, we show that if there is a sufficiently large cardinal above κ, then no, it cannot. Conversely, if one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility by stratified posets. or less level-by-level agreement, such as requiring it only on measure one sets, then yes. it can.

Combining techniques of the first author and Shelah with ideas of Magidor, we show how to get a model in which, for fixed but arbitrary finite n, the first n strongly compact cardinals k1..…kn are so that ki; for i = 1..…n is both the ith measurable cardinal and supercompact. This generalizes an unpublished theorem of Magidor and answers a question of Apter and Shelah.