Indestructibility and c(n)-supercompact cardinals

This thesis is split into two areas of interest. The first, a study of indestructibility results for two variants of supercompactness; the second, a discussion of double-membership graphs of models of Anti-Foundational set theory. In Chapter 3 we will consider alpha-subcompact cardinals — which can be viewed as a weakened version of supercompact cardinals — and we will show that, by defining a suitable preparatory forcing, an alpha-subcompact cardinal (with alpha a regular cardinal) can be made indestructible by all less than kappa-directed closed forcing. We will then turn our attention to stronger forms of supercompactness, namely C(n)- supercompacts, and answer (in part) open questions about their indestructibility, by showing that, for a C(2)-extendible, we can make its C(2)-supercompactness indestructible by less than kappa-directed closed forcing. We will then combine the concepts of alpha-subcompact cardinals and C(n)-cardinals and show that, below an alpha-subcompact cardinal where alpha is in C(n) with for n greater than equal to 1, there is a stationary set of partial extendibles below kappa, determined by alpha-subcompactness embeddings for kappa. Lastly, in joint work with Rosario Mennuni and John Howe, we consider reducts of countable models of Anti-Foundational set theory to the double-membership relation D, where D(x,y) if and only if x is a member of y and y is a member of x. We show that there are continuum many such graphs, and study their connected components. We describe their complete theories, and prove that each one has continuum-many countable models, some of which are not reducts of models of Anti-Foundation.