Self-similarity in the foundations

This thesis concerns embeddings and self-embeddings of foundational structures in both set theory and category theory. The first part of the work on models of set theory consists in establishing a refined version of Friedman's theorem on the existence of embeddings between countable non-standard models of a fragment of ZF, and an analogue of a theorem of Gaifman to the effect that certain countable models of set theory can be elementarily end-extended to a model with many automorphisms whose sets of fixed points equal the original model. The second part of the work on set theory consists in combining these two results into a technical machinery, yielding several results about non-standard models of set theory relating such notions as self-embeddings, their sets of fixed points, strong rank-cuts, and set theories of different strengths. The work in foundational category theory consists in the formulation of a novel algebraic set theory which is proved to be equiconsistent to New Foundations (NF), and which can be modulated to correspond to intuitionistic or classical NF, with or without atoms. A key axiom of this theory expresses that its structures have an endofunctor with natural properties

Infinite sequences in stability theory

We take a tour through some uses of independence in stability theory. The unifying theme is infinite sequences. In particular, we develop the basic theory of independent sequences, Morley sequences, indiscernible sequences, Shelah trees, F-isolation and stationary sets. The essay ends with a proof that a stable but not superstable theory has maximally many models in every sufficiently large regular cardinality. Our aim is to exhibit a variety of techniques involving the notion of independence.