Largest initial segments pointwise fixed by automorphisms of models of set theory

Given a model $\mathcal{M}$ of set theory, and a nontrivial automorphism $j$ of $\mathcal{M}$, let $\mathcal{I}_{\mathrm{fix}}(j)$ be the submodel of $\mathcal{M}$ whose universe consists of elements $m$ of $\mathcal{M}$ such that $j(x)=x$ for every $x$ in the transitive closure of $m$ (where the transitive closure of $m$ is computed within $\mathcal{M}$). Here we study the class $\mathcal{C}$ of structures of the form $\mathcal{I}_{\mathrm{fix}}(j)$, where the ambient model $\mathcal{M}$ satisfies a frugal yet robust fragment of $\mathrm{ZFC}$ known as $\mathrm{MOST}$, and $j(m)=m$ whenever $m$ is a finite ordinal in the sense of $\mathcal{M}$. We show that every structure in $\mathcal{C}$ satisfies $\mathrm{MOST}+\Delta_0^\mathcal{P}\textrm{-Collection}$. We also show that the following countable structures are in $\mathcal{C}$: (a) transitive models of $\mathrm{MOST}+\Delta_0^\mathcal{P}\textrm{-Collection}$, (b) recursively saturated models of $\mathrm{MOST}+\Delta_0^\mathcal{P}\textrm{-Collection}$, (c) models of $\mathrm{ZFC}$. It follows from (b) that the theory of $\mathcal{C}$ is precisely $\mathrm{MOST+\Delta}_{0}^{\mathcal{P}}$-Collection. We conclude by proving a refinement of a result due to Amir Togha.Comment: 44 page

Rank-initial embeddings of non-standard models of set theory

A theoretical development is carried to establish fundamental results about rank-initial embeddings and automorphisms of countable non-standard models of set theory, with a keen eye for their sets of fixed points. These results are then combined into a "geometric technique" used to prove several results about countable non-standard models of set theory. In particular, back-and-forth constructions are carried out to establish various generalizations and refinements of Friedman's theorem on the existence of rank-initial embeddings between countable non-standard models of the fragment $\mathrm{KP}^\mathcal{P}$ + $\Sigma_1^\mathcal{P}$-Separation of $\mathrm{ZF}$; and Gaifman's technique of iterated ultrapowers is employed to show that any countable model of $\mathrm{GBC}$ + "$\mathrm{Ord}$ is weakly compact" can be elementarily rank-end-extended to models with well-behaved automorphisms whose sets of fixed points equal the original model. These theoretical developments are then utilized to prove various results relating self-embeddings, automorphisms, their sets of fixed points, strong rank-cuts, and set theories of different strengths. Two examples: The notion of "strong rank-cut" is characterized (i) in terms of the theory $\mathrm{GBC}$ + "$\mathrm{Ord}$ is weakly compact", and (ii) in terms of fixed-point sets of self-embeddings