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

TL;DR

This paper develops a theoretical framework for rank-initial embeddings and automorphisms of countable non-standard models of set theory, providing new techniques and results about their structure, fixed points, and automorphism properties.

Contribution

It introduces a geometric technique for analyzing embeddings and automorphisms, generalizes Friedman's theorem, and characterizes strong rank-cuts via automorphism fixed points.

Findings

Generalized Friedman's theorem on rank-initial embeddings

Constructed models with automorphisms fixing specific sets

Characterized strong rank-cuts in terms of automorphisms

Abstract

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.