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

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}} + text {``}{mathrm {Ord}}text { 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}} + text {``}{mathrm {Ord}}text { is weakly compact''}, and (ii) in terms of fixed-point sets of self-embeddings.