Rankinitial embeddings of nonstandard models of set theory
Abstract
A theoretical development is carried to establish fundamental results about rankinitial embeddings and automorphisms of countable nonstandard 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 nonstandard models of set theory. In particular, backandforth constructions are carried out to establish various generalizations and refinements of Friedman's theorem on the existence of rankinitial embeddings between countable nonstandard 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 rankendextended to models with wellbehaved automorphisms whose sets of fixed points equal the original model. These theoretical developments are then utilized to prove various results relating selfembeddings, automorphisms, their sets of fixed points, strong rankcuts, and set theories of different strengths. Two examples: The notion of "strong rankcut" is characterized (i) in terms of the theory $\mathrm{GBC}$ + "$\mathrm{Ord}$ is weakly compact", and (ii) in terms of fixedpoint sets of selfembeddings.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.08724
 Bibcode:
 2021arXiv210608724G
 Keywords:

 Mathematics  Logic;
 03H99;
 03C15;
 03C20;
 03C62;
 03E30;
 03E55
 EPrint:
 Arch. Math. Logic 59, 517563 (2020)