Self-similarity in the Foundations

TL;DR

This thesis explores self-embeddings in set theory and category theory, establishing new theorems about models of set theory and introducing a novel algebraic set theory linked to New Foundations.

Contribution

It refines existing theorems on models of set theory and introduces a new algebraic set theory equivalent to New Foundations, with flexible logical variants.

Findings

Refined Friedman's theorem on embeddings between models of ZF

Analogues of Gaifman's theorem on automorphisms of models

A new algebraic set theory equiconsistent with New Foundations

Abstract

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.