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

TL;DR

This paper investigates the structure of fixed-point submodels of automorphisms in models of set theory satisfying a fragment of ZFC, characterizing their properties and showing their inclusion of various model types.

Contribution

It introduces the class of fixed-point submodels under automorphisms of models satisfying MOST, and characterizes their properties and the types of models they contain.

Findings

Every fixed-point submodel satisfies MOST plus Δ₀^P-Collection.

Countable fixed-point submodels include transitive, recursively saturated, and ZFC models.

The theory of the class of fixed-point submodels is exactly MOST plus Δ₀^P-Collection.

Abstract

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.