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

Archive for Mathematical Logic 57 (1-2):91-139 (2018)
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Abstract

Given a model \ of set theory, and a nontrivial automorphism j of \, let \\) be the submodel of \ whose universe consists of elements m of \ such that \=x\) for every x in the transitive closure of m ). Here we study the class \ of structures of the form \\), where the ambient model \ satisfies a frugal yet robust fragment of \ known as \, and \=m\) whenever m is a finite ordinal in the sense of \ Our main achievement is the calculation of the theory of \ as precisely \-\. The following theorems encapsulate our principal results:Theorem A. Every structure in \ satisfies \-\.Theorem B. Each of the following three conditions is sufficient for a countable structure \ to be in \: \ is a transitive model of \-\. \ is a recursively saturated model of \-\. \ is a model of \.Theorem C. Suppose \ is a countable recursively saturated model of \ and I is a proper initial segment of \ that is closed under exponentiation and contains \. There is a group embedding \ from \\) into \\) such that I is the longest initial segment of \ that is pointwise fixed by \ for every nontrivial \.\)In Theorem C, \\) is the group of automorphisms of the structure X, and \ is the ordered set of rationals.

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References found in this work

The strength of Mac Lane set theory.A. R. D. Mathias - 2001 - Annals of Pure and Applied Logic 110 (1-3):107-234.
Toward model theory through recursive saturation.John Stewart Schlipf - 1978 - Journal of Symbolic Logic 43 (2):183-206.
Automorphisms of models of set theory and extensions of NFU.Zachiri McKenzie - 2015 - Annals of Pure and Applied Logic 166 (5):601-638.
Recursively saturated models generated by indiscernibles.James H. Schmerl - 1985 - Notre Dame Journal of Formal Logic 26 (2):99-105.

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