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

Abstract

Given a model M of set theory, and a nontrivial automorphism j of M, let Ifix(j) be the submodel of M whose universe consists of elements m of M such that j(x)=x for every x in the transitive closure of m (where the transitive closure of m is computed within M). Here we study the class C of structures of the form Ifix(j), where the ambient model M satisfies a frugal yet robust fragment of ZFC known as MOST, and j(m)=m whenever m is a finite ordinal in the sense of M. Our main achievement is the calculation of the theory of C as precisely MOST+Δ0P Collection. The following theorems encapsulate our principal results: Theorem A. Every structure inC satisfies MOST+Δ0P Collection. Theorem B. Each of the following three conditions is sufficient for a countable structure (a) N is a transitive model of MOST+Δ0P Collection. (b) N is a recursively saturated model of MOST+Δ0P Collection. (c) N is a model of ZFC. Theorem C. Suppose M is a countable recursively saturated model of ZFC and I is a proper initial segment of OrdM that is closed under exponentiation and contains ωM. There is a group embedding j⟼j from Aut(Q) into Aut(M) such that I is the longest initial segment of OrdM that is pointwise fixed by jˇ for every nontrivial j∈Aut(Q). In Theorem C, Aut(X) is the group of automorphisms of the structure X, and Q is the ordered set of rationals.