Fixed points of self-embeddings of models of arithmetic

Abstract

We investigate the structure of fixed point sets of self-embeddings of models of arithmetic. Our principal results are Theorems A, B, and C below. In what follows M is a countable nonstandard model of the fragment IΣ1 of PA (Peano Arithmetic); N is the initial segment of M consisting of standard numbers of M; Ifix(j) is the longest initial segment of fixed points of j; Fix(j) is the fixed point set of j; K1(M) consists of Σ1-definable elements of M; and a self-embedding j of M is said to be a proper initial self-embedding if j(M) is a proper initial segment of M. Theorem AThe following are equivalent for a proper initial segment I ofM:(1)I=Ifix(j)for some self-embedding j ofM.(2)I is closed under exponentiation.(3)I=Ifix(j)for some proper initial self-embedding j ofM.Theorem BThe following are equivalent for a proper initial segment I ofM:(1)I=Fix(j)for some self-embedding j ofM.(2)I is a strong cut ofMandI≺Σ1M.(3)I=Fix(j)for some proper initial self-embedding j ofM.Theorem CThe following are equivalent:(1)Fix(j)=K1(M)for some self-embedding j ofM.(2)Nis a strong cut ofM.(3)Fix(j)=K1(M)for some proper initial self-embedding j ofM.