# Fixed Points of Self-embeddings of Models of Arithmetic

## TL;DR

This paper characterizes the fixed point sets of self-embeddings in models of arithmetic, revealing how initial segments relate to these embeddings and the conditions under which the standard cut is strong.

## Contribution

It provides complete characterizations of initial segments as fixed points or longest fixed points of self-embeddings in models of arithmetic, a novel analysis in the field.

## Key findings

- Initial segments as fixed points of self-embeddings are characterized.
- The standard cut's strength relates to specific self-embeddings that move non-definable elements.
- Conditions for the existence of certain self-embeddings are established.

## Abstract

We investigate the structure of fixed point sets of self-embeddings of models of arithmetic. In particular, given a countable nonstandard model M of a modest fragment of Peano arithimetic, we provide complete characterizations of (a) the initial segments of M that can be realized as the longest initial segment of fixed points of a nontrivial self-embedding of M onto a proper initial segment of M; and (b) the initial segments of M that can be realized as the fixed point set of some nontrivial self-embedding of M onto a proper initial segement of M. Moreover, we demonstrate the the standard cut is strong in M iff there is a self-embedding of M onto a proper initial segment of itself that moves every element that is not definable in M by an existential formula.