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Fixed points of self-embeddings of models of arithmetic

Journal article
Authors Saeideh Bahrami
Ali Enayat
Published in Annals of Pure and Applied Logic
Volume 169
Issue 6
Pages 487-513
ISSN 0168-0072
Publication year 2018
Published at Department of Philosophy, Linguistics and Theory of Science
Pages 487-513
Language en
Links front.math.ucdavis.edu/1703.02588
https://doi.org/10.1016/j.apal.2018...
Keywords Peano Arithmetic, Nonstandard model, Self-embedding, Fixed point, Strong cut
Subject categories Logic, Algebra and Logic

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 segment 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.

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