Automorphisms of models of arithmetic: a unified view

Annals of Pure and Applied Logic 145 (1):16-36 (2007)
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Abstract

We develop the method of iterated ultrapower representation to provide a unified and perspicuous approach for building automorphisms of countable recursively saturated models of Peano arithmetic . In particular, we use this method to prove Theorem A below, which confirms a long-standing conjecture of James Schmerl.Theorem AIf is a countable recursively saturated model of in which is a strong cut, then for any there is an automorphism j of such that the fixed point set of j is isomorphic to .We also fine-tune a number of classical results. One of our typical results in this direction is Theorem B below, which generalizes a theorem of Kaye–Kossak–Kotlarski .Theorem BSuppose is a countable recursively saturated model of in which is a strong cut. There is a group embedding from into such that for each that is fixed point free, moves every undefinable element of

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Citations of this work

Fixed points of self-embeddings of models of arithmetic.Saeideh Bahrami & Ali Enayat - 2018 - Annals of Pure and Applied Logic 169 (6):487-513.
Rank-initial embeddings of non-standard models of set theory.Paul Kindvall Gorbow - 2020 - Archive for Mathematical Logic 59 (5-6):517-563.
Iterated ultrapowers for the masses.Ali Enayat, Matt Kaufmann & Zachiri McKenzie - 2018 - Archive for Mathematical Logic 57 (5-6):557-576.

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References found in this work

Models and types of Peano's arithmetic.Haim Gaifman - 1976 - Annals of Mathematical Logic 9 (3):223-306.
Toward model theory through recursive saturation.John Stewart Schlipf - 1978 - Journal of Symbolic Logic 43 (2):183-206.
Some applications of iterated ultrapowers in set theory.Kenneth Kunen - 1970 - Annals of Mathematical Logic 1 (2):179.
Recursively saturated models generated by indiscernibles.James H. Schmerl - 1985 - Notre Dame Journal of Formal Logic 26 (2):99-105.

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