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SELF-EMBEDDINGS OF MODELS OF ARITHMETIC; FIXED POINTS, SMALL SUBMODELS, AND EXTENDABILITY

Part of: Model theory Nonstandard models Proof theory and constructive mathematics

Published online by Cambridge University Press:  22 December 2022

SAEIDEH BAHRAMI Open the ORCID record for SAEIDEH BAHRAMI [Opens in a new window]
SAEIDEH BAHRAMI*
Affiliation:
SCHOOL OF MATHEMATICS INSTITUTE FOR RESEARCH IN FUNDAMENTAL SCIENCES (IPM) P.O. BOX 19395-5764 TEHRAN, IRAN
*
E-mail: bahrami.saeideh@gmail.com
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Abstract

In this paper we will show that for every cut I of any countable nonstandard model $\mathcal {M}$ of $\mathrm {I}\Sigma _{1}$, each I-small $\Sigma _{1}$-elementary submodel of $\mathcal {M}$ is of the form of the set of fixed points of some proper initial self-embedding of $\mathcal {M}$ iff I is a strong cut of $\mathcal {M}$. Especially, this feature will provide us with some equivalent conditions with the strongness of the standard cut in a given countable model $\mathcal {M}$ of $ \mathrm {I}\Sigma _{1} $. In addition, we will find some criteria for extendability of initial self-embeddings of countable nonstandard models of $ \mathrm {I}\Sigma _{1} $ to larger models.

Keywords

Peano arithmeticnonstandard modelself-embeddingfixed pointstrong cutsmall submodel

MSC classification

Primary: 03C62: Models of arithmetic and set theory 03F30: First-order arithmetic and fragments 03H15: Nonstandard models of arithmetic

Information

Type
Article
Information
The Journal of Symbolic Logic , Volume 89 , Issue 3 , September 2024 , pp. 1044 - 1066
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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