## “School of Mathematics”

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Paper   IPM / M / 17090
 School of Mathematics Title: Self-embeddings of models of arithmetic; fixed points, small submodels, and extendability Author(s): Saeideh Bahrami Status: To Appear Journal: J. Symbolic Logic Supported by: IPM
Abstract:
In this paper we will show that for every cut $I$ of any countable nonstandard model $\M$ of $\I\Sigma_{1}$, each $I$-small $\Sigma_{1}$-elementary submodel of $\M$ is of the form of the set of fixed points of some proper initial self-embedding of $\M$ iff $I$ is a strong cut of $\M$. Especially, this feature will provide us with some equivalent conditions with the strongness of the standard cut in a given countable model $\M$ of $\I\Sigma_{1}$. In addition, we will find some criteria for extendability of initial self-embeddings of countable nonstandard models of $\I\Sigma_{1}$ to larger models.