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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
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.

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