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Paper IPM / M / 17090  


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 selfembedding 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 selfembeddings of countable nonstandard models of $ \I\Sigma_{1} $ to larger models.
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