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| Main Author: | |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2204.11284 |
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Table of Contents:
- In this paper we will show that for every cut $ I $ of any countable nonstandard model $ \mathcal{M} $ of $ \mathrm{I}Σ_{1} $, each $ I $-small $ Σ_{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}Σ_{1} $. In addition, we will find some criteria for extendability of initial self-embeddings of countable nonstandard models of $ \mathrm{I}Σ_{1} $ to larger models.