Self-embeddings of models of arithmetic; fixed points, small submodels, and extendability
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866910704546611200 |
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| author | Bahrami, Saeideh |
| author_facet | Bahrami, Saeideh |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_11284 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Self-embeddings of models of arithmetic; fixed points, small submodels, and extendability Bahrami, Saeideh Logic 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. |
| title | Self-embeddings of models of arithmetic; fixed points, small submodels, and extendability |
| topic | Logic |
| url | https://arxiv.org/abs/2204.11284 |