The Pentagon as a Substructure Lattice of Models of Peano Arithmetic
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2019
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| _version_ | 1866915496143618048 |
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| author | Schmerl, James H. |
| author_facet | Schmerl, James H. |
| contents | Wilke proved in 1977 that every countable model ${\mathcal M}$ of Peano Arithmetic has an elementary end extension ${\mathcal N}$ such that the interstructure lattice Lt(${\mathcal N} / {\mathcal M}$) is the pentagon lattice ${\mathbf N}_5$. This theorem implies that every countable nonstandard $\mathcal M$ has an elementary cofinal extension such that Lt(${\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$. It is proved here that if ${\mathcal M} \prec {\mathcal N}$ and Lt(${\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$, then ${\mathcal N}$ is either an end or a cofinal extension of ${\mathcal M}$. In contrast, there are ${\mathcal M}^* \prec {\mathcal N}^* \models {\mathsf PA}^*$ such that Lt(${\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$ and ${\mathcal N}^*$ is neither an end nor a cofinal extension of ${\mathcal M}^*$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_05284 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | The Pentagon as a Substructure Lattice of Models of Peano Arithmetic Schmerl, James H. Logic 03H15 Wilke proved in 1977 that every countable model ${\mathcal M}$ of Peano Arithmetic has an elementary end extension ${\mathcal N}$ such that the interstructure lattice Lt(${\mathcal N} / {\mathcal M}$) is the pentagon lattice ${\mathbf N}_5$. This theorem implies that every countable nonstandard $\mathcal M$ has an elementary cofinal extension such that Lt(${\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$. It is proved here that if ${\mathcal M} \prec {\mathcal N}$ and Lt(${\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$, then ${\mathcal N}$ is either an end or a cofinal extension of ${\mathcal M}$. In contrast, there are ${\mathcal M}^* \prec {\mathcal N}^* \models {\mathsf PA}^*$ such that Lt(${\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$ and ${\mathcal N}^*$ is neither an end nor a cofinal extension of ${\mathcal M}^*$. |
| title | The Pentagon as a Substructure Lattice of Models of Peano Arithmetic |
| topic | Logic 03H15 |
| url | https://arxiv.org/abs/1910.05284 |