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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.10282 |
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| _version_ | 1866915933185900544 |
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| author | Bahrami, Saeideh |
| author_facet | Bahrami, Saeideh |
| contents | In this paper, we extend the concept of a Lascar generic automorphism in the setting of models of Peano arithmetic ($\mathrm{PA}$) to the subgroup of the automorphism group of a countable recursively saturated model $\mathcal{M}$ of $\mathrm{PA}$ that fixes pointwise a strong cut $I$ of $\mathcal{M}$, denoted by $(\mathrm{Aut}(\mathcal{M}))_{(I)}$. Then, we prove that:
(1) $(\mathrm{Aut}(\mathcal{M}))_{(I)}$ has the small index property.
(2) The cofinality of $(\mathrm{Aut}(\mathcal{M}))_{(I)}$ is uncountable.
(3) Any nontrivial normal subgroup of $(\mathrm{Aut}(\mathcal{M}))_{(I)}$ is meagre in it. In particular, the infinite cyclic group $\mathbb{Z}$ is not a homomorphic image of $(\mathrm{Aut}(\mathcal{M}))_{(I)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10282 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The automorphism group of countable recursively saturated models of Peano arithmetic and strong cuts Bahrami, Saeideh Logic In this paper, we extend the concept of a Lascar generic automorphism in the setting of models of Peano arithmetic ($\mathrm{PA}$) to the subgroup of the automorphism group of a countable recursively saturated model $\mathcal{M}$ of $\mathrm{PA}$ that fixes pointwise a strong cut $I$ of $\mathcal{M}$, denoted by $(\mathrm{Aut}(\mathcal{M}))_{(I)}$. Then, we prove that: (1) $(\mathrm{Aut}(\mathcal{M}))_{(I)}$ has the small index property. (2) The cofinality of $(\mathrm{Aut}(\mathcal{M}))_{(I)}$ is uncountable. (3) Any nontrivial normal subgroup of $(\mathrm{Aut}(\mathcal{M}))_{(I)}$ is meagre in it. In particular, the infinite cyclic group $\mathbb{Z}$ is not a homomorphic image of $(\mathrm{Aut}(\mathcal{M}))_{(I)}$. |
| title | The automorphism group of countable recursively saturated models of Peano arithmetic and strong cuts |
| topic | Logic |
| url | https://arxiv.org/abs/2604.10282 |