An analogue of U-rank for atomic classes
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866912216855347200 |
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| author | Baldwin, John T. Laskowski, Michael C. Shelah, Saharon |
| author_facet | Baldwin, John T. Laskowski, Michael C. Shelah, Saharon |
| contents | For a countable, complete, first-order theory $T$, we study $At$, the class of atomic models of $T$. We develop an analogue of $U$-rank and prove two results. On one hand, if some tp(d/a) is not ranked, then there are $2^{\aleph_1}$ non-isomorphic models in $At$ of size $\aleph_1$. On the other hand, if all types have finite rank, then the rank is fully additive and every finite tuple is dominated by an independent set of realizations of pseudo-minimal types. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_00984 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An analogue of U-rank for atomic classes Baldwin, John T. Laskowski, Michael C. Shelah, Saharon Logic 03C50 For a countable, complete, first-order theory $T$, we study $At$, the class of atomic models of $T$. We develop an analogue of $U$-rank and prove two results. On one hand, if some tp(d/a) is not ranked, then there are $2^{\aleph_1}$ non-isomorphic models in $At$ of size $\aleph_1$. On the other hand, if all types have finite rank, then the rank is fully additive and every finite tuple is dominated by an independent set of realizations of pseudo-minimal types. |
| title | An analogue of U-rank for atomic classes |
| topic | Logic 03C50 |
| url | https://arxiv.org/abs/2502.00984 |