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| Format: | Preprint |
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2018
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| Online Access: | https://arxiv.org/abs/1811.09902 |
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| _version_ | 1866912035465330688 |
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| author | Ulrich, Danielle |
| author_facet | Ulrich, Danielle |
| contents | Malliaris and Shelah famously proved that Keisler's order $\trianglelefteq$ has infinitely many classes. In more detail, for each $2 \leq k < n < ω$, let $T_{n, k}$ be the theory of the random $k$-ary $n$-clique free hypergraph. Malliaris and Shelah show that whenever $k+1 < k'$, then $T_{k+1, k} \not \trianglelefteq T_{k'+1, k'}$. However, their arguments do not separate $T_{k+1, k}$ from $T_{k+2, k+1}$, and the model-theoretic properties detected by their ultrafilters are difficult to evaluate in practice.
We uniformize the relevant ultrafilter constructions and obtain sharper model-theoretic bounds. As a sample application, we prove the following: suppose $3 \leq k < \aleph_0$, and $T$ is a countable low theory. Suppose that every independent system $(M_s: s \subsetneq k)$ of countable models of $T$ can be independently amalgamated. Then $T_{k, k-1} \not \trianglelefteq T$. In particular, for all $k < k'$, $T_{k+1, k} \not \trianglelefteq T_{k'+1, k'}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1811_09902 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Amalgamation and Keisler's Order Ulrich, Danielle Logic 03C55 Malliaris and Shelah famously proved that Keisler's order $\trianglelefteq$ has infinitely many classes. In more detail, for each $2 \leq k < n < ω$, let $T_{n, k}$ be the theory of the random $k$-ary $n$-clique free hypergraph. Malliaris and Shelah show that whenever $k+1 < k'$, then $T_{k+1, k} \not \trianglelefteq T_{k'+1, k'}$. However, their arguments do not separate $T_{k+1, k}$ from $T_{k+2, k+1}$, and the model-theoretic properties detected by their ultrafilters are difficult to evaluate in practice. We uniformize the relevant ultrafilter constructions and obtain sharper model-theoretic bounds. As a sample application, we prove the following: suppose $3 \leq k < \aleph_0$, and $T$ is a countable low theory. Suppose that every independent system $(M_s: s \subsetneq k)$ of countable models of $T$ can be independently amalgamated. Then $T_{k, k-1} \not \trianglelefteq T$. In particular, for all $k < k'$, $T_{k+1, k} \not \trianglelefteq T_{k'+1, k'}$. |
| title | Amalgamation and Keisler's Order |
| topic | Logic 03C55 |
| url | https://arxiv.org/abs/1811.09902 |