$\leq_{SP}$ Can Have Infinitely Many Classes
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2018
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| _version_ | 1866909321311289344 |
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| author | Shelah, Saharon Ulrich, Danielle |
| author_facet | Shelah, Saharon Ulrich, Danielle |
| contents | Building off of recent results on Keisler's order, we show that consistently, $\leq_{SP}$ has infinitely many classes. In particular, we define the property of $\leq k$-type amalgamation for simple theories, for each $2 \leq k < ω$. If we let $T_{n, k}$ be the theory of the random $k$-ary, $n$-clique free random hyper-graph, then $T_{n, k}$ has $\leq k-1$-type amalgamation but not $\leq k$-type amalgamation. We show that consistently, if $T$ has $\leq k$-type amalgamation then $T_{k+1, k} \not \leq_{SP} T$, thus producing infinitely many $\leq_{SP}$-classes. The same construction gives a simplified proof of Shelah's theorem that consistently, the maximal $\leq_{SP}$-class is exactly the class of unsimple theories. Finally, we show that consistently, if $T$ has $<\aleph_0$-type amalgamation, then $T \leq_{SP} T_{rg}$, the theory of the random graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1804_08523 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | $\leq_{SP}$ Can Have Infinitely Many Classes Shelah, Saharon Ulrich, Danielle Logic 03C55, 03E35 Building off of recent results on Keisler's order, we show that consistently, $\leq_{SP}$ has infinitely many classes. In particular, we define the property of $\leq k$-type amalgamation for simple theories, for each $2 \leq k < ω$. If we let $T_{n, k}$ be the theory of the random $k$-ary, $n$-clique free random hyper-graph, then $T_{n, k}$ has $\leq k-1$-type amalgamation but not $\leq k$-type amalgamation. We show that consistently, if $T$ has $\leq k$-type amalgamation then $T_{k+1, k} \not \leq_{SP} T$, thus producing infinitely many $\leq_{SP}$-classes. The same construction gives a simplified proof of Shelah's theorem that consistently, the maximal $\leq_{SP}$-class is exactly the class of unsimple theories. Finally, we show that consistently, if $T$ has $<\aleph_0$-type amalgamation, then $T \leq_{SP} T_{rg}$, the theory of the random graph. |
| title | $\leq_{SP}$ Can Have Infinitely Many Classes |
| topic | Logic 03C55, 03E35 |
| url | https://arxiv.org/abs/1804.08523 |