A separation theorem for simple theories
HTML articles powered by AMS MathViewer
- by M. Malliaris and S. Shelah PDF
- Trans. Amer. Math. Soc. 375 (2022), 1171-1205 Request permission
Abstract:
This paper builds model-theoretic tools to detect changes in complexity among the simple theories. We develop a generalization of dividing, called shearing, which depends on a so-called context $\mathbf {c}$. This leads to defining $\mathbf {c}$-superstability, a syntactical notion, which includes supersimplicity as a special case. The main result is a separation theorem showing that for any countable context $\mathbf {c}$ and any two theories $T_1$, $T_2$ such that $T_1$ is $\mathbf {c}$-superstable and $T_2$ is $\mathbf {c}$-unsuperstable, and for arbitrarily large $\mu$, it is possible to build models of any theory interpreting both $T_1$ and $T_2$ whose restriction to $\tau (T_1)$ is $\mu$-saturated and whose restriction to $\tau (T_2)$ is not $\aleph _1$-saturated. (This suggests “$\mathbf {c}$-superstable” is really a dividing line.) The proof uses generalized Ehrenfeucht-Mostowski models, and along the way, we clarify the use of these techniques to realize certain types while omitting others. In some sense, shearing allows us to study the interaction of complexity coming from the usual notion of dividing in simple theories and the more combinatorial complexity detected by the general definition. This work is inspired by our recent progress on Keisler’s order, but does not use ultrafilters, rather aiming to build up the internal model theory of these classes.References
- Z. Chatzidakis and E. Hrushovski, Algebraically closed fields with an automorphism, Trans. AMS 351 (1999), no. 8, 2997–3071.
- A. Ehrenfeucht and A. Mostowski, Models of axiomatic theories admitting automorphisms, Fund. Math. 43 (1956), 50–68. MR 84456, DOI 10.4064/fm-43-1-50-68
- Rami Grossberg, José Iovino, and Olivier Lessmann, A primer of simple theories, Arch. Math. Logic 41 (2002), no. 6, 541–580. MR 1923196, DOI 10.1007/s001530100126
- Vincent Guingona, Cameron Donnay Hill, and Lynn Scow, Characterizing model-theoretic dividing lines via collapse of generalized indiscernibles, Ann. Pure Appl. Logic 168 (2017), no. 5, 1091–1111. MR 3620067, DOI 10.1016/j.apal.2016.11.007
- Ehud Hrushovski, Pseudo-finite fields and related structures, Model theory and applications, Quad. Mat., vol. 11, Aracne, Rome, 2002, pp. 151–212. MR 2159717
- J. Hubička and J. Nešetřil, All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms), Adv. Math. 356C (2019), 106791.
- A. S. Kechris, V. G. Pestov, and S. Todorcevic, Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups, Geom. Funct. Anal. 15 (2005), no. 1, 106–189. MR 2140630, DOI 10.1007/s00039-005-0503-1
- M. E. Malliaris, Hypergraph sequences as a tool for saturation of ultrapowers, J. Symbolic Logic 77 (2012), no. 1, 195–223. MR 2951637, DOI 10.2178/jsl/1327068699
- M. Malliaris and S. Shelah, Existence of optimal ultrafilters and the fundamental complexity of simple theories, Adv. Math. 290 (2016), 614–681. MR 3451934, DOI 10.1016/j.aim.2015.12.009
- Maryanthe Malliaris and Saharon Shelah, Keisler’s order has infinitely many classes, Israel J. Math. 224 (2018), no. 1, 189–230. MR 3799754, DOI 10.1007/s11856-018-1647-7
- M. Malliaris and S. Shelah, A new look at interpretability and saturation, Ann. Pure Appl. Logic 170 (2019), no. 5, 642–671. MR 3926500, DOI 10.1016/j.apal.2019.01.001
- M. Malliaris and S. Shelah, Shearing in some simple rank one theories. arXiv::2109.12642 (2021).
- Jaroslav Ne et il, Ramsey classes and homogeneous structures, Combin. Probab. Comput. 14 (2005), no. 1-2, 171–189. MR 2128088, DOI 10.1017/S0963548304006716
- Jaroslav Ne et il and Vojtěch Rödl, Ramsey classes of set systems, J. Combin. Theory Ser. A 34 (1983), no. 2, 183–201. MR 692827, DOI 10.1016/0097-3165(83)90055-9
- Michael O. Rabin, Universal groups of automorphisms of models, Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), North-Holland, Amsterdam, 1965, pp. 274–284. MR 0201307
- Lynn Scow, Characterization of NIP theories by ordered graph-indiscernibles, Ann. Pure Appl. Logic 163 (2012), no. 11, 1624–1641. MR 2959664, DOI 10.1016/j.apal.2011.12.013
- Saharon Shelah, Simple unstable theories, Ann. Math. Logic 19 (1980), no. 3, 177–203. MR 595012, DOI 10.1016/0003-4843(80)90009-1
- Saharon Shelah, Classification theory and the number of nonisomorphic models, Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 513226
- Saharon Shelah, General non structure theory, E59, 2016, http://shelah.logic.at/files/E59.pdf.
- Saharon Shelah, Toward classifying unstable theories, Ann. Pure Appl. Logic 80 (1996), no. 3, 229–255. MR 1402297, DOI 10.1016/0168-0072(95)00066-6
Additional Information
- M. Malliaris
- Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
- MR Author ID: 864805
- Email: mem@math.uchicago.edu
- S. Shelah
- Affiliation: Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem 91904, Israel; and Department of Mathematics, Hill Center - Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
- MR Author ID: 160185
- ORCID: 0000-0003-0462-3152
- Email: shelah@math.huji.ac.il
- Received by editor(s): October 22, 2018
- Received by editor(s) in revised form: June 18, 2020, and June 14, 2021
- Published electronically: December 2, 2021
- Additional Notes: The first author was partially supported by NSF CAREER award 1553653 and a Minerva research foundation membership at IAS
The second author was partially supported by European Research Council grant 338821 and ISF grant 1838/19. Both authors thank NSF grant 1362974 to the second author at Rutgers, ERC 338821, and NSF-BSF 2051825. This is paper 1149 in the second author’s list. - © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 1171-1205
- MSC (2020): Primary 03C45, 03C50, 03C68
- DOI: https://doi.org/10.1090/tran/8513
- MathSciNet review: 4369245