Categoricity transfer in simple finitary abstract elementary classes. (English) Zbl 1250.03055

Summary: We continue our study of finitary abstract elementary classes, defined in [the authors, Ann. Pure Appl. Logic 143, No. 1–3, 103–138 (2006; Zbl 1112.03026)]. In this paper, we prove a categoricity transfer theorem for a case of simple finitary AECs. We introduce the concepts of weak \(\kappa\)-categoricity and \(\mathrm{f}\)-primary models to the framework of \(\aleph_{0}\)-stable simple finitary AECs with the extension property, whereby we gain the following theorem: Let (\(\mathbb K, \curlyeqprec_{\mathbb {K}}\)) be a simple finitary AEC, weakly categorical in some uncountable \(\kappa\). Then (\(\mathbb K, \curlyeqprec_{\mathbb {K}}\)) is weakly categorical in each \(\lambda \geq \min\{\kappa ,\beth_{(2^{\aleph_{0}})^{+}\}}\). If the class (\(\mathbb K, \curlyeqprec_{\mathbb {K}}\)) is also LS(\(\mathbb K\))-tame, weak \(\kappa\)-categoricity is equivalent with \(\kappa\)-categoricity in the usual sense. We also discuss the relation between finitary AECs and some other non-elementary frameworks and give several examples.


03C48 Abstract elementary classes and related topics
03C35 Categoricity and completeness of theories


Zbl 1112.03026
Full Text: DOI


[1] DOI: 10.1007/BFb0082243
[2] DOI: 10.1016/S0168-0072(98)00016-5 · Zbl 0945.03049
[3] Classification theory and the number of nonisomorphic models (1978)
[4] DOI: 10.4064/fm195-3-3 · Zbl 1125.03027
[5] DOI: 10.1016/j.apal.2006.01.009 · Zbl 1112.03026
[6] DOI: 10.1007/BF02807869 · Zbl 0723.03017
[7] Shelah’s categoricity conjecture from a successor for tame abstract elementary classes 71 pp 553– (2006) · Zbl 1100.03023
[8] DOI: 10.1142/S0219061306000487 · Zbl 1107.03029
[9] DOI: 10.1142/S0219061306000554 · Zbl 1129.03019
[10] DOI: 10.1007/s11856-009-0035-8 · Zbl 1181.03038
[11] Categoricity 50 (2009) · Zbl 1183.03002
[12] Annals of Mathematical Logic 2 pp 293– (1970)
[13] Upward categoricity from a successor cardinal for tame abstract classes with amalgamation 70 pp 639– (2005) · Zbl 1089.03025
[14] An introduction to uncountable categoricity in abstract elementary classes 6 (2005)
[15] DOI: 10.1090/conm/380/07114
[16] DOI: 10.1305/ndjfl/1172787550 · Zbl 1130.03026
[17] DOI: 10.1016/j.apal.2008.07.002 · Zbl 1155.03016
[18] Model theory for infinitary logic (1971)
[19] DOI: 10.1142/S0219061305000390 · Zbl 1082.03033
[20] DOI: 10.1016/j.apal.2005.04.002 · Zbl 1093.03017
[21] A rank for the class of elementary submodels of a superstable homogeneous model 67 pp 1469– (2002) · Zbl 1039.03024
[22] DOI: 10.1007/BF02761954 · Zbl 0552.03019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.