Abstract
Analyzing diagrams forming generative classes, we describe definable sets and their links in generic structures as well as cardinality bounds for these definable sets, finite or infinite. Introducing basic characteristics for definable sets in generic structures, we compare them each others and with cardinalities of these sets.We introduce calculi for (type-)definable sets allowing to compare their cardinalities. In terms of these calculi, Trichotomy Theorem for possibilities comparing cardinalities of definable sets is proved. Using these calculi, we characterize the possibility to construct a generic structure of a given generative class.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Yu. L. Ershov, Definability and Computability (Consultants Bureau, New York, 1996).
E. R. Baisalov, K. A. Meirembekov, and A. T. Nurtazin, “Definably minimal models,” in Model Theory in Kazakhstan, Collection of papers in the memory of A. D. Taimanov (Eco Study, Almaty, 2006), 14.
B. S. Baizhanov, “Orthogonality of one types in weakly o-minimal theories,” in Algebra and Model Theory 2. Collection of papers (NSTU, Novosibirsk, 1999), 5.
B. S. Baizhanov and B. Sh. Kulpeshov, “On behaviour of 2-formulas in weakly o-minimal theories,” in Mathematical Logic in Asia, Proceedings of the 9th Asian Logic Conference, August 16-19, 2005, Novosibirsk, Russia (World Scientific, Singapore, 2006), 31.
B. S. Baizhanov, S. V. Sudoplatov, and V. V. Verbovskiy, “Conditions for nonsymmetric relations of semiisolation”, Siberian ElectronicMath. Reports 9, 161 ( 2012).
J. T. Baldwin and A. H. Lachlan, “On strongly minimal sets,” J. Symbolic Logic 36, 79 (1971).
W. Hodges, Model Theory (Cambridge Univ. Press, Cambridge, 1993).
D. Marker, Model Theory: An Introduction, Graduate texts in Mathematics 217 (Springer-Verlag, New York, 2002).
K. Zh. Kudaibergenov, “Cardinalities of definable sets in superstructures over models,” Siberian Adv. Math. 20, 58 (2010).
K. Zh. Kudaibergenov, “On FraïsséTheorem for an uncountable class of finitely generated structures,” Siberian Adv. Math. 28, 60 (2018).
A. Pillay, “Countable models of stable theories,” Proc. Amer. Math. Soc. 89, 666 (1983).
A. Pillay, “Stable theories, pseudoplanes and the number of countable models,” Ann. Pure and Appl. Logic 43, 147 (1989).
J. Reineke, “Minimalle Gruppen,” Zeitschr.Math. Logic 21, 357 (1975).
S. V. Sudoplatov, “Inessential combinations and colorings of models,” Siberian Math. J. 44 883 (2003).
S. V. Sudoplatov, “Syntactic approach to constructions of generic models,” Algebra and Logic 46, 134 (2007).
S. V. Sudoplatov, The Lachlan Problem (NSTU, Novosibirsk, 2009).
S. V. Sudoplatov, “Hypergraphs of prime models and distributions of countable models of small theories,” J. Math. Sciences 169, 680 (2010).
S. V. Sudoplatov, “Algebras of distributions of formulas with respect to generalized semi-isolation,” in Algebra and Model Theory 9, Collection of papers (NSTU, Novosibirsk, 2013), 67.
S. V. Sudoplatov, Classification of Countable Models of Complete Theories (NSTU, Novosibirsk, 2014).
S. V. Sudoplatov, “Generative and pre-generative classes,” Proceedings of the 10th Panhellenic Logic Symposium, June 11-15, 2015, Samos, Greece (Uni. of Aegean, Uni. of Crete, and Uni. of Athens, 2015), 30.
S. V. Sudoplatov, “Generative classes generated by sets of diagrams,” in Algebra and Model Theory 10, Collection of papers (NSTU, Novosibirsk, 2015), 163.
S. V. Sudoplatov, Y. Kiouvrekis, and P. Stefaneas, “Generic constructions and generic limits,” in Algebraic Modeling of Topological and Computational Structures and Applications, Springer Proceedings in Mathematics & Statistics (Springer Int. Publ., Berlin, 2017), 219.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © Y. Kiouvrekis, P. Stefaneas, and S.V. Sudoplatov, 2017, published in Matematicheskie Trudy, 2017, Vol. 20, No. 2, pp. 52–79.
About this article
Cite this article
Kiouvrekis, Y., Stefaneas, P. & Sudoplatov, S.V. Definable Sets in Generic Structures and their Cardinalities. Sib. Adv. Math. 28, 39–52 (2018). https://doi.org/10.3103/S1055134418010030
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1055134418010030