Geodesic
Cardinalities of definable sets in superstructures over models
Matematičeskie trudy, Tome 12 (2009) no. 2, pp. 97-110

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We introduce the notion of a superstructure over a model. This is a generalization of the notion of the hereditarily finite superstructure $\mathbb{HF}(\mathfrak M)$ over a model $\mathfrak M$. We consider the question on cardinalities of definable (interpretable) sets in superstructures over $\lambda$-homogeneous and $\lambda$-saturated models. 
@article{MT_2009_12_2_a3,
     author = {K. Zh. Kudaibergenov},
     title = {Cardinalities of definable sets in superstructures over models},
     journal = {Matemati\v{c}eskie trudy},
     pages = {97--110},
     publisher = {mathdoc},
     volume = {12},
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     year = {2009},
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     url = {http://geodesic.mathdoc.fr/item/MT_2009_12_2_a3/}
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K. Zh. Kudaibergenov. Cardinalities of definable sets in superstructures over models. Matematičeskie trudy, Tome 12 (2009) no. 2, pp. 97-110. http://geodesic.mathdoc.fr/item/MT_2009_12_2_a3/