Geodesic
Model Theory for Hereditarily Finite Superstructures
Algebra i logika, Tome 41 (2002) no. 2, pp. 199-222.

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We study model-theoretic properties of hereditarily finite superstructures over models of not more than countable signatures. A question is answered in the negative inquiring whether theories of hereditarily finite superstructures which have a unique (up to isomorphism) hereditarily finite superstructure can be described via definable functions. Yet theories for such superstructures admit a description in terms of iterated families $\mathcal{TF}$ and $\mathcal{SF}$. These are constructed using a definable union taken over countable ordinals in the subsets which are unions of finitely many complete subsets and of finite subsets, respectively. Simultaneously, we describe theories that share a unique (up to isomorphism) countable hereditarily finite superstructure.
Keywords: hereditarily finite superstructures, $\omega$-logic. 
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V. G. Puzarenko. Model Theory for Hereditarily Finite Superstructures. Algebra i logika, Tome 41 (2002) no. 2, pp. 199-222. http://geodesic.mathdoc.fr/item/AL_2002_41_2_a5/

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