Geodesic
The L\"owenheim--Skolem--Mal'tsev Theorem for $\mathbb{HF}$-Structures
Algebra i logika, Tome 43 (2004) no. 6, pp. 749-758.

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We deal with the problem asking whether hereditarily finite superstructures have elementary extensions of the form $\mathbb{HF}(\mathfrak M)$. In so doing, we settle the question whether a theory for some hereditarily finite superstructure have $\mathbb{HF}(\mathfrak M)$ models of arbitrarily large cardinality. A Hanf number is shown to exist, and we provide an exact bound for the countable case.
Keywords: hereditarily finite superstructure, Hanf number. 
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V. G. Puzarenko. The L\"owenheim--Skolem--Mal'tsev Theorem for $\mathbb{HF}$-Structures. Algebra i logika, Tome 43 (2004) no. 6, pp. 749-758. http://geodesic.mathdoc.fr/item/AL_2004_43_6_a5/

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