Model Theory for Hereditarily Finite Superstructures

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.