document: 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 -LRB- up to isomorphism -RRB- hereditarily finite superstructure can be described via definable functions . Yet theories for such superstructures admit a description in terms of iterated families TF and 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 -LRB- up to isomorphism -RRB- countable hereditarily finite superstructure
doc_bio_tags: B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O B I I I
extractive_keyphrases: model theory countable signatures iterated families definable union finitely many complete subsets countable hereditarily finite superstructure
abstractive_keyphrases: model-theoretic properties