Numberings and definability in the hereditarily finite superstructure of a model
A model \(M\) is said to be \(\Sigma\)-definable over an admissible set \(A\) provided it is isomorphic to a model whose basic set is a quotient of a \(\Sigma\)-subset of \(A\) modulo a \(\Delta\)-equivalence and all predicates and functions are \(\Sigma\)-definable modulo this quotient along with their complements. If a model possesses a \(\Sigma\)-definable numbering, it is called \(\Sigma\)-enumerable or internally enumerable. The author proves a natural criterion for \(\Sigma\)-definability of a model in the hereditarily finite superstructure over another internally enumerable model. He also proves that, if a countable superatomic Boolean algebra is \(\Sigma\)-definable over a hereditarily finite superstructure of an internally enumerable model, then its ordinal type is also \(\Sigma\)-definable over it. The same result is proven for reduced Abelian groups and their Ulm types. The article under review is a translation of Vychisl. Sist. 156, 44-58 (1996; Zbl 0902.03030).
scientific article; zbMATH DE number 1353848