Ash, Chris; Knight, Julia; Manasse, Mark; Slaman, Theodore Generic copies of countable structures. (English) Zbl 0678.03012 Ann. Pure Appl. Logic 42, No. 3, 195-205 (1989). There is a series of papers concerning intrinsical relations, stability and categoricity of recursive structures in hyperarithmetical degrees [e.g. C. J. Ash, Ann. Pure Appl. Logic 32, 113-135 (1986; Zbl 0631.03016)]. It was shown that for these properties there is a natural syntactic condition which is sufficient and, with additional assumptions, is also necessary. In this paper arbitrary structures whose domain is the set of natural numbers, not just recursive structures, are considered. Recursiveness of relations is replaced by recursiveness relative to the diagram of a structure. It is proved that for this generalization, the natural syntactic condition is both necessary and sufficient. This is done producing an isomorphic copy of a given structure by forcing with finite partial isomorphisms. Reviewer: S.S.Starchenko Cited in 2 ReviewsCited in 72 Documents MSC: 03C15 Model theory of denumerable and separable structures 03C25 Model-theoretic forcing 03C35 Categoricity and completeness of theories Keywords:denumerable models; categoricity; intrinsical relations Citations:Zbl 0631.03016 PDFBibTeX XMLCite \textit{C. Ash} et al., Ann. Pure Appl. Logic 42, No. 3, 195--205 (1989; Zbl 0678.03012) Full Text: DOI References: [1] Ash, C. J., Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees, Trans. Amer. Math. Soc., 298, 497-514 (1986) · Zbl 0631.03017 [2] Ash, C. J., Categoricity in hyperarithmetical degrees, Ann. Pure Appl. Logic, 34, 1-34 (1987) · Zbl 0617.03016 [3] Ash, C. J.; Nerode, A., Intrinsically recursive relations, (Crossley, J. N., Aspects of Effective Algebra (1981), U.D.A. Book Co: U.D.A. Book Co Yarra Glen, Australia), 26-41 · Zbl 0467.03041 [4] Barker, E., Intrinsically \(Σ^o_{α\) · Zbl 0651.03034 [5] J. Chisolm, Doctoral Dissertation, University of Wisconsin, Madison, to appear.; J. Chisolm, Doctoral Dissertation, University of Wisconsin, Madison, to appear. [6] Knight, J. F., Degrees coded in jumps of ordering, J. Symbolic Logic, 51, 1034-1042 (1986) · Zbl 0633.03038 [7] Rogers, H., Theory of Recursive Functions and Effective Computability (1967), McGraw-Hill: McGraw-Hill New York · Zbl 0183.01401 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.