Generic copies of countable structures. (English) Zbl 0678.03012

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


03C15 Model theory of denumerable and separable structures
03C25 Model-theoretic forcing
03C35 Categoricity and completeness of theories


Zbl 0631.03016
Full Text: DOI


[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
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