Generic copies of countable structures.

*(English)*Zbl 0678.03012There 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.

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

##### MSC:

03C15 | Model theory of denumerable and separable structures |

03C25 | Model-theoretic forcing |

03C35 | Categoricity and completeness of theories |

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\textit{C. Ash} et al., Ann. Pure Appl. Logic 42, No. 3, 195--205 (1989; Zbl 0678.03012)

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##### 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, (), 26-41 · Zbl 0467.03041 |

[4] | Barker, E., Intrinsically σoα relations, Ann. pure appl. logic, 39, 105-130, (1988) · Zbl 0651.03034 |

[5] | 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 New York · Zbl 0183.01401 |

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