Relatively recursive expansions II

C. Ash; J. Knight; Theodore Slaman

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Relatively recursive expansions

C. Ash, J. Knight (1992)

Fundamenta Mathematicae

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In this paper, we consider the following basic question. Let A be an L-structure and let ψ be an infinitary sentence in the language L∪R, where R is a new relation symbol. When is it the case that for every B ≅ A, there is a relation R such that (B,R) ⊨ ψ and $R \leq_{T} D (B)$ ? We succeed in giving necessary and sufficient conditions in the case where ψ is a “recursive” infinitary $Π_{2}$ sentence. (A recursive infinitary formula is an infinitary formula with recursive disjunctions and conjunctions.) We consider...