Relatively recursive expansions
Tom 140 / 1992
Fundamenta Mathematicae 140 (1992), 137-155
DOI: 10.4064/fm-140-2-137-155
Streszczenie
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 $R ≤_T D(B)$? 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 also some variants of the basic question, in which R is r.e., $Δ_α^0$, or $Σ_α$ instead of recursive relative to D(B).
