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Relatively recursive expansions. (English) Zbl 0809.03023

Summary: [Part II is reviewed below.]
We consider the following basic question. Let \(\mathbf A\) be an \(L\)- structure and let \(\psi\) be an infinitary sentence in the language \(L \cup \{R\}\), where \(R\) is a new relation symbol. When is it the case that for every \({\mathbf B} \cong {\mathbf A}\), there is a relation \(R\) such that \(({\mathbf B}, R) \models\psi\) and \(R \leq_ T D({\mathbf B})\)? We succeed in giving necessary and sufficient conditions in the case where \(\psi\) is a “recursive” infinitary \(\Pi_ 2\) 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., \(\Delta^ 0_ \alpha\), or \(\Sigma_ \alpha\) instead of recursive relative to \(D({\mathbf B})\).

MSC:

03C57 Computable structure theory, computable model theory
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