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Classifying model-theoretic properties. (English) Zbl 1160.03012
Summary: In [J. Symb. Log. 69, No. 4, 1117–1142 (2004; Zbl 1071.03021)], B. F. Csima, D. R. Hirschfeldt, J. F. Knight, and R. I. Soare showed that a set \(A\leq _{\text T} 0'\) is nonlow\(_{2}\) if and only if \(A\) is prime bounding, i.e., for every complete atomic decidable theory \(T\), there is a prime model \(\mathcal M\) computable in \(A\). The authors presented nine seemingly unrelated predicates of a set \(A\), and showed that they are equivalent for \(\Delta ^{0}_{2}\) sets. Some of these predicates, such as prime bounding, and others involving equivalence structures and abelian \(p\)-groups come from model theory, while others involving meeting dense sets in trees and escaping a given function come from pure computability theory.
As predicates of \(A\), the original nine properties are equivalent for \(\Delta ^{0}_{2}\) sets; however, they are not equivalent in general. This article examines the (degree-theoretic) relationship between the nine properties. We show that the nine properties fall into three classes, each of which consists of several equivalent properties. We also investigate the relationship between the three classes, by determining whether or not any of the predicates in one class implies a predicate in another class.

MSC:
03C57 Computable structure theory, computable model theory
Citations:
Zbl 1071.03021
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References:
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