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Computable trees, prime models, and relative decidability
 PROC. AMER. MATH. SOC
, 2005
"... We show that for every computable tree T with no dead ends and all paths computable, and every D>T ∅, there is a Dcomputable listing of the isolated paths of T. It follows that for every complete decidable theory T such that all the types of T are computable and every D>T ∅, there is a Ddec ..."
Abstract

Cited by 5 (3 self)
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We show that for every computable tree T with no dead ends and all paths computable, and every D>T ∅, there is a Dcomputable listing of the isolated paths of T. It follows that for every complete decidable theory T such that all the types of T are computable and every D>T ∅, there is a Ddecidable prime model of T. This result extends a theorem of Csima and yields a stronger version of the theorem, due independently to Slaman and Wehner, that there is a structure with presentations of every nonzero degree but no computable presentation.
INDUCTION, BOUNDING, WEAK COMBINATORIAL PRINCIPLES, AND THE HOMOGENEOUS MODEL THEOREM
, 2014
"... Abstract. Goncharov and Peretyat’kin independently gave necessary and sufficient conditions for when a set of types of a complete theory T is the type spectrum of some homogeneous model of T. Their result can be stated as a principle of second order arithmetic, which we call the Homogeneous Model Th ..."
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Abstract. Goncharov and Peretyat’kin independently gave necessary and sufficient conditions for when a set of types of a complete theory T is the type spectrum of some homogeneous model of T. Their result can be stated as a principle of second order arithmetic, which we call the Homogeneous Model Theorem (HMT), and analyzed from the points of view of computability theory and reverse mathematics. Previous computability theoretic results by Lange suggested a close connection between HMT and the Atomic Model Theorem (AMT), which states that every complete atomic theory has an atomic model. We show that HMT and AMT are indeed equivalent in the sense of reverse mathematics, as well as in a strong computability theoretic sense. We do the same for an analogous result of Peretyat’kin giving necessary and sufficient conditions for when a set of types is the type spectrum of some model.