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The theory of the metarecursively enumerable degrees
"... Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equival ..."
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Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equivalently, that of the truth set of L ω CK
THE THEORY OF THE α DEGREES IS UNDECIDABLE
"... Abstract. Let α be an arbitrary Σ1admissible ordinal. For each n, there is a formula ϕn(⃗x, ⃗y) such that for any relation R on a finite set F with n elements, there are αdegrees ⃗p such that the relation defined by ϕn(⃗x, ⃗p) is isomorphic to R. Consequently, the theory of αdegrees is undecidabl ..."
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Abstract. Let α be an arbitrary Σ1admissible ordinal. For each n, there is a formula ϕn(⃗x, ⃗y) such that for any relation R on a finite set F with n elements, there are αdegrees ⃗p such that the relation defined by ϕn(⃗x, ⃗p) is isomorphic to R. Consequently, the theory of αdegrees is undecidable. 1.