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**1 - 1**of**1**### Notre Dame Journal of Formal Logic Cuppability of Simple and Hypersimple Sets

"... Abstract An incomplete degree is cuppable if it can be joined by an incomplete degree to a complete degree. For sets fulfilling some type of simplicity property one can now ask whether these sets are cuppable with respect to a certain type of reducibilities. Several such results are known. In this p ..."

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Abstract An incomplete degree is cuppable if it can be joined by an incomplete degree to a complete degree. For sets fulfilling some type of simplicity property one can now ask whether these sets are cuppable with respect to a certain type of reducibilities. Several such results are known. In this paper we settle all the remaining cases for the standard notions of simplicity and all the main strong reducibilities. There are two sides to every question.--Protagoras, quoted in Diogenes Laertius, Lives of Eminent Philosophers. 1 Introduction In his approach to constructing an incomplete c.e. degree, Emil Post at-tempted to define structural properties of c.e. sets that would force their incompleteness. In his groundbreaking 1944 paper Recursively enumerablesets of positive integers and their decision problems ([24], reprinted in Davis's The Undecidable [1]) this goal led him to isolate many of the classical con-cepts of computability, including creativity, many-one reducibility, bounded and unbounded truth-table reducibility, simplicity, hypersimplicity, and hy-perhypersimplicity.