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Bounding Nonsplitting Enumeration Degrees

Thomas F. Kent & Andrea Sorbi

Journal of Symbolic Logic 72 (4):1405 - 1417 (2007)

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Enumeration 1-Genericity in the Local Enumeration Degrees. [REVIEW]Liliana Badillo, Charles M. Harris & Mariya I. Soskova - 2018 - Notre Dame Journal of Formal Logic 59 (4):461-489.details We discuss a notion of forcing that characterizes enumeration 1-genericity, and we investigate the immunity, lowness, and quasiminimality properties of enumeration 1-generic sets and their degrees. We construct an enumeration operator Δ such that, for any A, the set ΔA is enumeration 1-generic and has the same jump complexity as A. We deduce from this and other recent results from the literature that not only does every degree a bound an enumeration 1-generic degree b such that a'=b', but also that, (...)) Download Export citation Bookmark 1 citation
(1 other version)Branching in the $${\Sigma^0_2}$$ -enumeration degrees: a new perspective. [REVIEW]Maria L. Affatato, Thomas F. Kent & Andrea Sorbi - 2008 - Archive for Mathematical Logic 47 (3):221-231.details We give an alternative and more informative proof that every incomplete ${\Sigma^{0}_{2}}$ -enumeration degree is the meet of two incomparable ${\Sigma^{0}_{2}}$ -degrees, which allows us to show the stronger result that for every incomplete ${\Sigma^{0}_{2}}$ -enumeration degree a, there exist enumeration degrees x 1 and x 2 such that a, x 1, x 2 are incomparable, and for all b ≤ a, b = (b ∨ x 1 ) ∧ (b ∨ x 2 ). Download Export citation Bookmark
The automorphism group and definability of the jump operator in the $$\omega $$ ω -enumeration degrees.Hristo Ganchev & Andrey C. Sariev - 2021 - Archive for Mathematical Logic 60 (7):909-925.details In the present paper, we show the first-order definability of the jump operator in the upper semi-lattice of the \-enumeration degrees. As a consequence, we derive the isomorphicity of the automorphism groups of the enumeration and the \-enumeration degrees. Download Export citation Bookmark
A note on the enumeration degrees of 1-generic sets.Liliana Badillo, Caterina Bianchini, Hristo Ganchev, Thomas F. Kent & Andrea Sorbi - 2016 - Archive for Mathematical Logic 55 (3):405-414.details We show that every nonzero $${\Delta^{0}_{2}}$$ enumeration degree bounds the enumeration degree of a 1-generic set. We also point out that the enumeration degrees of 1-generic sets, below the first jump, are not downwards closed, thus answering a question of Cooper. Download Export citation Bookmark 1 citation

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