A note on the enumeration degrees of 1-generic sets

Badillo, Liliana; Bianchini, Caterina; Ganchev, Hristo; Kent, Thomas F.; Sorbi, Andrea

doi:10.1007/s00153-015-0471-6

Arch. Math. Logic

2015

DOI: 10.1007/s00153-015-0471-6

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A note on the enumeration degrees of 1-generic sets

Liliana Badillo

Caterina Bianchini

Hristo Ganchev

et al.

Abstract: We show that every nonzero $\Delta^0_2$ enumeration degree bounds a set $1$-generic degree

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2018

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Enumeration 1-Genericity in the Local Enumeration Degrees

Badillo¹,

Harris²,

Soskova³

2018

Notre Dame J. Formal Logic

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We discuss a notion of forcing that characterises enumeration 1genericity and we investigate the immunity, lowness and quasiminimality properties of enumeration 1-generic sets and their degrees. We construct an enumeration operator ∆ such that, for any A, the set ∆ A 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, if a is nonzero, then we can find such b satisfying 0e < b < a. We conclude by proving the existence of both a nonzero low and a properly Σ 0 2 nonsplittable enumeration 1-generic degree hence proving that the class of 1-generic degrees is properly subsumed by the class of enumeration 1-generic degrees.

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Enumeration 1-Genericity in the Local Enumeration Degrees

Badillo¹,

Harris²,

Soskova³

2018

Notre Dame J. Formal Logic

Self Cite

View full text Add to dashboard Cite

show abstract

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A note on the enumeration degrees of 1-generic sets

Abstract: We show that every nonzero $\Delta^0_2$ enumeration degree bounds a set $1$-generic degree

Cited by 1 publication

References 13 publications

Enumeration 1-Genericity in the Local Enumeration Degrees

Enumeration 1-Genericity in the Local Enumeration Degrees

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