An Application of 1-Genericity in the $\Pi^0_2$ Enumeration Degrees

Badillo, Liliana; Harris, Charles M.

doi:10.1007/978-3-642-29952-0_56

Cited by 5 publications

(8 citation statements)

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“…In other words the sets underlying D e (≤0 ′ e ) all have good Σ 0 2 approximations. Lemma 2.3 ( [BH12]). If a is a good enumeration degree then, for every A ∈ a, K A ≤ e K A .…”

Section: Preliminariesmentioning

confidence: 99%

“…Note firstly that Lemmas 5.4-5.6 are proved by inspection (only, for some of the statements involved) and straightforward induction arguments over the stages of the construction, using the observations above. (Detailed proofs are given in [Bad13]. )…”

Section: ) Environment Parametersmentioning

confidence: 99%

“…Enumeration 1-genericity, a form of 1-genericity appropriate for positive reducibilities, was introduced in [BH12] and used as a tool to show that there exists a properly Π 0 2 degree b such that any x ≤ b contains only Π 0 2 sets. In [BH12] various questions about the basic properties of enumeration 1-genericity in the enumeration and singleton degrees, as also its relationship with 1-genericity, were investigated. We continue this investigation in Section 3 where in particular we look at immunity and lowness properties of enumeration 1-generic sets.…”

Section: Introductionmentioning

confidence: 99%

“…We now see that that this is not the case. Proposition 3.13 ( [BH12]). There exists a properly Π 0 2 enumeration degree b such that the principal ideal D e (≤b) only contains Π 0 2 sets.…”

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confidence: 99%

“…Let A be a set such that deg e (A) is properly Σ 0 2 and A is enumeration 1generic. For example take A to be the 1-generic set with noncuppable enumeration degree constructed in the proof of Theorem 3.2 in [BH12]. Let B = A and b = deg e (B).…”

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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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“…In other words the sets underlying D e (≤0 ′ e ) all have good Σ 0 2 approximations. Lemma 2.3 ( [BH12]). If a is a good enumeration degree then, for every A ∈ a, K A ≤ e K A .…”

Section: Preliminariesmentioning

confidence: 99%

Section: ) Environment Parametersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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