Abstract
We study the cone avoidance and the upper cone avoidance of two substructures of m-introimmune Turing degrees. We show that the substructure of the m-introimmune Turing degrees satisfies the cone avoidance property, and that the substructure of the computably enumerable m-introimmune Turing degrees satisfies the upper cone avoidance property.
Highlights
The study of sets of natural numbers with no subsets of higher Turing degrees started with Soare (Soare 1969) and continued with Jockusch (Jockusch 1973) and Simpson (Simpson 1978)
We study the cone avoidance and the upper cone avoidance of two substructures of m-introimmune Turing degrees
We show that the substructure of the m-introimmune Turing degrees satisfies the cone avoidance property, and that the substructure of the computably enumerable m-introimmune Turing degrees satisfies the upper cone avoidance property
Summary
The study of sets of natural numbers with no subsets of higher Turing degrees started with Soare (Soare 1969) and continued with Jockusch (Jockusch 1973) and Simpson (Simpson 1978). From their works we know that such sets exist and that they cannot be arithmetic. There are mintroimmune sets in the class Π01 (Cintioli 2005), where m stands for the many-one reducibility ≤m This suggests to study which properties satisfies the substructure of the computably enumerable (c.e.) m-introimmune Turing degrees. We consider here the cone avoidance property, and we prove that Jm satisfies this property
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