Abstract
Sets with no subsets of higher weak truth-table degree
Highlights
The existence of sets without subsets of higher Turing degree was proved by Soare [11]
In this paper we close the gap by considering the weak truth-table reducibility ≤wtt, and we prove the existence of arithmetical wtt-introimmune sets, in particular wtt-introimmune ∆02 sets
We know of the existence of m-introimmune sets in the class Π01 [3, 4]
Summary
The existence of sets without subsets of higher Turing degree was proved by Soare [11]. The approach of to consider such reducibilities ≤r and to study the existence of arithmetical sets without subsets of higher r-degree was initiated in [5], in which r-introimmune sets have been introduced. In [5] it was proved the existence of arithmetical c-introimmne sets, where ≤c is the conjunctive reducibility, a particular truth-table reducibility. It was proved the existence of c-introimmune ∆04 sets. In this paper we close the gap by considering the weak truth-table reducibility ≤wtt, and we prove the existence of arithmetical wtt-introimmune sets, in particular wtt-introimmune ∆02 sets. Since we currently do not know intermediate reducibilities between ≤wtt and ≤T , we deduce that for all the known reducibilities ≤r strictly contained in ≤T there are arithmetical r-introimmune sets
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