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The degrees of bi-hyperhyperimmune sets. (English) Zbl 1323.03054
The study of the upward closure of collections of Turing degrees is an interesting topic of computability theory, beginning with the work of W. Miller and D. A. Martin [Z. Math. Logik Grundlagen Math. 14, 159–166 (1968; Zbl 0216.29102)] and several works of C. G. Jockusch jun. [Z. Math. Logik Grundlagen Math. 15, 135–140 (1969; Zbl 0184.02002); J. Symb. Log. 34, 489–493 (1969; Zbl 0181.30601); Z. Math. Logik Grundlagen Math. 18, 285–287 (1972; Zbl 0257.02033); Isr. J. Math. 15, 332–335, (1973; Zbl 0279.02024)]. Thanks to these works we know of the upward closure of immune, bi-immune, hyperimmune, bi-hyperimmune and hyperhyperimmune Turing degrees. The case of the bi-hyperhyperimmune degrees remained open. In the paper under review, the authors close this gap by proving the upward closure of the bi-hyperhyperimmune Turing degrees. The strategy is to prove the following characterizations of the bi-hyperhyperimmune Turing degrees, from which the upward closure follows. Given a Turing degree \(\mathbf d\):
(1) \(\mathbf d\) computes a \(\Delta^0_2\) escaping function,
(2) \(\mathbf d\) computes a weakly 2-generic sequence,
(3) \(\mathbf d\) contains a blockwise bi-hyperhyperimmune set,
(4) \(\mathbf d\) contains a blockwise hyperhyperimmune set,
(5) \(\mathbf d\) contains a bi-hyperhyperimmune set.

The paper ends by negatively answering to Question 6.6 posed in [B. F. Csima and I. S. Kalimullin, Math. Log. Q. 56, No. 1, 67–77 (2010; Zbl 1184.03025)].

MSC:
03D28 Other Turing degree structures
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References:
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