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Turing incomparability in Scott sets. (English) Zbl 1123.03039
From the introduction: H. Friedman and A. McAllister posed the question whether for every nonrecursive set $$X$$ of a Scott set $${\mathcal F}$$ there is a $$Y\in {\mathcal F}$$ such that $$X$$ and $$Y$$ are Turing incomparable (see Problem 3.2 and also Problem 3.3 in [D. Cenzer and C. G. Jockusch jun., Contemp. Math. 257, 33–59 (2000; Zbl 0962.03040)]). We present a positive solution to the question, using recent results in the area of algorithmic randomness and also results on $$\Pi^0_1$$ classes.

##### MSC:
 03D28 Other Turing degree structures 68Q30 Algorithmic information theory (Kolmogorov complexity, etc.)
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##### References:
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