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\(\Pi _1^0\) classes with complex elements. (English) Zbl 1155.03032

Summary: An infinite binary sequence is complex if the Kolmogorov complexity of its initial segments is bounded below by a computable function. We prove that a \(\Pi _1^0\) class \(P\) contains a complex element if and only if it contains a wtt-cover for the Cantor set. That is, if and only if for every \(Y\subseteq \omega \) there is an \(X\) in \(P\) such that \(X\geqslant _{\text{wtt}} Y\). We show that this is also equivalent to the \(\Pi_1^0 \) class’s being large in some sense. We give an example of how this result can be used in the study of scattered linear orders.

MSC:

03D80 Applications of computability and recursion theory
68Q30 Algorithmic information theory (Kolmogorov complexity, etc.)
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