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Π10 classes with complex elements

Published online by Cambridge University Press:  12 March 2014

Stephen Binns
Stephen Binns*
Affiliation:
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, PO BOX 5046 Dhahran 31261, Saudi Arabia, E-mail: binns@kfupm.edu.sa
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Abstract

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 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 ⊆ ω there is an X in P such that X ≥wttY. We show that this is also equivalent to the 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.

Keywords

Π10 classcomplexrandomnessrankedscattered linear ordercomputably growingMuchnik
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 4 , December 2008 , pp. 1341 - 1353
Copyright
Copyright © Association for Symbolic Logic 2008

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References

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