Π⁰₁ classes with complex elements

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 Π₁⁰ 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≥_wtt Y. We show that this is also equivalent to the Π₁⁰ 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.