Abstract
The present work investigates several questions from a recent survey of Miller and Nies related to Chaitin's Ω numbers and their dependence on the underlying universal machine. Furthermore, the notion Ω U [ X ] = ∑ p : U ( p ) ↓ ∈ X 2 - | p | is studied for various sets X and universal machines U. A universal machine U is constructed such that for all x, Ω U [ { x } ] = 2 1 - H ( x ) . For such a universal machine there exists a co-r.e. set X such that Ω U [ X ] is neither left-r.e. nor Martin-Löf random. Furthermore, one of the open problems of Miller and Nies is answered completely by showing that there is a sequence U n of universal machines such that the truth-table degrees of the Ω U n form an antichain. Finally, it is shown that the members of hyperimmune-free Turing degree of a given Π 1 0 -class are not low for Ω unless this class contains a recursive set.
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