What is the halting sequence? It isn't mentioned in your linked article anywhere (or on google), and makes the rest of your response a little hard to make sense of.
> any Martin-Lof random sequence that computes (i.e. allows computing from it) a consistent completion of PA also computes the Halting Problem H; and by deLeeuw et al. [1965], only a recursive sequence (which H is not) can be computed with a positive probability by randomized algorithms.
It's just the sequence of the solutions to the halting problem, i.e., the characteristic function of the halting set. Levin points out this sequence is not recursive/decidable.
But it seems interesting, thanks for the link =)