Arithmetic complexity via effective names for random sequences

Abstract

We investigate enumerability properties for classes of sets which permit recursive, lexicographically increasing approximations, or left-r.e. sets. In addition to pinpointing the complexity of left-r.e. Martin-L\"{o}f, computably, Schnorr, and Kurtz random sets, weakly 1-generics and their complementary classes, we find that there exist characterizations of the third and fourth levels of the arithmetic hierarchy purely in terms of these notions. More generally, there exists an equivalence between arithmetic complexity and existence of numberings for classes of left-r.e. sets with shift-persistent elements. While some classes (such as Martin-L\"{o}f randoms and Kurtz non-randoms) have left-r.e. numberings, there is no canonical, or acceptable, left-r.e. numbering for any class of left-r.e. randoms. Finally, we note some fundamental differences between left-r.e. numberings for sets and reals

Things that can be made into themselves

“…Martin-Löf random reals could be made into themselves, then the set of indices for Martin-Löf non-random reals would be ∆ 2 inside this numbering. This contradicts a theorem of Kjos-Hanssen, Stephan, and Teutsch [23] which says that the Martin-Löf non-randoms are never Π 0 3 in any universal left-r.e. numbering.…”

“…Numberings for left-r.e. sets were first studied by Brodhead and Kjos-Hanssen [7,23] and provide more expressive possibilities than the traditional numberings for r.e. sets.…”

“…reals, which admit an increasing, recursive sequence of rationals from below, however in the context of effective enumerations it makes more sense to consider left-r.e. sets, see [23,Section 2]. For example, the infinite left-r.e.…”