Definable properties of the computably enumerable sets Original Research Article

Post in 1944 began studying properties of a computably enumerable (c.e.) set A such as simple, h-simple, and hh-simple, with the intent of finding a property guaranteeing incompleteness of A. From the observations of Post (1943) and Myhill (1956), attention focused by the 1950s on properties definable in the inclusion ordering of c.e. subsets of ω, namely E = (Wnnεω, ⊂). In the 1950s and 1960s Tennenbaum, Martin, Yates, Sacks, Lachlan, Shoenfield and others produced a number of elegant results relating ∄-definable properties of A, like maximal, hh-simple, atomless, to the information content (usually the Turing degree, deg(A) of A. Harrington and Soare (1991) gave an answer to Postʹs program for definable properties by producing an ∄-definable property Q(A) which guarantees that A is incomplete and noncomputable, but developed a new Δ30-automorphism method to prove certain other properties are not ∄-definable. In this paper we introduce new ∄-definable properties relating the ∄-structure of A to deg(A), which answer some open questions. In contrast to Q(A) we exhibit here an ∄-definable property T(A) which allows such a rapid flow of elements into A that A must be complete even though A may possess many other properties such as being promptly simple. We also present a related property NL(A) which has a slower flow but fast enough to guarantee that A is not low, even though A may possess virtually all other related lowness properties (low2 and others) and A may simultaneously be promptly simple.