Non-coding Enumeration Operators

An enumeration operator maps each set A of natural numbers to a set [Formula: see text], in such a way that E(A) can be enumerated uniformly from every enumeration of A. The maximum possible Turing degree of E(A) is therefore the degree of the jump [Formula: see text]. It is impossible to have [Formula: see text] for all A, but possible to achieve this for all A outside a meager set of Lebesgue measure 0. We consider the properties of two specific enumeration operators: the HTP operator, mapping a set W of prime numbers to the set of polynomials realizing Hilbert’s Tenth Problem in the ring [Formula: see text]; and the root operator, mapping the atomic diagram of an algebraic field F of characteristic 0 to the set of polynomials in [Formula: see text] with roots in F. These lead to new open questions about enumeration operators in general.