Principal numerations of functionals on admissible sets (Q810015)

The main goal of this paper is to find necessary conditions for an admissible set to possess principal numerations of the computable functionals of finite types over it. It is well-known that there exist principal numerations of the \(\Sigma\)-predicates of finite types over any admissible set [\textit{Yu. L. Ershov}, Algebra Logika, 24, No.5, 499-536 (1985; Zbl 0615.03036)]. The same question concerning functionals appeared to be more complicated. The notion of a resolvent admissible set, which was introduced by Yu. L. Ershov [loc. cit.] plays an important role in the proofs. The author proves in particular that if \({\mathbb{A}}\) is a resolvent admissible set then there exist principal numerations of the computable functionals of finite types over \({\mathbb{A}}\). If \({\mathfrak M}\) is an atomic model of a decidable theory with recursive set of atomic formulae, and for some \(m_ 1,...,m_ k\in {\mathfrak M}\) every atomic formula is equivalent to a \(\exists\)-formula with parameters \(m_ 1,...,m_ k\), then there exist principal numerations of the computable functionals of finite types over HF(\({\mathfrak M})\). For an arbitrary model \({\mathfrak M}\) there exist principal numerations of the computable functionals of finite types over HYP(\({\mathfrak M})\).