Published online by Cambridge University Press: 12 March 2014
Let p be a set. A function Φ is uniformly Σ 1(p) in every admissible set if there is a Σ 1 formula ϕ in the parameter p so that ϕ defines Φ in every Σ 1-admissible set which includes p. A theorem of Van de Wiele states that if Φ is a total function from sets to sets then Φ is uniformly Σ 1 in every admissible set if and only if it is E-recursive. A function is ESp -recursive if it can be generated from the schemes for E-recursion together with a selection scheme over the transitive closure of p. The selection scheme is exactly what is needed to insure that the ESP -recursively enumerable predicates are closed under existential quantification over the transitive closure of p. Two theorems are established: a) If the transitive closure of p is countable then a total function on sets is ESp -recursive if and only if it is uniformly Σ 1(p) in every admissible set. b) For any p, if Φ is a function on the ordinal numbers then Φ is ESP -recursive if and only if it is uniformly Σ 1(p) in every admissible set.
The author was supported by NSF grant MCS-8404208.