∑1 definitions with parameters

Journal of Symbolic Logic 51 (2):453 - 461 (1986)
Abstract
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 1R in every admissible set if anly only if it is E-recursive. A function is ES p -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 ES p - 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 than a total function on sets is ES p -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 ES p -recursive if and only if it is uniformly ∑ 1 (p) in every admissible set
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Robert S. Lubarsky (1988). Another Extension of Van de Wiele's Theorem. Annals of Pure and Applied Logic 38 (3):301-306.
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