On external Scott algebras in nonstandard models of peano arithmetic

Journal of Symbolic Logic 61 (2):586-607 (1996)
Abstract
We prove that a necessary and sufficient condition for a countable set L of sets of integers to be equal to the algebra of all sets of integers definable in a nonstandard elementary extension of ω by a formula of the PA language which may include the standardness predicate but does not contain nonstandard parameters, is as follows: L is closed under arithmetical definability and contains 0 (ω) , the set of all (Gödel numbers of) true arithmetical sentences. Some results related to definability of sets of integers in elementary extensions of ω are included
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