Inductive definitions, models of comprehension and invariant definability

Abstract

The connections between inductive definability and models of comprehension are studied. Let = 〈A, R _l, ...,R _n 〉 be an infinite structure and letI _φ be a set inductively defined by a formulaφ of the second order language. We prove that if is a model of Δ ¹₁ -Comprehension relativized toφ, andφ is-absolute, then for everyη smaller than the height of (h()),I _φ is in. If is aβ-structure which satisfies Σ ¹₁ -Comprehension relativized toφ and WF(X), and φ is-absolute, thenI _φ is in and ‖φ| <h (). These results imply that Barwise-Grilliot theorem is false in the case of uncountable acceptable structures. We also study the notion of invariant definability over models₁ of Δ ¹₁ -Comprehension.