Uniform inductive definability and infinitary languages

Abstract

Introduction . The purpose of this paper is to show how results from the theory of inductive definitions can be used to obtain new compactness theorems for uncountable admissible languages. These will include improvements of the compactness theorem by J. Green [9]. In [2] Barwise studies admissible sets satisfying the Σ 1 -compactness theorem. Our approach is to consider admissible sets satisfying what could be called the abstract extended completeness theorem, that is, sets where the consequence relation of the admissible fragment L A is Σ 1 -definable over A . We will call such sets Σ 1 - complete . For countable admissible sets, Σ 1 -completeness follows from the completeness theorem for L A . Having restricted our attention to Σ 1 -complete sets we are led to a stronger notion also true on countable admissible sets, namely what we shall call uniform Σ 1 - completeness . We will see that this notion can be viewed as extending to uncountable admissible sets, properties related to both the “recursion theory” and “proof theory” of countable admissible sets. By following Barwise's recent approach to admissible sets allowing “urelements,” we show that there is a natural connection between certain structures arising from the theory of inductive definability, and uniformly Σ 1 -complete admissible sets . The structures we have in mind are called uniform Kleene structures .