Published online by Cambridge University Press: 12 March 2014
One of the main results of Barwise [2] (see also [7, Chapter VIII]) showed that the s – reflection principle for a set A is equivalent to Σ1-compactness of
. Here A is any transitive p.r. closed set, and
is the infinitary language on A which allows conjunction and disjunction over arbitrary sets Φ Є A, and finite quantification.
In this paper we consider languages , where B is a Δ0 subset of A, which is like
but we allow quantifiers ∀x and ∃x where x is any set of variables indexed by an element of B. A treatment similar to that of [2] for
establishes a sufficient, and in some cases necessary, condition for
to be Σ1-compact. The use of infinitary Skolem functions is intrinsic to the method, so to avoid a separate development of the rudiments of the Skolem language
we actually define
to have b-ary relation and function symbols for every b Є B.