The language Ln is obtained from the first order predicate calculus by adjoining the quantifier Qn which binds n variables. The formula Qnυ1 … υnΨ is given a κ-interpretation for each infinite cardinal κ, namely, “there is a set X of power κ such that Ψx1 … xn holds for all distinct x1 … xn ϵ X”. L<ω is the result of adjoining all the Qn quantifiers for each n ϵ ω to the first order predicate calculus.

In [4] we showed that under the assumption (cf. [3]) L<ω is countably compact under the ω1-interpretation, and that any sentence σ ϵ L<ω that has a model in some κ-interpretation where κ is a regular infinite cardinal has a model in the ω1 interpretation. However, compactness for L<ω in the κ-interpretation for κ an infinite successor cardinal other than ω1 and the transfer of satisfiability from ω1 to any higher power remain open questions under any set theoretic assumptions.

Here we restrict our attention to a small fragment L2− of L2 consisting of universal first order formulas along with formulas of the kind Q2υ1υ2∀υ3 … υnΨ and ¬Q2υ1υ2∀υ3 … υnφ where Ψ and φ are open and no function symbol of arity > 1 occurs in any formula. Assuming the existence of a κ-Souslin tree, this language is λ compact in the κ-interpretation when λ < κ.