This paper is concerned with the proof theoretic development of certain infinite languages. These languages contain the usual infinite conjunctions and disjunctions, but in addition to homogeneous quantifiers such as ∀x0x1x2 … and ∃y0y1y2 …, we shall investigate particular subclasses of the dependent quantifiers described by Henkin [1]. The dependent quantifiers of Henkin can be thought of as partially ordered quantifiers defined by a function from one set to the power set of another set. This function assigns to each existentially bound variable, the set of universally bound variables on which it depends.

The natural extension of Gentzen's first order predicate calculus to an infinite language with homogeneous quantifiers results in a system that is both valid and complete, and in which a cut elimination theorem can be proved [2]. The problem then arises of devising, if possible, a logical system dealing with general dependent quantifiers that is valid and complete. In this paper a system is presented that is valid and complete for an infinite language with homogeneous quantifiers and dependent quantifiers that are anti-well-ordered, for example, … ∀x2∃y2∀x1∃y1∀x0∃y0.

Certain notational conventions will be employed in this paper. Greek letters will be used for ordinal numbers. The ordinal ω is the set of all natural numbers, and 2ω is the set of all ω -sequences of elements of 2 = {0,1}. The power set of S is denoted by P(S). μα[A(α)] stands for the smallest ordinal α such that A (α) holds.