Large Infinitary Languages, Model Theory

From PREFACE of the book:

The present book grew out of my lecture notes “Model Theory of Infinitary Languages” published in January 1970 by the Mathematics Institute of Aarhus University, Denmark.

Indeed, it is a thoroughly modified and updated version of those notes, which in its present form has only a vague resemblance to the original.

This book presents a systematic and largely self-contained development of the model theory of the infinitary languages L(k, λ) and L(∞, λ) (where indices  k, λ are infinite cardinals and ∞> k > λ). The language L(k, λ) admits conjunctions and disjunctions of sets of formulas of power less than k and simultaneous quantifications over sets of variables of power less than λ.

L(∞, λ) is, of course, the “union” of the L(k, λ). The basic semantic notions are defined in
Chapter 1 as direct generalizations of those for finitary, first order languages.
The special case of L(ω(1),ω) is not treated here; the word “large” in the title is intended precisely to denote
this omission.

The model theory of this particular language is the subject matter of Keisler’s monograph
Model theory for infinitary logic.