Annals of MathematicalLogic 13 (1978) 225-265. ' North-Holland Publishing Company MODEL-THEORETIC FORCING IN LOGIC WITH A GENERALIZED QUANTIFIER Kim B. BRUCE* Williams College, Williamstown, MA, U.S.A. Received 17 December 1975 In this paper a new method of model-theoretic forcing is developedfor the infinitary language LA(::!) where A is a countable fragment and "::!x" is a new quantifier that is read as "there are few x." In particular we will be interested in the case where ::!x is interpreted as "there are at most countablymany x." In this casewe will write the quantifier as ::! 1 x. The methodof forcing that we develop is an extension of Abraham Robinson’s finite forcing for first-order model theory. Our main result is the Standard› Generic Model Theorem.This theorem allows us to build uncountable standard models (models in which "::!x" is actually interpreted as "there are at most countably many x") from a given classof models by the forcing construction. By examining the forcing construction more carefully we prove a new omitting types theorem for countable fragments LA(::! 1 ) of Lw,w(::! 1 ). This new omitting types theorem improves a result of Keisler [10]. A. major advantageof the forcing construction is that we can restrict our attention to formulas of low complexity. For exampleif we begin with the basic formulas of L we obtain a new basic omitting types theorem for L(::! 1 ) that only refers to formulas of the form (Sx) 1\m<n’Pm where each ’Pm is basic and Sx is a finite string of quantifiers of the form Ox; and 3xj. We can also restrict our attention to other subsets of formulas of LA(::!) and obtain similar results. When we are working with the language LA(::!) it is convenient to define another quantifier, Ox, from ::!x by having Oxcp abbreviate 1::!xcp. Traditionally the model theory of this language has been looked at from the point of view that Ox is the fundamental new quantifier, and the language has been denoted by LA(O). While the difference in point of view is obviously not crucial, we believe there are good reasons for looking at this language as being built from ::!x as opposed to Ox, so we adopt this convention in this paper. Generalized quantifierswere first studiedby Mostowski [16], who raisedseveral questions about the compactness and completeness of various logics obtainedby adding new types of quantifiers. In example (d) of his paper, Mostowski defines the quantifier ::! 1 x which he denotesas P. Fuhrken [9] proved the compactness * Part of the research for this paper was done while the author held an NSF Graduate Fellowship at the University of Wisconsin, Madison. 225