A New Omitting Types Theorem for L(Q)

For L a countable first-order language, let L(Q) be logic with the quantifier Qx which means “there exist uncountably many x ”. We assume a little familiarity with Keisler's paper [8]. One finds there completeness and compactness theorems for L(Q) , as well as an omitting types theorem: a syntactic condition is given for a consistent countable theory to have a model satisfying ∀ x ⋁Σ( x ), where Σ is a countable set of formulas of L(Q) . (See also Chang and Keisler [3] for the first-order omitting types theorem, due to Henkin and Orey.) An analogous theorem is proved in Barwise, Kaufmann, and Makkai [1] and in Kaufmann [6] for stationary logic. However, a more general theorem than just an anlaogue to Keisler's is proved there. Conditions are given which are sufficient for a theory T to have models satisfying sentences such as aas 1 aas 2 … aas n ⋁Σ( s 1 , … s n ), ∀ xaas ∨ Σ( x, s ), and so forth. Bruce [2] had asked whether such a theorem can be proved for L(Q) . with “ aa ” replaced by “ Q *”, where Q * is ¬ Q ¬ (“for all but countably many”).