Models in ω 1

Preliminary Version This is an account of Keisler’s work showing that if a sentence φ of Lω1,ω that has few models in א1 each model of φ realizes only countably many types over the emptyset. The proof has two main components. One rephrases the existence of an end extension of a model A that omits a type p as a sentence (in an expanded vocabulary and logic) that is true in A. The other applies the rephrasing. We will state the rephrasing, make the applications of it, and then prove the rephrasing holds. The last of these comprises Theorem 13 (p. 55), 28 (p. 111) and 42 (p. 163) of [?]. The first is the rest of chapters 30 and 31 of that book.