OMITTING CLASSES OF ELEMENTS

Publisher Summary This chapter focuses on omitting classes of elements. By a class of elements we mean a class defined by a set of formulas. It is fairly easy to show that there must be some cardinal, x, such that for any theory T and any class Σ the existence of a model of T of power K which omits Σ implies the existence of such models in each infinite power. The principal result of this paper is the determination of this cardinal. The proof depends upon a partition theorem of Erdos and Rado. The letter T will always denote a theory in a countable first-order language L, and Σ will denote a set of formulas in L having a common single free variable. In particular, Ehrenfeucht used it to show that for any theory T (having an infinite model) there will be some countable set of types of elements such that T has arbitrarily large models containing only elements of those types.