Descendingly Incomplete Ultrafilters and the Cardinality of Ultrapowers

Let D be an ultrafilter on I, and let k be a cardinal. D is said to be k-descendingly incomplete (k-d.i.) if there exists a chain Xα : α < k of elements of D such that α < β → Xα ⊆ Xβ and Xα = ϕ. Such a chain will be called a k-chain for D. The notion of k-descending incompleteness is due to Chang [3]. In this paper we explore the relationship between the cardinality of the ultrapower kI/D and the existence of certain chains on D. Since we deal so much with questions of size, we do not ordinarily make a notational distinction between a set and its cardinality. Where such a distinction is useful, the cardinality of a set A will be denoted by |A|.