Exact Upper Bounds and Their Uses in Settheorymenachem

The existence of exact upper bounds for increasing sequences of ordinal functions modulo an ideal is discussed. The main theorem (Theorem 18 below) gives a necessary and suucient condition for the existence of an exact upper bound f for a < I-increasing sequence f = hf : < i On A where > jAj + is regular: an eub f with lim inf I cff(a) = exists if and only if for every regular 2 (jAj;) the set of at points in f of coonality is stationary. Two applications of the main Theorem to set theory are presented. A theorem of Magidor's on covering between models of ZFC is proved using the main theorem (Theorem 22): If V W are transitive models of set theory with !-covering and GCH holds in V , then-covering holds between V and W for all cardinals. A new proof of a Theorem by Cummings on collapsing successors of singulars is also given (Theorem 24). The appendix to the paper contains a short proof of Shelah's trichotomy theorem, for the reader's convenience. 1. Introduction Shelah's work on Cardinal Arithmetic (see 7], 1] and 4]) introduced the theory of possible true coonalities of products of sets of regular cardinals modulo an ideal | pcf theory. The relevance of pcf theory to set theory and other branches of mathematics was demonstrated by a series of applications. In this paper the dual problem is addressed: suppose a set of ordinal