EXACT UPPER-BOUNDS AND THEIR USES IN SET-THEORY

The existence of exact upper bounds for increasing sequences of ordina l functions module an ideal is discussed. The main theorem (Theorem 18 below) gives a necessary and sufficient condition for the existence o f an exact upper bound f for a <(I)-increasing sequence (f) over bar = [f(alpha) : alpha < lambda] subset of or equal to On(A) where lambda > \A\(+) regular: an eub f with lim inf(I) cf f(a) = mu exists if and only if for every regular kappa is an element of(\A\, mu) the set of f lat points in (f) over bar of cofinality kappa is stationary. Two appl ications of the main Theorem to set theory are presented. A theorem of Magidor's on covering between models of set is proved using the main theorem (Theorem 22): If V subset of or equal to W are transitive mode ls of set theory with omega-covering and GCH holds in V, then kappa-co vering holds between V and W for all cardinals kappa. A new proof of a Theorem by Cummings on collapsing successors of singulars is also giv en (Theorem 24). The appendix to the paper contains a short proof of S helah's trichotomy theorem, for the reader's convenience. (C) 1998 Els evier Science B.V. All rights reserved.