#### Date of Original Version

1997

#### Type

Technical Report

#### Abstract or Table of Contents

Abstract: "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 sufficient condition for the existence of an exact upper bound f for a <[subscript I]-increasing sequence f╠ä = [f[subscript ╬▒] : ╬▒ < [lambda]] [subset] On[superscript A] where [lambda] > [absolute value of A]Γü║ is regular: an eub f with lim inf[subscript I]cf f(a) = [mu] exists if and only if for every regular k [element of] ([absolute value A], [mu]) the set of flat points in f╠ä of cofinality k 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 [element of] W are transitive models of set theory with [omega]-covering and GCH holds in V, then k-covering holds between V and W for all cardinals k. 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 reaser's [sic] convenience."