Exact upper bounds and their uses in set theory

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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 <1-increasing sequence f̄ = (fα: a < λ) ⊆ OnA where λ > \A\+ is regular: an eub f with lim inf1 cf f(a) = μ-exists if and only if for every regular κ ∈(\A\, μ) the set of flat points in f̄ of cofinality κ 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 ω-covering and GCH holds in V, then κ-covering holds between V and W for all cardinals κ. 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.

Original languageEnglish
Pages (from-to)267-282
Number of pages16
JournalAnnals of Pure and Applied Logic
Volume92
Issue number3
DOIs
StatePublished - 21 Aug 1998
Externally publishedYes

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