TY - JOUR

T1 - Exact upper bounds and their uses in set theory

AU - Kojman, Menachem

N1 - Funding Information:
* E-mail: kojman@cs.bgu.ac.il. ’ Research was partially supported by an NSF grant number 9622579.

PY - 1998/8/21

Y1 - 1998/8/21

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0032555367&partnerID=8YFLogxK

U2 - 10.1016/S0168-0072(98)00011-6

DO - 10.1016/S0168-0072(98)00011-6

M3 - Article

AN - SCOPUS:0032555367

SN - 0168-0072

VL - 92

SP - 267

EP - 282

JO - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

IS - 3

ER -