On splitting stationary subsets of large cardinals

Journal of Symbolic Logic 42 (2):203-214 (1977)


Let κ denote a regular uncountable cardinal and NS the normal ideal of nonstationary subsets of κ. Our results concern the well-known open question whether NS fails to be κ + -saturated, i.e., are there κ + stationary subsets of κ with pairwise intersections nonstationary? Our first observation is: Theorem. NS is κ + -saturated iff for every normal ideal J on κ there is a stationary set $A \subseteq \kappa$ such that $J = NS \mid A = \{X \subseteq \kappa:X \cap A \in NS\}$ . Turning our attention to large cardinals, we extend the usual (weak) Mahlo hierarchy to define "greatly Mahlo" cardinals and obtain the following: Theorem. If κ is greatly Mahlo then NS is not κ + -saturated. Theorem. If κ is ordinal Π 1 1 -indescribable (e.g., weakly compact), ethereal (e.g., subtle), or carries a κ-saturated ideal, then κ is greatly Mahlo. Moreover, there is a stationary set of greatly Mahlo cardinals below any ordinal Π 1 1 -indescribable cardinal. These methods apply to other normal ideals as well; e.g., the subtle ideal on an ineffable cardinal κ is not κ + -saturated

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References found in this work

The Fine Structure of the Constructible Hierarchy.R. Björn Jensen - 1972 - Annals of Mathematical Logic 4 (3):229.
Some Applications of Iterated Ultrapowers in Set Theory.Kenneth Kunen - 1970 - Annals of Mathematical Logic 1 (2):179.

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Adding a Nonreflecting Weakly Compact Set.Brent Cody - 2019 - Notre Dame Journal of Formal Logic 60 (3):503-521.
Stationary Cardinals.Wenzhi Sun - 1993 - Archive for Mathematical Logic 32 (6):429-442.

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