Journal of Symbolic Logic

On Splitting Stationary Subsets of Large Cardinals

James E. Baumgartner, Alan D. Taylor, and Stanley Wagon

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Let $\kappa$ denote a regular uncountable cardinal and $NS$ the normal ideal of nonstationary subsets of $\kappa$. Our results concern the well-known open question whether $NS$ fails to be $\kappa^+$-saturated, i.e., are there $\kappa^+$ stationary subsets of $\kappa$ with pairwise intersections nonstationary? Our first observation is: Theorem. $NS$ is $\kappa^+$-saturated iff for every normal ideal $J$ on $\kappa$ 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 $\kappa$ is greatly Mahlo then $NS$ is not $\kappa^+$-saturated. Theorem. If $\kappa$ is ordinal $\Pi^1_1$-indescribable (e.g., weakly compact), ethereal (e.g., subtle), or carries a $\kappa$-saturated ideal, then $\kappa$ is greatly Mahlo. Moreover, there is a stationary set of greatly Mahlo cardinals below any ordinal $\Pi^1_1$-indescribable cardinal. These methods apply to other normal ideals as well; e.g., the subtle ideal on an ineffable cardinal $\kappa$ is not $\kappa^+$-saturated.

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J. Symbolic Logic, Volume 42, Issue 2 (1977), 203-214.

First available in Project Euclid: 6 July 2007

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Baumgartner, James E.; Taylor, Alan D.; Wagon, Stanley. On Splitting Stationary Subsets of Large Cardinals. J. Symbolic Logic 42 (1977), no. 2, 203--214.

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