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On splitting stationary subsets of large cardinals

  • James E. Baumgartner (a1), Alan D. Taylor (a2) and Stanley Wagon (a3)

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 isκ+-saturated iff for every normal ideal J on κ there is a stationary set Aκsuch that J = NS∣A = {Xκ: XANS}.

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 notκ+-saturated.

Theorem. If κ is ordinal Π11-indescribable (e.g., weakly compact), ethereal (e.g., subtle), or carries aκ-saturated ideal, thenκis greatly Mahlo. Moreover, there is a stationary set of greatly Mahlo cardinals below any ordinal Π11-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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[6] Thomas J. Jech , Some combinatorial problems concerning uncountable cardinals, Annals of Mathematical Logic, vol. 5 (1973), pp. 165198.

[7] Thomas J. Jech and Karel L. Prikry , On ideals of sets and the power set operation, Bulletin of the American Mathematical Society, vol. 82 (1976), pp. 593595.

[8] Ronald B. Jensen , The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229308.

[10] Jussi Ketonen , Some combinatorial principles, Transactions of the American Mathematical Society, vol. 188 (1974), pp. 387394.

[11] Kenneth Kunen , Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 1 (1970), pp. 179227.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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