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On splitting stationary subsets of large cardinals

Published online by Cambridge University Press:  12 March 2014

James E. Baumgartner ,
Alan D. Taylor  and
Stanley Wagon
James E. Baumgartner
Affiliation:
Dartmouth College, Hanover, New Hampshire 03755
Alan D. Taylor
Affiliation:
Union College, Schenectady, New York 12308
Stanley Wagon
Affiliation:
Smith College, Northampton, Massachusetts 01060
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Abstract

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 A ⊆ κsuch that J = NS∣A = {X ⊆ κ: X ∩ A ∈ 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 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 42 , Issue 2 , June 1977 , pp. 203 - 214
Copyright
Copyright © Association for Symbolic Logic 1977

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References

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