Extensions Of Ideals On Large Cardinals.

Abstract

It is well-known that the nonstationary subsets of a regular uncountable cardinal form a normal ideal. By restating a theorem of Neumer we can characterize stationariness at (kappa) in terms of unbound- edness at (alpha) (LESSTHEQ) (kappa). Iterating this characterization, we develop a new family of large cardinals--the n-stationary cardinals. In the same way that an ineffable cardinal has, for every (1,(kappa))sequence, a stationary homogeneous set, we define (kappa) as n-hyperineffable if every (1,(kappa))-sequence has an (n + 1)-stationary homogeneous set. Also, (kappa) is ineffable('n) if every (1,(kappa))-sequence has an ineffable('(n-1)) homogeneous set. We show that if (kappa) is ineffable('n) or (n - 1)-hyperineffable then (kappa) is (n + 1)-stationary. In a result that parallels work of Baumgartner, we show (kappa) is ineffable('n) if and only if (kappa) is (PI)(,2n)('1)-indescribable and subtle('n) and the associated ideals cohere. Baumgartner, Taylor and Wagon examined the saturation of large cardinal ideals by studying a Mahlo-type operation. For example, given X (L-HOOK EQ) (kappa), they look at K(X) (,def) (alpha) < (kappa)(VBAR)X(INTERSECT)(alpha) is not subtle in (alpha) . We examine the ideal J('Subtle) (,def) the normal closure of the subtle ideal (UNION) K(X)(VBAR)X is a subtle subset of K . We show J('Subtle) is proper when (kappa) is almost ineffable and using forcing techniques of Kunen we show that it is consistent relative to a measurable cardinal that J('Subtle) be proper at a (kappa) that is not almost ineffable. A similar construction can be carried out beginning with the almost ineffable ideal, but the results here are scarcer. Similarly, we can begin with the normal ideal that is dual to the intersection of all normal nonprincipal ultrafilters on (kappa) and relate the ideal J('Meas0) to the Mitchell hierarchy. For example we show J('Meas0) is not proper if the Mitchell order of (kappa) is less than (kappa)('+).