Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Any behaviour of the Mitchell ordering of normal measures is possible

Author(s): Jirí Witzany
Journal: Proc. Amer. Math. Soc. 124 (1996), 291-297.
MSC (1991): Primary 03E35, 03E55
MathSciNet review: 1286010
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Abstract: Let $U_0,U_1$ be two normal measures on $\kappa$ . We say that $U_0$ is in the Mitchell ordering less than $U_1,$ $U_0\vartriangleleft U_1,$ if $U_0 \in \mathrm{Ult}(V,U_1) .$ The relation is well-known to be transitive and well-founded. It has been an open problem to find a model where $\vartriangleleft$ embeds the four-element poset $\includegraphics{proc3019e-fig-a}$ . We find a generic extension where all well-founded posets are embeddable. Hence there is no structural restriction on the Mitchell ordering. Moreover we show that it is possible to have two $\vartriangleleft$ -incomparable measures that extend in a generic extension into two $\vartriangleleft$ -comparable measures.

References:

Ba85: S. Baldwin, The $\vartriangleleft$ -ordering on normal ultrafilters, J. Symbolic Logic 51 (1985), 936--952, MR 87d:03124.
Cu93: J. Cummings, Possible behaviors for the Mitchell ordering, Ann. Pure Appl. Logic (to appear).
KuP71: K. Kunen and J. B. Paris, Boolean extensions and measurable cardinals, Ann. of Math. Logic 2 (1971), 359--377, MR 43:3114.
Mi83: W.J. Mitchell, Sets constructible from sequences of measures: revisited, J. Symbolic Logic 48 (1983), 600--609, MR 85j:03052.
W94a: J. Witzany, Possible behaviours of the reflection ordering of stationary sets, J. Symbolic Logic (to appear).
W94b: J. Witzany, Reflection of stationary sets and the Mitchell ordering of normal measures, Ph.D. thesis, Pennsylvania State University, 1994.

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