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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Any behaviour of the Mitchell ordering
of normal measures is possible


Author: 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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03019-5
PII: S 0002-9939(96)03019-5
Keywords: Stationary sets, reflection, measurable cardinals, repeat points
Communicated by: Andreas R. Blass
Article copyright: © Copyright 1996 American Mathematical Society