Any behaviour of the Mitchell ordering of normal measures is possible

Witzany, Jiří

doi:10.1090/S0002-9939-96-03019-5

Any behaviour of the Mitchell ordering of normal measures is possible
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by Jiří Witzany PDF

Proc. Amer. Math. Soc. 124 (1996), 291-297 Request permission

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 $U_0 \in an{Ult}(V,U_1) .$ 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 \epssymbol{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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