Any behaviour of the Mitchell Ordering of Normal Measures Is Possible

Abstract

Let $U_0,U_1$ be two normal measures on $\kappa .$ We say that $U_0$ is in the Mitchell ordering less then $U_1,$ $U_0\vartriangleleft U_1,$ if $U_0 \in Ult(V,U_1) .$ The ordering is well-known to be transitive and well-founded. It has been an open problem to find a model where the Mitchell ordering embeds the four-element poset $|\; | .$ We show that in the Kunen-Paris extension 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.