Any Behaviour of the Mitchell Ordering of Normal Measures Is Possible Preliminary Version

Let U0, U1 be two normal measures on κ. We say that U0 is in the Mitchell oredering less then U1, U0 ⊳ U1, if U0 ∈ Ult(V, U1). The ordering is wellknown to be transitive and well-founded. It has been an open problem to find a model where ⊳ embeds the four-element poset | |. 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 ⊳-incomparable measures that extend in a generic extension into two ⊳-comparable measures. We say that a well-founded poset P embeds into the Mitchell ordering of normal measures on κ if there are measures {Up; p ∈ P} so that Up ⊳ Uq iff p <P q. In the well-known Mitchell’s model L[ −→ U ] the ordering is linear (cf.[Mi83]). S.Baldwin has constructed a model where ⊳ is a prewellordering (cf.[Ba85]). Recently J. Cummings has described the Mitchell ordering in a particular generic extension where it embeds any well-founded poset that does not embed the four element poset | | (cf.[Cu93]). It is actually an open problem of [Ba84] to find a model where the Mitchell ordering embeds the four-element poset | |. We show that in a generic extension any well-founded poset is embeddable. More specifically: 1991 Mathematics Subject Classification. 03E35, 03E55.