Any Behaviour of the Mitchell Ordering of Normal Measures Is Possible

Let U0, U1 be two normal measures on κ. We say that U0 is in the Mitchell ordering less than U1, U0 C U1, if U0 ∈ Ult(V,U1). The relation is well-known to be transitive and well-founded. It has been an open problem to find a model where C 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 C-incomparable measures that extend in a generic extension into two C-comparable measures. We address the question of possible behaviours of the Mitchell ordering of normal measures. In the well-known Mitchell’s model L[ −→U ] the ordering of measures on a cardinal κ is linear [Mi83]. S. Baldwin constructed a model where C is a pre-well-ordering [Ba85] (a well-founded poset is pre-well-ordered iff ∀p, q ∈ P : p <P q iff oP (p) < oP (q) where oP (p) is the rank of p in P ). Recently J. Cummings [Cu93] described the Mitchell ordering in a particular generic extension where it embeds any wellfounded poset that does not embed the four-element poset: We say that a well-founded poset P embeds into the Mitchell ordering of normal measures on κ if there are different measures {Up; p ∈ P} on κ so that Up C Uq iff p <P q. I show that there is a generic extension where any well-founded poset (with certain cardinality restrictions) embeds into C . It proves that there is no structural restriction on the Mitchell ordering. However it still remains open to construct a model where all measures on κ ordered by the Mitchell ordering are isomorphic to a given poset, e.g. there are only four measures ordered according to the figure above. That would certainly need to go into inner models, possibly Received by the editors February 11, 1994 and, in revised form, July 18, 1994; some results of this paper were presented at the annual meeting of the Association for Symbolic Logic at the University of Florida, March 5–8, 1994. 1991 Mathematics Subject Classification. Primary 03E35, 03E55.