Any behaviour of the Mitchell Ordering of Normal Measures Is Possible

Jiří Witzany

paper

Any behaviour of the Mitchell Ordering of Normal Measures Is Possible

arXiv:math/9312203

Abstract

Let $U_{0}, U_{1}$ be two normal measures on $κ .$ We say that $U_{0}$ is in the Mitchell ordering less then $U_{1},$ $U_{0} ⊲ U_{1},$ if $U_{0} \in U l t (V, U_{1}) .$ 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 $v a r t r ian g l e l e f t$ -incomparable measures that extend in a generic extension into two $⊲$ -comparable measures.