Mitchell order

In mathematical set theory, the Mitchell order is a well-founded preorder on the set of normal measures on a measurable cardinal κ.

In fact, the Mitchell order can be defined on the set (or proper class, as the case may be) of extenders for κ; but if it is so defined it may fail to be transitive, or even well-founded, provided κ has sufficiently strong large cardinal properties.

Well-foundedness fails specifically for rank-into-rank extenders; but Itay Neeman showed in 2004 that it holds for all weaker types of extender.

The Mitchell rank of a measure is the order type of its predecessors under ◅; since ◅ is well-founded this is always an ordinal.

Mitchell constructed an inner model for a measurable cardinal of rank