Mitchell order
Encyclopedia
In mathematical set theory
, the Mitchell order is a well-founded preorder
on the set of normal measure
s on a measurable cardinal
κ. It is named for William Mitchell. We say that M ◅ N (this is a strict order) if M is in the ultrapower model defined by N. Intuitively, this means that M is a weaker measure than N (note, for example, that κ will still be measurable in the ultrapower for N, since M is a measure on it).
In fact, the Mitchell order can be defined on the set (or proper class, as the case may be) of extender
s 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 ordertype of its predecessors under ◅; since ◅ is well-founded this is always an ordinal.
A cardinal which has measures of Mitchell rank α for each α < β is said to be β-measurable.
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, the Mitchell order is a well-founded preorder
Preorder
In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...
on the set of normal measure
Normal measure
In set theory, a normal measure is a measure on a measurable cardinal κ such that the equivalence class of the identity function on κ maps to κ itself in the ultrapower construction...
s on a measurable cardinal
Measurable cardinal
- Measurable :Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ...
κ. It is named for William Mitchell. We say that M ◅ N (this is a strict order) if M is in the ultrapower model defined by N. Intuitively, this means that M is a weaker measure than N (note, for example, that κ will still be measurable in the ultrapower for N, since M is a measure on it).
In fact, the Mitchell order can be defined on the set (or proper class, as the case may be) of extender
Extender (set theory)
In set theory, an extender is a set which represents an elementary embedding having large cardinal properties. A nonprincipal ultrafilter is the most basic case of an extender....
s for κ; but if it is so defined it may fail to be transitive
Transitive relation
In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
, or even well-founded, provided κ has sufficiently strong large cardinal properties. Well-foundedness fails specifically for rank-into-rank
Rank-into-rank
In set theory, a branch of mathematics, a rank-into-rank is a large cardinal λ satisfying one of the following four axioms :...
extenders; but Itay Neeman showed in 2004 that it holds for all weaker types of extender.
The Mitchell rank of a measure is the ordertype of its predecessors under ◅; since ◅ is well-founded this is always an ordinal.
A cardinal which has measures of Mitchell rank α for each α < β is said to be β-measurable.