In mathematical
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
, the Mitchell order is a
well-founded
In mathematics, a binary relation ''R'' is called well-founded (or wellfounded) on a class ''X'' if every non-empty subset ''S'' ⊆ ''X'' has a minimal element with respect to ''R'', that is, an element ''m'' not related by ''s ...
preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special c ...
on the set of
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. Equivalently, if f:κ→κ is such that f(α)<α for most α<� ...
s on a
measurable cardinal
In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal , or more generally on any set. For a cardinal , it can be described as a subdivis ...
''κ''. It is named for
William Mitchell William Mitchell may refer to:
People
Media and the arts
* William Mitchell (sculptor) (1925–2020), English sculptor and muralist
* William Frederick Mitchell (1845–1914), British naval artist
* William M. Mitchell, American writer, ministe ...
. We say that ''M'' ◅ ''N'' (this is a
strict order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
) if ''M'' is in the
ultrapower
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factor ...
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
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map f ...
, 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
In mathematics, a binary relation ''R'' is called well-founded (or wellfounded) on a class ''X'' if every non-empty subset ''S'' ⊆ ''X'' has a minimal element with respect to ''R'', that is, an element ''m'' not related by ''s ...
, provided ''κ'' has sufficiently strong
large cardinal
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least ...
properties. Well-foundedness fails specifically for
rank-into-rank
In set theory, a branch of mathematics, a rank-into-rank embedding is a large cardinal property defined by one of the following four axioms given in order of increasing consistency strength. (A set of rank < λ is one of the elements of the set V ...
extenders; but
Itay Neeman
Itay Neeman (born 1972) is a set theorist working as a professor of mathematics at the University of California, Los Angeles. He has made major contributions to the theory of inner models, determinacy and forcing.
Early life and education
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. Using the method of coherent sequences, Mitchell proved that for any rank
, there is a measurable cardinal of rank
.
[W. Mitchell]
Inner models for large cardinals
(2012, p.8). Accessed 2022-12-07.
A cardinal that has measures of Mitchell rank ''α'' for each ''α'' < ''β'' is said to be ''β''-measurable.
References
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{{DEFAULTSORT:Mitchell Order
Measures (set theory)
Large cardinals