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 conce ...
, a strong cardinal is a type of
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 Î ...
. It is a weakening of the notion of a
supercompact cardinal In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties.
Formal definition
If ''λ'' is any ordinal, ''κ'' is ''λ''-supercompact means that there exists an elementary ...
.
Formal definition
If λ is any ordinal, κ is λ-strong means that κ is a
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
and there exists an
elementary embedding In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences.
If ''N'' is a substructure of ''M'', one often ...
''j'' from the universe ''V'' into a transitive
inner model
In set theory, a branch of mathematical logic, an inner model for a theory ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''.
Definition
Let L = \langle \in \rangle be ...
''M'' with critical point κ and
:
That is, ''M'' agrees with ''V'' on an initial segment. Then κ is strong means that it is λ-strong for all ordinals λ.
Relationship with other large cardinals
By definitions, strong cardinals lie below
supercompact cardinal In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties.
Formal definition
If ''λ'' is any ordinal, ''κ'' is ''λ''-supercompact means that there exists an elementary ...
s and above
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 subdivisio ...
s in the consistency strength hierarchy.
κ is κ-strong if and only if it is measurable. If κ is strong or λ-strong for λ ≥ κ+2, then the
ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter o ...
''U'' witnessing that κ is measurable will be in ''V''κ+2 and thus in ''M''. So for any α < κ, we have that there exist an ultrafilter ''U'' in ''j''(''V''κ) − ''j''(''V''α), remembering that ''j''(α) = α. Using the elementary embedding backwards, we get that there is an ultrafilter in ''V''κ − ''V''α. So there are arbitrarily large measurable cardinals below κ which is regular, and thus κ is a limit of κ-many measurable cardinals.
Strong cardinals also lie below
superstrong cardinal In mathematics, a cardinal number κ is called superstrong if and only if there exists an elementary embedding ''j'' : ''V'' → ''M'' from ''V'' into a transitive inner model ''M'' with critical point κ and V_ ⊆ ''M''.
Similarl ...
s and
Woodin cardinal In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number \lambda such that for all functions
:f : \lambda \to \lambda
there exists a cardinal \kappa < \lambda with
:
and an s in consistency strength. However, the least strong cardinal is larger than the least superstrong cardinal.
Every strong cardinal is strongly unfoldable and therefore totally indescribable.