In set theory

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...

, a strong cardinal is a type of large cardinal. It is a weakening of the notion of a supercompact cardinal.

Formal definition

If λ is any ordinal

Ordinal number

In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...

, κ is λ-strong means that κ is a cardinal number

Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...

 and there exists an elementary embedding j from the universe V into a transitive inner model

Inner model

In mathematical logic, suppose T is a theory in the languageL = \langle \in \rangleof set theory.If M is a model of L describing a set theory and N is a class of M such that \langle N, \in_M, \ldots \rangle...

 M with critical point

Critical point (set theory)

In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is the smallest ordinal which is not mapped to itself....

 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

It is obvious from the definitions that strong cardinals lie below supercompact cardinals and above 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...

s in the consistency strength hierarchy.

They also lie below superstrong cardinals and Woodin cardinal

Woodin cardinal

In set theory, a Woodin cardinal is a cardinal number λ such that for all functionsthere exists a cardinal κ In set theory, a Woodin cardinal is a cardinal number λ such that for all functions...

s. However, the least strong cardinal is larger than the least superstrong cardinal.