ⓘ Strong cardinal
If λ is any ordinal, κ is λstrong means that κ is a cardinal number and there exists an elementary embedding j from the universe V into a transitive inner model M with critical point κ and
V λ ⊆ M {\displaystyle V_{\lambda }\subseteq M}That is, M agrees with V on an initial segment. Then κ is strong means that it is λstrong for all ordinals λ.
1. Relationship with other large cardinals
By definitions, strong cardinals lie below supercompact cardinals and above measurable cardinals in the consistency strength hierarchy.
κ is κstrong if and only if it is measurable. If κ is strong or λstrong for λ ≥ κ+2, then the ultrafilter 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 κmany measurable cardinals.
Strong cardinals also lie below superstrong cardinals and Woodin cardinals 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.
 1930 Every strongly inaccessible cardinal is also weakly inaccessible, as every strong limit cardinal is also a weak limit cardinal If the generalized
 inaccessible cardinals less than κ is stationary in κ. The term Mahlo cardinal now usually means strongly Mahlo cardinal though the cardinals originally
 . See also strong cardinal A Woodin cardinal is preceded by a stationary set of measurable cardinals and thus it is a Mahlo cardinal However, the
 arranges cardinals in order of consistency strength, with size of the cardinal used as a tiebreaker. In a few cases such as strongly compact cardinals the
 mathematical set theory, a strongly compact cardinal is a certain kind of large cardinal number. A cardinal κ is strongly compact if and only if every
 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
 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 κ, or more
 In mathematics, a partition cardinal is either: An Erdos cardinal or A Strong partition cardinal
 Cardinals in the family Cardinalidae, are passerine birds found in North and South America. They are also known as cardinal  grosbeaks and cardinal  buntings
 with the Cardinal cutting Friday editions and the Herald publishing print issues once a week. The 1970s saw the Cardinal maintain its strong issue advocacy
 northern cardinal Cardinalis cardinalis is a bird in the genus Cardinalis it is also known colloquially as the redbird, common cardinal or just cardinal which
 In mathematics, a subcompact cardinal is a certain kind of large cardinal number. A cardinal number κ is subcompact if and only if for every A H κ

Large cardinal 
Axiom of determinacy 

Berkeley cardinal 
Core model 
Critical point (set theory) 
Extender (set theory) 

Extendible cardinal 
Grothendieck universe 
Huge cardinal 
Indescribable cardinal 
Ineffable cardinal 
Iterable cardinal 
Kunens inconsistency theorem 
Mahlo cardinal 
Measurable cardinal 

Rankintorank 
Reinhardt cardinal 
Remarkable cardinal 
Shelah cardinal 
Shrewd cardinal 
Strongly compact cardinal 
Quasicompact cardinal 
Unfoldable cardinal 
Weakly compact cardinal 
Wholeness axiom 
Woodin cardinal 
Zero sharp 

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