Strong cardinal
In
mathematics, a
cardinal number κ is called
strong iff for all
ordinal numbers λ there exists an elementary embedding
j :
V →
M from
V into a transitive inner model
M with critical point κ and
Vλ ⊆
M. κ is called
λ-strong iff there exists an elementary embedding
j :
V →
M from
V into a transitive inner model
M with critical point κ and
Vλ ⊆
M; thus, κ is strong iff it is λ-strong for all λ.
It should be noted that the least strong cardinal is larger than the least Woodin, superstrong, etc. cardinals, but that the consistency strength of strong cardinals is lower: For example, if κ is Woodin, then Vκ is a model of "ZFC + there is a proper class of strong cardinals".