strong cardinal

In mathematics, a strong cardinal is a cardinal number κ such that 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

V_λ ⊆ M.

A λ-strong cardinal is a cardinal number κ such that exists an elementary embedding

j : V → M

from V into a transitive inner model M with critical point κ and

V_λ ⊆ 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 V_κ is a model of "ZFC + there is a proper class of strong cardinals".

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