Strong cardinal
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In mathematical set theory, a strong cardinal or λ-strong cardinal is a type of large cardinal.
Specifically, a λ-strong cardinal is a cardinal number κ such that exists an elementary embedding j from V into a transitive inner model M with critical point κ and
- Vλ ⊆ M;
κ is said to be strong iff it is λ-strong for all ordinals λ.
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". This phenomenon also occurs with other cardinal types, such as unfoldable, supercompact and extendible cardinals.Template:Mathlogic-stub