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.
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". This phenomenon also occurs with other cardinal types, such as unfoldable, supercompact and extendible cardinals.Template:Math-stub