Editing Extendible cardinal (section)

==Variants and relation to other cardinals==
A cardinal ''κ'' is called ''η-C<sup>(n)</sup>''-extendible if there is an elementary embedding ''j'' witnessing that ''κ'' is ''η''-extendible (that is, ''j'' is elementary from ''V<sub>κ+η</sub>'' to some ''V<sub>λ</sub>'' with critical point ''κ'') such that furthermore, ''V<sub>j(κ)</sub>'' is ''Σ<sub>n</sub>''-correct in ''V''.  That is, for every [[Lévy hierarchy#Definitions|''Σ<sub>n</sub>'']] formula ''φ'', ''φ'' holds in ''V<sub>j(κ)</sub>'' if and only if ''φ'' holds in ''V''.  A cardinal ''κ'' is said to be '''C<sup>(n)</sup>-extendible''' if it is ''η-C<sup>(n)</sup>''-extendible for every ordinal ''η''.  Every extendible cardinal is ''C<sup>(1)</sup>''-extendible, but for ''n≥1'', the least ''C<sup>(n)</sup>''-extendible cardinal is never ''C<sup>(n+1)</sup>''-extendible (Bagaria 2011).

[[Vopěnka's principle]] implies the existence of extendible cardinals; in fact, Vopěnka's principle (for definable classes) is equivalent to the existence of ''C<sup>(n)</sup>''-extendible cardinals for all ''n'' (Bagaria 2011).  All extendible cardinals are [[supercompact cardinal]]s (Kanamori 2003).