Extendible cardinal
In mathematics, extendible cardinals are large cardinals introduced by , who was partly motivated by reflection principles. Intuitively, such a cardinal represents a point beyond which initial pieces of the universe of sets start to look similar, in the sense that each is elementarily embeddable into a later one. Definition. For every ordinal "η", a cardinal κ is called η-extendible if for some ordinal "λ" there is a nontrivial elementary embedding "j" of "V" into "V", where "κ" is the critical point of "j", and as usual "V" denotes the "α"th level of the von Neumann hierarchy. A  cardinal "κ" is called an extendible cardinal if it is "η"-extendible for every nonzero ordinal "η" (Kanamori 2003). Variants and relation to other cardinals. A cardinal "κ" is called "η-C"-extendible if there is an elementary embedding "j" witnessing that "κ" is "η"-extendible (that is, "j" is elementary from "V" to some "V" with critical point "κ") such that furthermore, "V" is "Σ"-correct in "V". That is, for every "Σ" formula "φ", "φ" holds in "V" if and only if "φ" holds in "V". A cardinal "κ" is said to be C-extendible if it is "η-C"-extendible for every  ordinal "η". Every extendible cardinal is "C"-extendible, but for "n≥1", the least "C"-extendible cardinal is never "C"-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"-extendible cardinals for all "n" (Bagaria 2011). All extendible cardinals are supercompact cardinals (Kanamori 2003).