# Extendible cardinal

In mathematics, extendible cardinals are large cardinals introduced by Reinhardt (1974), 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 Vκ+η into Vλ, where κ is the critical point of j, and as usual Vα denotes the αth 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(n)-extendible if there is an elementary embedding j witnessing that κ is η-extendible (that is, j is elementary from Vκ+η to some Vλ with critical point κ) such that furthermore, Vj(κ) is Σn-correct in V. That is, for every Σn formula φ, φ holds in Vj(κ) if and only if φ holds in V. A cardinal κ is said to be C(n)-extendible if it is η-C(n)-extendible for every ordinal η. Every extendible cardinal is C(1)-extendible, but for n≥1, the least C(n)-extendible cardinal is never C(n+1)-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(n)-extendible cardinals for all n (Bagaria 2011). All extendible cardinals are supercompact cardinals (Kanamori 2003).