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 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.

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 such that furthermore, Vj is Σn-correct in V. That is, for every Σn formula φ, φ holds in Vj 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.
Vopěnka's principle implies the existence of extendible cardinals; in fact, Vopěnka's principle is equivalent to the existence of C-extendible cardinals for all n. All extendible cardinals are supercompact cardinals.