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 (Kanamori 2003).

Properties
For a cardinal $$\kappa$$, say that a logic $$L$$ is $$\kappa$$-compact if for every set $$A$$ of $$L$$-sentences, if every subset of $$A$$ or cardinality $$<\kappa$$ has a model, then $$A$$ has a model. (The usual compactness theorem shows $$\aleph_0$$-compactness of first-order logic.) Let $$L_\kappa^2$$ be the infinitary logic for second-order set theory, permitting infinitary conjunctions and disjunctions of length $$<\kappa$$. $$\kappa$$ is extendible iff $$L_\kappa^2$$ is $$\kappa$$-compact.

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).