Extendible cardinal

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_κ+η into V_λ, where κ is the critical point of j, and as usual V_α 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).

For a cardinal ${\displaystyle \kappa }$ , say that a logic ${\displaystyle L}$  is ${\displaystyle \kappa }$ -compact if for every set ${\displaystyle A}$  of ${\displaystyle L}$ -sentences, if every subset of ${\displaystyle A}$  or cardinality ${\displaystyle <\kappa }$  has a model, then ${\displaystyle A}$  has a model. (The usual compactness theorem shows ${\displaystyle \aleph _{0}}$ -compactness of first-order logic.) Let ${\displaystyle L_{\kappa }^{2}}$  be the infinitary logic for second-order set theory, permitting infinitary conjunctions and disjunctions of length ${\displaystyle <\kappa }$ . ${\displaystyle \kappa }$  is extendible iff ${\displaystyle L_{\kappa }^{2}}$  is ${\displaystyle \kappa }$ -compact.^[1]

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 V_κ+η to some V_λ with critical point κ) such that furthermore, V_j(κ) is Σ_n-correct in V. That is, for every Σ_n formula φ, φ holds in V_j(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).

• List of large cardinal properties
• Reinhardt cardinal