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.

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).