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 ''V''_κ+η into ''V''_λ, where ''κ'' is the critical point of ''j'', and as usual ''V_α'' denotes the ''α''th 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 ''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).

See also

* List of large cardinal properties
* Reinhardt cardinal

References

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Large cardinals

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