ⓘ Extendible cardinal. In mathematics, extendible cardinals are large cardinals introduced by Reinhardt, who was partly motivated by reflection principles. Intuit ..


ⓘ Extendible cardinal

In mathematics, extendible cardinals are large cardinals introduced by Reinhardt, 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.


1. 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.


2. 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 jκ is Σ n -correct in V. That is, for every Σ n formula φ, φ holds in V jκ 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.

Vopenkas principle implies the existence of extendible cardinals; in fact, Vopenkas 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.

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