ⓘ 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 ηextendible 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 ηC 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 ηC 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.
 version of 1  extendibility Existence of subcompact cardinals implies existence of many 1  extendible cardinals and hence many superstrong cardinals Existence
 supercompact, hypercompact cardinals η  extendible extendible cardinals Vopenka cardinals Shelah for supercompactness, high jump cardinals n  superstrong n 2
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Axiom of determinacy 

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Shrewd cardinal 
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Unfoldable cardinal 
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Wholeness axiom 
Woodin cardinal 
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