Extendible cardinal

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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.

Definition

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For every ordinal η, a cardinal κ is called η-extendible if for some ordinal λ there is a nontrivial elementary embedding j of Vκ+η into Vλ, where κ is the critical point of j, and as usual Vα 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).

Properties

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For a cardinal  , say that a logic   is  -compact if for every set   of  -sentences, if every subset of   or cardinality   has a model, then   has a model. (The usual compactness theorem shows  -compactness of first-order logic.) Let   be the infinitary logic for second-order set theory, permitting infinitary conjunctions and disjunctions of length  .   is extendible iff   is  -compact.[1]

Variants and relation to other cardinals

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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 Vκ+η to some Vλ with critical point κ) such that furthermore, Vj(κ) is Σn-correct in V. That is, for every Σn formula φ, φ holds in Vj(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

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References

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  1. ^ Magidor, M. (1971). "On the Role of Supercompact and Extendible Cardinals in Logic". Israel Journal of Mathematics. 10 (2): 147–157. doi:10.1007/BF02771565.