# 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.

## Definition[edit]

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[edit]

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*to some

_{κ+η}*V*with critical point

_{λ}*κ*) such that furthermore,

*V*is

_{j(κ)}*Σ*-correct in

_{n}*V*. That is, for every

*Σ*formula

_{n}*φ*,

*φ*holds in

*V*if and only if

_{j(κ)}*φ*holds in

*V*. A cardinal

*κ*is said to be

**C**if it is

^{(n)}-extendible*η-C*-extendible for every ordinal

^{(n)}*η*. Every extendible cardinal is

*C*-extendible, but for

^{(1)}*n≥1*, the least

*C*-extendible cardinal is never

^{(n)}*C*-extendible (Bagaria 2011).

^{(n+1)}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[edit]

## References[edit]

- Bagaria, Joan (23 December 2011). "
*C*-cardinals".^{(n)}*Archive for Mathematical Logic*.**51**(3–4): 213–240. doi:10.1007/s00153-011-0261-8. - Friedman, Harvey. "Restrictions and Extensions" (PDF).
- Kanamori, Akihiro (2003).
*The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings*(2nd ed.). Springer. ISBN 3-540-00384-3. - Reinhardt, W. N. (1974), "Remarks on reflection principles, large cardinals, and elementary embeddings.",
*Axiomatic set theory*, Proc. Sympos. Pure Math., XIII, Part II, Providence, R. I.: Amer. Math. Soc., pp. 189–205, MR 0401475

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