# Extendible cardinal

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In [mathematics](/source/Mathematics), **extendible cardinals** are [large cardinals](/source/Large_cardinal) introduced by Reinhardt (1974), who was partly motivated by [reflection principles](/source/Reflection_principle). Intuitively, such a cardinal represents a point beyond which initial pieces of the [universe of sets](/source/Von_Neumann_universe) start to look similar, in the sense that each is [elementarily embeddable](/source/Elementary_embedding) into a later one.

## Definition

For every [ordinal](/source/Ordinal_number) *η*, a [cardinal](/source/Cardinal_number) κ is called **η-extendible** if for some ordinal *λ* there is a nontrivial [elementary embedding](/source/Elementary_embedding) *j* of *V*κ+η into *V*λ, where *κ* is the [critical point](/source/Critical_point_(set_theory)) of *j*, and as usual *Vα* denotes the *α*th level of the [von Neumann hierarchy](/source/Von_Neumann_universe). A cardinal *κ* is called an **extendible cardinal** if it is *η*-extendible for every nonzero ordinal *η* (Kanamori 2003).

## Properties

For a cardinal \kappa, say that a logic L is \kappa-compact if for every set A of L-sentences, if every subset of A or cardinality <\kappa has a model, then A has a model. (The usual [compactness theorem](/source/Compactness_theorem) shows \aleph_0-compactness of first-order logic.) Let L_\kappa^2 be the [infinitary logic](/source/Infinitary_logic) for second-order set theory, permitting infinitary conjunctions and disjunctions of length <\kappa. \kappa is extendible iff L_\kappa^2 is \kappa-compact.[1]

## 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 *Vκ+η* to some *Vλ* with critical point *κ*) such that furthermore, *Vj(κ)* is *Σn*-correct in *V*. That is, for every [*Σn*](/source/L%C3%A9vy_hierarchy#Definitions) 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](/source/Vop%C4%9Bnka'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](/source/Supercompact_cardinal) (Kanamori 2003).

## See also

- [List of large cardinal properties](/source/List_of_large_cardinal_properties)
- [Reinhardt cardinal](/source/Reinhardt_cardinal)

## References

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

- Bagaria, Joan (23 December 2011). "*C(n)*-cardinals". *Archive for Mathematical Logic*. **51** (3–4): 213–240. [doi:10.1007/s00153-011-0261-8](https://doi.org/10.1007/s00153-011-0261-8). [S2CID 208867731](https://api.semanticscholar.org/CorpusID:208867731)
- Friedman, Harvey. ["Restrictions and Extensions"](http://u.osu.edu/friedman.8/files/2014/01/ResExt021703-1t4vsx4.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", Vol. XIII, Part II, Proc. Sympos. Pure Math., Providence, R. I.: Amer. Math. Soc., pp. 189–205, MR 0401475

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