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

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 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

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

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_j(κ) is Σ_n-correct in V. That is, for every Σ_n formula φ, φ holds in V_j(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

• List of large cardinal properties
• Reinhardt cardinal

References

• Bagaria, Joan (23 December 2011). "C⁽ⁿ⁾-cardinals". Archive for Mathematical Logic 51 (3–4): 213–240. doi:10.1007/s00153-011-0261-8. 
• 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. 
• "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., 1974, pp. 189–205