Subtle cardinal

In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number. A cardinal κ is called subtle if for every closed and unbounded C ⊂ κ and for every sequence A of length κ for which element number δ (for an arbitrary δ), A_δ ⊂ δ, there exist α, β, belonging to C, with α < β, such that A_α = A_β ∩ α.

A cardinal κ is called ethereal if for every closed and unbounded C ⊂ κ and for every sequence A of length κ for which element number δ (for an arbitrary δ), A_δ ⊂ δ and A_δ has the same cardinal as δ, there exist α, β, belonging to C, with α < β, such that card(α) = card(A_β ∩ A_α).

Subtle cardinals were introduced by (Jensen Kunen). Ethereal cardinals were introduced by (Ketonen 1974). Any subtle cardinal is ethereal, and any strongly inaccessible ethereal cardinal is subtle.

Relationship to Vopěnka's Principle

Subtle cardinals are equivalent to a weak form of Vopěnka cardinals. Namely, an inaccessible cardinal [math]\displaystyle{ \kappa }[/math] is subtle if and only if in [math]\displaystyle{ V_{\kappa+1} }[/math], any logic has stationarily many weak compactness cardinals.

Vopenka's principle itself may be stated as the existence of a strong compactness cardinal for each logic.

Theorem

There is a subtle cardinal ≤ κ if and only if every transitive set S of cardinality κ contains x and y such that x is a proper subset of y and x ≠ Ø and x ≠ {Ø}. An infinite ordinal κ is subtle if and only if for every λ < κ, every transitive set S of cardinality κ includes a chain (under inclusion) of order type λ.

Extensions

A hypersubtle cardinal is a subtle cardinal which has a stationary set of subtle cardinals below it.^p.1014

See also

• List of large cardinal properties

References

• Friedman, Harvey (2001), "Subtle Cardinals and Linear Orderings", Annals of Pure and Applied Logic 107 (1–3): 1–34, doi:10.1016/S0168-0072(00)00019-1 
• Jensen, R. B.; Kunen, K. (1969), Some Combinatorial Properties of L and V, Unpublished manuscript, http://www.mathematik.hu-berlin.de/~raesch/org/jensen.html 
• Ketonen, Jussi (1974), "Some combinatorial principles", Transactions of the American Mathematical Society (Transactions of the American Mathematical Society, Vol. 188) 188: 387–394, doi:10.2307/1996785, ISSN 0002-9947