Subtle cardinal

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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), Aδ ⊂ δ, there exist αβ, belonging to C, with α < β, such that Aα = Aβ ∩ 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), Aδ ⊂ δ and Aδ has the same cardinal as δ, there exist αβ, belonging to C, with α < β, such that card(α) = card(Aβ ∩ Aα).

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.[1]

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.[2]p.1014

See also

References

Citations

  1. https://victoriagitman.github.io/files/largeCardinalLogics.pdf
  2. C. Henrion, "Properties of Subtle Cardinals. Journal of Symbolic Logic, vol. 52, no. 4 (1987), pp.1005--1019."