In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number.

A cardinal \kappa is called subtle if for every closed and unbounded C\subset\kappa and for every sequence (A_\delta)_{\delta<\kappa} of length \kappa such that A_\delta\subset\delta for all \delta<\kappa (where A_\delta is the \deltath element), there exist \alpha,\beta, belonging to C, with \alpha<\beta, such that A_\alpha=A_\beta\cap\alpha.

A cardinal \kappa is called ethereal if for every closed and unbounded C\subset\kappa and for every sequence (A_\delta)_{\delta<\kappa} of length \kappa such that A_\delta\subset\delta and A_\delta has the same cardinality as \delta for arbitrary \delta<\kappa, there exist \alpha,\beta, belonging to C, with \alpha<\beta, such that \textrm{card}(\alpha)=\mathrm{card}(A_\beta\cup A_\alpha).[1]

Subtle cardinals were introduced by Jensen & Kunen (1969). Ethereal cardinals were introduced by Ketonen (1974). Any subtle cardinal is ethereal,[1]p. 388 and any strongly inaccessible ethereal cardinal is subtle.[1]p. 391

Characterizations

Some equivalent properties to subtlety are known.

Relationship to Vopěnka's Principle

Subtle cardinals are equivalent to a weak form of Vopěnka cardinals. Namely, an inaccessible cardinal \kappa is subtle if and only if in V_{\kappa+1}, any logic has stationarily many weak compactness cardinals.[2]

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

Chains in transitive sets

There is a subtle cardinal \leq\kappa if and only if every transitive set S of cardinality \kappa contains x and y such that x is a proper subset of y and x\neq\varnothing and x\neq\{\varnothing\}.[3]Corollary 2.6 If a cardinal \lambda is subtle, then for every \alpha<\lambda, every transitive set S of cardinality \lambda includes a chain (under inclusion) of order type \alpha.[3]Theorem 2.2

Extensions

A hypersubtle cardinal is a subtle cardinal which has a stationary set of subtle cardinals below it.[4]p.1014

See also

References

Citations

  1. ^ Ketonen, Jussi (1974), "Some combinatorial principles", Transactions of the American Mathematical Society. 188: 387–394, Transactions of the American Mathematical Society, Vol. 188, doi:10.2307/1996785. ISSN 0002-9947. JSTOR 1996785. MR 0332481
  2. ^ W. Boney, S. Dimopoulos, V. Gitman, M. Magidor "Model Theoretic Characterizations of Large Cardinals Revisited" (2023).
  3. ^ H. Friedman, "Primitive Independence Results" (2002). Accessed 18 April 2024.
  4. ^ C. Henrion, "Properties of Subtle Cardinals. Journal of Symbolic Logic, vol. 52, no. 4 (1987), pp.1005--1019."