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Subtle cardinal

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

A cardinal is called subtle if for every closed and unbounded and for every sequence of length such that for all (where is the th element), there exist , belonging to , with , such that .

A cardinal is called ethereal if for every closed and unbounded and for every sequence of length such that and has the same cardinality as for arbitrary , there exist , belonging to , with , such that .[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

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Some equivalent properties to subtlety are known.

Relationship to Vopěnka's Principle

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Subtle cardinals are equivalent to a weak form of Vopěnka cardinals. Namely, an inaccessible cardinal is subtle if and only if in , 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

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There is a subtle cardinal if and only if every transitive set of cardinality contains and such that is a proper subset of and and .[3]Corollary 2.6 An infinite ordinal is subtle if and only if for every , every transitive set of cardinality includes a chain (under inclusion) of order type .

Extensions

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

See also

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References

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  • 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  • Jensen, R. B.; Kunen, K. (1969), Some Combinatorial Properties of L and V, Unpublished manuscript

Citations

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  1. ^ a b c Ketonen, Jussi (1974), "Some combinatorial principles" (PDF), Transactions of the American Mathematical Society, 188, Transactions of the American Mathematical Society, Vol. 188: 387–394, 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"
  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."