# Subtle cardinal

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In [mathematics](/source/Mathematics), **subtle cardinals** and **ethereal cardinals** are closely related kinds of [large cardinal](/source/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](/source/Vopenka's_principle). Namely, an [inaccessible cardinal](/source/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

- [List of large cardinal properties](/source/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](https://doi.org/10.1016/S0168-0072(00)00019-1)
- Jensen, R. B. & Kunen, K. (1969), ["Some Combinatorial Properties of L and V"](http://www.mathematik.hu-berlin.de/~raesch/org/jensen.html), Unpublished manuscript

### Citations

1. Ketonen, Jussi (1974), ["Some combinatorial principles"](https://www.ams.org/journals/tran/1974-188-00/S0002-9947-1974-0332481-5/S0002-9947-1974-0332481-5.pdf), *[Transactions of the American Mathematical Society](/source/Transactions_of_the_American_Mathematical_Society)*. **188**: 387–394, Transactions of the American Mathematical Society, Vol. 188, [doi:10.2307/1996785](https://doi.org/10.2307/1996785). [ISSN 0002-9947](https://www.worldcat.org/issn/0002-9947). [JSTOR 1996785](https://www.jstor.org/stable/1996785). MR 0332481

1. W. Boney, S. Dimopoulos, V. Gitman, M. Magidor "[Model Theoretic Characterizations of Large Cardinals Revisited](https://web.archive.org/web/20231220180708/https://victoriagitman.github.io/files/LargeCardinalLogics.pdf)" (2023).

1. [H. Friedman](/source/Harvey_Friedman_(mathematician)), "[Primitive Independence Results](https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/PrimitiveIndResults071302-189vmn0.pdf)" (2002). Accessed 18 April 2024.

1. C. Henrion, "[Properties of Subtle Cardinals](https://www.jstor.org/stable/2273834). Journal of Symbolic Logic, vol. 52, no. 4 (1987), pp.1005--1019."

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