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Relationship to Vopěnka's Principle

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

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

A cardinal <math>\kappa</math> is called ethereal if for every closed and unbounded <math>C\subset\kappa</math> and for every sequence <math>(A_\delta)_{\delta<\kappa}</math> of length <math>\kappa</math> such that <math>A_\delta\subset\delta</math> and <math>A_\delta</math> has the same cardinality as <math>\delta</math> for arbitrary <math>\delta<\kappa</math>, there exist <math>\alpha,\beta</math>, belonging to <math>C</math>, with <math>\alpha<\beta</math>, such that <math>\textrm{card}(\alpha)=\mathrm{card}(A_\beta\cup A_\alpha)</math>.<ref name="Ketonen74">{{Citation | last1=Ketonen | first1=Jussi | title=Some combinatorial principles |mr=0332481 | year=1974 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=188 | pages=387–394 | doi=10.2307/1996785 | publisher=Transactions of the American Mathematical Society, Vol. 188 | jstor=1996785| doi-access=free| url=https://www.ams.org/journals/tran/1974-188-00/S0002-9947-1974-0332481-5/S0002-9947-1974-0332481-5.pdf }}</ref>

Subtle cardinals were introduced by {{harvtxt|Jensen|Kunen|1969}}. Ethereal cardinals were introduced by {{harvtxt|Ketonen|1974}}. Any subtle cardinal is ethereal,<ref name="Ketonen74" />p. 388 and any strongly inaccessible ethereal cardinal is subtle.<ref name="Ketonen74" />p. 391

==Characterizations==
Some equivalent properties to subtlety are known.

===Relationship to Vopěnka's Principle===
Subtle cardinals are equivalent to a weak form of [[Vopenka's principle|Vopěnka cardinals]]. Namely, an [[inaccessible cardinal]] <math>\kappa</math> is subtle if and only if in <math>V_{\kappa+1}</math>, any logic has stationarily many weak compactness cardinals.<ref>W. Boney, S. Dimopoulos, V. Gitman, M. Magidor "[https://web.archive.org/web/20231220180708/https://victoriagitman.github.io/files/LargeCardinalLogics.pdf Model Theoretic Characterizations of Large Cardinals Revisited]" (2023).</ref>

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 <math>\leq\kappa</math> if and only if every transitive set <math>S</math> of cardinality <math>\kappa</math> contains <math>x</math> and <math>y</math> such that <math>x</math> is a proper subset of <math>y</math> and <math>x\neq\varnothing</math> and <math>x\neq\{\varnothing\}</math>.<ref name="Friedman02">[[Harvey Friedman (mathematician)|H. Friedman]], "[https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/PrimitiveIndResults071302-189vmn0.pdf Primitive Independence Results]" (2002). Accessed 18 April 2024.</ref>Corollary 2.6 If a cardinal <math>\lambda</math> is subtle, then for every <math>\alpha<\lambda</math>, every transitive set <math>S</math> of cardinality <math>\lambda</math> includes a chain (under inclusion) of order type <math>\alpha</math>.<ref name="Friedman02" />Theorem 2.2

==Extensions==
A hypersubtle cardinal is a subtle cardinal which has a stationary set of subtle cardinals below it.<ref name="Henrion87">C. Henrion, "[https://www.jstor.org/stable/2273834 Properties of Subtle Cardinals]. Journal of Symbolic Logic, vol. 52, no. 4 (1987), pp.1005--1019."</ref>p.1014

==See also==
* [[List of large cardinal properties]]

== References ==
{{refbegin}}
*{{citation|first=Harvey |last=Friedman|authorlink=Harvey Friedman (mathematician)|title=Subtle Cardinals and Linear Orderings|journal= Annals of Pure and Applied Logic |year= 2001|volume=107|issue=1–3|pages=1–34
|doi=10.1016/S0168-0072(00)00019-1|doi-access=free}}
*{{citation|title=Some Combinatorial Properties of L and V |url=http://www.mathematik.hu-berlin.de/~raesch/org/jensen.html|first=R. B. |last=Jensen|first2=K.|last2=Kunen|publisher=Unpublished manuscript|year=1969}}
{{refend}}

===Citations===
{{reflist}}

[[Category:Large cardinals]]

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Subtle cardinal - Revision history

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